An Invitation to Higher Gauge Theory

An In vitatio n to Higher Gauge Theory John C. Baez and John Hue rt a Departmen t of Mathematics, Univ ersit y of California Riv erside, Califo rnia 92521 USA email: baez@math.ucr.edu h uerta@math.ucr.edu Marc h 24, 201 0 Abstract In this easy introduction to higher gauge theory , w e describ e parallel trans- p ort for particles and strings in terms of 2-conn ections on 2-bundles. Just as ordinary gauge theory inv olves a gauge group, th is generalization in - vol ves a gauge ‘2-group’. W e focus on 6 examples. Firs t , every abelian Lie group giv es a Lie 2 - group; the case of U(1) y ields the theory of U ( 1) gerbes, whic h p la y an imp ortant role in string theory and multisymplec- tic geometry . Second, every group representation gives a Lie 2-group; the representa t ion of the Loren tz group on 4d Minko wski spacetime give s the P oincar´ e 2-group, which leads to a spin foa m mo del for Minko wski space- time. Third , taking the adjoint represen tation of an y Lie group on its o wn Lie algebra giv es a ‘tangent 2-group’, which serves as a gauge 2-group in 4d B F theory , whic h has top ological gra vity as a sp ecial case. F ourth, every Lie g roup has an ‘inner automorphism 2-group’, which serv es as th e gauge group in 4d B F th eory with cosmological constan t term. Fifth, ev- ery Lie group has an ‘automorphism 2-group’, whic h plays an imp ortant role in th e th eory of nonab elian gerbes. And sixth, ev ery compact simple Lie group gives a ‘string 2-group’. W e also touch up on higher structures such as the ‘gravit y 3-group’, and the Lie 3-sup eralg ebra that go verns 11-dimensional sup ergravit y . 1 In tro d uction Higher g auge theory is a g eneraliza tion o f g auge theory that describ es parallel transp ort, not just for p oint particles, but also for higher- dimensional extended ob jects. It is a b eautiful new branch of mathematics, with a lo t of ro om left for explor ation. It has a lready b een applied to string theory and lo op qua nt um gravit y—o r more sp eciﬁcally , spin fo am mo de ls . This should not be surpris ing , since while thes e riv al approa ches to quan tum gravit y dis agree ab out almost 1 everything, they both agree that point particles are not eno ugh: we nee d higher- dimensional extended ob jects to build a theory suﬃciently rich to descr ib e the quantum geo metry of spacetime. Indeed, man y existing ideas fro m string theory and s uper gravity hav e recently b een cla riﬁed b y hig her gauge theory [82, 83]. But we may also hop e for applications of higher gauge theory to other less sp eculative branches of physics, such as condensed matter physics. Of course, for this to happ e n, more physicists nee d to lea rn highe r ga uge theory . It would b e grea t to hav e a comprehens ive introduction to the sub ject which started from scr atch a nd le d the reader to the fro nt ier s o f knowledge. Un- fortunately , ma thematical work in this sub ject uses a wide ar ray of to ols, such as n -categor ies, stacks, ge r b es, Deligne cohomolo gy , L ∞ algebras , Kan complexes , and ( ∞ , 1)-catego r ies, to name just a few. While these to o ls are b eautiful, im- po rtant in their own r ight, a nd p er haps necessar y for a deep under standing o f higher gauge theory , lear ning them takes time—and explaining them all would be a ma jor pro ject. Our goa l he r e is far more mo dest. W e shall sketc h how to g eneralize the theory of parallel transp or t fro m po int particles to 1 -dimensional ob jects, such as strings. W e shall do this starting with a bar e minimum of prerequisites: manifolds, diﬀerential forms, L ie gr oups, Lie alge br as, and the tra ditional theory of parallel transp ort in terms of bundles and connections. W e shall give a small taste of the applications to physics, and po int the reader to the literature for more details. In Section 2 we star t by explaining c a tegories , functors, and how parallel transp ort for par ticles ca n b e seen as a functor taking any path in a manifold to the op eratio n o f parallel trans p o rt along that path. In Section 3 we ‘add one’ a nd explain how parallel transp ort for particles and strings ca n be s een a s ‘2-functor’ b etw een ‘2-catego ries’. This r equires that we genera lize Lie gro ups to ‘Lie 2-gro ups’. In Section 4 we describe many examples of L ie 2-g roups, and sketc h s ome o f their a pplications: • Section 4.1: shifted a be lia n g roups, U(1) gerb es, and their role in string theory and mult is ymplectic geometry . • Section 4 .2: the Poincar´ e 2-g roup and the spin foam mo del for 4d Minko wsk i spacetime. • Section 4.3: tangen t 2-gr o ups, 4d B F theory and top olo gical g r avit y . • Section 4.4: inner automorphism 2-groups and 4d B F theory with cosmo- logical constant term. • Section 4.5: automor phism 2-gro ups, nonab elia n gerb es , a nd the gr avit y 3-gro up. • Section 4.6 : string 2-g roups, string structures, the pass age from Lie n - algebras to Lie n -gro ups, a nd the Lie 3- sup eralgebr a g ov erning 11-dimensio na l sup e rgravity . 2 Finally , in Sectio n 5 w e discuss gauge tra nsformations , curv ature and nontrivial 2-bundles. 2 Categories and Connections A category cons ists o f ob jects, which we draw as dots: • x and mo r phisms betw een o b jects, which we draw as arrows betw ee n do ts: x • f ' ' • y Y ou s hould think of ob jects a s ‘things’ and morphisms as ‘pro c esses’. The main thing you can do in a category is take a mor phism from x to y a nd a morphism from y to z : x • f ' ' • y g ' ' • z and ‘co mpo se’ them to g et a morphis m from x to z : x • gf $ $ • z The most famo us example is the categor y Set, which ha s sets as ob jects and functions as morphisms. Most of us know how to co mp o se functions, a nd we hav e a pretty g o o d intuition of how this works. So, it ca n b e helpful to think of morphisms as b eing like functions. But as we sha ll s o on see, there are so me very impor tant categories where the mo rphisms are not functions. Let us give the for mal deﬁnitio n. A category consists of: • A collection of ob jects , and • for any pair of o b jects x, y , a set of morphism s f : x → y . Given a morphism f : x → y , we call x its source and y its target . • Given tw o mo rphisms f : x → y and g : y → z , there is a comp osite morphism g f : x → z . Composition satisﬁes the asso ciativ e law : ( hg ) f = h ( g f ) . • F or any o b ject x , there is an identit y morphism 1 x : x → x . These identit y morphisms satisfy the l eft and right un i t l a ws : 1 y f = f = f 1 x for any morphism f : x → y . 3 The hardest thing ab o ut ca tegory theo ry is getting your arrows to p oint the right wa y . It is standard in mathematics to use f g to denote the result of doing ﬁrst g and then f . In pictures, this backw ards conv ention can be annoying. But rather than tr ying to ﬁght it, let us give in and draw a morphism f : x → y as an a rrow from right to left: y • • x f w w Then comp osition lo oks a bit b etter: z • • y f w w • x g w w = z • • x f g w w An imp or tant ex a mple of a categor y is the ‘path group oid’ o f a space X . W e give the precise deﬁnition b elow, but the basic idea is to take the diagra ms we hav e b een drawing ser iously! The ob jects ar e p oints in X , and morphisms are paths: y • • x γ   W e also g et examples fr om gro ups. A group is the sa me as a ca tegory with one ob ject where all the morphisms are in vertible. The mor phisms of this catego ry are the elements of the gro up. The ob ject is there just to provide them w ith a source and target. W e co mpo se the morphis ms using the multiplication in the group. In b oth these examples, a mo rphism f : x → y is not a function from x to y . And these tw o examples have s o mething else in common: they are impor tant in gauge theor y! W e c a n use a path gr oup oid to des crib e the p ossible motions of a particle thro ugh spacetime. W e can use a g roup to descr ib e the symmetries of a particle. And when we combine these t wo examples, we get the co ncept of c onne ction —the basic ﬁeld in a ny gauge theo r y . How do we co mbine these examples? W e do it using a ma p b etw een cate- gories. A map b etw een ca tegories is called a ‘functor’. A functor fr om a path group oid to a gr oup will se nd every ob ject o f the pa th gro upo id to the same ob ject of our g roup. After a ll, a group, rega rded as a catego ry , has o nly one ob ject. But this functor will also s end any mor phism in our path gr oup oid to a gro up element. In o ther words, it will ass ign a g roup element to each path in our space. This g roup element describ es how a p article tr ansforms as it mov es along that pa th. But this is precise ly what a connection do es ! A connection lets us compute for any path a group element describing parallel transp or t along that path. So, the la nguage of ca teg ories and functors quickly leads us to the concept o f connection—but with an emphasis o n pa r allel transp ort. 4 The following theorem makes these ideas precise. L et us ﬁrst state the theorem, then deﬁne the terms involv ed, and then give some idea of how it is prov ed: Theorem 1. F or any Lie gr oup G and any smo oth manifold M , ther e is a one-to-one c orr esp ondenc e b etwe en: 1. c onne ctions on the trivial princip al G -bund le over M , 2. g -value d 1-forms on M , wher e g is the Lie algebr a of G , and 3. smo oth funct ors hol: P 1 ( M ) → G wher e P 1 ( M ) is the p ath gr oup oid of M . W e ass ume y o u are familiar with the ﬁrst tw o items. Our goal is to explain the third. W e must sta rt by expla ining the pa th g roup oid. Suppo se M is a manifold. Then the path gr o up oid P 1 ( M ) is ro ughly a category in whic h ob jects a re p o int s of M and a morphism fr om x to y is a path from x to y . W e comp ose paths by gluing them end to end. So, given a path δ from x to y , and a path γ from y to z : z • y • γ   x • δ   we w o uld like γ δ to b e the path from x to z built from γ and δ . How ever, we need to b e ca reful abo ut the details to make sur e that the comp osite pa th γ δ is w ell- de ﬁned, and that comp osition is a sso ciative! Since w e are studying paths in a smo oth manifold, we wan t them to b e smo oth. B ut the path γ δ may no t b e smo oth: there could b e a ‘kink’ at the p o int y . There are diﬀerent wa y s to get a r ound this pro blem. One is to work with piecewise smo oth pa ths. But her e is a nother appr oach: say that a path γ : [0 , 1] → M is lazy if it is smo oth a nd also constant in a neighbor ho o d of t = 0 a nd t = 1. The ide a is that a lazy hiker takes a rest b efor e sta rting a hike, and a lso after completing it. Supp o se γ a nd δ a re smo o th paths and γ s tarts where δ ends. Then we deﬁne their comp osite γ δ : [0 , 1] → M in the usual way: ( γ δ )( t ) =  δ (2 t ) if 0 ≤ t ≤ 1 2 γ (2 t − 1) if 1 2 ≤ t ≤ 1 5 In other w ords, γ δ sp ends the ﬁrst ha lf of its time moving alo ng δ , and the second half moving along γ . In g eneral the pa th γ δ may not b e smo oth at t = 1 2 . Howev er, if γ and δ a re lazy , then their comp osite is smo oth—and it, to o, is lazy! So, lazy paths are c losed under comp osition. Unfortunately , comp osition of lazy paths is not a sso ciative. The paths ( αβ ) γ and α ( β γ ) diﬀer by a smo oth repara metr ization, but they are not equal. T o solve this pro blem, we can take certain e quivalenc e classes of lazy paths as morphisms in the path gr oup oid. W e might try ‘homotopy cla sses’ of paths. Remem b er, a homotopy is a wa y of interpo lating betw ee n pa ths: y • • x δ g g γ w w Σ   More pr ecisely , a homotop y from the path γ : [0 , 1] → M to the path δ : [0 , 1] → M is a smo oth map Σ: [0 , 1] 2 → M such that Σ(0 , t ) = γ ( t ) and Σ(1 , t ) = δ ( t ). W e say tw o pa ths are homo topic , or lie in the same homotopy class , if ther e is a homotopy b etw een them. There is a well-deﬁned categ ory where the morphisms are homo topy cla s ses of lazy paths. Unfortunately this is not right for gaug e theo r y , since for mo st connections, parallel transpo r t alo ng homotopic paths g ives diﬀerent results. In fact, parallel transp ort g ives the same result fo r all homo to pic paths if and only if the co nnection is ﬂat . So, unless we are willing to settle for ﬂat co nnections, we need a more delicate equiv alence relation betw een paths. Here the conc e pt of ‘thin’ homotopy comes to our rescue. A homoto py is thin if it sweeps out a s ur face that has zero area . In other words, it is a homotopy Σ such that the r ank of the diﬀerential d Σ is less than 2 at every p oint. If tw o paths diﬀer b y a smo oth r e parametriza tion, they ar e thinly homotopic. But there are other examples, to o . F or exa mple, suppo se we hav e a path γ : x → y , and let γ − 1 : y → x b e the rev erse path, deﬁned as follows: γ − 1 ( t ) = γ (1 − t ) . Then the comp os ite path γ − 1 γ , which go es from x to itself: y • • x   γ − 1 γ   is thinly homotopic to the c o nstant pa th tha t sits at x . The re a son is that we can s hrink γ − 1 γ down to the cons tant pa th without sweeping out a ny area. W e deﬁne the path group oid P 1 ( M ) to b e the categ o ry wher e: • Ob jects a r e p o int s of M . 6 • Morphisms are thin homotopy classes of la zy paths in M . • If we write [ γ ] to denote the thin homotopy class of the path γ , comp osition is deﬁned by [ γ ][ δ ] = [ γ δ ] . • F or any p oint x ∈ M , the identit y 1 x is the thin homotopy cla ss of the constant path at x . With these rule s , it is ea sy to chec k that P 1 ( M ) is a categ ory . The most impo rtant p oint is that since the comp osite paths ( αβ ) γ and α ( β γ ) diﬀer by a smo oth repar ametrization, they are thinly ho motopic. This gives the ass o ciativ e law when we work with thin homotopy cla sses. But a s its name sugg ests, P 1 ( M ) is b etter than a mer e catego ry . It is a group oid : that is, a ca tegory where every morphis m γ : x → y ha s an inv e rse γ − 1 : y → x satisfying γ − 1 γ = 1 x and γ γ − 1 = 1 y In P 1 ( M ), the inv ers e is deﬁned using the concept of a re verse path: [ γ ] − 1 = [ γ − 1 ] . The rules fo r a n inverse only hold in P 1 ( M ) after we take thin homotopy class es. After a ll, the comp osites γ γ − 1 and γ − 1 γ are n ot co nstant paths, but they are thinly ho motopic to co nstant paths. But henceforth, we will relax and w r ite simply γ for the morphism in the path groupo id corresp o nding to a path γ , instead o f [ γ ]. As the name s uggests, gr oup oids a re a bit like groups. Indeed, a group is secretly the same as a g roup oid with one ob ject! In other words, suppo se we hav e gr oup G . Then there is a categ ory where: • There is only one o b ject, • . • Morphisms from • to • ar e e le ments of G . • Comp osition of morphisms is m ultiplicatio n in the g roup G . • The identit y morphism 1 • is the ident ity element o f G . This category is a gr oup oid, since every gr oup element has a n inv ers e. Con- versely , any group oid with o ne ob ject gives a g roup. Henceforth we will fre e ly switch back and forth b etw een thinking of a gro up in the traditional wa y , a nd thinking of it as a one-ob ject group oid. How can we use group oids to descr ib e connections? It should not b e sur- prising that we ca n do this, now that w e have o ur path gr oup oid P 1 ( M ) and our one-o b ject group oid G in ha nd. A connection gives a map from P 1 ( M ) to G , which says ho w to transfor m a particle when w e mov e it along a path. Mo re precisely: if G is a Lie g roup, any connection on the trivial G -bundle over M 7 yields a map, calle d the par allel transp or t map or ho lonomy , that assigns an element of G to each path: hol: • • γ v v 7→ hol( γ ) ∈ G In ph ysics no ta tion, the holonomy is de ﬁned as the path-ordered exp onential o f some g -v alued 1- form A , where g is the Lie algebra of G : hol( γ ) = P exp  Z γ A  ∈ G. The holonomy map sa tisﬁes certa in rules, most of which ar e summarized in the word ‘functor’. Wha t is a functor? It is a map betw ee n catego ries tha t preserves all the structure in sight! More pre c isely: given categories C and D , a functor F : C → D consists of: • a map F sending ob jects in C to ob jects in D , and • another map, also called F , s ending mo rphisms in C to morphisms in D , such that: • given a morphism f : x → y in C , we hav e F ( f ): F ( x ) → F ( y ), • F preser ves c omp osition: F ( f g ) = F ( f ) F ( g ) when either side is well-deﬁned, and • F preser ves ide ntities: F (1 x ) = 1 F ( x ) for every ob ject x of C . The last pro p erty actually follows fro m the r est. The second to last—preserving comp osition—is the most impor tant prop erty of functor s. As a test of your understanding, c heck that if C and D a re just gr oups (that is, one-ob ject group oids) then a functor F : C → D is just a homomorphi sm . Let us see w ha t this deﬁnition says ab out a functor hol: P 1 ( M ) → G where G is some Lie gro up. T his functor hol must send all the p oints of M to the one ob ject o f G . More interestingly , it must send thin homotopy cla sses of paths in M to ele ments of G : hol: • • γ v v 7→ hol( γ ) ∈ G 8 It m ust preser ve comp ositio n: hol( γ δ ) = hol( γ ) hol( δ ) and identities: hol(1 x ) = 1 ∈ G. While they may b e stated in unfamiliar language, these ar e actually well- known prop erties o f connec tio ns! First, the holo no my of a connection alo ng a path hol( γ ) = P e xp  Z γ A  ∈ G only dep ends on the thin ho motopy class of γ . T o see this, c o mpute the v a riation of hol( γ ) as we v ary the path γ , and show the v a riation is zero if the homotopy is thin. Second, to compute the gr oup element for a compo site of paths, we just m ultiply the gr oup elements for each one : P exp  Z γ δ A  = P ex p  Z γ A  P exp  Z δ A  And third, the pa th-ordered exp onential along a constant path is just the iden- tit y: P exp  Z 1 x A  = 1 ∈ G. All this information is nea tly ca ptur ed by saying hol is a functor. And Theorem 1 says this is almost all there is to being a connection. The only additional condition required is that hol b e smo oth . This mea ns, ro ughly , that hol( γ ) dep ends smoo thly o n the pa th γ —more on that later. But if we dr op this condition, we can generalize the concept of connectio n, and deﬁne a generalized connection o n a s mo oth ma nifold M to b e a functor hol: P 1 ( M ) → G . Generalized connections have long play ed an imp or tant role in lo op qua n- tum gravit y , ﬁrst in the co nt ex t of real-a nalytic manifolds [3], a nd later for smo oth manifolds [17, 65]. The reaso n is that if M is any manifo ld and G is a connec ted compact Lie group, there is a natural meas ure on the spa ce of generalized connections. This means that y ou can deﬁne a Hilbert space of complex-v alued squa re-integrable functions o n the space of generalized c o nnec- tions. In lo op quantum gravity these ar e used to descr ib e q ua ntum s tates b efor e any co nstraints hav e been impo sed. The switch from connections to g eneralized connections is crucial here—and the lack of smo othness gives lo op q ua ntu m gravit y its ‘discr ete’ ﬂavor. But suppos e w e are interested in ordinar y connectio ns. Then w e really w ant hol( γ ) to dep end smo othly on the path γ . How can we make this precise? One w ay is to use the theory of ‘smooth group oids’ [16]. An y Lie gro up is a smo oth group oid, and so is the path g roup oid of any smo o th ma nifo ld. W e can deﬁne smo oth functors betw een smo oth gro up o ids, and then smo oth functors hol: P 1 ( M ) → G are in one-to- one co rresp ondence with connections o n 9 the trivial principa l G -bundle over M . W e can g o e ven further: there are more general maps betw e e n smo oth group oids, and maps hol: P 1 ( M ) → G of this more genera l so rt corresp o nd to connections on not n e c essarily trivial principal G -bundles ov er M . F or details, see the work o f Bartels [25], Schreiber and W aldorf [87]. But if this sounds like to o muc h work, we can take the following sho rtcut. Suppo se we hav e a s mo oth function F : [0 , 1] n × [0 , 1] → M , which we think of as a parametrized family of paths. And supp ose that for each ﬁxed v alue of the parameter s ∈ [0 , 1 ] n , the path γ s given by γ s ( t ) = F ( s, t ) is la zy . The n our functor hol: P 1 ( M ) → G gives a function [0 , 1] n → G s 7→ hol( γ s ) . If this function is smo o th whenever F has the a b ov e prop er ties, then the functor hol: P 1 ( M ) → G is smo oth . Starting fro m this deﬁnition one can prove the following lemma, which lies at the hear t o f Theo rem 1: Lemma. Ther e is a one-to-one c orr esp ondenc e b etwe en smo oth fun ctors hol: P 1 ( M ) → G and Lie( G ) -value d 1-forms A on M . The idea is that given a Lie( G )-v alued 1-form A on M , we can deﬁne a holonomy for a ny smo oth pa th as follows: hol( γ ) = P exp  Z γ A  , and then chec k tha t this deﬁnes a smo oth functor hol: P 1 ( M ) → G . Con versely , suppo se we have a s mo oth functor hol of this s o rt. Then we can deﬁne hol( γ ) for smo oth paths γ that are no t lazy , using the fact that every smo oth path is thinly homotopic to a lazy one. W e ca n even do this for paths γ : [0 , s ] → M where s 6 = 1 , since a ny such path can be repar ametrized to give a path of the usual s ort. Given a smo oth path γ : [0 , 1] → M we can trunca te it to o bta in a path γ s that go es along γ until time s : γ s : [0 , s ] → M . By what we have s a id, hol( γ s ) is well-deﬁned. Using the fact that hol: P 1 ( M ) → G is a s mo oth functor, one ca n chec k that hol( γ s ) v arie s smo othly with s . So, we can diﬀerentiate it and deﬁne a Lie( G )-v alued 1-form A a s fo llows: A ( v ) = d ds hol( γ s )   s =0 10 where v is an y tangent vector at a point x ∈ M , and γ is any s mo oth path with γ (0) = x, γ ′ (0) = v. Of c ourse, we need to check that A is well-deﬁned and smo o th. W e also need to chec k that if we star t with a smo oth functor hol, constr uct a 1-fo r m A in this wa y , and then turn A ba ck into a smo oth functor, we wind up back where we started. 3 2-Categories and 2-Connections Now we wan t to climb up one dimension, and talk ab out ‘2-connections’. A connection tells us how par ticles transform as they move along paths. A 2 - connection will a lso tell us how strings tra nsform a s they sw eep out surfac es . T o make this idea precise, we need to take everything we sa id in the previo us section and bo ost the dimensio n by o ne. Ins tead of categories , we need ‘2 -catego ries’. Instead of gr oups, we nee d ‘2-gro ups’. Instead of the path gro upo id, we need the ‘path 2-group oid’. And instead of functors, we need ‘2-functors’. When we unders tand all these things, the analog ue of Theor em 1 will lo ok strikingly similar to the or iginal version: Theorem. F or any Lie 2-gr oup G and any smo oth manifold M , t her e is a one-to-one c orr esp ondenc e b etwe en: 1. 2-c onne ctions on the trivial princip al G -2-bund le over M , 2. p airs c onsisting of a smo oth g -value d 1-form A and a smo oth h -value d 2-form B on M , such that t ( B ) = dA + A ∧ A wher e we u se t : h → g , the diﬀer ential of the m ap t : H → G , to c onvert B into a g -value d 2-form, and 3. smo oth 2-funct ors hol: P 2 ( M ) → G wher e P 2 ( M ) is the p ath 2-gr oup oid of M . What do es this say? In brief: there is a wa y to extra ct from a Lie 2- group G a pair of Lie gr oups G and H . Supp os e we hav e a 1-fo r m A taking v alues in the Lie algebr a o f G , and a 2-for m B v alued in the Lie alg ebra of H . Suppo se furthermore that these forms ob ey the equation a b ov e. Then we can use them to consis tent ly deﬁne par allel transp o r t, or ‘holonomies’, for paths and surfaces. They thus deﬁne a ‘2- connection’. That is the idea . But to make it precise, we need 2-categ ories. 11 3.1 2-Categories Sets hav e elements. Categ ories have elemen ts, usua lly called ‘ob jects’, but also morphisms b e tw e e n these. In a n ‘ n -catego ry’, we g o further and include 2- morphisms b etw een morphisms, 3 -morphisms b etw een 2-mor phisms,... and so on up to the n th level. W e ar e b eginning to see n - c ategorie s provide an a lgebraic language for n - dimensional structure s in physics [1 2]. Higher gauge theory is just one plac e where this is happ ening. An yone learning n -catego ries needs to star t with 2-categorie s [63]. A 2- category co nsists of: • a collection of ob jects, • for any pair of ob jects x and y , a set of morphisms f : x → y : y • • x f u u • for any pair of morphisms f , g : x → y , a set of 2-mor phisms α : f ⇒ g : y • • x g g g f w w α   W e call f the source of α and g the target of α . Morphisms can be co mpo sed just as in a categor y: z • • y f w w • x g w w = z • • x f g w w while 2-morphisms can b e co mpo sed in tw o distinct wa ys , vertically: y • • x f   f ′ o o f ′′ ` ` α   α ′   = y • • x f x x f ′′ f f α ′ · α   and ho rizontally: z • y • f 1 v v f ′ 1 h h α 1   • x f 2 w w f ′ 2 g g α 2   = z • • x f 1 f 2 x x f ′ 1 f ′ 2 f f α 1 ◦ α 2   Finally , these laws m ust hold: 12 • Comp osition of morphisms is a sso ciative, a nd every o b ject x has a mor - phism x • • x 1 x u u serving as an ide ntit y for comp o s ition, just as in an ordinar y catego ry . • V ertical comp os ition is ass o ciative, and every morphism f has a 2 - morphism y • • x f g g f w w 1 f   serving as an ide ntit y for vertical comp os ition. • Horizontal comp osition is asso ciativ e, a nd the 2- morphism x • • x 1 x h h 1 x v v 1 1 x   serves as an ident ity for horiz o ntal compos itio n. • V ertical and horizo nt a l composition o f 2-morphis ms obey the interc hange la w : ( α ′ 1 · α 1 ) ◦ ( α ′ 2 · α 2 ) = ( α ′ 1 ◦ α ′ 2 ) · ( α 1 ◦ α 2 ) so that diagrams o f the form x • y • f 1 | | f ′ 1 o o f ′′ 1 b b α 1   α ′ 1   • z f 2 | | f ′ 2 o o f ′′ 2 c c α 2   α ′ 2   deﬁne unambiguous 2-mor phisms. The in ter change law is the truly new thing here. A ca tegory is a ll ab out attaching 1-dimensiona l ar rows end to end, and we need the a s so ciative law to do that unambiguously . In a 2-categ ory , we visualize the 2-mor phisms a s little pieces o f 2-dimensional surface: • • f f x x   W e can attach these to gether in t wo w ays: vertically and horizontally . F or the result to b e unambiguous, we need not only asso ciativ e laws but also the int er change law. In what follows we will se e this law tur ning up all over the place. 13 3.2 P ath 2-Groupoids Path gr oup oids play a big though often neglected ro le in physics: the path group oid of a spacetime manifold descr ib e s all the p ossible motions of a p oint particle in that spacetime. The path 2-gr o up oid do e s the same thing for particles and strings . First of a ll, a 2-group oi d is a 2-catego r y where: • Every morphism f : x → y has an i n v e rse , f − 1 : y → x , such that: f − 1 f = 1 x and f f − 1 = 1 y . • Every 2 -morphism α : f ⇒ g has a vertical in verse , α − 1 vert : g ⇒ f , s uch that: α − 1 vert · α = 1 f and α · α − 1 vert = 1 f . It actually follows fro m this deﬁnition that every 2-mo rphism α : f ⇒ g also has a ho rizon tal in verse , α − 1 hor : f − 1 ⇒ g − 1 , such that: α − 1 hor ◦ α = 1 1 x and α ◦ α − 1 hor = 1 1 x . So, a 2- g roup oid ha s every kind o f inv er se your heart could desire . An example of a 2-gr oup is the ‘path 2 - group oid’ of a smo o th manifold M . T o deﬁne this, we can start with the path g roup oid P 1 ( M ) as deﬁned in the previous section, and then throw in 2-mor phisms. Just a s the mo r phisms in P 1 ( M ) were thin homotopy c la sses of lazy pa ths, these 2-morphisms will b e thin ho motopy classes of la z y sur faces. What is a ‘laz y surface’ ? First, r ecall that a hom o top y betw een lazy paths γ , δ : x → y is a smo o th map Σ: [0 , 1 ] 2 → M w ith Σ(0 , t ) = γ ( t ) Σ(1 , t ) = δ ( t ) W e say this homo topy is a lazy surface if • Σ( s, t ) is indep endent of s nea r s = 0 and near s = 1, • Σ( s, t ) is constant near t = 0 and constant near t = 1. An y homotopy Σ yields a one-par ameter family of paths γ s given by γ s ( t ) = Σ( s, t ) . If Σ is a la zy surface, each of these pa ths is lazy . F urthermore, the path γ s equals γ 0 when s is suﬃcient ly close to 0 , and it equals γ 1 when s is suﬃcient ly close to 1. This allows us to comp ose lazy homotopies either vertically or horiz o ntally and o btain new la zy ho motopies! How ever, vertical and horizontal comp osition will only ob ey the 2-gro up oid axioms if we ta ke 2-morphisms in the path 2-gr oup oid to be e quivalenc e cla sses 14 of lazy surfaces. W e saw this kind of issue alr eady when discussing the path group oid, so we we will allow ourselves to b e a bit sketch y this time. The key idea is to deﬁne a concept of ‘thin homotopy’ be tw e e n la zy sur faces Σ and Ξ. F or starter s, this should b e a smo oth map H : [0 , 1] 3 → M s uch that H (0 , s, t ) = Σ( s, t ) and H (1 , s, t ) = Ξ( s, t ). But we also wan t H to b e ‘thin’. In other words, it should sweep out no volume: the rank of the diﬀerential dH should b e les s than 3 at every p oint. T o make thin ho mo topies well-deﬁned b etw een thin homotopy class es of paths, some more tec hnical co nditio ns are also useful. F or these, the reader can turn to Section 2.1 of Sc hreib er and W aldorf [87]. The upsho t is that w e obtain for any smo oth ma nifo ld M a path 2-group oi d P 2 ( M ), in which: • An ob ject is a p oint of M . • A mor phism fro m x to y is a thin homotopy class of lazy paths from x to y . • A 2- morphism b etw een equiv alence classe s of lazy paths γ 0 , γ 1 : x → y is a thin homotopy class of la zy s urfaces Σ: γ 0 ⇒ γ 1 . As we alr eady did with the concept of ‘lazy path’, we will often use ‘lazy surface’ to mea n a thin ho motopy class of lazy surfaces. But now let us hasten on to another imp or tant class of 2-gro upo ids, the ‘2-gr oups’. Just as groups des crib e symmetries in gauge theo r y , thes e descr ib e s ymmetries in higher gaug e theo ry . 3.3 2-Groups Just as a group was a gr oup oid with one ob ject, we deﬁne a 2- group to b e a 2-gro upo id with one ob ject. This deﬁnition is so eleg ant that it may be hard to understand at ﬁrst! So, it will b e useful to take a 2-group G and chop it into four bite-sized pieces of data , g iving a ‘crossed mo dule’ ( G, H , t, α ). Indeed, 2- groups were or iginally introduce d in the guise of crossed mo dules b y the fa mo us top ologist J. H. C. Whitehead [93]. In 195 0, with help from Ma c L ane [66], he used cr ossed mo dules to generalize the fundamental gr o up of a space to wha t we might now call the ‘fundamental 2 -group’. But only la ter did it become clear that a cro ssed module w a s another wa y of talking a bo ut a 2- group oid with just one o b ject! F or more of this history , and muc h mor e on 2-gr oups, see [13]. Let us sta rt by seeing what it means to say a 2- g roup is a 2- group oid with one o b ject. It means that a 2-group G has: • one ob ject: • • morphisms: • • g v v 15 • and 2-mor phisms: • • g   g ′ \ \ α   The mor phisms fo r m a group under co mpo sition: • • g v v • g ′ v v = • • gg ′ v v The 2- morphisms for m a g roup under ho rizontal comp osition: • • g 1 x x g ′ 1 f f α 1   g 2 x x g ′ 2 f f α 2   = • • g 1 g 2 z z g ′ 1 g ′ 2 d d α 1 ◦ α 2   In addition, the 2- morphisms ca n b e co mp o sed vertically: • • g   g ′ o o g ′′ ] ] α   α ′   = • • g z z g ′′ d d α ′ · α   V ertical comp os ition is also ass o ciative with iden tity and in verses. But the 2- morphisms do not form a group under this op eration, b ecause a given pa ir may not be comp os a ble: their sour ce and tar g et may no t matc h up. Finally , vertical and hor izontal comp ositio n are tied toge ther b y the interc hange law, whic h s ays the t wo ways one can read this diagra m are consistent. • • g 1 } } g ′ 1 o o g ′′ 1 a a α   α ′   • g 2 } } g ′ 2 o o g ′′ 2 a a β   β ′   Now let us crea te a crossed mo dule ( G, H , t, α ) from a 2-gr oup G . T o do this, ﬁrst note that the morphisms of the 2 -gro up form a gro up b y themse lves, with compo sition as the g roup op er ation. So: • Let G b e the set o f morphis ms in G , made into a gro up with comp osition as the g roup op eration: • • g v v • g ′ v v = • • gg ′ v v 16 How ab out the 2-morphis ms ? These also form a gro up, with horizontal com- po sition as the group op eratio n. But it turns out to b e eﬃcient to fo cus on a subgroup o f this: • Let H b e the set of all 2-mor phisms whose source is the ident ity: • • 1 • z z t ( h ) d d h   W e make H into a group with horizo ntal comp osition as the group op er- ation: • • 1 • x x t ( h ) f f h   • 1 • x x t ( h ′ ) f f h ′   = • • 1 • z z t ( hh ′ ) d d hh ′   Above we use hh ′ as an abbr eviation for the hor iz o ntal compos ite h ◦ h ′ of t wo elements of H . W e will use h − 1 to denote the horizo ntal inv erse of an element of H . W e use t ( h ) to denote the target of an element h ∈ H . The deﬁnition of a 2 -catego r y implies that t : H → G is a group homomorphism: t ( hh ′ ) = t ( h ) t ( h ′ ) . This ho mo morphism is o ur thir d piece o f data : • A g roup homomor phis m t : H → G sending ea ch 2-morphism in H to its target: • • 1 • z z t ( h ) d d h   The fourth piece of data is the subtlest. Ther e is a way to ‘horizontally co nju- gate’ any element h ∈ H by an element g ∈ G , or mor e precisely by its identit y 2-morphism 1 g : • • g x x g f f 1 g   • 1 • x x t ( h ) f f h   • g − 1 x x g − 1 f f 1 g − 1   The r esult is a 2-mor phism in H w hich we call α ( g )( h ). In fact, α ( g ) is an automorphism of H , meaning a o ne - to-one a nd onto function with α ( g )( hh ′ ) = α ( g )( h ) α ( g )( h ′ ) . 17 Comp osing tw o automo rphisms g ives another a utomorphism, a nd this makes the automorphisms of H into a gro up, s ay Aut ( H ). Even b etter, α gives a group ho momorphism α : G → Aut( H ) . Concretely , this mea ns that in addition to the ab ov e equation, we hav e α ( g g ′ ) = α ( g ) α ( g ′ ) . Checking these tw o e quations is a nice w ay to test your understa nding of 2- categorie s. A group homo morphism α : G → Aut ( H ) is also called an action of the gro up G on the gr oup H . So, the fourth and ﬁnal piece of data in our crossed module is : • An action α of G o n H given by: • • g x x g f f 1 g   • 1 • x x t ( h ) f f h   • g − 1 x x g − 1 f f 1 − 1 g   = • • 1 x x t ( α ( g )( h )) f f α ( g )( h )   A cros s ed mo dule ( G, H , t, α ) m ust also sa tisfy tw o mor e equatio ns which follow from the deﬁnition of a 2-gr o up. First, examining the ab ov e diagram, we se e that t is G - equiv arian t , b y which we mean: • t ( α ( g ) h ) = g ( t ( h )) g − 1 for all g ∈ G and h ∈ H . Second, the Peiﬀer identit y holds: • α ( t ( h )) h ′ = hh ′ h − 1 for all h, h ′ ∈ H . The Peiﬀer ident ity is the least obvious thing ab out a crossed mo dule. It follows from the in ter change law, and it is w o rth s e eing how. First, we have: hh ′ h − 1 = • • 1 • z z t ( h ) d d h   • 1 • z z t ( h ′ ) d d h ′   • 1 • z z t ( h − 1 ) d d h − 1   where—b eware!—we a re now using h − 1 to mean the hori zont al in verse o f h , since this is its inv erse in the gr oup H . W e ca n pad out this equation by vertically comp osing with some iden tity mo rphisms: hh ′ h − 1 = • • 1 • ~ ~ t ( h ) o o t ( h ) ` ` h   1 t ( h )   • 1 • ~ ~ 1 • o o t ( h ) ` ` 1 1 •   h   • 1 • ~ ~ t ( h − 1 ) o o t ( h − 1 ) ` ` h − 1   1 t ( h − 1 )   18 This diagr am desc rib es a n unambiguous 2-mo rphism, thanks to the interc hang e law. So, we ca n do the horizontal comp ositions ﬁrst and get: hh ′ h − 1 = • • 1 • z z 1 • o o t ( hh ′ h − 1 ) d d 1 1 •   α ( t ( h ))( h ′ )   But vertically comp osing with an identit y 2-morphism ha s no eﬀect. So, we obtain the Peiﬀer iden tity: hh ′ h − 1 = α ( t ( h ))( h ′ ) . All this le ads us to deﬁne a crossed mo dule ( G, H , t, α ) to co nsist o f: • a group G , • a group H , • a homomorphism t : H → G , and • an action α : G → Aut ( H ) such that: • t is G - e quiv ariant: t ( α ( g ) h ) = g ( t ( h )) g − 1 for all g ∈ G and h ∈ H , and • the Peiﬀer identit y holds for a ll h, h ′ ∈ H : α ( t ( h )) h ′ = hh ′ h − 1 . In fac t, we can rec over a 2-gro up G from its cro s sed mo dule ( G, H, t, α ), so crossed modules are just another wa y of thinking ab out 2- g roups. The trick to seeing this is to notice that 2-mor phisms in G a re the sa me a s pair s ( g , h ) ∈ G × H . Such a pair gives this 2 -morphism: • • 1 • x x t ( h ) f f h   • g x x g f f 1 g   W e leave it to the reader to chec k that every 2- morphism in G is of this for m. Note that this 2-mo rphism go es from g to t ( h ) g . So, when we construct a 2-gro up from a cr ossed mo dule, we get a 2-mor phis m ( g , h ): g → t ( h ) g 19 from any pa ir ( g , h ) ∈ G × H . Horizontal comp osition of 2-morphisms then makes G × H in to a gr oup, as follows: ( g , h ) ◦ ( g ′ , h ′ ) = • • 1 • } } t ( h ) a a h   • g } } g a a 1 g   • 1 • } } t ( h ′ ) a a h ′   • g ′ } } g ′ a a 1 g ′   = • • 1 • } } t ( h ) a a h   • g } } g a a 1 g   • 1 • } } t ( h ′ ) a a h ′   • g − 1 } } g − 1 a a 1 g − 1   • g } } g a a 1 g   • g ′ } } g ′ a a 1 g ′   = • • 1 • } } t ( h ) a a h   • 1 • u u t ( α ( g )( h ′ )) i i α ( g )( h ′ )   • gg ′ x x gg ′ f f 1 gg ′   = • • 1 • w w t ( hα ( g )( h ′ )) g g hα ( g )( h ′ )   • gg ′ x x gg ′ f f 1 gg ′   = ( g g ′ , hα ( g )( h ′ )) . So, the gro up of 2-mor phisms of G is the semidirect pro duct G ⋉ H , deﬁned using the a ction α . F ollowing this line o f thought, the r eader c a n chec k the following: Theorem 2. Given a cr osse d mo dule ( G, H , t, α ) , ther e is a u nique 2-gr oup G wher e: • the gr oup of morphisms is G , • a 2-morphism α : g ⇒ g ′ is the same as a p air ( g , h ) ∈ G × H with g ′ = t ( h ) g , • the vertic al c omp osite of ( g , h ) and ( g ′ , h ′ ) , when they ar e c omp osable, is given by ( g , h ) · ( g ′ , h ′ ) = ( g ′ , hh ′ ) , • the horizontal c omp osite of ( g , h ) and ( g ′ , h ′ ) is given by ( g , h ) ◦ ( g ′ , h ′ ) = ( gg ′ , hα ( g )( h ′ )) . Conversely, given a 2-gr oup G , t her e is a unique cr osse d mo dule ( G, H, t, α ) wher e: • G is the gr oup of m orphisms of G , 20 • H is the gr oup of 2-morphisms with sour c e e qual t o 1 • , • t : H → G assigns t o e ach 2-morphism in H its tar get, • the action α of G on H is given by α ( g ) h = 1 g ◦ h ◦ 1 g − 1 . Indeed, thes e tw o pro cesses set up an eq uiv alence betw e e n 2-g roups and crossed mo dules, as describ ed more for mally elsewhere [13, 51]. It th us makes sense to deﬁne a Lie 2-group to b e a 2-gr o up for whic h the gr oups G and H in its c r ossed mo dule are Lie g roups, with the maps t : H → G and α : G → Aut( H ) being smo oth. It is worth emphasiz ing that in this context we use Aut( H ) to mean the gr oup of smo oth automor phisms of H . This is a Lie group in its own right. In Section 4 we will use Theo rem 2 to construct many examples of Lie 2- groups. But ﬁrst we should ﬁnish explaining 2-co nnections. 3.4 2-Connections A 2 -connection is a r ecip e for para llel tra nsp orting b o th 0-dimensio nal a nd 1-dimensional ob jects—sa y , pa r ticles and strings . Just as we ca n des crib e a connection on a trivial bundle using a Lie-alg ebra v alued diﬀerential for m, we can descr ib e a 2 -connection using a p air o f diﬀer e nt ia l forms. But there is a deep e r way of under standing 2-co nnections. J ust as a co nnection was revealed to b e a smo oth functor hol: P 1 ( M ) → G for some Lie gro up G , a 2-co nnec tion will turn out to be a smo o th 2- functor hol: P 2 ( M ) → G for some L ie 2 -group G . Of course, to make sense of this we need to deﬁne a ‘2-functor’, and s ay what it mea ns for such a thing to b e s mo oth. The deﬁnition of 2 -functor is utterly straightforward: it is a map b etw een 2-catego ries that pre s erves everything in sight. So, g iven 2- categor ies C a nd D , a 2 -functor F : C → D consis ts of: • a map F sending ob jects in C to ob jects in D , • another map called F sending morphisms in C to morphisms in D , • a third map called F sending 2-morphisms in C to 2-morphisms in D , such that: • given a morphism f : x → y in C , we hav e F ( f ): F ( x ) → F ( y ), 21 • F preser ves c omp osition for morphisms, and ident ity morphisms: F ( f g ) = F ( f ) F ( g ) F (1 x ) = 1 F ( x ) , • given a 2-morphism α : f ⇒ g in C , we hav e F ( α ): F ( f ) ⇒ F ( g ), • F prese r ves vertical and hor izontal comp os itio n for 2-morphis ms , and ident ity 2-morphisms: F ( α · β ) = F ( α ) · F ( β ) F ( α ◦ β ) = F ( α ) ◦ F ( β ) F (1 f ) = 1 F ( f ) . There is a gener al theory of smo oth 2-g roup oids and smo oth 2-functors [16, 88]. But here we prefer to take a mor e elementary appr oach. W e already k now that for any Lie 2-group G , the morphisms and 2-morphisms each for m Lie groups. Given this, we ca n say that for any smo oth manifold M , a 2 -functor hol: P 2 ( M ) → G is s m o oth if: • F or any smo o thly parametriz e d family of laz y paths γ s ( s ∈ [0 , 1 ] n ) the morphism hol( γ s ) depends s mo othly on s , and • F or any smo othly parametr ized family of lazy surfaces Σ s ( s ∈ [0 , 1 ] n ) the 2-morphism hol(Σ s ) depends s mo othly on s . With these deﬁnitions in hand, we are ﬁnally ready to under stand the basic result abo ut 2-connections. It is completely ana logous to Theor em 1: Theorem 3. F or any Lie 2-gr oup G and any smo oth manifold M , ther e is a one-to-one c orr esp ondenc e b etwe en: 1. 2-c onne ctions on the trivial princip al G -2-bund le over M , 2. p airs c onsisting of a smo oth g -value d 1-form A and a smo oth h -value d 2-form B on M , such that t ( B ) = dA + A ∧ A wher e we u se t : h → g , the diﬀer ential of the m ap t : H → G , to c onvert B into a g -value d 2-form, and 3. smo oth 2-funct ors hol: P 2 ( M ) → G wher e P 2 ( M ) is the p ath 2-gr oup oid of M . 22 This result was announced by Ba e z and Schreiber [1 6], and a pro of can b e be found in the w ork of Schreiber and W aldorf [88]. This work w as deeply inspired b y the ideas of Breen and Mes sing [29, 30], who considered a s pe cial class of 2-g roups, and omitted the equation t ( B ) = dA + A ∧ A , since their sort of connection did no t a ssign holonomies to surfaces. One should also compar e the closely related work of Ma ck aay , Martins, and Pick en [67, 6 9], and the work of P feiﬀer a nd Gir e lli [76, 58]. In the ab ov e theorem, the ﬁrs t item mentions ‘2-connections’ and ‘2 - bundles’— concepts that we have not deﬁned. But since we are only talking ab out 2- connections o n trivial 2-bundles, we do not need these general concepts yet. F or no w, we can take the third item as the deﬁnition of the ﬁrst. Then the con- ten t of the theor em lies in the diﬀer ential form description of smo oth 2-functors hol: P 2 ( M ) → G . This is wha t we need to understand. A 2-functor of this so rt must assign holono mies b oth to paths and sur- faces. As y ou migh t exp ect, the 1-form A is primarily r e sp onsible for deﬁn- ing holonomies alo ng paths, while the 2-form B is r e sp onsible for deﬁning holonomies for surfaces . But this is a bit of a n oversimpliﬁcation. When co m- puting the holonomy of a sur face, we need to use A as well a s B ! Another s urprising thing is that A and B need to b e r elated by an e quation for the holono m y to b e a 2- functor. If w e ponder how the ho lonomy of a s urface is a ctually computed, we can see why this is s o. W e shall not b e at a ll rigo r ous here. W e just w ant to give a ro ugh intuitiv e idea o f how to compute a holo nomy for a surface, and where the equation t ( B ) = dA + A ∧ A comes fro m. O f c o urse dA + A ∧ A = F is just the curvatur e of the co nnection A . This is a big clue. Suppo se we ar e trying to co mpute the holono m y for a surface star ting from a g -v alued 1-form A a nd an h -v alued 2- form B . Then following the ideas o f calculus, we can try to chop the sur fa ce into man y small pieces, compute a holonomy for each one, and multiply these together s omehow. It is easy to chop a surface in to small squa res. Unfor tunately , the deﬁnition of 2- c ategory doesn’t seem to know anything ab out squar es! But this is not a serious problem. F or example, w e can interpret this square : • • f o o • h O O • k o o g O O α {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ as a 2- morphism α : f g ⇒ hk . W e can then co mpo se a bunch of such 2 - 23 morphisms: • • o o • o o • o o • O O • o o O O • o o O O • o o O O {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ • O O • o o O O • o o O O • o o O O {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ with the help o f a trick called ‘whiskering’. Whiskering is a wa y to co mpo se a 1 - morphism and a 2-mo rphism. Supp ose we wan t to co mp ose a 2-mor phism α and a morphism f that sticks out lik e a whisker on the left: z • y • f o o • x g w w g ′ g g α   W e can do this by taking the horizontal comp osite 1 f ◦ α : z • y • f v v f h h 1 f   • x g w w g ′ g g α   W e ca ll the result f ◦ α , or α left whiskered by f . Similarly , if we hav e a whisker stic k ing out on the right: z • y • g v v g ′ h h α   x • f o o we can take the hor izontal compo site α ◦ 1 f : z • y • g v v g ′ h h α   • x f w w f g g 1 f   and ca ll the re s ult α ◦ f , or α right w hi sk ered by f . With the help of whiskering, we can comp os e 2-mor phisms sha pe d like ar - bitrary p olyg o ns. F or example, suppo se we wan t to horizontally c omp ose tw o squares: • • f o o • f ′ o o • h O O • k o o ℓ O O • k ′ o o g O O α {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ β {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ 24 T o do this, we can left whisker β by f , o btaining this 2 -morphism: f ◦ β : f f ′ g ⇒ f ℓk ′ • • f o o • f ′ o o • ℓ O O • k ′ o o g O O β {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ Then we can r ig ht whisker α by k ′ , obtaining α ◦ k ′ : f ℓ k ′ ⇒ hk k ′ • • f o o • h O O • k o o ℓ O O • k ′ o o α {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ Then we can vertically co mpo se thes e to ge t the desir ed 2 -morphism: ( α ◦ k ′ ) · ( g ◦ β ): f f ′ g ⇒ h k k ′ • • f o o • f ′ o o • h O O • k o o ℓ O O • k ′ o o g O O α {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ β {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ The same sort of tric k lets us v er tically compo s e squares. By iterating these pro cedures w e can deﬁne more complicated comp osites, like this: • • o o • o o • o o • o o • O O • o o O O • o o O O • o o O O • o o O O {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ • O O • o o O O • o o O O • o o O O • o o O O {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ Of course, one ma y wonder if these more co mplicated comp osites are unambigu- ously deﬁned! Luckily they are, tha nk s to as so ciativity and the interch a nge law. This is a nontrivial res ult, ca lled the ‘pasting theore m’ [77]. By this metho d, we can reduce the task of computing hol(Σ) for a large surface Σ to the task o f computing it for lots of small s q uares. Ultimately , o f course, we should take a limit as the squares b ecome sma ller and smaller . But 25 for our nonrigor ous discussion, it is enough to co ns ider a very sma ll sq ua re like this: • • Σ {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ o o O O γ 2 γ 1 W e can think of this square as a 2-morphism Σ: γ 1 ⇒ γ 2 where γ 1 is the path that go es up and then a cross, while γ 2 go es across and then up. W e wish to compute hol(Σ): hol( γ 1 ) ⇒ hol( γ 2 ) . On the one hand, hol(Σ) inv olves the 2 - form B . On the other hand, its sour ce and tar get depend o nly on the 1-fo rm A : hol( γ 1 ) = P exp  Z γ i A  , hol( γ 2 ) = P exp  Z γ 2 A  . So, hol(Σ) canno t hav e the rig ht source a nd target unless A and B ar e related by an equa tion! Let us try to guess this equation. Recall from Theorem 2 that a 2-morphis m α : g 1 ⇒ g 2 in G is determined by a n element h ∈ H with g 2 = t ( h ) g 1 . Using this, we may think of hol(Σ): ho l( γ 1 ) → hol( γ 2 ) as determined by an element h ∈ H with P exp  Z γ 2 A  = t ( h ) P exp  Z γ 1 A  , or in other words t ( h ) = P exp  Z ∂ Σ A  (1) where the loo p ∂ Σ = γ 2 γ − 1 1 go es around the square Σ. F or a very sma ll square, we can appr oximately compute the right hand side us ing Sto kes’ theorem: P exp  Z ∂ Σ A  ≈ exp  Z Σ F  . On the other hand, ther e is an obvious guess for the appr oximate v alue o f h , which is suppo sed to b e built using the 2-for m B : h ≈ exp  Z Σ B  . F or this gues s to yield Equation (1), at lea st to ﬁrst or der in the size of our square, we need t (exp  Z Σ B  ) ≈ exp  Z Σ F  . 26 But this will b e true if t ( B ) = F . And this is the equation that relates A and B ! What have we lear ne d her e? First, for a ny surfac e Σ: γ 1 ⇒ γ 2 , the holonomy hol(Σ) is deter mined b y an element h ∈ H with P exp  Z γ 2 A  = t ( h ) P exp  Z γ 1 A  In the limit wher e Σ is very small, this element h dep ends only on B : h ≈ exp  Z Σ B  . But for a ﬁnite-sized sur face, this formula is no g o od, since it inv olves a dding up B at diﬀerent p oints, which is not a smart thing to do. F or a ﬁnite-s ized surface, h dep ends on A as w ell as B , sinc e we can approximately compute h by c ho pping this surfac e into sma ll sq uares, whiskering them with paths, a nd comp osing them—and the holono mies along these paths are computed using A . T o get the ex act holo nomy ov er a ﬁnite-sized sur face by this metho d, we need to take a limit where we sub divide the sur face into ever smaller squar es. This is the Lie 2 -group analo gue o f a Riemann sum. But for actual ca lculations, this pro ce s s is not very convenien t. Mo re pr actical formulas for computing holonomies ov er surfaces can b e found in the work of Sc hreib er and W aldorf [88], Martins and P icken [69]. 4 Examples and Applications Now let us give so me examples of Lie 2-g roups, and see what higher ga uge theory can do with these ex a mples. W e will build these examples using crossed mo dules. Throughout what follo ws, G is a Lie 2-gr oup whose co rresp onding crossed module is ( G, H , t, α ). 4.1 Shifted A b elian Groups An y g roup G automatically gives a 2 -group where H is tr ivial. Then higher gauge theor y reduces to o rdinary g auge theory . But to see what is n ew ab out higher g a uge theor y , let us instead supp ose that G is the trivia l group. Then t and α are forc e d to b e trivia l, and t is automatically G -e q uiv ariant. On the other hand, the Peiﬀer identit y α ( t ( h )) h ′ = hh ′ h − 1 is no t a utomatic: it holds if and only if H is ab elian! 27 There is also a nice picture pro of tha t H must be ab elian when G is triv ial. W e simply move tw o elements of H around each other using the interc hange law: • • 1 Z Z 1   h   • • 1 Z Z 1   h ′   = • • o o Y Y   h   1   • • o o Y Y   1   h ′   = • • Y Y o o   h   h ′   = • • o o Y Y   1   h ′   • • o o Y Y   h   1   = • • Z Z   h ′   • • Z Z   h   As a side-beneﬁt, we s ee that horizontal and v er tical comp osition m ust be equal when G is trivial. This pro of is called the ‘Eckmann–Hilton argument’, sinc e Eckmann and Hilton used it to show that the se c ond homotopy gr oup o f a space is a be lia n [4 3]. So, we can build a 2-g roup wher e : • G is the trivial group, • H is a ny ab e lian Lie g roup, • α is trivial, and • t is tr ivial. This is ca lled the shi fted version o f H , and denoted b H . In a pplica tions to physics, w e often se e H = U(1). A principal bU(1)-2- bundle is usually called a U(1) gerb e , and a 2 -connection on suc h a thing is usually just called a connection. By Theorem 3, a connection on a trivia l U(1) gerb e is just an o r dinary r eal-v alued 2 - form B . Its holo nomy is g iven b y: hol: • • γ v v 7→ 1 hol: • • Z Z   Σ   7→ exp  i Z Σ B  ∈ U(1) . 28 The bo ok by Brylins ki [31] gives a rather extensiv e introduction to U(1) gerb es and their applications. Murray’s theory of ‘bundle gerb es ’ gives a dif- ferent viewp oint [74, 89]. Here let us discuss t wo places where U(1) gerb es show up in ph ysics. One is ‘multisymplectic geo metry’; the other is ‘2-for m electromagne tis m’. The tw o are closely related. First, let us remember ho w 1-for ms show up in symplectic geometry and electromagne tis m. Suppos e we hav e a po int particle moving in s o me manifold M . At any time its p o sition is a p oint q ∈ M and its momentum is a cotangent vector p ∈ T ∗ q M . As time pass e s, its po s ition and momentum trace out a curve γ : [0 , 1] → T ∗ M . The ac tion o f this path is given by S ( γ ) = Z γ  p i ˙ q i − H ( q, p )  dt where H : T ∗ M → R is the Hamiltonian. But no w suppose the Hamiltonian is zero! Then there is still a nontrivial action, due to the ﬁr s t term. W e can rewrite it as fo llows: S ( γ ) = Z γ α where the 1 -form α = p i dq i is a ca nonical structure o n the c o tangent bundle. W e can think of α as connec- tion o n a trivial U(1)-bundle over T ∗ M . Physically , this connection de s crib es how a quantum par ticle c ha nges phase e ven when the Hamiltonian is zero ! The change in pha se is computed by e x po nentiating the action. So, we have: hol( γ ) = ex p  i Z γ α  . Next, s upp o s e we carr y our pa r ticle around a sma ll lo op γ which b ounds a disk D . Then Stokes’ theorem gives S ( γ ) = Z γ α = Z D dα Here the 2 -form ω = dα = dp i ∧ dq i is the curv ature of the co nnection α . It ma kes T ∗ M in to a s ymplectic man- ifold , that is, a manifold with a closed 2 -form ω satisfying the nondeg e neracy condition ∀ v ω ( u, v ) = 0 = ⇒ u = 0 . 29 The s ub ject of symplectic g eometry is v ast a nd deep, but sometimes this simple po int is neg le cted: the s y mplectic structure de s crib es the change in phase of a quantum particle a s we move it a round a lo op: hol( γ ) = exp  i Z D ω  . Perhaps this justiﬁes calling a symplectic manifold a ‘phase spa ce’, tho ugh his- torically this s eems to be just a coincidence. It may seem str ange to ta lk ab out a q ua ntu m par ticle tra cing o ut a lo op in pha se space, since in quantum mechanics we canno t simultaneously know a particle’s position and momentum. How ever, there is a lo ng line o f work, begin- ning with F eynman, which computes time evolution by an int eg ral over paths in phase spa ce [40]. This idea is a lso implicit in geometric quantization, where the ﬁrst step is to equip the phas e spa ce with a principal U(1)-bundle having a connection whose cur v ature is the symplectic structur e. (Our discussio n s o far is limited to trivial bundles, but everything w e say generalize s to the nont r ivial case.) Next, co nsider a charged particle in an elec tromagnetic ﬁeld. Supp ose that we ca n descr ibe the elec tr omagnetic ﬁeld using a vector p otential A which is a connectio n on tr iv ial U(1) bundles ov er M . Then w e can pull A ba ck via the pro jection π : T ∗ M → M , o bta ining a 2-for m π ∗ A on phase space. In the absence o f any other Hamiltonian, the pa rticle’s a ction a s we mov e it alo ng a path γ in phas e space will be S ( γ ) = Z γ α + e π ∗ A if the particle has charge e . In sho rt, the ele ctr omagnetic ﬁ eld changes the c onne ction on phase sp ac e from α to α + e π ∗ A . Similarly , when the path γ is a lo op bo unding a disk D , we hav e S ( γ ) = Z D ω + e π ∗ F where F = dA is the electro magnetic ﬁeld strength. So, ele c tromagnetism also changes the symplectic structure on phase space from ω to ω + e π ∗ F . F or more on this, see Guillemin and Sternberg [6 0], who also treat the cas e o f nonab elian gauge ﬁelds . All of this has an analog where par ticles a re replaced b y strings. It has been known for some time that just as the electro magnetic vector potential natura lly couples to p o int particles, there is a 2 -form B called the Kalb– Ramond ﬁeld which naturally couples to strings. The action for this c oupling is o btained simply b y integrating B over the s tr ing w orlds hee t. In 1986, Gawedski [54] show ed that the B ﬁeld should b e see n a s a connection on a U(1) gerb e. Later F reed and Witten [52] show ed this viewp oint was crucial for understa nding anomaly cancellation. Howev er, these a uthors did not actually use the w ord 30 ‘gerb e’. The r ole of g erb es was later made explicit by Carey , J ohnson and Murray [34], and even mor e s o by Gawedski and Reis [55]. In short, electro magnetism has a ‘hig he r version’. What ab out sy mplec tic geometry? This also has a higher version, whic h dates back to 1 935 work by DeDonder [39] and W eyl [92]. The idea here is that an n -dimensio nal classica l ﬁeld theory has a kind of ﬁnite-dimensional phase space equipped with a closed ( n + 2)-for m ω which is nondege n e rate in the following sense: ∀ v 1 , . . . , v n +1 ω ( u, v 1 , . . . , v n ) = 0 = ⇒ u = 0 . Such a n -form is called a m ulti symplectic structure , or mor e speciﬁcally , an n -plectic structure F or a nice introduction to m ultisymplectic ge o metry , see the pap e r by Gotay , Isenber g, Ma rsden, a nd Mo nt g omery [5 7]. The link betw een multisymplectic geometry a nd higher e le ctromagnetism was made in a pa pe r by Baez, Hoﬀnung and Roger s [10]. E verything is clos ely analogo us to the stor y for p oint particles. F o r a cla ssical b osonic str ing pro p- agating on Minkowski spacetime of an y dimension, s ay M , there is a ﬁnite- dimensional ma nifold X whic h serves as a kind of ‘pha se spa c e ’ for the string. There is a pr o jection π : X → M , a nd there is a go d-giv en wa y to take any ma p from the string’s w or lds heet to M and lift it to an em b edding o f the w orldshe e t in X . So, let us write Σ for the string worldsheet consider ed as a surface in X . The pha se space X is equipp ed with a 2 -plectic structure: that is, a closed nondegenera te 3 -form, say ω . But in fact, ω = dα for so me 2- form α . Even when the string ’s Hamiltonian is zero , there is a term in the action of the string coming from the integral of α : S (Σ) = Z Σ α. W e may also consider a charged string c o upled to a Kalb–Ramond ﬁeld. This beg ins life as a 2-form B on M , but we may pull it back to a 2-form π ∗ B o n X , and then S (Σ) = Z Σ α + e π ∗ B . In particular, s upp o se Σ is a 2-sphere bo unding a 3-ball D in X . Then b y Stokes’ theorem we hav e S (Σ) = Z D ω + e π ∗ Z where the 3 -form Z = dB is the Kalb–Ra mo nd analo g of the ele c tr omagnetic ﬁeld s trength, and e is the string’s charge. (The Kalb–Ramo nd ﬁeld strength is usually called ‘ H ’ in the ph y sics literature, but that conﬂicts with our usag e o f H to mean a Lie gr oup, so w e shall call it ‘ Z ’.) In s umma ry: the Kalb–R amond ﬁeld mo diﬁes the 2-ple ctic struct u r e on the phase sp ac e of the string . The rea der will note that we hav e coyly refused to 31 describ e the phase space X or its 2-fo r m α . F or this, see the pap er by Ba ez, Hoﬀn ung and Rog ers [10]. In this pap er, we explain how the us ua l dynamics of a classical b osonic str ing coupled to a Kalb–Ramo nd ﬁeld can b e describ ed using m ultisy mplectic g eometry . W e als o expla in how to generaliz e Poisson brack ets from symplectic geometry to multisymplectic geometry . Just as Poisson brackets in symplec tic geometry make the functions on phase space in to a Lie algebr a, Poisson brack ets in mult is ymplectic g eometry g ive rise to a ‘Lie 2- algebra ’. Lie 2-algebr as ar e also imp or tant in higher g auge theory in the sa me wa y tha t Lie algebras a re impor tant for g auge theor y . Indeed, the ‘string 2-g roup’ describ ed in Section 4.6 was co nstructed only after its Lie 2 -algebr a w as found [7]. Later , this L ie 2-algebra w a s see n to arise na turally from m ultisy mplectic geometr y [15]. 4.2 The P oincar´ e 2-Group Suppo se we have a representation α o f a Lie g roup G on a ﬁnite-dimensional vector space H . W e can rega rd H as an ab elian Lie gr o up with addition as the group op eratio n. This lets us regard α a s an a ction of G on this ab elian Lie group. So, we can build a 2-gr oup G where: • G is any Lie g roup, • H is a ny vector space, • α is the representation of G o n H , and • t is tr ivial. In particular, note that the Peiﬀer identit y holds . In this way , w e see that any group repre s entation giv e s a crossed mo dule—s o group repr esentations are secretly 2-gro ups! F or example, if we let G be the Lorentz g roup and let α b e its obvious representation on R 4 : G = SO (3 , 1) H = R 4 we obtain the so-c alled P oincar ´ e 2-group , w hich has the Lo rentz g roup as its group o f morphisms, and the Poincar´ e group as its g roup o f 2-morphisms [1 3]. What is the Poincar´ e 2- group go o d for? It is not cle ar, but there are some clues. Just a s we can study representations of groups on vector spa ces, we can study representations of 2-g roups on ‘2-vector spaces’ [6, 2 4, 38, 48]. The re p- resentations of a g roup are the ob jects o f a catego ry , a nd this so rt of categ ory can be used to build ‘spin foam models’ of bac k g round-free quant um ﬁeld theo- ries [5]. This endeavor has b een most successful with 3 d quantum gravity [53], but e very one working on this sub ject dreams of doing so mething similar for 4d quantum gravit y [8 0]. Going from gro ups to 2-g roups b o os ts the dimensio n o f everything: the representations of a 2-g roup are the ob jects of a 2-c ate gory , and Crane and Shepp ear d outlined a pr ogra m for building a 4-dimensional spin foam 32 mo del starting from the 2-catego ry of r e pr esentations of the Poincar´ e 2-g roup [36]. Crane and Shepp ea r d hoped their model would b e related to qua nt um grav- it y in 4 spacetime dimensions. This ha s not come to pass, at least not y et—but this spin fo a m mo del do es have interesting connections to 4 d physics. The spin foam mo del of 3d quantum gravit y automatica lly includes p oint par ticle s , and Baratin and F reidel have shown that it reduces to the usual theor y of F eynman diagrams in 3d Mink owski spacetime in the limit where the gravitational con- stant G Newton go es to zero [21]. This line of tho ught led Baratin and F r eidel to construct a spin foam mo del tha t is equiv alent to the usual theo ry of F eynman diagrams in 4d Minkowski spacetime [22]. At ﬁrst the ma thema tics underlying this mo del was a bit mysterious—but it now seems clear tha t this mo del is based on the r e presentation theory of the Poincar´ e 2-group! F or a preliminary rep or t on this fas cinating research, see the paper by Bar atin a nd Wise [23]. In shor t, it app ears that the 2-catego ry of repres entations of the Poincar´ e 2-gro up g ives a s pin foam description of quantum ﬁeld theory on 4d Minko ws k i spacetime. Unfortunately , while s pin foam mo dels in 3 dimensions can be ob- tained by quantizing g auge theories, we do not see how to obta in this 4d spin foam mo del by quan tizing a higher g auge theory . Indeed, we kno w of no class i- cal ﬁeld theory in 4 dimensio ns whose solutions a re 2 - connections on a pr inc ipa l G -2-bundle wher e G is the Poincar´ e 2-group. How ever, if we r eplace the Poincar´ e 2-gr oup by a closely related 2- group, this puzzle do es have a nice solution. Namely , if we take G = SO (3 , 1) H = so (3 , 1) and take α to be the adjoint representation, we obtain the ‘tangent 2-gr o up’ of the Lorentz gr oup. As we shall se e , 2-co nnections for this 2 -group a rise naturally as solutions of a 4d ﬁeld theory called ‘top olog ical gr avity’. 4.3 T a ngent 2-Groups W e have seen that an y group repr esentation gives a 2-gro up. But any Lie group G has a repr esentation o n its own Lie algebra: the a djoint repre sentation. This lets us build a 2-g roup from the cross ed mo dule where: • G is any Lie g roup, • H is g r egarded as a vector spac e and thus an ab elian Lie group, • α is the adjoint representation, and • t is tr ivial. W e call this the tangent 2-group T G of the L ie group G . Why? W e hav e already seen that for any Lie 2-gro up, the gro up of all 2-morphisms is the semidirect pro duct G ⋉ H . In the ca se at hand, this semidirect pro duct is just 33 G ⋉ g , with G acting on g via the adjoint repres ent a tion. But a s a manifo ld, this semidirect pro duct is nothing other than the tangent bundle T G of the Lie group G . So, the tangent bundle T G beco mes a group, and this is the gr oup of 2-morphisms of T G . By Theorem 3, a 2 -connection on a trivial T G -2-bundle consists of a g -v alued 1-form A and a g - v alued 2-form B such that the cur v ature F = dA + A ∧ A satisﬁes F = 0 , since t ( B ) = 0 in this case. Where can w e ﬁnd such 2 -connections? W e can ﬁnd them as s o lutions of a ﬁeld theory called 4-dimensiona l B F theory! B F theor y is a clas sical ﬁeld theory that works in a ny dimension. So, take an n -dimensional oriented ma nifold M as o ur s pacetime. The ﬁelds in B F theo r y are a connection A on the trivial principal G -bundle over M , together with a g -v alued ( n − 2)-form B . The action is given by S ( A, B ) = Z M tr( B ∧ F ) . Setting the v a riation of this action equal to zero , we obtain the following ﬁeld equations: dB + [ A, B ] = 0 , F = 0 . In dimension 4, B is a g -v alued 2 -form—and thank s to the second equa tion, A and B ﬁt tog ether to deﬁne a 2-co nnection on the tr ivial T G -2-bundle ov e r M . It may seem dull to study a ga uge theo ry where the equations o f motion imply the connection is ﬂat. But there is still ro o m for some fun. W e see this already in 3-dimensional B F theory , where B is a g -v alued 1-form rather than a 2-form. This lets us pa ck age A and B in to a connection o n the tr ivial T G -bundle ov er M . The ﬁeld equations dB + [ A, B ] = 0 , F = 0 then say precisely tha t this connection is ﬂat. When the g roup G is the Lo rentz group SO(2 , 1 ), T G is the cor resp onding Poincar´ e gro up. With this choice of G , 3d B F theory is a version of 3d gener al relativity . In 3 dimensions, unlike the mor e ph ysic a l 4 d case, the equa tions of general rela tivity say that spacetime is ﬂat in the a bsence of matter. And at ﬁrst glance, 3d B F theory only descr ib es genera l rela tivit y without matter . After all, its so lutions are ﬂat connec tio ns. Nonetheless, we c a n consider 3d B F theory o n a manifold fr om which the worldline of a p oint particle has b een remov ed. In the B ohm–Aharonov e ﬀect, if we carry a charged ob ject around a solenoid, we obtain a nont r ivial phas e even though the U(1) co nnection A is ﬂat outside the so lenoid. Similarly , in 3d B F theory , the connection ( A, B ) will b e ﬂat a wa y from the particle’s worldline, but 34 it can hav e a nontrivial holo no my around a lo op γ that encircles the worldline: γ This holonom y says wha t ha ppe ns when we pa rallel transpor t an ob ject around our point particle. The holonomy is an element o f the P oinca r´ e group. Its conjugacy cla ss describ es the mass and spin of our particle. So, massive spinning po int particles are lurking in the for malism of 3d B F theory! Even b etter, this theor y predicts an upp er bound on the particle’s mass, roughly the P lanck mass. T his is true even cla s sically . This may seem strange , but unlike in 4 dimensions, where we need c , G Newton and ~ to build a quantit y with dimensio ns of length, in 3 -dimensional spa cetime we can do this using o nly c and G Newton . So, ironically , the ‘Planck mass’ do es no t dep end on Planck’s constant. F urthermore, in this theory , particles have ‘ex otic statistics’, meaning that the interc hang e of identical par ticles is g ov erned by the bra id gro up instead o f the s y mmetric group. Particles with exotic sta tistics are also known as ‘any o ns’. In the simplest examples, the a ny ons in 3d gravit y reduce to boso ns or fermions in the G Newton → 0 limit. There is thus a wealth o f interesting phenomena to b e studied in 3d B F theory . See the pap er b y Baez , C r ans a nd Wise [9] for a q uick ov erview, and the work of F reidel, Loua pre and Baratin for a deep treatment of the details [21, 53]. The ca se of 4d B F theo ry is just as interesting, a nd not a s fully explor e d. In this c a se the ﬁeld equa tions imply that A and B deﬁne a 2 -connection on the trivial T G -2 -bundle ov er M . But in fact they say mor e: they s ay precisely that this 2- c onnection is ﬂat . By this we mean t wo things. First, the holo nomy hol( γ ) alo ng a path γ do es no t change when we change this path by a ho motopy . Second, the holonomy ho l(Σ) a long a sur face Σ do es not change when we change this s urface by a homotopy . The ﬁrst fact here is eq uiv alen t to the equation F = 0 . The second is equiv alent to the eq uation dB + [ A, B ] = 0 . When the gr oup G is the Lor entz g roup SO(3 , 1 ), 4 d B F theory is sometimes called ‘top ologica l g ravit y’. W e can think of it a s a simpliﬁed version o f gener al relativity that a c ts mo r e like gravity in 3 dimensions. In particular, w e can copy wha t we did in 3 dimensions, and consider 4d B F theory on a manifold from which the w o rldlines of par ticles and t he worldshe ets of strings hav e been remov ed. Some of w ha t we will do here works for more g eneral groups G , but let us take G = SO(3 , 1) just to be sp ec iﬁc . First consider strings. T ake a 2- dimensional manifold X embedded in a 4- dimensional ma nifold M , and think of X a s the w o rldsheet o f a string. Supp ose 35 we can ﬁnd a sma ll lo op γ that encircles X in such a w ay that γ is co ntractible in M but no t in M − X . If we do 4d B F theory on the spacetime M − X , the holonomy hol( γ ) ∈ SO(3 , 1 ) will not change when we a pply a homotopy to γ . This holonomy describ es the ‘mass densit y ’ of our string [9, 20]. Next, consider par ticle s . T ake a curve C embedded in M , and think of C as the worldline of a particle. Supp os e w e ca n ﬁnd a small 2- sphere Σ in M − C that is contractible in M but no t M − C . W e can think of this 2-s phere as a 2-morphism Σ: 1 x ⇒ 1 x in the path 2 - group oid of M . If w e do 4d B F theory on the spa cetime M − C , the holo nomy hol(Σ) ∈ so (3 , 1) will no t change whe n we apply a homotopy to Σ. So, this holonomy describ es some information a bo ut the pa r ticle—but so far as we know, the physical mean- ing o f this information has no t b een worked out. What if we had a ﬁeld theory whose solutions were ﬂa t 2 -connections for the Poincar´ e 2 -group? Then we would hav e hol(Σ) ∈ R 4 and there would b e a tempting in ter pretation of this quantit y: namely , as the energy-mo mentum of our p o int particle. So , the puzzle p o sed a t the end o f the previous s ection is a ta ntalizing one. One may r ig htly ask if the ‘string s’ describ ed a bove bear any relation to tho se of str ing theory . If they are mer e ly surfaces cut out of spacetime, they la ck the dynamical deg rees o f freedo m nor mally as so ciated to a string. Certainly they do not hav e an a c tion pro po rtional to their s urface ar e a, as for the Polyak ov string. I ndee d, one may ask if ‘area ’ even makes sens e in 4 d B F theory . After all, there is no metric on spacetime: the closes t substitute is the so (3 , 1)-v alued 2-form B . Some of these problems may hav e solutions. F or starter s, when we r emov e a s urface X from our 4-manifold M , the action S ( A, B ) = Z M − X tr( B ∧ F ) is no lo nger gauge- inv a riant: a gauge transfo rmation changes the action by a bo undary term which is an in tegr al over X . W e ca n remedy this b y in tro ducing ﬁelds that live o n X , and adding a term to the action which is an integral ov er X inv olving these ﬁelds. There a re a n umber of ways to do this [14, 49, 59, 50, 73]. F or some, the integral ov er X is prop or tional to the area o f the string worldsheet in the sp ecial case where the B ﬁeld ar ises from a c o tetrad (that is, a n R 4 -v alued 1-form) a s follows: B = e ∧ e 36 where w e us e the is omorphism Λ 2 R 4 ∼ = so (3 , 1). In this case there is close rela- tion to the Nambu–Goto s tring, which has b een ca refully e xamined by F airbairn, Noui and Sa rdelli [50]. This is esp ecia lly intriguing b ecause when B takes the a b ov e form, the B F action beco mes the usua l Palatini action fo r g eneral r elativity: S ( A, e ) = Z M tr( e ∧ e ∧ F ) where ‘tr’ is a suitable nondegenera te bilinea r form o n so (3 , 1). Unfortunately , solutions of Palatini gravit y typically fail to ob ey the condition t ( B ) = F when we tak e B = e ∧ e . So, we cannot construct 2-co nnections in the sense of Theor em 3 from these solutions! If we w ant to treat g eneral relativity in 4 dimensions as a higher gauge theory , w e need other ideas. W e des crib e tw o possibilities at the end of Section 4 .5. 4.4 Inner Automorphism 2-Groups There is als o a Lie 2- group where: • G is any Lie g roup, • H = G , • t is the identit y map, • α is conjugation: α ( g ) h = g hg − 1 . F ollowing Ro b er ts and Schreiber [7 9] we call this the inner automorphi sm 2-group of G , and deno te it by I N N ( G ). W e explain this terminolo gy in the next section. A 2-c o nnection on the trivia l I N N ( G )-2-bundle ov er a ma nifold co ns ists of a g -v alued 1- form A and a g -v alued 2-form B such that B = F since t is now the identit y . Int r iguingly , 2-connections o f this s ort show up as solutions of a slight v a r iant of 4d B F theory . In a move that he later c a lled his biggest blunder , Einstein to ok gener al r elativity and threw an extra term into the equatio ns: a ‘c osmolog ic al constant’ term, which gives the v acuum nonzer o energy . W e can do the same for top olog ical gravit y , or indeed 4 d B F theo ry for any group G . After all, wha t counts as a blunder for E instein mig ht co unt as a go o d idea fo r le s ser mortals s uch as o urselves. So, ﬁx a 4-dimensional oriented manifold M a s o ur spacetime. As in ordinary B F theor y , take the ﬁelds to b e a connection A o n the trivial principal G -bundle ov er M , together with a g -v alued 2-for m B . The action for B F theor y ‘with cosmolog ical constant’ is deﬁned to be S ( A, B ) = Z M tr( B ∧ F − λ 2 B ∧ B ) . 37 Setting the v aria tion of the action equal to zero , we obtain these ﬁeld equations: dB + [ A, B ] = 0 , F = λB . When λ = 0 , these are just the equations we saw in the pr e v ious s e c tion. But let us consider the ca se λ 6 = 0. The n these equatio ns hav e a drastically diﬀeren t character! The Bia nchi identit y dF + [ A, F ] = 0, together with F = λB , automatically implies that dB + [ A, B ] = 0. So, to get a solutio n of this theo ry we simply take any connection A , compute its curv ature F and set B = F /λ . This may seem b oring : a ﬁeld theory where a ny co nnection is a solution. But in fact it has an interesting relation to higher g auge theor y . T o see this, it helps to c ha nge v ariables a nd work with the ﬁeld β = λB . Then the ﬁeld equations b eco me dβ + [ A, β ] = 0 , F = β . An y solution of these equations gives a 2-co nnection on the trivial principal I N N ( G )-2-bundle ov er M ! There is a lso a tantalizing r elation to the cosmolog ic a l constant in genera l relativity . If the B ﬁeld arises from a c o tetrad as explained in the previous section: B = e ∧ e, then the a b ov e ac tio n b ecomes S = Z M tr( e ∧ e ∧ F − λ 2 e ∧ e ∧ e ∧ e ) . When we choose the bilinea r form ‘tr’ co rrectly , this is the actio n for general relativity with a cosmologic a l constant prop ortio na l to λ . There is some e v idence [4] that B F theo ry with nonzer o co smologica l con- stant can b e qua nt ize d to obtain the so-ca lled Crane–Y etter model [35, 37], which is a spin foam mo del based on the category of representations of the quantum g roup asso ciated to G . Indeed, in some cir cles this is taken almost as an article of faith. But a rigo rous arg umen t, or even a fully convincing argument, seems to b e miss ing . So, this issue deserves more study . The λ → 0 limit of B F theory is fascina ting but highly s ingular, s ince fo r λ 6 = 0 a solution is just a c o nnection A , while for λ = 0 a so lution is a ﬂat connection A together w ith a B ﬁeld s uch that dB + [ A, B ] = 0 . At least in some ro ug h in tuitive s e nse, as λ → 0 the group H in the crossed mo dule corres p o nding to I N N ( G ) ‘expands and ﬂattens out’ from the g roup G to its tangent space g . Thus, I N N ( G ) degenerates to the tangent 2- group T G . It would b e nice to make this precise using a 2-gr oup version of the theory of gr oup contractions. 4.5 Automorphism 2-Group s The inner automorphis m gr oup of the previous se ction is closely related to the automorphism 2-group AU T ( H ), deﬁned using the cr ossed mo dule wher e: 38 • G = Aut( H ), • H is a ny Lie group, • t : H → Aut( H ) sends a ny gro up element to the op era tio n of conjugating by that e le ment , • α : Aut( H ) → Aut( H ) is the identit y . W e use the ter m ‘automor phism 2- group’ b ecaus e AU T ( H ) rea lly is the 2- group of symmetries of H . Lie g roups form a 2-categ ory , any ob ject in a 2-categ ory has a 2 -group of symmetries, and the 2-g roup of s ymmetries o f H is naturally a Lie 2-g roup, which is none o ther than AU T ( H ). See [13] for details. A principa l AU T ( H )-2-bundle is usually called a nonab el ian gerb e [28]. Nonab elian ger b es ar e a ma jor test ca s e for ideas in hig her g auge theory . Indeed, almost the who le formalism of 2-co nnections was worked out ﬁrst fo r nonab elian gerb es by Breen and Messing [30]. The one asp ect they did no t consider is the one we hav e fo cused on here: par allel transp ort. Thus, they did not impose the equation t ( B ) = F , which we need to obtain ho lonomies sa tisfying the c onditions of Theorem 3. Nonetheless, the quantit y F − t ( B ) plays an imp o r tant ro le in Breen and Messing’s formalis m: they call it the fak e curv ature . Generalizing their idea s slightly , for any Lie 2-gro up G , we may deﬁne a connection on a trivial pr incipal G -2-bundle to b e a pair consisting of a g - v alued 1-for m A a nd an h -v alued 2-form. A 2-connec tio n is then a connection with v anishing fake curv ature. The relation betw een the automor phism 2-group a nd the inner automor- phism 2-group is nicely explained in the work of Rob erts and Schreib er [79]. As they discuss, for a ny group G there is an exact sequence of 2-gro ups 1 → Z ( G ) → I N N ( G ) → AU T ( G ) → OU T ( G ) → 1 where Z ( G ) is the center of G and O U T ( G ) is the group of o uter a utomorphisms of G , b oth reg arded a s 2-g roups with o nly identit y 2-mor phis ms. Rob erts and Schreiber go on to consider a n analogo us sequence of 3-gr oups constructed starting from a 2-gr oup. Among these, the ‘inner automor phism 3-gro up’ I N N ( G ) of a 2-g roup G plays a sp ecial r ole [86]. The reaso n is that any connectio n on a pr incipal G -2-bundle, not necessarily ob eying t ( B ) = F , gives a ﬂat 3-connection on a principa l I N N ( G )-3-bundle! This in turn allows us to deﬁne a version of parallel transp o r t for particles, strings and 2-br anes . This may give a way to unders tand general relativity in terms of higher gauge theory . As w e hav e already s e en in Section 4.3, Palatini gravity in 4d spacetime inv olves an s o (3 , 1)-v alued 1 -form A and a n so (3 , 1)-v alued 2-form B = e ∧ e . This is precisely the data w e exp ect for a connection on a principal G -2-bundle where G is the tangent 2-group of the Lor entz gr oup. Typically this connectio n fails to ob ey the equation t ( B ) = F . So, it is not a 2-co nnection. But, it gives a ﬂat 3-connection on an I N N ( T SO(3 , 1))-3-bundle. So, we may optimistica lly call I N N ( T SO(3 , 1 )) the grav i t y 3-group . 39 Do e s the gravit y 3-gro up actually shed a ny light on general relativity? The work of Martins and P ick en [70] establishes a useful framew o rk for studying these issues . They deﬁne a path 3-gro upo id P 3 ( M ) for a smo oth manifold M . Given a Lie 3 -gro up G , they describ e 3-connections on the triv ial G -3- bundle ov er M as 3 - functors hol: P 3 ( M ) → G Moreov er , they show how to construct these functors fro m a 1-fo r m, a 2-form, and a 3-form taking v alues in three Lie alg ebras asso cia ted to G . In the ca se where G = I N N ( T SO(3 , 1)) and hol is a ﬂat 3-connection, this data re duce s to a n so (3 , 1)-v alued 1 -form A and an so (3 , 1)-v alued 2-form B . 4.6 String 2-Groups The L ie 2 -groups dis cussed so far are easy to construct. The string 2 -gro up is considerably more subtle. Ultimately it force s up on us a deep er co nception of what a Lie 2-g roup rea lly is, and a more sophisticated appro a ch to hig he r gauge theory . T reated in prop er detail, these topics would c arry us far beyond the limits of this quic k intro ductio n. But it would b e a s hame not to men tion them at a ll. Suppo se w e have a cen tral extension of a Lie gro up G b y a n ab elian Lie group A . In other words, suppose we hav e a short ex act sequence of Lie gr oups 1 → A → H t → G → 1 where the ima ge of A lie s in the c e nt er of H . Then we can constr uct an a ction α o f G on H as follows. The map t : H → G descr ib es H as a ﬁb er bundle ov er G , so choo s e a section of this bundle: that is, a function s : G → H with t ( s ( g )) = g , not necessarily a ho momorphism. T he n s e t α ( g ) h = s ( g ) hs ( g ) − 1 . Since A is included in the cen ter of H , α is indep endent of the choice of s . Thanks to this, we do not need a g lobal smo oth section s to chec k that α ( g ) depe nds s mo othly on g : it suﬃces that there ex ist a lo ca l smo oth section in a neigh b orho od o f ea ch g ∈ G , and indeed this is always true. W e can us e these lo cal sections to deﬁne α g lobally , s inc e they must give the same α on ov erlapping neig hborho o ds. Given all this, we can chec k that t is G -equiv ariant and that the Peiﬀ er ident ity holds. So, we obtain a Lie 2- group where: • G is any Lie g roup, • H is a ny Lie group, • t : H → G makes H into a c e nt r al extensio n of G , • α is given by α ( g ) h = s ( g ) hs ( g ) − 1 where s : G → H is any section. 40 W e call this the centr al e xtension 2-group C ( H t → G ). T o get co ncrete examples, we nee d examples o f central extensions. F o r any choice of G and A , we can alwa ys take H = G × A and use the ‘trivia l’ central extension 1 → A → A × G → G → 1 . F or mor e interesting examples, we need non trivia l central extensions. These tend to arise from problems in quan tiza tion. F or example, suppose V is a ﬁnite- dimensional symple ctic v ector space : that is, a vector space equipp ed with a no ndegenerate antisymmetric bilinear form ω : V × V → R . Then w e can make H = V × R into a Lie g roup called the Heisenberg grou p , with the pr o duct ( u, a )( v , b ) = ( u + v , a + b + ω ( u , v )) . The Heis enberg group plays a fundamental role in quan tum mec hanics , b eca use we can think of V as the phas e space o f a classical p oint particle. If w e let G stand for V r egarde d as an ab elian Lie gro up, then elements of G describ e translations in phase spa ce: that is, transla tions of b oth p o s ition and momen- tum. The Heisenberg gr oup H descr ibe s how these tr a nslations commute only ‘up to a phase’ when we take q uantum mechanics into account: the phase is given by exp( iω ( u , v )). There is a homomorphism t : H → G that forgets this phase information, given b y t ( u, a ) = u . This exhibits H as a central extension o f G . W e thus obtain a central extension 2-gro up C ( H t − → G ), calle d the He isenberg 2-group o f the s ymplectic vector space V . The applications of Heisenberg 2-g roups se e m lar g ely unex plored, and sho uld be worth studying. So far, m uch more work has b een put in to understanding 2- groups arising from central extens ions o f lo op gr oups. The r eason is tha t cen tra l extensions of lo op gro ups play a basic role in str ing theor y and conforma l ﬁeld theory , as nicely expla ined by Pressley and Sega l [7 8]. Suppo se that G is a co nnected and s imply-connected co mpact simple Lie group G . Deﬁne the lo op group Ω G to b e the set of all smo oth paths γ : [0 , 1] → G that sta rt and end at the identit y o f G . This b eco mes a g roup under p oint wise m ultiplica tion, and in fact it is a kind of inﬁnite-dimensional Lie group [71]. F or each in teg er k , ca lled the le v el , the lo o p gr oup has a central extension 1 → U(1) → [ Ω k G t − → Ω G → 1 . These extens io ns ar e all diﬀerent, and all nontrivial except for k = 0 . In physics, they arise b eca use the 2 d ga uge theory calle d the W ess– Zumino–Witten mo del has an ‘anomaly’. The lo op gro up Ω G acts as g a uge transfor mations in the classical version of this theo ry . How ever, when we quantize the theory , we obtain 41 a represe nt a tion o f Ω G only ‘up to a phas e’—that is, a pr o jective represe ntation. This can be understo o d as an hones t repres e ntation of the cent r al extens io n [ Ω k G , wher e the in teger k app ear s in the Lagrang ia n for the W ess–Zumino– Witten mo del. Starting from this cen tra l extension w e can construct a central extensio n 2-gro up called the lev el- k lo op 2-group of G , L k ( G ). This is an inﬁnite- dimensional Lie 2-gro up, mea ning that it comes fr om a crossed mo dule where the gro ups inv olved are inﬁnite-dimensional L ie gro ups, and all the maps are smo oth. Mor eov er, it ﬁts into an exact sequence 1 → L k ( G ) → S T RI N G k ( G ) − → G → 1 where the middle term, the lev el- k string 2-group of G , has very interesting prop erties [8 ]. Since the s tring 2 -group S T RI N G k ( G ) is an inﬁnite-dimensional Lie 2- group, it is a top olog ical 2 -group. There is a wa y to take any topo logical 2- group and squash it down to a top o lo gical gr oup [8, 18]. Applying this trick to S T RI N G k ( G ) when k = 1, we obtain a top olo gical group whose homotopy groups match thos e of G —except for the third ho motopy gro up, which has b een made trivial. In the s p ec ia l case wher e G = Spin( n ), this top ologica l g roup is called the ‘string gro up’, s ince to consistently deﬁne sup ers trings pr opagating on a spin manifold, w e must reduce its structure g roup from Spin( n ) to this group [94]. The string group also plays a r ole in Stolz and T eichner’s work on elliptic co homolog y , which inv olves a notion of para llel tra nsp ort over sur faces [90]. There is a lot o f so phisticated mathematics in volved here , but ultimately m uch o f it should arise from the wa y string 2 -groups a re in volved in the parallel transp ort of strings! The work o f Sa ti, Schreiber and Stasheﬀ [83] pr ovides go o d evidence fo r this, a s do es the work of W aldorf [91]. In fac t, the string Lie 2-group ha d lived through man y previo us incar nations befo re b eing co nstructed as an inﬁnite-dimensional Lie 2 - group. Br ylinski a nd McLaughlin [33] thought of it as a U(1) gerb e ov er the gr oup G . The fact that this ger be is ‘multiplicativ e’ makes it so mething like a group in its own right [32]. This viewp oint ha s also b een explo r ed by Murray and Stevenson [75]. Later, B aez a nd Cr ans [7] constructed a Lie 2-a lgebra s tring k ( g ) corr esp ond- ing to the str ing Lie 2-gr oup. F or pe dagogic a l purp oses, our dis c us sion of Lie 2-gro ups has fo cused solely on ‘s tr ict’ 2-g roups, wher e the 1-mo rphisms satisfy the group axio ms s tr ictly , as equations. How ever, there is also an extensiv e theory of ‘weak’ 2-gro ups, where the 1-morphisms ob ey the gro up ax ioms only up to inv er tible 2-morphisms [1 3]. F ollowing this line of thought, we may also deﬁne weak Lie 2-alg ebras [81], and the Lie 2-a lgebra string k ( g ) is one of these where only the Ja cobi identit y fails to hold strictly . The b eauty of weak Lie 2 -algebr as is that string k ( g ) is very easy to describ e in these terms. In particular , it is ﬁnite-dimensiona l. The har d part is constructing a weak Lie 2-g roup corr esp onding to this weak Lie 2-alg ebra. It is easy to chec k that a ny str ict Lie 2- algebra has a cor resp onding str ict Lie 2-gro up. W eak Lie 2 - algebras are more tricky [13]. Bae z, Crans, Schreiber and Stevenson [8] do dg ed 42 this problem b y showing that the string Lie 2-alg e bra is e qu ivalent (in some precise sense) to a strict Lie 2-a lg ebra, which how ever is inﬁnite-dimensiona l. They then c o nstructed the inﬁnite-dimensional strict Lie 2-gro up cor resp onding to this str ict Lie 2 - algebra . This is just S T RI N G k ( G ) a s des c rib ed a b ove. On the o ther hand, a ﬁnite-dimensional mo del of the string 2 -gro up w as recently introduced by Schommer-Pries [8 5]. This uses an improv ed deﬁnition of ‘weak Lie 2-group’, bas ed on an impo rtant realization: the corr ect maps betw e e n smo oth gr oup oids ar e not the smo oth functors, but s o mething mor e general [64]. W e have a lready mentioned this in our discuss io n o f connection: a smo oth functor ho l: P 1 ( M ) → G is a connection o n the trivial principal G - bundle ov er M , while one of these mo re genera l maps is a connection on a n arbitr ary principal G -bundle over M . If w e take this les son to heart, w e ar e led int o the world o f ‘stacks’—and in that w o rld, we can ﬁnd a ﬁnite-dimensio nal version of the string 2- group. There has also b een pro gress on co nstructing weak Lie n -gro ups from weak Lie n -a lgebras fo r n > 2. Getzler [56] and Henriques [61] hav e developed an approach that w o rks for all n , even n = ∞ . Their a pproach is able to ha ndle weak Lie ∞ -algebr a s of a sort known as ‘ L ∞ -algebra s’. Quite r oughly , the idea is that in an L ∞ -algebra , the Ja cobi identit y holds o nly weakly , while the antisymmetry of the br ack et still holds strictly . In fact, L ∞ -algebra s were develop ed by Stasheﬀ and collab orato rs [68, 84] befo re higher gauge theory b ecame r e c ognized as a sub ject of study . But more recently , Sati, Schreib er, Stas heﬀ [8 2, 83] hav e develop ed a lot o f hig her gauge theory with the help of L ∞ -algebra s. T ha nks to their w ork, it is becoming clear that super string theo r y , sup erg r avit y and even the mysterious ‘ M -theory ’ hav e strong ties to higher ga uge theory . F or example, they a rgue that 11 - dimensional sup ergravity can be seen as a higher gauge theo ry g overned by a certain ‘L ie 3-sup eralgebr a’ whic h they call sugra (10 , 1 ). The n umber 3 here relates to the 2 -brane solutions of 11-dimensio nal sup erg ravit y: just as para lle l transp ort of strings is de s crib ed by 2-connec tio ns, the parallel transp ort of 2- branes is describ ed by 3-connections, which in the supe rsymmetric ca se inv olve Lie 3 - sup eralgebr as. In fa c t, su gra (10 , 1) is o ne of a family o f four Lie 3-s upe r algebra s that ex tend the Poincar ´ e Lie sup eralg ebra in dimensions 4, 5, 7 and 11. These can b e built via a sy stematic construction starting from the four normed division algebr as: the real n umbers, the complex n umbers, the quater nions and the octo nions [11]. These four algebras also g ive rise to Lie 2 -sup eralg ebras extending the Poincar´ e Lie sup era lg ebra in dimensions 3, 4, 6, and 10. The Lie 2 - sup eralgebr as ar e related to sup erstring theories in dimensions 3, 4, 6, a nd 10, while the Lie 3- sup e ralgebr as a re related to sup er-2- brane theories in dimensions 4, 5, 7 and 11. All these theories, and even their relation to division alg ebras, have b een kno wn since the late 19 80s [42]. Higher gauge theor y provides new insights in to the geometry of these theories . In particular, the work of D’Auria , Ca stellani a nd F r´ e [41] can be seen as impli citly making extensive use of Lie n - s upe r algebra s— but this only b ecame clear later, thro ug h the work of Sati, Schreiber a nd Stasheﬀ [83]. 43 Alas, expla ining these fascinating issues in detail would v astly expa nd the scop e of this pa pe r . W e should instea d return to s impler things: ga uge tra ns- formations, c ur v ature, a nd nontrivial 2-bundles. 5 F urther T opics So far our intro duction to higher gaug e theor y has neglected the mos t imp ortant topic of all: gauge tr ansformations! W e hav e also said nothing ab out cur v ature or nontrivial 2-bundles. Now it is time to beg in correc ting these ov ers ights. 5.1 Gauge T ransformations First consider ordinary gauge theor y . Supp ose that M is a manifold a nd G is a Lie group. Then a g auge transformation o n the trivial principal G -bundle ov er M simply amounts to a smo oth function g : M → G, while a connectio n on this bundle ca n b e seen a s a g -v alued 1-form. A g auge transformatio n g acts on a connection A to giv e a new connec tio n A ′ as follo ws : A ′ = g Ag − 1 + g dg − 1 . This form ula mak es literal sens e if G is a gr oup of matrices: then g also consists of ma tr ices, so w e can freely multiply elemen ts of G with elements of g . If G is an arbitra ry Lie group the formula r equires a bit more ca r eful in ter pr etation, but it still makes sense . A w ell- known ca lculation says the curv ature F ′ = dA ′ + A ′ ∧ A ′ of the gauge- transformed connection is just the cur v ature of the original connection conjugated b y g : F ′ = g F g − 1 . In hig her gauge theory the formulas are similar, but a bit mor e complica ted. Suppo se M is a manifold and G is a Lie 2-gr oup with cr o ssed mo dule ( G, H , t, α ). It will be helpful to take everything in this cro ssed mo dule and diﬀerentiate it. Doing this, we ge t: • the Lie algebra g o f G , • the Lie algebra h of H , • the Lie alg ebra homomorphis m t : h → g obtained by diﬀerentiating t : H → G , and • the Lie algebra homomo rphism α : g → aut ( H ) obtained by diﬀeren tiating α : G → Aut( H ). 44 Here aut ( H ) is the Lie algebra of Aut( H ). It is b est to think of this as the Lie algebra of deri v ations of h : tha t is, linear maps D : h → h such that D [ x, y ] = [ D x, y ] + [ x, D y ] . If we diﬀerentiate the tw o equations in the deﬁnition of a crossed mo dule, we obtain the g -equi v ariance of t : t ( α ( x )( y )) = [ x, t ( y )] and the inﬁ ni tesimal Peiﬀer ident i t y : α ( t ( y ))( y ′ ) = [ y, y ′ ] where x ∈ g and y , y ′ ∈ h . In c ase the reader is curio us: we wr ite t and α instead of dt and dα b ecause la ter we will do computations involving these maps and also diﬀerential forms, where d s tands for the exter io r deriv atve. A quadruple ( g , h , t , α ) of Lie algebr as and homomor phis ms ob eying these t wo equatio ns is called a n i nﬁnitesim al cross ed mo dule . Just as crossed mo dules are a way of working with 2 -gro ups, inﬁnitesimal cross ed mo dules a re a w ay of w o rking with Lie 2-alg ebras [7]. Any inﬁnitesimal crossed mo dule comes from a Lie 2-g roup, and this Lie 2-gr oup is unique if we demand that G and H b e connected and simply connected. But we digr ess! W e have in tro duced inﬁnitesimal cross e d mo dules in order to say ho w gaug e transfor mations act o n 2- c o nnections. A gauge transformation of the trivia l G -2-bundle over M consists of t wo pieces of data: • a smo oth function g : M → G , • an h -v alued 1 -form a o n M . Wh y two pieces of data? P er haps this should no t be s o surprising. Remember, a 2 -connection a lso consists of tw o pieces of data: • a g -v alued 1-form A on M , • an h -v alued 2 -form B on M satisfying t ( B ) = F . Breen and Messing [3 0] work ed out how gaug e transformations act on connec- tions o n nonab elian gerb es, a nd their work was later g eneralized to 2-co nnections on a rbitrary pr inc ipa l 2 -bundles [1 6, 88]. Here we merely present the formulas. A gaug e trans formation ( g , a ) ac ts on a 2 -connection ( A, B ) to g ive a new 2- connection ( A ′ , B ′ ) as follows: A ′ = gAg − 1 + g dg − 1 + t ( a ) B ′ = α ( g )( B ) + α ( A ′ ) ∧ a + da − a ∧ a The second for mula requires a bit of expla nation. In the ﬁrs t term w e comp ose α : G → Aut( H ) with g : M → G and obtain an Aut( H )-v alued function α ( g ), 45 which then acts o n the h -v alued 2-form B to give a new h -v alued 2-fo rm α ( g )( B ). In the sec ond term we start by comp osing A ′ with α to obtain an aut ( H )-v a lued 1-form α ( A ′ ). Then we wedge this with a , letting aut ( H ) a c t on h a s par t of this pro cess, a nd obtain an h -v alued 2- fo rm. As a kind of co nsistency c heck and test of our understanding, let us see why the gaug e-transfor med 2 -connection ( A ′ , B ′ ) satisﬁes the equatio n t ( B ′ ) = F ′ . Fir st let us compute the curv ature 2-form F ′ of the gauge-tr ansformed 2-connection: F ′ = dA ′ + A ′ ∧ A ′ = d ( gAg − 1 + g dg − 1 + t ( a )) + ( g Ag − 1 + g dg − 1 + t ( a )) ∧ ( g Ag − 1 + g dg − 1 + t ( a )) This looks lik e a mess—but except for the terms containing t ( a ), this is just the usual mes s we get in o rdinary gauge theory when we compute the curv ature o f a g auge tra nsformed connection. So , we hav e: F ′ = g F g − 1 + d ( t ( a )) + t ( a ) ∧ A ′ + A ′ ∧ t ( a ) − t ( a ) ∧ t ( a ) = g F g − 1 + t ( da ) + [ t ( a ) , A ′ ] − t ( a ) ∧ t ( a ) where we use the fact that d ( t ( a )) = t ( da ) and rewrite A ′ ∧ t ( a ) + t ( a ) ∧ A ′ as a g raded co mm uta to r. On the other hand, we hav e t ( B ′ ) = t ( α ( g )( B )) + t ( α ( A ′ ) ∧ a ) + t ( da − a ∧ a ) The G -equiv ariance of t implies tha t t ( α ( g )( B )) = g t ( B ) g − 1 , and the g - equiv ariance of t implies that t ( α ( A ′ ) ∧ a ) = [ A ′ , t ( a )]. So, we see that t ( B ′ ) = gt ( B ) g − 1 + [ A ′ , t ( a )] + t ( da − a ∧ a ) and thus t ( B ′ ) = F ′ , as desire d. 5.2 Curv ature Suppo se G is a Lie 2-gr oup whose cro s sed module is ( G, H , t, α ), a nd let ( g , h , t , α ) be the cor resp onding diﬀerential cro s sed mo dule. Suppo se we have a connec- tion on the tr ivial G -2-bundle over M : that is, a g -v alued 1-for m A and an h -v alued 2- fo rm B . As in o rdinary gaug e theor y , we may deﬁne the curv ature of this connection to b e the g -v alued 2-for m given by: F = dA + A ∧ A. W e als o hav e a nother g - v alued 2-for m, the fak e curv ature F − t ( B ). Recall from Section 3 that only a connection with v anishing fake curv ature coun ts as a 2-connection. In other words, we need t ( B ) = F to o bta in well-deﬁned parallel transp ort over surfaces. 46 W e may also deﬁne the 2-curv ature of a connection in higher gauge theor y . This is the h -v alued 3-form given by: Z = dB + α ( A ) ∧ B . In the seco nd term here, w e compo se α : g → a ut ( H ) with the g -v alued 1-form A and obtain an aut ( H )-v alued function α ( g ). Then we wedge this with B , letting aut ( H ) act on h as par t of this pro cess, and obtain an h -v alued 2- form. The int uitive idea o f 2-curv ature is this: just as the cur v ature des c r ib es the holonomy of a co nnection around an inﬁnitesimal lo op, the 2-curv ature describ es the holono m y of a 2-co nnection ov er an inﬁnitesimal 2-spher e. This ca n be made precise using formulas for holo nomies ov er surfaces [69, 88] If the 2-curv ature o f a 2 -connection v a nishes, the holonomy ov er a surface will not change if we apply a smo oth homotopy to that surfac e while keeping its edges ﬁxed. A 2 -connection whos e curv ature and 2-curv ature b oth v anish truly deserves to be calle d ﬂat . W e hav e seen ﬂat 2- connections already in our discussio n of 4-dimensional B F theory in Section 4 .3: the solutions of this theory are ﬂat 2 -connections. 5.3 Non trivial 2-Bundles So far w e ha ve implicitly b een look ing at 2-connections on trivia l 2 -bundles. This is ﬁne lo cally . But there are also interesting issues in volving nontrivial 2-bundles, which become cr ucial when we work g lobally . A car eful tr eatment of 2 -bundles would r equire some work, and the reader int er ested in this topic would do well to start with Mo erdijk’s intro ductory pap er on ‘sta cks’ and ‘gerb es ’ [72]. Here we take a less sophisticated approa ch: we simply describ e how to build a principal 2-bundle and put a 2 - connection on it. Since we do not say when tw o principal 2- bundles built this wa y are the ‘same’, our treatment is incomplete. The reader can ﬁnd more details e lsewhere [18, 19, 25, 28, 29, 30, 88]. W e warn the reader that almos t every pap er in the literature uses diﬀerent notation, sign conv entions, and so forth. First r ecall ordinar y gauge theo ry: supp ose G is a Lie group and M a ma n- ifold. In this case we can build a principal G -bundle ov er M using tr ansition functions. Fir st, wr ite M as the union of op en sets o r patc hes U i ⊆ M : M = [ i U i . Then, cho ose a smooth transition function on each double intersection o f patches: g ij : U i ∩ U j → G. These transition functions giv e g auge tra ns formations. W e can build a principal G -bundle ov er all of M by gluing tog ether trivial bundles ov er the patc hes with the help o f these gauge tra nsformations . Howev er, this pro cedure will only 47 succeed if the tra nsition functions sa tisfy a consistency condition on each triple int er section: g ij ( x ) g j k ( x ) = g ik ( x ) for all x ∈ U i ∩ U j ∩ U k . This eq uation is called a co cycle condition . W e can visualize it as a tria ngle: • • • g ik o o g jk X X ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ g ij   ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ where we suppres s the v ariable x for the sake o f reada bilit y . The idea is that this tr iangle should ‘commute’: the dir e ct way o f ide ntifying po ints in the trivial bundle in the i th pa tch to points in the tr ivial bundle ov er the k th patch should match the indir ect wa y which pr o ceeds via the j th patch. A similar but more elab o r ate recip e works for higher gaug e theory . Now let G b e a Lie 2 -group with crossed mo dule ( G, H, t, α ). T o build a G -2-bundle, we start b y choos ing transitio n functions on double intersections of patches: g ij : U i ∩ U j → G How ever, now it makes sense to replace the equation in the co cy c le co ndition by a 2- morphism! So , for each triple intersection we cho ose 2- morphisms in G : γ ij k ( x ): g ij ( x ) g j k ( x ) ⇒ g ik ( x ) depe nding smo othly on x ∈ U i ∩ U j ∩ U k . W e can again visualize these as triangles: • • • g ik o o g jk X X ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ g ij   ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ γ ijk   But now we de ma nd that these 2-mor phisms themselves ob ey a co c ycle condition on quadruple in ter sections o f patches. As we ascend the ladder of hig her ga uge theory , triangles b ecome tetrahedr a a nd then higher-dimensiona l simplexes. In 48 this case, the co cyc le co ndition says that this tetrahedron co mm utes : • • • • g kl O O g jk o o g ij   g il o o g ik   g jl _ _ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ γ ijl   ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ γ jk l {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ γ ijk  # γ ikl   By sa ying that this tetrahedron ‘comm utes’, we mean that the comp os ite o f the front t wo sides equals the comp osite of the back t wo s ides : • • • • g kl O O g jk o o g ij   g il o o g jl _ _ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ γ ijl   ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ γ jk l {  ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ = • • • • g kl O O g jk o o g ij   g il o o   ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ g ik γ ijk  # ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ γ ikl   ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ W e need whiskering to comp ose the 2 -morphisms in this diagr am, as e xplained near the e nd of Sectio n 3. So in equations, the tetr a hedral co cy cle condition says that: γ ij l · ( g ij ◦ γ j kl ) = γ ikl · ( γ ij k ◦ g kl ) where · stands for vertical comp os ition a nd ◦ stands for whiskering. W e can des crib e this co cy cle co ndition in a more down-to-earth manner if we use Theor em 2 , which s ays that a 2- morphism γ ij k : g ij g j k ⇒ g ik is the sa me as an element h ij k ∈ H such that t ( h ij k ) g ij g j k = g ik . This theorem also gives formulas for vertical and horizontal comp os ition in terms of the groups G a nd H . Since whiskering by a morphism is horizo nt a l comp osition with its iden tity 2-mor phis m, we can also ex pr ess whiskering in these terms. So, a little calculation—a wonderful exercise for the would-be higher gauge theo rist—shows that: h ij l α ( g ij )( h j kl ) = h ikl h ij k where α is the action of G on H . There is no nee d to have g ii = 1 in this for malism; we should really choo se a 2-morphism from g ii to 1. How ever, without lo s s of g enerality , we can as sume 49 that g ii = 1 and set this 2-mo rphism equal to the identit y . W e can also assume that h ij k = 1 whenever tw o or more of the indices i, j, k ar e equal. The reason is that Ba rtels [2 5] ha s shown that any principal 2-bundle is equiv alent to o ne for whic h these simplifying ass umptions ho ld. W e will make these as sumptions in wha t follows. F o r the full story without these assumptions, see Schreiber and W aldorf [88]. Now conside r connections. Aga in, it helps to beg in by r eviewing the story for ordinar y ga uge theor y . Suppo se we have a manifold M wr itten as a union of patches U i , a nd suppo se we hav e principal G -bundle ov er M built using transition functions g ij . T o put a connection on this bundle, we ﬁrst put a connection on the trivial bundle over each patch: that is, for each i , we choo s e a g -v alued 1 -form A i on U i . But then we must c heck to see if thes e ﬁt together to give a well-deﬁned co nnection on all of M . F or this, we need the ga uge transform of the connection on the j th patc h to equal the connection on the i th patch: A i = g ij A j g − 1 ij + g ij dg − 1 ij on ea ch double intersection U i ∩ U j . The s to ry is similar for 2-connections. Suppose w e ha ve a pr incipal G -2 - bundle over M built using tra ns ition functions g ij and h ij k as descr ibe d ab ov e. T o equip this 2-bundle with a 2-connection, we ﬁrst put a 2 -connection o n the trivial 2-bundle ov er each patch. So, on each op en set U i we choose a g -v alued 1-form A i and an h -v alued 2-for m B i with t ( B i ) = F i . But then we must ﬁt these tog ether to get a 2-c onnection on a ll of M . F or this, we should follo w the ideas from Section 5 .1 on how ga uge transfor- mations work in higher g auge theory . So, we cho o se an h -v alued 1 -form a ij on each double intersection U i ∩ U j , and r equire that A i = g ij A j g − 1 ij + g ij dg − 1 ij + t ( a ij ) B i = α ( g ij )( B j ) + α ( A i ) ∧ a ij + da ij − a ij ∧ a ij . These e q uations say that the 2-connection ( A i , B i ) is a g auge-tra nsformed v er - sion o f ( A j , B j ). The a pp earance o f A i on the r ight-hand side of the second equation is not a typo! Finally , the 1-fo rms a ij m us t ob ey a consistency condi- tion o n tr iple intersections: h − 1 ij k α ( A i )( h ij k ) + h − 1 ij k dh ij k + α ( g ij )( a j k ) + a ij = h − 1 ij k a ik h ij k (2) Where do es this consistency co ndition c o me fr om? Indeed, what do es it even mean? W e have not yet deﬁned ‘ h − 1 α ( A )( h )’ when A ∈ g and h is a n ele ment of the gr oup H . W e could systematically derive this condition fro m a more conceptual ap- proach to 2 -connections [1 6, 88], but it will be mar ginally less str essful to mo- tiv a te it as follows. F or every tr iple in ter section U i ∩ U j ∩ U k we hav e three 50 equations r elating A i , A j and A k : A i = g ij A j g − 1 ij + g ij dg − 1 ij + t ( a ij ) A j = g j k A k g − 1 j k + g j k dg − 1 j k + t ( a j k ) A k = g ki A i g − 1 ki + g ki dg − 1 ki + t ( a ki ) The ﬁrst equation ex presses A i in terms of A j . W e can substitute the second equation in the ﬁrst to get a fo rmula for A i in terms o f A k . Then we can use the third equation to get a for mula for A i in ter ms of itself ! W e would like to b e able to simplify this formula to get simply A i = A i . The consis tency condition, Equation (2 ), ensur es that w e can do this. This calculatio n is a bit of a work o ut; le t us see how it go es. W e b egin by doing the s ubstitutions: A i = g ij A j g − 1 ij + g ij dg − 1 ij + t ( a ij ) = g ij  g j k A k g − 1 j k + g j k dg − 1 j k + t ( a j k )  g − 1 ij + g ij dg − 1 ij + t ( a ij ) = g ij  g j k  g ki A i g − 1 ki + g ki dg − 1 ki + t ( a ki )  g − 1 j k + g j k dg − 1 j k + t ( a j k )  g − 1 ij + g ij dg − 1 ij + t ( a ij ) . Then we do a bit of simpliﬁcation: A i = g ij g j k g ki A i ( g ij g j k g ki ) − 1 + g ij g j k g ki d ( g ij g j k g ki ) − 1 + g ij g j k t ( a ki )( g ij g j k ) − 1 + g ij t ( a j k ) g − 1 ij + t ( a ij ) . Since t ( h ij k ) g ij g j k = g ik and by our assumptions g − 1 ik = g ki , we hav e g ij g j k g ki = t ( h ij k ) − 1 so 0 = t ( h ij k ) − 1 A i t ( h ij k ) − A i + t ( h ij k ) − 1 dt ( h ij k ) + t ( h ij k ) − 1 g ik t ( a ki ) g − 1 ik t ( h ij k ) + g ij t ( a j k ) g − 1 ij + t ( a ij ) . (3) Now, if G is a matrix group, we can freely multip ly group elements and Lie algebra elements. Then for any h ∈ H and A ∈ g we have t ( h ) − 1 At ( h ) − A = t ( h ) − 1 [ A, t ( h )] . (4) This will allow us to simplify the ﬁrst tw o terms in Equation (3). Mo reov er , for any g ∈ G , h ∈ H we hav e t ( h − 1 α ( g ) h ) = t ( h ) − 1 t ( α ( g )( h )) = t ( h ) − 1 g t ( h ) g − 1 . T aking g = exp( sA ) for A ∈ g and diﬀerentiating this equation with resp ect to s at s = 0 , we get t ( h − 1 α ( A )( h )) = t ( h ) − 1 At ( h ) − A (5) 51 where o n the left side α means the deriv ative o f the map α : G × H → H with resp ect to its ﬁr st argument, while t is the deriv ative of t . Here are we extending our previous deﬁnitions o f α and t . Combining E quations (4) and (5) we see that t ( h ) − 1 [ A, t ( h )] = t  h − 1 α ( A )( h )  . In fact this re s ult holds even when G is a non-matrix Lie group, as long as we carefully make sense of b oth sides. Using this result, we can re w r ite E quation (3) as follows: 0 = t ( h − 1 ij k α ( A i )( h ij k )) + t ( h ij k ) − 1 dt ( h ij k ) + t ( h ij k ) − 1 g ik t ( a ki ) g − 1 ik t ( h ij k ) + g ij t ( a j k ) g − 1 ij + t ( a ij ) Each term in the ab ove equation is t applied to a n h -v alued 1-for m. W riting down all these h -v alued 1-for ms, we see that the a b ove equation will b e true if this condition ho lds: 0 = h − 1 ij k α ( A i )( h ij k ) + h − 1 ij k dh ij k + h − 1 ij k α ( g ik )( a ki ) h ij k + α ( g ij )( a j k ) + a ij This is o ur consistency condition in disguise! T o remov e the disguise, let us simplify it a bit further. When i = j this condition r educes to a ik + α ( g ik )( a ki ) = 0 . Reinserting this result we o bta in 0 = h − 1 ij k α ( A i )( h ij k ) + h − 1 ij k dh ij k − h − 1 ij k a ik h ij k + α ( g ij )( a j k ) + a ij V oil` a! This is clear ly equiv alent to the cons istency condition we stated in the ﬁrst pla ce, E quation (2): h − 1 ij k α ( A i )( h ij k ) + h − 1 ij k dh ij k + α ( g ij )( a j k ) + a ij = h − 1 ij k a ik h ij k Now let us consider some ex a mples. Reca ll from Section 4.1 that bU(1) is 2-gro up with one morphism a nd U(1) as its group o f 2-mor phisms. A U(1) gerb e is pr incipal bU(1)-2 - bundle. Let’s lo ok a t principal U(1)-bundles a nd then U(1) g erb es, to g et a feel for how they are similar and how they diﬀer. T o build a principal U(1)-bundle with a connection on it, we choose transi- tion functions g ij : U i ∩ U j → U(1) such that g ij g j k = g ik on each triple intersection. T o put a connection on this bundle, we then choos e a 1 -form A i on ea ch patch suc h that A i = A j + g ij dg − 1 ij 52 on ea ch double intersection U i ∩ U j . The cur v ature o f this connectio n is then F i = dA i on the i th patch. Note that F i = F j on U i ∩ U j , so we get a well-deﬁned curv ature 2-fo rm F on all o f M . T o build a U(1) gerb e, w e choose transition functions h ij k : U i ∩ U j ∩ U k → U(1) such that h j kl h ij l = h ikl h ij k on each quadruple in tersection. Remem b er, the 2-group bU(1) ha s only one morphism, the iden tity , so the transitio n functions g ij are trivial and can b e ignored. T o put a 2 -connection o n this g e rb e, we must ﬁrst choose a 2 - form B i on each patc h. Then w e m us t choose a 1-forms a ij on each double intersection. W e require that B i = B j + da ij on ea ch double intersection, and a ij + a j k = a ik + h ij k dh − 1 ij k on ea ch triple intersection. The 2-curv ature of this 2- c onnection is then Z i = dB i on the i th patc h. Note that Z i = Z j on U i ∩ U j , so we g e t a well-deﬁned 2-curv ature 3 -form Z on all o f M . There is a nice link b e t ween U(1) gerb es and cohomolog y , which in fact is the reason they were in vented in the ﬁrst place. F or any principa l U(1)-bundle with connection, the cur v ature F is i n tegral : Z Σ F ∈ 2 π Z for any closed surface Σ mapp ed into M . In additio n, F is closed: dF = 0 . Conv ersely , any clo sed, integral 2-for m F on M is the curv ature of so me con- nection o n a pr incipal U(1)-bundle over M . Two diﬀerent connections on the same bundle hav e cur v ature 2-for ms that diﬀer by a n ex act 2-for m, s o we get a well-deﬁned element of the de Rham cohomolo gy H 2 ( M , R ) fr om a principal U(1)-bundle. This idea can b e reﬁned further, and the upshot is that principal U(1)-bundles ov er M are clas s iﬁed by the cohomo logy g roup H 2 ( M , Z ). Similarly , for any U(1) gerb e, the c ur v ature 3-for m Z is closed and integral, where the la tter ter m now means that Z Y Z ∈ 2 π Z 53 for a ny closed 3 -manifold Y mapp ed in to M . Conv er sely , any such 3- form is the 2-curv ature of some 2-co nnec tio n o n a U(1) gerb e ov er M —and in fact, U(1) gerb es over M are class iﬁed by H 3 ( M , Z ). This is just the beginning o f a long er tale : namely , the s to ry of characteristic classes in hig he r gauge theory [18 , 83]. Indeed, thoug h higher ga uge theory is only in its infancy , there is muc h more to say . But our sto ry ends here. W e invite the r eader to g o further. Ac knowledge ments This paper is base d on J H’s notes of lectur es given by JB at the 2nd Sc ho o l and W orkshop on Quantum Gravit y and Quantum Geometry , held as pa rt of the 2010 Corfu Summer Institute. W e thank George Zoupanos, Harald Grosse, and every one else inv olved with the Corfu Summer Institute for making our sta y a ple a sant a nd pro ductive o ne. W e thank Urs Schreiber for many discussions of hig her gauge theo ry , and thank Tim v an Beek, David Ro be r ts and E llio t Schn e ide r for catching some errors . This r esearch was partially supp orted by an FQXi grant. References [1] P . Aschieri, L. Cantini, and B. Jurˇ co, Nonab elian bundle gerb es, their diﬀerential geometry and gauge theory , Comm. Math. Phys. 254 (2005 ), 367–4 00. Also av ailable as a rXiv:hep-th/03 12154. [2] P . Aschieri and B. Jurˇ co , Ger b es , M5-br ane a nomalies a nd E 8 gauge theory , JHEP 10 (2004) 0 6 8. Also av ailable as arXiv:hep-th/ 0409 2 00 . [3] A. Asht e k ar and J. Le wandowski, Diﬀeren tial geometr y on the space of connections via gr aphs a nd pro jective limits, J. Geom. Phys. 17 (199 5), 191–2 30. Also av ailable as a rXiv:hep-th/94 12073. [4] J. Ba ez, F our -dimensional B F theory as a top olog ic al qua ntum ﬁeld theory , Lett. Math. Phys. 38 (1996), 1 2 9–14 3. [5] J. Bae z, Spin foam mo dels, Class. Qua ntum Grav. 15 (199 8), 1827– 1858 . Also av ailable as arXiv:gr-q c/970 9052 . J. Baez, An in tro ductio n to s pin foam mo dels of B F theory and quan- tum gravit y , in Geometry and Quantum Physics , eds. H. Gausterer and H. Grosse, Springer, Berlin, 200 0, pp. 25–93 . [6] J. Baez , A. Baratin, L. F reidel and D. Wise, Inﬁnite-dimensional repr esen- tations of 2-gro ups . Av ailable a s a rXiv:081 2.496 9 . [7] J. Baez and A. Crans, Higher dimensional algebra VI: Lie 2-algebra s, The- ory and Applications of Categories 12 (2004), 4 92–5 3 8. Also av a ilable as arXiv:math/03 0726 3 . 54 [8] J. Ba ez, A. Crans, U. Schreiber and D. Stevenson F rom lo o p g roups to 2- groups, HHA 9 (200 7), 101– 135. Also av ailable a s a rXiv:math.QA/05 0412 3 . [9] J. Ba ez, A. Cra ns and D. Wise, Exotic s tatistics for string s in 4d B F theory , Adv. Theor. Math. P hys. 11 (2007), 70 7–74 9. Also av ailable as arXiv:gr- qc/06 0 3085 . [10] J . Baez, A. Hoﬀn ung, C. Rogers , Categoriﬁed symplectic geometry and the classical string, Co mm. Ma th. Phys. 293 (2010), 701– 7 15. Also av a ilable as arXiv:080 8.0246. [11] J . Baez a nd J. Huerta, Division algebra s and sup er symmetry II. Av ailable as arXiv:100 3.3436. [12] J . Baez a nd A. Lauda , A prehistor y of n -catego rical physics. Av a ilable as arXiv:090 8.246 9 . [13] J . B a ez and A. Lauda, Higher dimensional alge br a V: 2-gro ups, The- ory and Applications of Categories 12 (2004), 4 23–4 9 1. Also av a ilable as arXiv:math/03 0720 0 . [14] J . Baez and A. Perez, Quantization of strings and bra nes c oupled to B F theory , Adv. Theo r. Ma th. Phys. 11 (2007 ), 1–19. Also av aila ble as arXiv:gr- qc/06 0 5087 . [15] J . Baez and C. Rog ers, Ca tegoriﬁed symplectic geo metry and the str ing Lie 2-algebr a, to app ea r in Homology , Homotopy and Applica tions . Av a ilable as arXiv:090 1.4721. [16] J . Baez and U. Sc hreib er, Higher gauge theory , in Catego ries in Al- gebra, Geo metry and Mathematica l P hysics , eds. A. Davydo v et al, Contemp. Math. 431, AMS, P rovidence, 2007 , pp. 7 –30. Also av a ilable arXiv:math/05 1171 0 . [17] J . Baez and S. Sawin, F unctional integration on spa c es of connections , J. F unct. Analysis 150 (1997 ), 1- 26. Also av a ila ble as arXiv:q- a lg/95 0702 3 . [18] J . Baez and D. Stevenson, The classifying space o f a top ologica l 2-g roup, in Algebraic T op olog y: The Ab el Symp os ium 2 007 , eds. Nils Baas , Eric F riedlander, Bjørn Jahren and Paul Arne Ø stvær, Springer, Be rlin, 2009 . [19] I. Bakovic and B. Jurˇ co, The c la ssifying top os o f a top ologica l bica tegory . Av a ilable as arXiv:09 02.17 5 0 . [20] A. P . Balachandran, F. Lizzi and G. Spara no, A new appr oach to string s and sup erstrings , Nucl. Phys. B277 (1986 ), 359- 3 87. [21] A. Bara tin and L. F reidel, Hidden quantum gravit y in 3d F eynman di- agrams , Class . Quant. Gra v. 24 (2 007), 1993–2 026. Also a v ailable as arXiv:gr- qc/06 0 4016 . 55 [22] A. Baratin and L. F reidel, Hidden q ua ntum gravit y in 4d F eynman dia- grams: emergence of spin foa ms, Class . Qua nt . Gr av. 24 (2 007), 20 27–2 060. Also av ailable as arXiv:hep-th/06 11042 . [23] A. Bar atin and D. Wise, 2-Group representations for spin foams. Av ailable as arXiv:091 0.1542. [24] J . W. Barrett and M. Mac k aay , Categorica l repr esentations of cate- gorica l gro ups, Th. Appl. Ca t. 16 (20 06), 5 29–5 57 Also a v ailable as arXiv:math/04 0746 3 . [25] T. B artels, Higher gauge theo ry: 2 - bundles. Av ailable a s arXiv:math.CT/0 4 1032 8 . [26] J ean B´ ena bo u, Int r o duction to bicategor ie s , in Re p o rts of the Midwest Category Seminar , Springer, Berlin, 1967, pp. 1–77. [27] F. Borceux, Handbo o k of Catego rical Algebr a 1: B asic Ca tegory Theory , Cambridge U. Press , Cambridge, 1994 . [28] L . Breen, Notes on 1- a nd 2-ger b es, in T ow ards Higher Ca tegories , eds. J . Baez and P . May , Spring er, Berlin, 20 09, pp. 193–235 . Also av ailable as arXiv:math/06 1131 7 . [29] L . Br een, Diﬀerential ge ometry of gerb es a nd diﬀerential forms. Av a ilable as arXiv:080 2.1833. [30] L . Breen and W. Messing, Diﬀeren tial geometry of g e rb es, Adv. Math. 198 (2005), 732– 846. Av a ilable as a rXiv:math.AG/0106083. [31] J .-L. Brylinsk i, Lo op Spa c es, Character istic Class e s and Geometr ic Quan- tization , Birkh¨ auser, Bo ston, 19 93. [32] J .-L. Brylinski, Diﬀerentiable cohomology of g auge groups. Av ailable a s arXiv:math.DG/001 1069. [33] J .-L. Brylinski and D. A. McL a ughlin, The geometr y of degree-four char- acteristic clas ses and of line bundles on lo op s pa ces I, Duk e Ma th. J. 75 (1994), 603– 638. J.-L. Brylinski and D. A. Mc L a ughlin, The geo metr y of degree-four char- acteristic class e s a nd o f line bundles on lo op spac e s II, Duk e Ma th. J. 83 (1996), 105– 139. [34] A. L. Carey , S. Jo hnson and M. K. Murray , Holono my on D-branes, Jour. Geom. Phys. 52 (2004), 186– 216. Also av ailable as arXiv:hep-th/020 4199. [35] L . Cr ane, L. Kauﬀman and D. Y etter, State-sum inv aria nt s of 4 -manifolds I. Av a ilable as a rXiv:hep-th/94 09167. [36] L . Crane a nd M. D. Shepp eard, 2-Categorica l Poincar´ e repr esentations a nd state sum applications, av ailable as arXiv:math/03 0644 0 . 56 [37] L . Crane and D. Y etter, A categ orical co nstruction o f 4 d TQFTs, in Quan- tum T op o lo gy , eds. L. Kauﬀman and R. Baadhio, W orld Scient iﬁc, Singa- po re, 1993, pp. 1 20-1 30. Also av ailable a s a rXiv:hep-th/93 0106 2 . [38] L . Crane and D. N. Y etter, Measurable ca tegories and 2- g roups, Appl. Ca t. Str. 13 (2005), 501–5 16. Also av ailable as arXiv:math/ 03051 76 . [39] T. DeDonder, The orie Invariantive du Calcul des V ariations , Gauthier– Villars, Paris, 1935 . [40] C. DeWitt-Morette, A. Mahesh wari, and B. Nelson, P a th in tegra tion in phase space, Gen. R elativity Gr avitation 8 (1 977), 5 81–5 93. [41] L . Castellani, R. D’Auria and P . F r´ e, Sup erg ravit y and Sup ers tr ings: A Geometric Perspective , W orld Scientiﬁc, Singap o r e, 199 1. [42] M. J . Duﬀ, Sup ermembranes: the ﬁrst ﬁfteen w eeks, Class. Quantum Gr av. 5 (1988), 189–20 5. Also av ailable a t h h ttp:// www-lib.kek.jp/cgi-bin/img index?87084 25 i . [43] B . Eckmann and P . Hilton, Group-like structures in ca tegories, Ma th. Ann. 145 (1962), 227-2 55. [44] S. Eilenberg and S. Mac Lane, General theo ry o f na tural e q uiv alences, T rans. Amer. Math. So c . 58 (1 945), 231– 294. [45] C. Ehr e s mann, Ca t´ egories top ologiq ues et cat´ egories diﬀerentiables, Coll. G ´ eom. Diﬀ. Globale (Br uxelles, 1958 ) , Cen tre Belge Rec h. Math. Louv ain, 1959, pp. 137–15 0. [46] C. Ehre s mann, Cat´ egor ies structur´ ees, Ann. Ecole Norm. Sup. 80 (19 63), 349–4 26. [47] C. E hresmann, Cat´ egorie s et Structures , Duno d, Paris, 1965 . [48] J . Elgueta, Representation theor y of 2-gr oups on K apranov and V o evo d- sky 2-vector spa ces, Adv. Math. 213 (2007 ), 53–92. Also av ailable as arXiv:math/04 0812 0 . [49] W. F airbairn a nd A. Perez, E x tended matter coupled to B F theory , Adv. Theor. Ma th. P hys. D78 :02 4013 (2 008). Also av ailable a s ar Xiv:0709 .4235 . [50] W. F air bairn, K. Noui and F. Sardelli, Canonical a nalysis of a lgebraic string actions. Av a ilable as arXiv:09 08.09 53 . [51] M. F orr ester-Ba r ker, Group ob jects and in ter nal categories, av ailable as arXiv:math.CT/0 2 1206 5 . [52] D. S. F reed a nd E. Witten, Anomalies in string theory with D-B r anes, § 6: Additional r emarks, Asian J. Math. 3 (19 9 9) 819– 852. Also a v aila ble a s arXiv:hep-th/99 0718 9 . 57 [53] L . F reidel and D. Lo uapre: Ponzano–Regg e mo de l r evisited. I: Gauge ﬁx- ing, obser v ables and interacting spinning particles, Class . Quan t. Grav. 21 (2004), 5685 –572 6. Also av aila ble a s a rXiv:hep-th/04 0107 6 . L. F reidel and D. Louapre: Ponzano–Regg e mo de l revis ited. I I: E quiv alence with Chern–Simons. Also av ailable as a rXiv:gr- qc/041 0141 . L. F re idel and E . Livine: P onza no–Regg e mo del rev is ited. I I I: F eynman diagrams a nd eﬀective ﬁeld theor y . Also av aila ble as a rXiv:hep-th/05 0210 6 . [54] K . Gaw edzki, T op ologica l actions in tw o-dimensiona l quantum ﬁeld theo- ries, in Nonper turbative Quantum Field Theory , eds. G. t’Hoo ft, A. Jaﬀe, G. Ma ck P . K. Mitter and R. Stora, Plenum, New Y ork, 19 88, pp. 10 1–14 1. [55] K . Gaw edzki and N. Reis, WZW br anes and gerb es, Rev. Math. Phys. 14 (2002), 1281 –133 4. Also av aila ble a s a rXiv:hep-th/02 0523 3 . [56] E . Getzler, Lie theory for nilpo tent L ∞ -algebra s. Av aila ble as arXiv:math/04 0400 3 . [57] M. Gotay , J. Isenberg , J. Mar sden, and R. Montgomery , Momentu m maps and classical r elativistic ﬁelds. Part I: cov ariant ﬁeld theory , av ailable as a rXiv:physics/9801 019 . [58] F. Girelli and H. Pfeiﬀer, Higher g auge theory - diﬀerential v er sus integral formulation, Jour. Math. Phys. 45 (200 4), 3949 –3971 . Also av ailable as arXiv:hep-th/03 0917 3 . [59] F. Girelli, H. P feiﬀer and E. M. Pop escu, T o po logical higher g auge theory - from B F to B F C G theory , J . Math. Phys. 4 9 :0325 03, 20 08. Also av ailable as arXiv:070 8.3051. [60] V. Guillemin and S. Sternberg, Symple ct ic T e chniques in Physics , Cam- bridge U. Press, Ca mbridge, 1984 . [61] A. Henriq ues, Integrating L ∞ -algebra s. Av a ilable as a rXiv:math/06 03563. [62] M. Johnson, The combinatorics of n -c ategoric a l pasting, Jour. Pure Appl. Alg. 62 (1989), 21 1–22 5. [63] S. La ck, A 2- c ategorie s companion, in T ow ards Higher Categor ies , eds. J. Bae z and P . May , Spring er, 2009, pp. 105–19 1. Also a v ailable a s arXiv:math/07 0253 5 . [64] E . Lerman, O rbifolds as stacks?, av ailable as a rXiv:080 6.4160 . [65] J . Lewandowski and T. Thiemann, Diﬀeomor phism inv ar iant quantum ﬁeld theories of co nnections in terms of webs, Class. Quant. Grav. 16 (1999), 2299– 2322 . Also av a ilable as ar Xiv:g r-qc/ 9 9010 15 . [66] S. Mac Lane and J. H. C. Whitehead, On the 3-type of a co mplex, Pro c. Nat. Acad. Sci. 36 (1 950), 4 1–48 . 58 [67] M. Mack aay a nd R. Picken, Holonomy a nd parallel tr ansp ort for Abelia n gerb es, Adv. Math. 170 (200 2), 287– 339. Also av a ilable as arXiv:math.DG/000 7053. [68] M. Mar kl, S. Schnider and J. Stasheﬀ, Op erads in Algebra, T op olo gy and Physics , AMS, Providence, Rho de Island, 2002 . [69] J . F. Martins and R. P ick en, On t wo-dimensional holonomy . Av ailable a s arXiv:071 0.431 0 . J. F. Ma rtins and R. Pick en, A cubical set approach to 2-bundles with connection and Wilson s urfaces. Av a ilable as arXiv:0808 .3964. [70] J . F. Mar tins and R. Picken, The fundamental Gr ay 3- g roup oid o f a smo oth manifold and lo cal 3- dimens io nal ho lo nomy based on a 2- c rossed mo dule. Av a ilable as arXiv:09 07.25 6 6 [71] J . Milnor , Rema rks on inﬁnite dimensional Lie g roups, Relativity , Groups and T op ology I I, (Les Houches, 19 83) , North- Ho lland, Amster dam, 1984 . [72] I. Mo erdijk, Intro duction to the languag e of sta cks a nd gerb es. Av ailable as arXiv:math.A T/021 2266. [73] M. Montesinos a nd A. Perez, Tw o- dimensional top o logical ﬁeld theor ies coupled to four-dimensio nal B F theory , Phys. Rev. D77 :10 4020 (2 008). Also av ailable as arXiv:070 9 .4235 . [74] M. K. Murray , Bundle gerb es, J . London Math. So c. 5 4 (1996), 403 –416 . Also av ailable as arXiv:dg-ga /940 7015 . M. K. Murray , An introductio n to bundle gerb es. Av ailable as arXiv:071 2.165 1 . [75] M. K. Murray and D. Stev enson, Hig g s ﬁelds, bundle gerb es and string structures, Commun . Math. Phys. 243 (2003 ), 541 –555. Also av ailable as arXiv:math.DG/010 6179. [76] H. P feiﬀer, Higher gauge theory and a non-Ab elian generaliza tion of 2- form electr omagnetism, Ann. Phys. 308 (2 003), 447– 4 77. Also av ailable as arXiv:hep-th/03 04074. [77] A. J. Po wer, A 2- categor ical pasting theore m, J our. Alg. 129 (19 90), 439- 445. [78] A. P ressley and G. Segal, Lo o p Groups , Oxfor d U. Pr ess, Oxfo rd, 1 986. [79] D. Rob er ts a nd U. Schreib er, The inner automor phism 3 -group of a strict 2-gro up. Also av ailable as arXiv:07 0 8.174 1 . [80] C. Rov elli, Qua nt um Gr avit y , Ca mbridge U. Press , 2004 . Also av ailable at h h ttp:// www.cpt.univ-mrs.fr/ ∼ rov elli/ b o ok.pdf i . 59 [81] D. Royten b erg, On weak Lie 2-a lgebras . Av ailable as arXiv:07 1 2.346 1 . [82] H. Sati, Geo metr ic and topo logical structures re lated to M-branes. Av a il- able as arXiv:10 0 1.502 0 . [83] H. Sati, U. Schreiber and J. Stas heﬀ, L ∞ -algebra s and applications to string- and C he r n–Simons n -transp or t. Av ailable as a rXiv:080 1.3480. [84] M. Schlessinger and J . Stasheﬀ, The Lie a lgebra structure of tang ent coho- mology a nd deformation theor y , Jour. Pure App. Alg. 38 (19 85), 313– 322. [85] C. Sc hommer-P ries, A ﬁnite-dimensional string 2-group. Av ailable as arXiv:091 1.248 3 . [86] U. Sc hreib er , comments at the n -Catego ry Ca f´ e. Av aila ble at h h ttp:// golem.ph.utexas.edu/ categor y/200 9/09/questions on ncurv ature.html i . [87] U. Schreiber and K. W aldorf, Parallel transp ort and functor s, J. Homoto py Relat. Struct. 4 (2 009), 187– 244. Also av ailable as a rXiv:070 5.0452 . [88] U. Schreiber and K. W aldorf, Smo o th functors vs. diﬀerential forms. Av ail- able as arXiv:08 0 2.066 3 . U. Schreiber and K. W aldorf, Connections on non-ab e lian gerbe s and their holonomy . Av ailable as a rXiv:080 8.1923 . [89] D. Stevenson, The Geometr y of Bundle Gerbes , Ph.D. thesis , University of Adelaide, 2000 . Also av ailable a s a rXiv:math.DG/000 4117. [90] S. Stolz and P . T eichner, What is an elliptic ob ject?, in T op o logy , Geometry and Qua nt um Field Theory: Pro ceedings o f the 2 002 Oxfor d Sympo sium in Honour o f the 60th Birthday o f Grae me Segal , ed. U. Tillmann, Cambridge U. Press, Cambridge, 2004 . [91] K . W aldor f, String co nnections a nd Chern– Simons theor y . Av ailable as arXiv:090 6.011 7 . [92] H. W eyl, Geo desic ﬁelds in the c a lculus of v ariation for multiple in teg rals, Ann. Math. 3 6 (1935 ), 607–6 29. [93] J . H. C. Whitehead, Note on a previo us pap er en titled ‘On adding re lations to homotopy groups’, Ann. Math. 47 (1 946), 8 06–81 0. J. H. C. Whitehea d, Co mbinatorial homotopy I I, Bull. Amer. Math. Soc . 55 (1949 ), 453– 496 [94] E . Witten, The index of the Dirac op erator in lo op space, in Elliptic Curves and Mo dular F orms in Algebraic T op olog y , ed. P . S. Landweber, Lecture Notes in Mathematics 1326 , Springer, Be rlin, 1 988, pp. 1 61–1 81. 60

An Invitation to Higher Gauge Theory

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