Chern-Simons theory in mathematics, condensed matter theory and cosmology

Chern-Simons theory in mathematics, condensed matter theory and cosmology ∗ Jürg F röhlic h A ugust 2025 Dedicated to the memory of Jim Simons Abstract V arious applications of Chern-Simons theory in algebraic top ology , in particular knot theory , condensed matter ph ysics and cosmology are reviewed. Sp ecial attention is paid to app earances of Chern-Simons actions in the theory of the (in teger and fractional) quantum Hall effect. A mec hanism related to fiv e-dimensional abelian Chern-Simons theory that ma y be at the origin of the observ ed intergalactic magnetic fields in the universe is describ ed. 1 The Chern-Simons forms in mathematics It is widely kno wn and appreciated that the Chern-Simons forms and the corresp onding Chern- Simons actions play an interesting and imp ortant role in algebraic top ology . Let G b e a compact Lie group. Consider a principal G -bundle with connection ∇ whose base space is a p 2 n ` 2 q -dimensional manifold, M . Let A b e the connection 1-form (gauge p otential) with v alues in the Lie algebra of G , and let F denote its curv ature. Let T r denote a trace on the Lie algebra of G that is inv ariant under the adjoint action of G . Lo cally , the Chern-Simons p 2 n ` 1 q -form, ω 2 n ` 1 , on M is defined b y the equation dω 2 n ` 1 p A q “ T r ` F ^p n ` 1 q ˘ . If M is a compact manifold obtained by gluing together t wo oriented manifolds, M ` and M ´ , along their common b oundary , Λ “ B M ` “ B M op ´ , the Chern-Simons action on Λ is defined by C S ˘ 2 n ` 1 p A q : “ 1 z n π n ż M ˘ T r ` F ^p n ` 1 q ˘ “ “ 1 z n π n ż Λ ω 2 n ` 1 p A q ” , (1) where z n is a certain integer ( z 1 “ 4 , z 2 “ 24 , . . . ), and the right side is only defined mo dulo in tegers. One observes th at C S ` 2 n ` 1 p A q ´ C S ´ 2 n ` 1 p A q “ I P Z , ∗ This review is based on the w ork of a v ariet y of p eople who profited from mathematical insights and the generosity of Jim Simons, including w ork by this author and his collab orators. It does not contain original results; but I hop e it will still b e somewhat useful and entertaining. I do not not provide an exhaustive list of references to original pap ers on topics men tioned; how ever, these pap ers can b e easily found by tracing the literature quoted in the bibliography . I ap ologize to colleagues whose work should b e quoted in this pap er but is not. 1 where I is the “ins tan ton num b er” asso ciated with the connection ∇ on the principal G-bundle o v er M . It follo ws that the “Chern-Simons functional,” exp “ 2 π i k C S 2 n ` 1 p A q ‰ , k P Z , is a gauge- in v ariant, single-v alued functional of the connection 1-form A ˇ ˇ Λ ; k is called its “level” . Cho osing an auxiliary Riemannian metric, g , on M , one may attempt (at least for n “ 1 ) to make mathematical sense of the functional integral Z Λ p k q : “ ż A exp “ 2 π i k C S 2 n ` 1 p A q ‰ D g A , (2) where A is the affine space of gauge fields, and D g A is a formal volume form on A that dep ends on the c hoice of the metric g . The dep endence of the formal expression ( 2 ) on g is univ ersal and can b e eliminated by dividing this functional by a standard one (e.g., a gravitational Chern- Simons partition function). F or n “ 1 , the functional integral in ( 2 ) can b e given a fairly precise mathematical meaning and shown to define a top ological inv ariant of the framed 3-manifold Λ . Three-dimensional Chern-Simons theory turns out to hav e interesting applications in the theory of knots and links em b edded in Λ . Let χ R b e the c haracter of an irreducible representation, R , of the gauge group G , and let K b e a framed knot embedded in Λ . A Wilson lo op “observ able,” W R p K q , asso ciated with K is defined by W R p K ; A q : “ χ R ´ P exp ” i ¿ K A ı¯ , (3) where P denotes path ordering. One attempts to make sense of the expression D W R p K q E k : “ Z Λ p k q ´ 1 ż A W R p K ; A q ¨ exp “ 2 π i k C S 3 p A q ‰ D g A , (4) whic h formally defines an in v arian t of the framed knot K embedded in Λ . There are v arious approaches to pro viding a precise mathematical meaning to this expression; see [ 1 , 2 , 3 ] (the b est known one b eing the one in [ 1 ]). The approach describ ed in [ 2 ] emplo ys the follo wing ingredien ts: Cho osing Λ “ R 3 , one may define the in tegrals in ( 2 ) and ( 4 ) b y fixing a sp ecial gauge, called “light-c one gauge. ” Let x “ p x 1 , x 2 , τ q b e Cartesian co ordinates on R 3 . W e set x ` “ z : “ x 1 ` ix 2 , x ´ “ z : “ x 1 ´ ix 2 , and write the gauge p oten tial as A p x q “ a ` p x q dz ` a ´ p x q dz ` a 0 p x q dτ . One c ho oses the gauge condition a ´ p x q “ 0 . The Chern-Simons action then b ecomes a quadratic functional of a ` and a 0 , and the functional in tegrals in ( 2 ) and ( 4 ) b ecome formal Gaussian integrals. The Wilson lo op exp ectation D W R p K q E k can b e calculated by integration of K nizhnik-Zamolo dchikov e quations [ 4 ] (which are systems of ordinary differential equations in the v ariable τ ), or by expanding the path-ordered exp onen tial defining W R p K ; A q in a p ow er series and ev aluating the formal Gaussian integrals. These are the metho ds employ ed in [ 2 ] to make sense of ( 4 ); they yield mathematically precise expressions for D W R p K q E k . F or a somewhat more detailed review of these ideas see [ 5 ]. I will not present further details of applications of Chern-Simons theory to problems in algebraic top ology . Instead, I prop ose to outline applications of three- and five-dimensional ab elian Chern- Simons theories ( G “ R ) to problems in condensed-matter ph ysics and cosmology related to the 2 quan tum Hall effect and to ph ysical systems in 3 ` 1 dimensions. In these applications, the Chern- Simons 3-form and 5-form are in tegrated ov er three- and five-dimensional manifolds, Λ , resp ectively , with non-empt y b oundaries, B Λ “ H . The resulting actions are anomalous , i.e., they are not gauge- in v ariant. Their gauge v ariation is given by certain b oundary terms, whic h are cancelled by the gauge v ariation of effectiv e actions of chiral degrees of freedom attached to the b oundary B Λ that are coupled to the gauge field A ˇ ˇ B Λ . These are insigh ts going back to the analysis of anomalies in quan tum gauge theories. They hav e in teresting consequencs for chiral conformal field theory . 2 Three-dimensional Chern-Simons theory and the Quan tum Hall Effect Let Λ b e an op en domain in R 3 with a non-empty b oundary B Λ ; (later, Λ will b e an infinitely extended full cylinder whose axis is parallel to the time axis). Let A b e the electromagnetic vector p oten tial restricted to Λ . W e consider the ab elian Chern-Simons action C S Λ p A q : “ σ 2 ż Λ A ^ F , with F “ dA , (5) where σ is a real constan t, which, later, will turn out to b e the Hal l c onductivity. R emark: If Λ “ R 3 and F falls off rapidly at infinit y then C S R 3 p A q is the so-called helicity of the magnetic field,  B , dual to F . Helical magnetic fields (and helical fluid flo ws, with A the v elo city v ector field of a fl uid and F dual to its v orticit y) hav e b een studied for a long time. Consider a gauge transformation, A ÞÑ A ` dχ , where χ is a smo oth function on R 3 ; one finds that C S Λ p A ` dχ q “ C S Λ p A q ` σ 2 ż B Λ χ F , (6) i.e., the Chern-Simons action on a manifold with b oundary fails to b e gauge-in v ariant; it is said to b e “anomalous. ” Let a : “ A ˇ ˇ B Λ b e the restriction of the v ector p otential A to the b oundary , B Λ , of a full cylinder Λ Ă R 3 with axis parallel to the time axis, which we equip with a Lorentzian metric; and let F “ E } b e the dual of da , i.e, the dual of the restriction of the field strength of the electromagnetic v ector p oten tial to the b oundary of Λ . W e consider the functional Γ B Λ p a q : “ 1 2 ż B Λ d v ol “` F ´ div a ˘ l ´ 1 ` F ´ div a ˘ ` a µ a µ ‰ (7) where l is the d’Alem b ertian on B Λ . This is the anomalous effectiv e action of a chiral current lo calized on B Λ coupled to the electromagnetic v ector p otential a . The reader is invited to verify that the combination C S Λ p A q ´ σ 2 Γ B Λ p a “ A ˇ ˇ B Λ q is gauge-in v ariant. (8) This observ ation implies that the gauge anomaly of the Chern-Simons action describ ed in ( 6 ) is cancelled by the gauge anomaly of certain charged chiral degrees of freedom attac hed to the b oundary of Λ when they are coupled to the electromagnetic v ector p otential. (This will b e made more precise b elow.) Next, w e prop ose to sk etc h how these functionals come up in the theory of the quantum Hal l effe ct (QHE). W e consider a system consisting of a t wo-dimensional (2D) electron gas forming at 3 the planar in terface b etw een a semi-conductor and an insulator when a gate voltage p erp endicular to the interface is turned on pushing electrons from the bulk of the semi-conductor to the interface. The “space-time” of such a gas is given b y the full cylinder Λ “ Ω ˆ R , where Ω is the (b ounded op en) planar region the gas is confined to, and R is the time axis. Let h denote Planck’s constan t, and e the elemen tary electric c harge; the densit y of the 2D electron gas is denoted b y n . W e imagine that the gas is put into a uniform magnetic field,  B 0 , of strength B 0 , measured in units of the flux quan tum, h { e , p erp endicular to the plane of the electron gas. The dimensionless quan tity ν : “ n B 0 is called the “filling factor” of the gas; it is equal to the n um b er of Landau lev els filled by a non-inter acting gas of electrons freely moving in Ω of densit y n . W e prop ose to consider the p 2 ` 1 q -dimensional electro dynamics of this system. W e assume that the filling factor ν is suc h that the ground-state energy of the gas is separated from the energy sp ectrum corresp onding to conducting states by a strictly p ositiv e so-called mobility gap, so that the longitudinal (Ohmic) c onductivity of the gas vanishes (at zero temp erature). 1 A 2D electron gas with these prop erties is called “inc ompr essible. ” W e study the resp onse of an incompressible 2D electron gas to turning on a tiny , smo oth, p erturbing external electromagnetic field, F , where the co efficients, ␣ F µν ˇ ˇ µ, ν “ 0 , 1 , 2 ( , of the 2-form F are giv en by ` F µν ˘ “ ¨ ˝ 0 E 1 E 2 ´ E 1 0 ´ B ´ E 2 B 0 ˛ ‚ . (9) Here E “ p E 1 , E 2 q is the comp onent of the external electric field parallel to the plane of the 2D gas, and B “ B tot ´ B 0 , where B tot is the comp onent of the total external magnetic induction p erp endicular to the plane of the system. (Note that the orbital motion of electrons confined to Ω only dep ends on the comp onen ts E and B tot .) The basic equations of the electro dynamics of a 2D incompressible electron gas (in units where the vel o city of light c “ 1 ) are as follo ws. (i) Hall’s La w – a phenomenolo gic al law j k p x q “ σ H ε kℓ E ℓ p x q , k , ℓ “ 1 , 2 , (10) where j “ p j 1 , j 2 q denotes the electric current densit y of the gas, ε 12 “ ´ ε 21 “ 1 , ε ii “ 0 , and σ H is the so-called Hal l c onductivity. 2 Since the electron gas is assumed to b e incompress- ible, an Ohmic contribution to the current density do es not app ear on the right side of this equation; the longitudinal c onductivity of the gas vanishes . Hall’s la w sho ws that if σ H “ 0 the symmetries of reflections in lines ( P ) and time rev ersal ( T ) are broken. These symmetries are broken explicitly by the presence of the external magnetic induction. (ii) Charge conserv ation – a fundamental law B B t ρ p x q ` ∇ ¨ j p x q “ 0 , (11) where ρ is the electric c harge densit y of the gas. 1 T o show that, for a 2D gas of inter acting electrons, there are filling factors, ν , where a mobility gap op ens, is a difficult problem of quantum-mec hanical many-bo dy theory that is not particularly well understo o d, yet. 2 Einstein’s summation conv ention is used here and everywhere else in this text. 4 (iii) F arada y’s induction law – a fundamental law B B t B p x q ` ∇ ^ E p x q “ 0 , (12) and we hav e assumed that B 0 is constant. Com bining Eqs. (i) through (iii), w e obtain B B t ρ p ii q “ ´ ∇ ¨ j p i q “ ´ σ H ∇ ^ E p iii q “ σ H B B t B . (13) (iv) Chern-Simons Gauss la w In tegrating Eq. ( 13 ) in time t , with integration constants chosen as follows j 0 p x q : “ ρ p x q ` e ¨ n, B p x q “ B tot p x q ´ B 0 , with B tot the comp onen t of the (total) external magnetic induction and B 0 the comp onen t of the uniform magnetic induction in the direction p erp endicular to Ω , we find the so-called “Chern-Simons Gauss law” j 0 p x q “ σ H B p x q . (14) This equation must b e understo o d as saying that a small c hange, ∆ B , of the external magnetic induction p erp endicular to the plane of the gas induces a small change, ∆ j 0 , in the electric c harge density of the gas. Eq.( 14 ) is also known under the name of Str e da formula ; see [ 6 ]. In tegrating b oth sides of Eq. ( 14 ) in time from t “ t i to t “ t f and ov er Ω , we conclude that ∆ Q “ σ H ∆Φ , (15) where ∆ Q is the change of the electric charge stored in the system, and ∆Φ is the change of the magnetic flux through Ω during the time interv al r t i , t f s . Quan tization of the Hall conductivit y : In order to explain the surprizing quantization of the Hall conductivity σ H , we consider an incompressible electron gas confined to a 2D torus, T 2 “ : T , and imagine threading a magnetic flux tub e through the interior of T that do es not cross/in tersect T and is parallel to a non-con tractible cycle, γ , of T . One may ask how muc h electric c harge, ∆ Q r γ , crosses a cycle, r γ , on T conjugate to γ (i.e., γ and r γ in tersecting eac h other in a single p oint of T ) when the magnetic flux in the interior of the torus in the direction of γ changes b y an amount ∆Φ . Com bining F araday’s induction law with Hall’s law ( 10 ), we readily find that the quotient ∆ Q r γ { ∆Φ is giv en b y the Hall conductivity σ H . Assuming that all quasi-particles con tributing to the electric current j in the system are ele ctr ons or holes, i.e., particles with electric c harge ¯ e , one can inv oke the theory of the Ahar onov-Bohm effe ct to conclude that the state of this system is unchange d if the magnetic flux in the interior of T is slo wly increased b y an inte ger multiple of the flux quantum h { e . This implies that, in the pro cess of increasing ∆Φ by an amount h { e , an integer num b er, N , of electrons with a total electric charge - N e, N P Z , m ust cross the cycle r γ . Hence ∆ Q r γ “ N e “ σ H p h { e q ô σ H “ e 2 h N , N P Z , whic h shows that, in an incompressible 2D electron gas without fractionally charged quasi-particles con tributing to the electric current of the system, the Hall conductivity σ H m ust b e an integer 5 m ultiple of e 2 h . This reasoning pro cess do es not explain wh y the “Hal l fr action,” h e 2 σ H , is often observ ed not to b e an integer, but a r ational numb er (most often, but not alwa ys, with an o dd denominator). It suggests, how ever, that if it is not an integer then there m ust exist fr actional ly char ge d quasi-p articles contributing to the electric curren t in the system, as suggested by T sui and L aughlin [ 7 ]. The sign of σ H is determined by the sign of the electric charge of the quasi-particles carrying the Hall current, namely whether these quasi-particles are electrons or holes. Historically , this fact has led to the discov ery of holes in nearly full conduction bands of semi-conductors. Quan tum Hall effect and Chern-Simons action : Let A b e the v ector p otential on Λ “ Ω ˆ R with field strength (curv ature) F “ dA . Hall’s la w ( 10 ) and the Chern-Simons Gauss law ( 14 ) yield the equation δ S ef f p A q δ A µ ” j µ p x q “ σ H ε µν λ F ν λ p x q , (16) where S ef f p A q is the so-called effe ctive action of the 2D electron gas; (the first equation in ( 16 ) follo ws from the definition of effectiv e actions). Using that F “ dA , we can in tegrate ( 16 ) to conclude that, up to an irrelev an t constant, S ef f p A q “ σ H 2 ż Λ A ^ F , (17) whic h is the Chern-Simons action asso ciated with the electromagnetic vector p otential A . Eq. ( 16 ) is a gener al ly c ovariant relation b et w een the current density , j µ , of the electron gas and the elec- tromagnetic field tensor F ν λ . R emark: It should b e noted that the Chern-Simons action is not the exact effectiv e action of an incompressible 2D electron gas. But it is the leading term in an expansion of its effective action in a series of terms that are increasingly “irrelev an t” at large distance scales and lo w energies. 3 The problem of predicting the form of effectiv e actions of incompressible (“gapp ed”) electron gases in t w o and three space dimensions has b een dealt with in [ 8 ]. Eq. ( 16 ) seems to lead to a contradiction . 0 p ii q “ B µ j µ p iii q , ( 16 ) “ ε µν λ pB µ σ H q F ν λ . (18) W e observe that, while the left side of this equation v anishes b y charge conserv ation, the right side do es not v anish in general wherever the v alue of σ H jumps, as, for example, at the b oundary , B Ω , of the sample region Ω con taining the 2D electron gas. Resolution of con tradiction : In Eq. ( 16 ), j µ is the bulk current density , j µ bulk , whic h is apparen tly not conserved, b ecause it is not the total current densit y! The c onserve d total ele ctric curr ent density, j µ tot , can b e decomp osed as j µ tot “ j µ bulk ` j µ edg e , (19) where j µ edg e is an e dge curr ent density whose supp ort is equal to the supp ort of ∇ σ H . W e then hav e that B µ j µ tot “ 0 , but B µ j µ bulk “ ´B µ j µ edg e ( 18 ) “ 0 , 3 The term “irrelev ant” is understo o d in the sense of standard dimensional analysis. 6 where supp j µ edg e “ supp t ∇ σ H u Ě B Ω , with j edg e K ∇ σ H . Equations ( 18 ) (setting j µ “ j µ bulk !) and ( 19 ) imply that B µ j µ edg e ( 19 ) “ ´B µ j µ bulk | supp t ∇ σ H u ( 18 ) “ ´ ε j k ` B j σ H ˘ E k . When restricted to the supp ort of ∇ σ H this equation tells us that B µ j µ edg e “ ´p ∆ σ H q E } ˇ ˇ supp t ∇ σ H u , (20) where ∆ σ H is the jump of the Hall conductivit y across the edge in question, and E } is the comp onen t of the electric field parallel to the edge. This equation is an expression of the chir al anomaly in 1+1 dimensions [ 9 ]. In the following, we assume that σ H is constan t throughout the interior of the spatial domain Ω and v anishes outside Ω (so that ∆ σ H “ σ H in ( 20 )). This is not a realistic assumption; but it simplifies our discussion without introducing misleading concepts. Equation ( 20 ) shows that j µ edg e is an anomalous chir al curren t density in 1 ` 1 dimensions. It can b e viewed as a manifestation of “holo gr aphy” : The degrees of freedom propagating along the edge of the 2D electron gas store as m uc h information ab out the Hall conductivity and other quantities characteristic of the system as the degrees of freedom in the bulk. (A prediction of c hiral edge currents in 2D electron gases exhibiting the quantum Hall effect first app eared in [ 10 ].) Not surprisingly , the observ ations made here are closely related to ones that w e hav e made b efore, namely in Eqs. ( 6 ), ( 7 ) and ( 8 ): The Chern-Simon s action ( 5 ), with σ “ σ H the Hall conductivit y , on a three-dimensional space-time manifold Λ “ Ω ˆ R with a non-empty b oundary , B Ω ˆ R , is not gauge-in v ariant. Its gauge anomaly is cancelled by the one of the action functional ´ σ H 2 Γ B Λ p a q introduced in ( 7 ), which is the effective action of c hiral c harged degrees of freedom propagating along the boundary B Ω of the sample domain Ω of the 2D gas, whic h give rise to the c hiral edge current j µ edg e . (Related observ ations ha v e b een made in [ 11 ].) The theory of chiral degrees of freedom in 1+1 dimensions leads to predictions of the p ossible v alues of the Hall conductivity σ H and of the sp ectrum of quasi-particles exhibited by the electron gas. A general approac h to this topic (based on the theory of sup er-selection sectors of a class of c hiral conformal field theories in 1+1 dimensions) has b een presented in [ 12 ]. It predicts that σ H is a r ational multiple of e 2 h and that a 2D electron gas with a Hall conductivit y that is not an in teger m ultiple of e 2 h exhibits quasi-particles with fractional electric charge and exotic quantum statistics (called fr actional or br aid (goup) statistics , [ 13 ]). 4 Here I only mention a sp ecial case of the general theory treated in [ 5 ] and [ 8 ] (and references giv en there), which is relev ant for the in terpretation of a very large set of exp erimen tal data. I supp ose that the c hiral electric edge current densit y , j µ edg e , is a linear combination of sev eral (canonically normalized) chiral current densities, j µ α , α “ 1 , . . . , N , asso ciated with “emergent” Kac-Mo o dy symmetries (at level 1) acting on the space of edge states, j µ edg e “ N ÿ α “ 1 Q α j µ α , (21) where Q 1 , . . . , Q N are real num b ers. The current density Q α j µ α is the contribution of chiral degrees of freedom in an “edge c hannel,” lab elled α , to the electric edge curren t densit y j µ edg e , for α “ 1 , . . . , N . (F or an incompressible 2D gas of non-inter acting electrons, these c hannels corresp ond to 4 Suc h quasi-particles, which, in the bulk, are quasi-static, might be of interest for topological quantum computing. 7 the edge states of N filled Landau lev els, with Q α “ 1 , for α “ 1 , . . . , N .) The theory of the c hiral anomaly in tw o dimensions [ 9 ] tells us that B µ j µ edg e “ N ÿ α “ 1 Q α B µ j µ α “ ´ e 2 h “ ÿ α p Q α q 2 ‰ E } , (22) where E } is the comp onent of the external electric field parallel to B Ω . T ogether with ( 20 ) (setting ∆ σ H “ σ H ), this equation yields a form ula for σ H , namely σ H “ e 2 h ÿ α p Q α q 2 . (23) One can show (see [ 5 ], [ 8 ] and references given there) that the quantum n um b ers of chiral edge states describing configurations of electrons and holes propagating along the edge, B Ω , of the 2D electron gas clo ckwise or anti-clockwise (dep ending on the sign of B 0 ) form an N -dimensional o dd inte gr al lattic e, L , and that Q : “ ` Q 1 , . . . , Q N ˘ is a visible ve ctor in the dual lattice L ˚ . This implies that ř α p Q α q 2 is a rational num b er , i.e., σ H is a rational multiple of e 2 h . The sites in the lattice L ˚ are the quantum n umbers of configurations of quasi-particles some of which are fractionally charged and hav e fractional statistics; (except when L “ L ˚ , whic h holds, e.g., for non-in teracting 2D electron gases). The theory sketc hed here can also b e dev elop ed by fo cusing on the physics in the bulk of the 2D electron gas. In 2+1 dimensions, a conserved curren t density , j , is dual to a closed 2-form, J . If the space-time, Λ , of the gas is a full cylinder then J is exact, i.e., J “ d Z (P oincaré’s lemma), where Z is a 1-form, the “ve ctor p otential” of the conserved current density . In a 2D incompressible electron gas (which is a so-called “top ological insulator”), the quantum the ory of a conserved curren t at very large distance scales and lo w energies can b e describ ed by functional integrals with a “top ological” action functional, S Λ p Z q , dep ending on the vector p oten tial Z , given by S Λ p Z q “ h 2 ! ż Λ Z ^ d Z ´ Γ B Λ p z “ Z ˇ ˇ B Λ q ) , i.e., by a Chern-Simons action. Here Γ B Λ is a b oundary term restoring in v ariance under “gauge transformations,” Z ÞÑ Z ` dζ , with ζ a real-v alued function on Λ ; see ( 7 ). F or an incompressible gas with N conserved current densities, J α “ d Z α , α “ 1 , . . . , N , the action functional is given by S Λ p Z q “ h 2 N ÿ α “ 1 ! ż Λ Z α ^ d Z α ´ Γ B Λ p Z α ˇ ˇ B Λ q ) , with Z “ ` Z 1 , . . . , Z N ˘ . Coupling the electric curren t density j µ “ ř N α “ 1 Q α j µ α to the vector p oten tial, A , of an external electromagnetic field by adding the term e J ^ A “ e N ÿ α “ 1 Q α d Z α ^ A , to the action, we find the effe ctive action of the system b y carrying out the functional in tegral e iS ef f p A q{ ℏ : “ const. ż N ź α “ 1 D Z α exp ” i ℏ ´ S Λ p Z q ` ż Λ eJ ^ A ¯ı (24) 8 (with the constant chosen such that S ef f p A “ 0 q “ 0 ) and find that S ef f p A q “ e 2 2 h N ÿ α “ 1 p Q α q 2 ! ż Λ A ^ F ´ Γ B Λ ` A ˇ ˇ B Λ ˘ ) . This repro duces form ula ( 17 ), with σ H giv en b y ( 23 ), and it shows that one obtains the same expression for the Hall conductivity indep enden tly of whether one considers the physics of edge states or the bulk of an incompressible 2D electron gas (an instance of “holography”). R emark: An inte resting duality b etw een 2D insulators and 2D sup er c onductors and a self-duality of 2D incompressible electron gases exhibiting the quantum Hall effect is revealed b y functional F ourier transformation, as in ( 24 ), with the electromagnetic vector potential A and the vector p oten tial Z of the conserved electric curren t densit y J playing the role of conjugate v ariables [ 8 ]. What we hav e outlined here is how the theory of “ab elian ” quantum Hal l fluids is intimately related to ab elian Chern-Simons the ory of the vector p otentials of exact current 2-forms; (a more general theory , implicitly related to non-ab elian Chern-Simons theory , is describ ed in [ 12 ]). The facts that quantum num bers of configurations of quasi-particles corresp ond to sites in the dual, L ˚ , of an o dd integral lattice L , and that Q “ ` Q 1 , . . . , Q N ˘ is a “visible” v ector in L ˚ lead to plent y of v ery specific predictions of prop erties of 2D incompressible electron gases exhibiting the quan tum Hall effect that match exp erimental data with astounding accuracy; (this theme is dev elop ed in w ork reviewed in [ 5 ], [ 8 ], and refs.). 3 The fiv e-dimensional Chern-Simons action and a cousin of the quan tum Hall effect This section closely follo ws ideas describ ed in [ 14 ] and review ed in [ 15 ]. W e consider a cousin of the quantum Hall effect in systems of c harged matter on a five-dimensional (5D) space-time slab, Λ “ Ω ˆ r 0 , L s , with tw o four-dimensional “b oundary branes,” B ˘ Λ » Ω , parallel to the p x 0 , x 1 , x 2 , x 3 q -plane in 5D Mink owski space, where x 0 is the time co ordinate. The slab Λ is assumed to b e filled with very he avy, four-c omp onent Dir ac fermions coupled to the 5D electromagnetic v ector p otential, p A . F unctional integration ov er configurations of fermion degrees of freedom yields an effectiv e action dep ending on p A that has the form S Λ p p A q “ 1 2 Lα ż Λ d 5 x p F M N p x q p F M N p x q ` C S Λ p p A q ´ Γ ℓ p p A | B ` Λ q ´ Γ r p p A | B ´ Λ q ` ¨ ¨ ¨ , (25) where M , N “ 0 , . . . , 4 , α is a dimensionless constant, L is the width of the 5D slab Λ , and C S Λ p p A q : “ κ H 24 π 2 ż Λ p A ^ p F ^ p F (26) is the 5D Chern-Simons action ; (henceforth Planc k’s constant ℏ is set to 1); the dots on the righ t side of ( 25 ) stand for terms that are “irrelev ant” on large distance scales and at low energies. The v ector p otential p A is treated as a classical external field. The action ( 26 ) yields a formula for a 5D analogue of the Hall current (see ( 10 )): j M “ κ H 8 π 2 ε M N J K L F N J F K L , (27) 9 where κ H is a dimensionless constant, with κ H P Z (assuming that all fermions hav e electric charges that are in teger multiples of the elementary electric charge); it pla ys the role of the Hall fraction h e 2 σ H . In the following κ H is set to 1. Equation ( 27 ) is the five-dimensional analogue of equation ( 16 ) in Sect. 2. The action functional in equation ( 25 ), with C S Λ p p A q as in ( 26 ), is the effective action describing a 5D analogue of the quantum Hall effect. Apparen tly , such effects can be observ ed in certain systems of condensed-matter ph ysics with some “virtual dimensions” pla ying the role of space dimensions; see [ 16 ] and references giv en there. In ( 25 ), the functionals Γ ℓ { r are the anomalous effective actions of left-handed/right-handed Dirac-W eyl fermions propagating along the b oundary branes, B ˘ Λ . The gauge anomaly of the actions ´ Γ ℓ and ´ Γ r on the right side of ( 25 ) cancels the one of the Chern-Simons action C S Λ p p A q ; see [ 9 ]. It is of interest to study the dimensional r e duction of the 5D theory , with effectiv e action giv en in ( 25 ) and ( 26 ), to a four-dimensional space-time. W e consider gauge fields p A with the prop erty that, for an appropriate choice of gauge, the comp onents p A M are indep endent of x 4 , for M “ 0 , 1 , 2 , 3 . A field φ of scaling dimension 0, henceforth called axion field, is defined by φ p x q : “ ż γ x p A , where γ x is a straight curve parallel to the x 4 -axis connecting B ´ Λ to B ` Λ , with x : “ p x 0 , x 1 , x 2 , x 3 q k ept fixed. The action functional in ( 25 ) then b ecomes S Ω p A ; φ q “ 1 2 α ż Ω d 4 x “ F µν p x q F µν p x q ` 1 L 2 B µ φ p x qB µ φ p x q ‰ ` 1 8 π 2 ż Ω φ p F ^ F q ´ Γ Ω p A q ` ¨ ¨ ¨ , µ, ν “ 0 , . . . , 3 . (28) Under the present assumptions on the vector p oten tial p A the b oundary effective action Γ Ω p A q “ Γ ℓ p A q ` Γ r p A q , with A “ p A ˇ ˇ B Λ , is not anomalous and can b e ignored in the following. Expression ( 28 ) shows that the pseudo-scalar field φ can indeed b e interpreted as an axion field; see [ 17 ]. One can add a self-interaction term, U p φ q , to the Lagrangian densit y in ( 28 ), requiring that U p φ q b e p erio dic in φ with p erio d 2 π ; such self-in teraction terms can b e induced by coupling the axion φ to massive non-ab elian gauge fields, which are subsequen tly integrated out. F rom ( 28 ) w e derive the equations of motion for the electromagnetic field tensor F µν and the axion φ , B µ F µν “ ´ α 4 π 2 B µ ` φ r F µν ˘ , L ´ 2 l φ “ α 8 π 2 F µν r F µν ´ δ U p φ q δ φ , (29) where r F µν is the dual field tensor. These are the field equations of “axion-ele ctr o dynamics” in four dimensions. In terms of electric and magnetic fields,  E and  B , these equations are given by  ∇ ¨  B “ 0 ,  ∇ ^  E ` 9  B “ 0 ,  ∇ ¨  E “ α 4 π 2 `  ∇ φ ˘ ¨  B , (30)  ∇ ^  B “ 9  E ` α 4 π 2 t 9 φ  B `  ∇ φ ^  E u , and L ´ 2 l φ “ α 4 π 2  E ¨  B ´ δ U p φ q δ φ , (31) 10 (w e use units where the velocity of ligh t is set to 1). Comparing the last equation in ( 30 ) with Maxw ell’s generalization of Amp ère’s la w, w e realize that  j : “ α 4 π 2 ␣ 9 φ  B `  ∇ φ ^  E ( (32) can b e interpreted as the electric current density of the system (ignoring a term corresp onding to an Ohmic current). W e conclude this review b y mentioning some interesting applications of axion-electro dynamics in condensed matter physics and cosmology . 3.1 Chiral magnetic effect and 3D quan tum Hall effect W e b egin by considering sp ecial axion field configurations where φ only dep ends on time t “ x 0 . W e set µ 5 ” µ ℓ ´ µ r : “ 9 φ . In the original form ulation of Eqs. ( 25 ), ( 26 ) and ( 27 ) the quantities µ ℓ and µ r can b e interpreted as the chemical p otentials of the b oundary branes B ` Λ and B ´ Λ , resp.; i.e., µ 5 is the “voltage drop” b etw een the t wo b oundary branes. The electric current density is then giv en by  j “ α 4 π 2 µ 5  B , (33) whic h is the equation describing the so-called chir al magnetic effe ct [ 18 ]. In condensed-matter theory , the equation of motion of 9 φ ” µ 5 tak es the form of a diffusion e quation , 9 µ 5 ` τ ´ 1 µ 5 ´ D △ µ 5 “ L 2 α 4 π 2  E ¨  B , (34) where τ is a r elaxation time asso ciated with pro cesses mixing left- and right-handed quasi-particles in certain solids, dubb ed W eyl semi-metals, D is a diffusion c onstant , and α “ 2 π e 2 h is the fine structure constant; (it is assumed here that U p φ q ” 0 ). Eq. ( 34 ) implies that µ 5 approac hes µ 5 » τ L 2 α 4 π 2  E ¨  B , as time t tends to 8 (  E ¨  B is assumed to b e slowly v arying in space, so that µ 5 is nearly space-indep endent). This expression for µ 5 can b e plugged into equation ( 33 ) for the v ector curren t densit y  j predicted by the chiral magnetic effect. By comparing the resulting equation with Ohm’s law we find an expression for the c onductivity tensor, σ “ ` σ kℓ ˘ , namely σ kℓ “ τ ` Lα 4 π 2 ˘ 2 B k B ℓ . (35) This expression is relev ant for studies of charge transp ort in W eyl semi-metals. Next, I consider a 3D sp atial ly p erio dic (crystalline) system of matter exhibiting (among others) magnetic degrees of freedom that can b e describ ed b y an “emergen t” axion field, φ . The crystal lattice is denoted by L . The effective action, S Ω ` A ; φ ˘ , of the system is given b y ( 28 ). Imp osing p erio dic b oundary conditions at the b oundary , B Ω , of space-time Ω , the functional exp “ iS Ω ` A ; φ ˘‰ turns out to b e p erio dic in φ under shifts φ ÞÑ φ ` 2 nπ , n P Z . W e consider a stationary state of the system, with φ time-indep endent, i.e., µ 5 “ 0 , and we assume that the state of the system is in v arian t under (lattice) translations lea ving L inv ariant. Then the “axion field” φ must b e in v ariant under lattice translations, modulo 2 π ; hence it is giv en by φ p  x q “ 2 π `  K ¨  x ˘ ` ϕ p  x q , (36) 11 where  K is a v ector in the dual lattic e, L ˚ , and ϕ is a function that is in v ariant under lattice translations, whic h w e neglect. Using that  ∇ φ “ 2 π  K , w e find that Eqs. ( 30 ) and ( 32 ) imply the follo wing formulae for the electric c harge densit y , ρ , and curren t densit y ,  j , ρ “ e 2 h  K ¨  B ,  j “ e 2 h  K ˆ  E ,  K P L ˚ . This is Halp erin’s 3D quantum Hall effect [ 19 ]. There are numerous further applications of the ideas sketc hed here in condensed matter physics; (see, e.g., [ 15 ] for a review with plent y of references to orginal pap ers). 3.2 The generation of primordial magnetic fields in the universe It is interesting to study mo dels of the universe featuring an axion field φ , b esides the electromag- netic field (and other degrees of freedom describing visible matter). Ultraligh t axions may give rise to dark matter (see, e.g., [ 20 ] and references given there), whic h pro duces observ able gravitational effects, such as anomalies in rotation curv es or gravitational lensing. In the following w e study axion electro dynamics for the purp ose of sk etc hing a mechanism that migh t give rise to the inter- galactic magnetic fields observed in the universe. The equations giv en in ( 30 ) and ( 31 ) must then b e mo dified so as to account for the curv ature of space-time. T o ke ep our discussion simple, w e supp ose that the large-scale structure of the universe is describ ed by a conformally flat F rie dmann- L emaîtr e sp ac e-time, Λ , with a metric, p g µν q , given by g 00 ” 1 , g 0 j ” 0 , and g ij “ ´ a p t q 2 δ ij , for i, j “ 1 , 2 , 3 , where t is cosmological time, and a p t q is the scale factor. The Lorentzian geometry of Λ is enco ded in the Hubble “c onstant,” H “ 9 a a , a function of cosmological time expressing the observ ed expansion of the univ erse. F or a F riedmann-Lemaître universe it is straightforw ard to find the mo difications due to curv a- ture of the equations of axion electro dynamics given in ( 30 ) and ( 31 ). The electric field  E can b e eliminated by using the first three equations in ( 30 ) and plugging the result into the last equation. F or simplicity we assume that the axion field φ is very slowly v arying in space, so that terms pro- p ortional to  ∇ φ can b e neglected. Then the field equation for the magnetic induction  B is given b y :  B ´ ∆  B ` 3 H 9  B ` r 3 2 9 H ` p 3 2 H q 2 s  B ´ α 4 π 2 9 φ  ∇ ^  B “ 0 . (37) Here and in the last equation of ( 30 ) we hav e neglected an Ohmic con tribution to the vector curren t densit y . This is justified by noticing that, after recom bination of charged particles in to neutral atoms and molecules, Ohmic currents are tin y . W e solve Eq. ( 37 ) by F ourier transformation in the spatial v ariables  x “ p x 1 , x 2 , x 3 q ,  B p  x, t q “ ż R 3 d 3 k  B p  k q e i p  k ¨  x ´ ω p k q t q , (38) where  k ¨  B p  k q “ 0 , b ecause  ∇ ¨  B “ 0 , and k : “ |  k | . Without loss of generality , we ma y supp ose that µ 5 “ 9 φ " H ě 0 during a certain in terv al I of cosmological time in the evolution of the univ erse; ( µ 5 ă 0 can b e treated similarly). T o simplify matters, we consider an approximate solution of ( 37 ) v alid under the assumptions that H and µ 5 are slowly v arying functions of time t P I , for a range of  k -vectors indicated in ( 40 ) b elo w. Plugging ( 38 ) into ( 37 ), we find that ω p k q “ ´ i 3 2 H ˘ d k 2 ` 3 9 H 2 ˘ µ 5 k . (39) 12 This formula shows that the expansion of the Universe ( H ą 0 ) leads to p ow er-law (actually , for a constan t H , exp onential) damping of  B in time if its F ourier amplitude,  B , is supp orted outside of the shell in  k -space given by Σ : “ !  k ˇ ˇ µ 5 ´ b µ 2 5 ´ K ă 2 k ă µ 5 ` b µ 2 5 ´ K ) , with K : “ 9 H 2 ` 6 9 H . (40) Ho w ever, if the supp ort of the F ourier amplitude  B of the magnetic induction in tersects the shell Σ sp ecified in ( 40 ) then  B gro ws exp onential ly in time t . The magnetic induction dev elops a non-v anishing helicity , ş  A ¨  B , where  A is the electromagnetic vector p otential. A ctually , µ 5 “ 9 φ is of course not constant in time, but dies out, µ 5 Ñ 0 , at large times t , so that the exp onen tial growth of  B comes to an end. The con v ersion of axion oscillations, µ 5 “ 0 , into a helical magnetic induction with an ever gro wing wa ve length λ (initially giv en by λ “ 2 π |  k | ´ 1 , with  k P Σ ) could explain the observed presence of tiny , very homogeneous intergalactic magnetic fields in the universe. More details on these matters can b e found in [ 21 , 22 , 14 , 23 ]. T o conclude, I hop e to hav e con vinced readers that, besides the role they play in algebraic top ology , Chern-Simons forms and -actions app ear prominently in the theoretical in terpretation of fascinating effects in condensed matter ph ysics and cosmology . A c kno wledgemen ts . I ha v e greatly profited from numerous discussions with my colleagues and friends, the late V. F. R. Jones (braid groups and knot theory), L. Michel (integral lattices) and R. Morf (quan tum Hall effect). I hav e had the great privilege of joint efforts with many p eople, including my friend Chr. King, who passed a wa y to o so on , my former PhD studen ts T. Kerler, B. Pedrini and U. M. Studer, my p ostdo cs E. Thiran and Chr. Sch weigert, and my colleague A. Zee, b esides further p eople. I thank J.-P . Bourguignon for a very careful reading of the manuscript and for suggesting corrections. I gladly recall that I was invited to present material closely related to what is describ ed in Sections 1 and 2 in a lecture at Stony Bro ok on the o ccasion of Jim Simons’ 60 th birthda y , back in 1998. – I am grateful to J. Cheeger and B. Lawson for giving me the opp ortunity to contribute this pap er to the Memorial Collection in honor of Jim Simons. References [1] E. Witten, Commun. Math. Phys. 121 , 351 (1989) [2] J. F röhlich and Chr. King, Comm un. Math. Phys. 126 , 167 (1989) [3] M. K ontsevic h, Adv. in So vj. Math. 16 (2), 137 (1993) [4] V. G. 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