A matrix solution to pentagon equation with anticommuting variables

A MA TRIX SOLUTION TO PENT A GON EQUA TION WITH ANTICOMMUTING V ARIABLES S. I. BEL’KO V, I. G. KOREP AN OV Abstract. W e construct a solution to pentagon equation with ant i commu ting v ariables l iving on tw o-dim ensional faces of tetrahedra. In this solution, matrix coordinates are ascrib ed to tetrahedron v ertices. As matrix multiplication is noncomm utative , this provides a “more quan tum” top ological field theory than in our previous works. 1. Introduction Pen tagon equa tio n deals with a Pachner mov e 2 → 3 a nd is a fundamental co n- stituen t of many top ologica l qua ntu m field theorie s (TQFT’s) for three-dimensiona l manifolds. Pac hner moves a re ele mentary r ebuildings o f a manifold triangulation whose imp or tance is due to the theorem o f Pachner [11, 7 ]: it sta tes (in pa rticular) that, for a giv en three-dimensio nal manifo ld, any triangulation ca n b e obtained from any other by a finite seq ue nc e of Pac hner moves of just four kinds: 2 ↔ 3 and 1 ↔ 4. Here “2 ↔ 3” means the following. Let ther e be in the tria ngulation, among others, tw o tetrahedra having a common tw o-dimensio na l face. W e denote them 1234 and 51 2 3, wher e 1 , . . . , 5 are their vertices, so 123 is their common 2 -face. The 2 → 3 mov e, by definition, repla ces these tw o tetrahedra with three tetra hedra 1254, 235 4 and 3154, o ccupying the same domain in the ma nifold. The 3 → 2 mov e is the inv e rse op eration. As fo r the mov e 1 → 4, it inserts a new v er tex 5 int o a tetra hedron 1 234 and replaces it with four tetrahedra 12 35, 1425, 134 5, a nd 32 4 5. Mov e 4 → 1 is again the inv er se op eration. Usually , how ever, the central r ole is play ed by mov es 2 ↔ 3 : if we hav e managed to do something meaningful for them, then it so happ ens that our cons truction works for mov es 1 ↔ 4 “automatica lly ” — and we will mee t with this exactly situation b elow when proving Theorem 4 of the present work. As Pachner mov es relate any tw o triangulations of a given manifold, a quantit y inv ariant under a ll Pachner mov e s do es not dep end on a sp ecific tr iangulation and is thus a manifold invariant . T o b e exact, this applies to piecewise- linear (PL) manifolds. In three dimensions, how ever, the piecewise-linear category co incides with the top ologica l categor y [10], so we get a to p o logical inv aria nt as well. The sp ecific mathematical s ense of “p entagon e quation” can b e different in dif- ferent scientific pap ers, the situation tha t is well-kno wn also, e.g ., for Y ang– Baxter equation: in b o th cases, an equation “ with v aria ble s on the edges”, or o n the 2-face s, etc., can be considered, as w ell as functional (or “set-theoretic” ) version of equa- tion, and so on. W e call solution t o p entagon e quation any alg ebraic relation that can b e r easona bly sa id to cor resp ond to a 2 → 3 Pachner mov e and from which o ne can exp ect that manifold inv aria nt s ca n b e constructed. In this pap er, o ur so lution to p entagon equation is formulated in terms of Gra ssmann–Ber ezin anticomm uting Key wor ds and phr ases. Pe ntagon equation, topological quant um field theory , algebraic com- plex, torsion. 1 2 S. I. BEL’KO V, I. G. K ORE P ANO V v ariables and is, in this resp ect, similar to the solution in [2], where it app ea red as the first s tep to co nstructing a simple TQFT 1 related to g roup PSL(2 , C ). Note that our TQFT’s are finite-dimensional in the sens e tha t they in volve no functional (infinite-dimensional) integrals: they deal with finite triang ulations and ascrib e to them a finite num b er of quantities. The disting uis hing fea ture o f the pres e nt paper is that the par ameters o f the the- ory — so-called “c o ordinates” ascr ibe d to triangulation vertices — are matric e s, and the noncommutativ e matr ix m ultiplication plays essential role in our p enta- gon equation here . This may b e imp ortant, b ecause, putting it a bit informally , any new no ncommutativit y makes o ur theor ie s “ more quantum” and thus remov es “classica l” features which might be present in our previous TQFT’s. There ar e some co nsiderations s howing that this “mor e quantum” c ha r acter will manifest its e lf pr o p erly only in the context inv o lving nontrivial — a nd even non- ab elian — representation of the manifold’s fundamental group, like it is describ ed in pap ers [8, 9]. The aim of the pr esent shor t pap er is, howev er, just to construct the solutio n to pentagon equa tion and show that it works also for mov es 1 ↔ 4, so we le ave thos e “quantum” calculations for further work. Below, we b egin in Section 2 with presenting our constructions in the scala r (matrices 1 × 1) case, which alrea dy gives a new and elegant s o lution to penta- gon equa tion. The genera lization to ma trices n × n is not so straig htf o rward, it arises from some s pe c ific a lg ebraic complexe s , in tr o duced in Section 3. In Section 4 , we establish the connectio n b ewteen matrices of linear mappings in a lgebraic com- plexes and expre ssions in an tico mm uting v ar ia bles. The actual so lution to p entagon equation with matrice s n × n is pr esented in Section 5. Then we construct in Sec- tion 6 the (simplest version of ) r e la ted manifold inv a riants; here mov es 1 ↔ 4 co me int o play . Finally , in Section 7 we discuss some miracles enco untered in previous sections. 2. S olution to pent ag o n equa tion: the scalar case 2.1. Grassmann algebras and Berezin in tegral. Reca ll [1] that Gr assmann al- gebr a over field F = R or C is a n asso ciative alg ebra with unit y , having generator s a i and r elations a i a j = − a j a i , including a 2 i = 0 . Thu s , any element o f a Grassmann a lgebra is a po lynomial of degree ≤ 1 in each a i . The Ber ezin inte gr al [1] is an F -linear op era tor in a Gr assmann algebr a defined by e q ualities Z da i = 0 , Z a i da i = 1 , Z g h da i = g Z h da i , (1) if g do es not dep end on a i (that is, generator a i do es not enter the ex pr ession for g ); m ultiple integral is understo o d as iterated one. 2.2. Sol ution to p en tagon equation. Cons ide r a tetrahedr on with vertices i 1 , i 2 , i 3 , i 4 , or simply tetrahedron i 1 i 2 i 3 i 4 . W e introduce a complex parameter ζ i for ev er y vertex i , c a lled its “co or dinate”. These pa rameters are arbitr a ry , with the only condition that any t wo different vertices i 6 = j hav e differ e nt co ordinates ζ i 6 = ζ j . W e will also use the notation ζ ij def = ζ i − ζ j . 1 Note also a long wa y f orm for mula (5) in pap er [4], which gav e the origin to our research on pentagon equations, to a TQFT in papers [5, 6]. MA TRIX SOLUTION TO PENT A GON WITH ANTICOMMUTING V ARIABLES 3 Then, we put in corres po ndence to any (unoriented) 2-face ij k a Gra ssmann generator a ij k (= a ikj = · · · = a kj i ), and to the tetrahedron i 1 i 2 i 3 i 4 — its weight f i 1 i 2 i 3 i 4 def = ζ i 1 i 2 a i 1 i 2 i 3 a i 1 i 2 i 4 − ζ i 1 i 3 a i 1 i 3 i 2 a i 1 i 3 i 4 + ζ i 1 i 4 a i 1 i 4 i 2 a i 1 i 4 i 3 + ζ i 2 i 3 a i 2 i 3 i 1 a i 2 i 3 i 4 − ζ i 2 i 4 a i 2 i 4 i 1 a i 2 i 4 i 3 + ζ i 3 i 4 a i 3 i 4 i 1 a i 3 i 4 i 2 . (2) Note that, in each summand in (2), the ζ b elo ngs to a n edge, while the tw o a ’s — to the tw o adjac e nt faces. The weigh t (2) is the simplest example of a gener ating funct ion of in v ar iants of manifold with triangula ted b oundary ; the in v aria nt s are the co efficients at the pro ducts o f anticomm uting v ariables . This makes, of course, little sense w he n the manifold is just one tetra hedron, but b ecomes nontrivial alrea dy in the cas e of clusters of tw o and thr ee tetra hedra in Theo rem 1 b elow. R emark 1 . E xpression (2) c ha nges its sig n under an o dd pe rmutations of indices i 1 , i 2 , i 3 , i 4 , i.e., it b elongs to an oriente d tetrahedron i 1 i 2 i 3 i 4 ; o rientation is under- sto o d as an orde r ing of tetrahedron vertices up to even per mut a tions. It will be conv enient for us , how e ver, mostly to ignore the orientations in this pa p er and sim- ply write the vertices in the increa sing order of their num b e r s, like in the following Theorem 1. Theorem 1. The function f i 1 i 2 i 3 i 4 define d by (2) satisfies the fol lowing p entagon e quation (de aling with two t etr ahe dr a 123 4 and 1235 in its l.h.s. and thr e e tetr ahe dr a 1245 , 2 345 and 13 4 5 in its r.h.s.): Z f 1234 f 1235 da 123 = 1 ζ 45 Z Z Z f 1245 f 2345 f 1345 da 145 da 245 da 345 . (3) Pr o of. F or mula (3) can b e pr ov ed, e.g., by a computer ca lc ulation.  R emark 2 . The integration in both sides o f (3) go es in the Grassmann v ariables living a t the inner 2- faces o f the cor r esp onding cluster of tw o or three tetrahedra. The spe cial role of edge 45 in (3), manifested in the factor 1 /ζ 45 , corr esp onds to the fact that 45 is the only inner edge amo ng the ten edges of the r.h.s . tetra hedra. 2.3. A ten tative state-sum in v arian t in the scalar case. If there is a trian- gulated orie nted manifold M with b oundary , then one can construct the following function of a nticomm uting v a riables a ij k living o n b oun dary faces (and para me- ters ζ i in vertices): 1 Q ′ ζ ij Z . . . Z Y f klmn Y ′ da ij k , (4) where the pro duct Q ′ ζ ij go es over all inner edges ij , the pro duct Q f klmn — over all tetrahedra k l mn , and Q ′ da ij k — ov er all inner faces. The expres sion (4) is determined up to an overall sign which may change if with change the order o f the th vertices (and/o r tetrahedra, differentials, etc.). It is a quite obvious consequence from Theor em 1 and Remar k following it that (4) is at least inv aria nt under all Pac hner mov es 2 ↔ 3 not changing the b oundary . It turns out that (4) is already , in s ome cases, a w o r king multicomponent (that is, incorp ora ting ma ny co efficients at v arious monomia ls in anticomm uting v ariables) inv ariant. It can b e called a state su m for manifold M ; fr om a physical v ie wpo int, the ant ico mmut ing v ariables mean that this state sum is of fermionic na ture. It can be shown, how ever, that there are t wo difficulties with direct a pplication of (4): • if the triangulatio n ha s at lea st one inner (no t b oundary) vertex, (4) yields zero, • if the boundar y of a connected manifold has more than one connected comp onent, (4) also y ie lds zero. 4 S. I. BEL’KO V, I. G. K ORE P ANO V These ar e tw o r e asons for intro ducing mor e p ow e r ful technique for obta ining manifold inv ar ia nts. The third r eason is that the noncommutativ e (matr ix) gener- alization of weigh t (2) is neither stra ightforw ar d nor o bvious. It turns out that these problems are so lved by in tro ducing new v ar iables, united in an algebr aic (chain) c omplex . 3. Algebraic complexes with ma trix “coordina tes” 3.1. Expli cit form ulas. W e consider a triangulated thre e-dimensional compact oriented connected ma nifold M with one-c omp onent 2 bo undary ∂ M . W e will even- tually present, b elow in Section 6, a set of inv ariants, constructed for the g iven b oundary triangula tion and dep ending on n × n complex matrices ζ i assigned to each b oundary vertex i ; every individual in v ariant fro m the set corre s po nds to a certain coloring of bo undary faces. Here color ing means choo sing some set C of certain differentials, this will b e explained so on after formula (5). W e present (a simple version o f ) our constr uction of algebra ic complexes, pro- viding, in particular, the matrix generaliza tion of weigh t (2). In this subsection, we present the formulas defining our a lg ebraic co mplexes in the explicit fo r m: es- sentially , as a sequence o f three matrices f 2 , f 3 , f 4 . These formulas a r e well suited for computer calcula tions, although their for m can hardly e x plain ho w they w er e found and for wha t reas on o ur seq ue nce (5) of vector spa ces a nd linear mappings is indeed a n algebra ic complex. This is expla ined in the next Subse ction 3 .2. W e denote by N k , k = 0 , 1 , 2 , 3, the nu mber o f k -simplexes in the triangula tion, and by N ′ k — the num be r o f inner k -simplexes. Then we n umber a ll vertices, in some ar bitrary order , by n umber s i = 1 , . . . , N 0 . Our inv ariants come out from algebra ic (chain) complexes of the following for m: 0 − → C n · N ′ 0 f 2 − → C 2 n · N 3 C f 3 − → C 2 n · N 3 f 4 − → C n · N ′ 0 − → 0 . (5) W e cons ider each vector spa ce in (5) a s consisting o f column vectors of the height equal to the exp onent at C . All vector s paces have thus natura l distinguishe d b ases consisting of vectors with one coo rdinate unit y and all o ther zero 3 ; thus w e ca n, and do , ident ify them with their matrices. R emark 3 . The fir st nonzero mapping in (5) is denoted f 2 , and not f 1 , in order to match our no tations here with other paper s, where similar but long e r complexes app ear, including tw o more mappings called f 1 and f 5 . See also the next Subsec- tion 3.2. A column vector — element of the first (fro m the left) spa c e C n · N ′ 0 is made, by definition, of N ′ 0 vectors dz i corres p o nding to each inner vertex i and ea ch having n co mpo nents. The next space, C 2 n · N 3 C , r equires a longe r explanation. Let there b e a 2 -face ij k , with i < j < k . T o such a face corres p o nds, by definition, a co lumn vector dϕ ij k of height n . An element of the vector space in question co nsists, by definition, of all elements of a ll dϕ ij k corres p o nding to N ′ 2 inner faces, and of so me set C of cardinality # C = n · (2 N 3 − N ′ 2 ) of comp onents of dϕ ij k corres p o nding to b oundar y faces ij k . W e would like, how ever, to define some more quantities for o ur further needs. W e denote by b any ordered triple i j k of tr iangulation vertices corr e sp onding to 2 The case where ∂ M has exactly one connec ted component i s the easiest tec hnically and seems to be enough for the presen t short pap er. The complications ar i sing when ∂ M is allow ed to ha ve arbi tr ary num b er 0 , 1 , 2 , . . . of component s are not very bi g, and such situation for a simil ar construction has b een considered in [2]. 3 This is imp ortan t when we are dealing with sub ject related to Reidemeister-style torsions. These will appear b elow i n Section 6. MA TRIX SOLUTION TO PENT A GON WITH ANTICOMMUTING V ARIABLES 5 some 2-fac e in the triang ulation. Here “order ed” means that we take them in this exact order : i, j, k , ignoring which o f n umber s i , j and k is smaller or greater. Now, if i < j < k , we set by definition dϕ i,b def = dϕ b . (6) Then we define dϕ i,b for any pair i, b with i ∈ b by the following co nditions: • if b 2 is o btained from b 1 by an o dd per mut a tion of i, j, k , then dϕ i,b 2 = − dϕ i,b 1 (th us, for an even permutation, the tw o dϕ i,b are of course eq ual), • the following relations hold: dϕ i,b + dϕ j,b + dϕ k,b = 0 , (7) ζ i dϕ i,b + ζ j dϕ j,b + ζ k dϕ k,b = 0 . (8) W e now pas s on to the following space, also C 2 n · N 3 . Let there b e a tetrahe- dron ij k l , with i < j < k < l , also denoted by a single letter a . T o such a tetrahedron corresp o nd, by definition, two column vectors dψ i,a and dψ j,a , each o f height n . An element of the vector space consists, by definition, o f all such co lumn vectors together. W e would like, howev er, to define a g ain some more quantities, namely , dψ i,a for any vertex i and any tetrahedron a ∋ i , r e gar d less o f condition i < j < k < l . W e do it in analo gy w ith what we have done for fa ces, by imp osing the following conditions: • if a 2 is obtained from a 1 by an o dd p ermutation of i, j, k , then dψ i,a 2 = − dψ i,a 1 , • the following relations hold: dψ i,a + dψ j,a + dψ k,a + dψ l,a = 0 , (9) ζ i dψ i,a + ζ j dψ j,a + ζ k dψ k,a + ζ l dψ l,a = 0 . (10) Finally , a n element of the last spac e C n · N ′ 0 is similar to that in the fir st space: it consis ts of N ′ 0 vectors dχ i corres p o nding to each inner vertex i and each having n co mpo nents. W e define linea r mappings f 2 , f 3 and f 4 as follows. • f 2 , by definitio n, makes the fo llowing dϕ ij k from g iven dz i : dϕ ij k = ( ζ i − ζ j ) − 1 ( dz i − dz j ) − ( ζ i − ζ k ) − 1 ( dz i − dz k ) . (11) • f 3 , by definition, makes the following dψ i,a and dψ j,a , where a = ij k l , i < j < k < l , from g iven dϕ ’s: dψ i,a = dϕ i,ijk + dϕ i,ikl + dϕ i,ilj , dψ j,a = dϕ j,ijk + dϕ j,ilj + dϕ j,j lk , (12) where the dϕ ’s in the r.h.s. are o f course ca lculated using (6), (7) and (8). • f 4 , by definitio n, makes the fo llowing dχ i from g iven dϕ ij k : dχ i = X a dψ i,a , (13) with the sum taken over a ll tetr ahedra a surrounding the given vertex i and taken all with p ositive o rientation; the dψ ’s in (13) a r e calculated , if necessary , using for mulas (9) a nd (10 ). R emark 4 . Ma trix f 3 depe nds th us o n the chosen set C of comp onents of dϕ ij k corres p o nding to bo undary faces ij k , as explained a b ove. All such matrices ar e, obviously , submatrices of matr ix f full 3 incorp ora ting all rows cor resp onding to al l comp onents of dϕ ij k . Matrix f full 3 acts thus from C n · N 2 to C 2 n · N 3 , we will make use o f it below in Se c tion 5. 6 S. I. BEL’KO V, I. G. K ORE P ANO V Theorem 2. The se quenc e (5) of ve ctor sp ac es and line ar mapp ings is inde e d an algebr aic c omplex, i.e., f 3 ◦ f 2 = 0 and f 4 ◦ f 3 = 0 . (14) Pr o of. Theore m 2 can b e prov ed by a direct calculation.  3.2. The mathematical orig ins of complex (5). The proo f o f Theo rem 2 by means of dir ect calculation do es no t ma ke clear the mathematica l reasons ensuring that (5 ) is a complex. So, in this subsection we briefly explain the mathematical origins 4 of complex (5). Namely , the linear mappings f 2 , f 3 and f 4 app ear as differ entials dF 2 , dF 3 and dF 4 (with some mo difications/r efinements if necessar y) of so me mapping s F i forming a s ort of “nonlinear complex” in the sense that F i +1 ◦ F i = const . (15) Then it obviously follows fr om (15) that dF i +1 dF i = 0 , to b e co mpared with (14). There are ac tually five mappings F i : i = 1 , . . . , 5. But, as T he o rem 2 is alr eady prov ed, and her e we just wan t to give an idea of where the for mulas in Subsection 3.1 come fro m, we restrict ourselves to prese nting firs t thre e of F i , leaving F 4 and F 5 as an ex ercise for an in ter ested r eader. Let there b e a triangulated thr e e -dimensional manifold, with fixed n × n matri- ces ζ i assigned to each tria ngulation vertex i = 1 , . . . , N 0 . • By definition, F 1 takes a pair ( a, b ) of n × n ma trices to z i = ζ i a + b (16) for each vertex i in the triangulation. Thus, a and b are, es sentially , pa- rameters of the gr oup of affine tr ansformations o f n × n matrix algebra. The co rresp o nding ta ngent ma pping f 1 = dF 1 do es not app ear in our co m- plex (5), but will a pp ear in its mo re general versions. • Mapping F 2 takes, by definition, matrices z 1 in vertices to matr ic es ϕ ij k = ζ − 1 ij z ij z − 1 ik ζ ik (17) for all tw o -faces ij k with i < j < k . Here and b elow we use notations ζ ij def = ζ i − ζ j , z ij def = z i − z j , etc. Note that for “ initial” v alues z ... = ζ ... , we have ϕ = 1. Obviously , the first of equa lities (1 5 ), namely F 2 ◦ F 1 = c onst, holds. The linear mapping f 2 in our sp ecific complex (5) is obtained, first, by differentiating formula (15) with res pe c t to z i for inner vertices i at their “initial” v alue s z i = ζ i , while z i = ζ i for bounda r y i stay co ns tant. This gives formula (11), where b oth dz i and dϕ ij k are, at this momen t, n × n matrices. As, how ever, (11) obviously o pe r ates with each column of dz i and corres po nding column of dϕ ij k separately , we then consider just one (e.g., first) column in b oth matrices, leaving for this column, a bit lo osely , the same resp ective no tation dz i or dϕ ij k . • Mapping F 3 , by definition, ta kes matrices ϕ ij k to matr ic es ψ i,a asso ciated with every tetrahedro n a = ij k l and its vertex i , a ccording to the following formula: ψ i,a = ϕ ij k ϕ ikl ϕ ilj . (18) 4 In this connect i on, see Ackno wledgemen ts in the end of the pap er . MA TRIX SOLUTION TO PENT A GON WITH ANTICOMMUTING V ARIABLES 7 Here, the ϕ ’s for any o rder of their vertices are ca lculated accor ding to formulas ϕ ikj = ϕ − 1 ij k , ζ − 1 ij ζ ik − ϕ ij k = − ζ − 1 j i ζ j k ϕ − 1 kij , which is in agreement with (17). Again, it is quite obvious that F 3 ◦ F 2 = c onst. F ormula (12) is obtained from (18) by differe nt ia ting, ag a in at the initial v alues z i = ζ i , ϕ ij k = 1, and then taking single co lumns in place of matrix differentials, like we did it for matrix f 2 . 4. Genera ting functions of Grassm ann v ariables for rect angular ma trices W e now wan t to link matrices (having in mind mostly matrix f 3 from Subsec- tion 3.1) to functions of Grassma nn v ariables. Let A be an arbitra ry ma trix whose en tries are complex num ber s or complex- v alued expr essions, with the only condition that the num be r of columns is no t smaller than the num b er of r ows 5 . With each column k o f A , we ass o ciate a Grassmann generator a k , while with the whole ma trix A — the gener ating fu n ction defined as f A = X C det A | C Y k ∈C a k , (19) where C runs ov er all s ubsets o f the set of columns of the cardinality equal to the nu mber o f r ows; A | C is the sq uare submatrix of A containing all columns in C ; the order of a k in the pro duct is the same as the order of columns in A | C (e.g., the most natural — incr easing — o rder o f k ’s in b oth). Lemma 1. L et C b e the vertic al c onc atenation of matric es A and B having the e qual n umb er of r ows: C =  A B  . Then f C = f A f B . Pr o of. The lemma easily follows fro m the expans ion of the form minor C = X ± minor A mino r B , (20) known from linear alg ebra, for every minor of C having the full n umber of columns.  Let there b e now a subset I in the set of all columns of A . W e ca ll the columns in I inner , the res t of them — outer , and we define the gener ating fun ction of matrix A with the set I of inner c olumn s a s I f A = X C ⊃I det ′ A | C Y k ∈C \I a k . (21) Here det ′ means that, unlike in (19), we are changing the or der of A ’s co lumns in the following wa y: all inner columns are br o ught to the right of the ma tr ix; the order of columns within the set I and its complement is conser ved; the order o f a k ’s in the pro duct (wher e k b elong s to the mentioned c o mplement) is the s a me a s the order o f columns k . 5 Note that here we hav e swapped the roles of rows and columns with resp ect to pap er [ 2]! 8 S. I. BEL’KO V, I. G. K ORE P ANO V Lemma 2. The gener ating function of matrix A with the set I of inner c olumns is t he fol lowing Ber ezin inte gr al of the usual gener ating funct ion: I f A = Z . . . Z f A Y l ∈I da l , (22) wher e the differ entials ar e written in the same 6 or der as r ows in A . Pr o of. First, we note that only those terms in f A survive the integration in the r.h.s. of (22) which contain all the a k for k ∈ I . W e take the function f A as defined in (19), leav e o nly the mentioned terms in it, and note that none of them is changed if we br ing both the columns k in A for all k ∈ I to the right of the matrix and the corres po nding generator s a k to the r ight in the pro duct 7 , neither changing the order within I nor within its complement. The n, the int e g ration in (22) just ta kes aw ay the a k for k ∈ I , as requir ed.  5. Pent a gon equa tion with ma trix co ordina tes 5.1. The p ent agon equation. It turns out that the matrix version of tetra hedron weigh t (2), satisfying the (matrix version o f ) pentagon equation, can b e co ns tructed as the g enerating function fo r matrix f full 3 (see Remark 4 ) corresp o nding to just one tetrahedron a = i 1 i 2 i 3 i 4 considered as a manifold with b oundary 8 . This matr ix f full 3 can b e c alculated using (1 2), (7) a nd (8), a nd rea ds: f full 3 =  1 − 1 1 0 ζ − 1 i 2 i 3 ζ i 3 i 1 − ζ − 1 i 2 i 4 ζ i 4 i 1 0 − 1  . (23) Matrix (2 3) is of cour se a blo ck matrix, with 0 and 1 meaning the n × n zero and unity matrices, resp ectively . The blo ck rows of ma trix (23) corr e s p o nd to differentials dψ i 1 ,a and dψ i 2 ,a , while the columns — to dϕ i 1 i 2 i 3 , dϕ i 1 i 2 i 4 , dϕ i 1 i 3 i 4 , and dϕ i 2 i 3 i 4 (in the na tur al or der in b oth cas es). W e denote f i 1 i 2 i 3 i 4 the g e nerating function (19) for matrix (23); we do not w r ite it out here beca use, as computer calculation shows, it contains 60 nonzero monomials already in the case n = 2. T o each dϕ ij k corres p o nds thus a ve ctor a ij k of n a nticomm uting v ar iables. Theorem 3 . The gener ating functions f i 1 i 2 i 3 i 4 for matric es (23) satisfy the fol low- ing p ent agon e quation: det ζ 23 det ζ 34 det ζ 35 Z . . . Z | {z } n f 1234 f 1235 D a 123 = 1 det ζ 45 Z . . . Z | {z } 3 n f 1245 f 2345 f 1345 D a 145 D a 245 D a 345 , (24) wher e D a i 1 i 2 i 3 me ans t he pr o duct of al l c omp onents of da i 1 i 2 i 3 , taken in their natur al or der. Pr o of. W e first prove the following lemma. Lemma 3. T he n -fold and 3 n -fold int e gr als in the l.h.s. and r.h.s. of (24) ar e gen- er ating functions for matric es f full 3 c orr esp onding to the l.h.s. and r.h.s. of Pachner move 2 → 3 , r esp e ctively, c onsider e d as triangulate d m anifolds with b oundary. 6 W e adopt the conv ention that the multiple Berezin integral is calculated following the rule RR f ( a ) g ( b ) da db = R f ( a ) da · R g ( b ) db . This con ven tion seems most commonly accepted. Note that we we r e using a different co ven tion in [2], wi th differen tial s in a multiple integral written in the rev ers e order — hence the di ffer ence betw een (22) and [2, f ormula (52)]. 7 because any elemen tary p ermutation of columns brings a minus sign which cancels out with the minus br ought b y the corresp onding p ermut ation of a k ’s 8 Note that, as it has no inner vertice s , there are no matrices f 2 and f 4 in complex (5) written for a si ngle tetrahedron, in the sense that one of dimensions in b oth matrices is zero. The same applies to the l.h.s. and r.h.s. of Pac hner mov e 2 → 3 below in Lemma 3. MA TRIX SOLUTION TO PENT A GON WITH ANTICOMMUTING V ARIABLES 9 Pr o of. Both these matrice s f full 3 are vertical concatenatio ns o f matrices (2 3) ex- tended with nec essary co lumns, co rresp onding to 2 -faces absent fro m the g iven tetrahedron and filled with zeros. F or instance, here is the matr ix f full 3 for the l.h.s. of move 2 → 3: ( f full 3 ) l . h . s . =     1 − 1 0 1 0 0 0 ζ − 1 23 ζ 31 − ζ − 1 24 ζ 41 0 0 0 − 1 0 1 0 − 1 0 1 0 0 ζ − 1 23 ζ 31 0 − ζ − 1 25 ζ 51 0 0 0 − 1     . (25) Like in (23 ), every element of matrix (2 5) is a matrix of sizes n × n . The fir s t blo ck column in matrix (25) corresp onds to the ( n compo nents of ) differential dϕ 123 at the inner face 1 2 3, the rest of columns — to the following b oundary faces, from left to rig ht: dϕ 124 , dϕ 125 , dϕ 134 , dϕ 135 , dϕ 234 , a nd dϕ 235 . The r ows co rresp ond to dψ 1 , 1234 , dψ 2 , 1234 , dψ 1 , 1235 , and dψ 2 , 1235 . W e do not write out her e the matrix corre s p o nding to the r.h.s . o f the Pac hner mov e. It is made in the same obvious manner and contains 6 × 9 blo ck entries. The sta tement o f the lemma fo llows now fro m Lemmas 1 a nd 2.  Now we contin ue with the pro of of Theorem 3. It remains to prov e that the minors of ma trix ( f full 3 ) l . h . s . and the similar 6 × 9 blo ck ma trix ( f full 3 ) r . h . s . , corr e- sp onding to the r.h.s. of the Pac hner mov e 2 → 3, are pro po rtional with the same ratio as the integrals in b oth sides of (24), provided these minors contain all the rows of the cor resp onding matr ix, all the columns co rresp onding to dϕ ’s a t inner faces (or simply “inner dϕ ”), and the o ther co lumns in tw o minors corres p o nd to the same dϕ ’s at b ounda ry faces (or simply “b oundary dϕ ”). Let f 3 denote, in the rest of this pro of, a ny of ( f full 3 ) l . h . s . and ( f full 3 ) r . h . s . . Co nsider the following question: wha t conditions must be imp osed on b oundary dϕ in order that there exist some inner dϕ suc h that the vector co mp o sed of all these (inner and b o undary) dϕ b elong to the kernel of f 3 ? F o r mulas (1 2 ), together with (9) and (10), make it clear that these co nditions on dϕ 124 , . . . , dϕ 235 can b e written a s dϕ 1 , 124 + dϕ 1 , 143 + dϕ 1 , 135 + dϕ 1 , 152 = 0 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dϕ 5 , 512 + dϕ 5 , 523 + dϕ 5 , 531 = 0 .    (26) The skipp ed lines in (26) corr e sp ond to going a round vertices 2, 3 and 4 along the bo undary faces in the sa me obvious way a s the fir s t and the last lines corr esp ond to g oing around vertices 1 and 5 . It follows fro m (7) a nd (8) that there are 3 n indep endent co nditions a mong the 5 n conditions in (26). Thus, there is a lso a 3 n -dimensiona l space of b oundar y dϕ lying in the kernel o f f 3 mo dulo inner dϕ . Th us, any of t he 6 n “b oundary” c olumns of matrix f 3 is a line ar c ombination of just 3 n of them mo dulo “inner” c olumns, and the c o efficients in this line ar c ombination ar e the same for ( f full 3 ) l . h . s . and ( f full 3 ) r . h . s . . This yields immediately the desir ed prop or tionality o f minors. T o calculate the co efficient, it is enoug h to take any specific minor in ( f full 3 ) l . h . s . and the co rresp ond- ing minor in ( f full 3 ) r . h . s . ; this b ecomes an ea sy exe r cise if we ta ke minors containing our n × n blo cks only as a who le.  5.2. T en tative i n v arian t. It folows fro m (24) that the following function of anti- commuting v ariables at b o undary faces is inv ariant under moves 2 ↔ 3 : ± Q (f ) det ζ j 2 j 3 Q (e) det ζ i 1 i 2 · Q (t) det ζ k 3 k 4 Z . . . Z Y (t) f k 1 k 2 k 3 k 4 · Y (f ) D a j 1 j 2 j 3 (27) Here: • the pro duct denoted Q (e) is taken ov er all inner edg es i 1 i 2 , 10 S. I. BEL’KO V, I. G. K ORE P ANO V • b oth pro ducts denoted Q (f ) are ta ken o ver all inner 2 -faces j 1 j 2 j 3 , j 1 < j 2 < j 3 , • b oth pro ducts denoted Q (t) are tak en o ver all tetrahedra k 1 k 2 k 3 k 4 , k 1 < k 2 < k 3 < k 4 . The sign “ ± ” in (27) co rresp onds to the fact that it is a separa te problem to order the weigh ts f in their pro duct, as w e ll as the in tegra tion in different v ariables; so we just leave (27) defined up to a sig n 9 . 5.3. The case n = 1 : repro ducing formula (2). T ake matrix (23) with n = 1 and do the following: multiply the first column, corr esp onding to dϕ i 1 i 2 i 3 , by ζ i 2 i 3 , and similar ly the other columns b y ζ i 2 i 4 , ζ i 3 i 4 and aga in ζ i 3 i 4 , resp ectively; then divide the se c ond r ow by ( − ζ i 3 i 4 ). The gener ating function (19 ) for such “gauge transformed” matrix is nothing but the “s c a lar” weight (2). It is now an e asy exercise to deduce equation (3) from the genera l matrix equation (24). 6. Arbitrar y manifold with on e-component boundar y: torsio n and a set of inv ariants As already mentioned in Subsection 3.1, we ar e co nsidering a triangula ted three- dimensional compa c t o riented connected manifold M with one-c omp onent b ound- ary ∂ M . It can b e shown that if the triangulation do es co ntain inner vertices, the ten ta tive inv ariant (27) just tur ns int o zer o. More ov er, (27) is obviously inv aria nt only with resp ect to mov e s 2 ↔ 3, a nd nothing is known a priori ab out mov es 1 ↔ 4. This is why we are going to construct in this section the inv ariants in the case where inner vertices are allow ed, and pr ov e their inv ariants under al l Pac hner mov es. W e define the to rsion of c omplex (5) as τ = minor f 3 minor f 2 minor f 4 , (2 8) where the mino rs corresp ond to so me nonde gener ate τ -ch ain according to the usual rules [12]; if such τ -chain do es not exist, then τ = 0. Theorem 4. The expr ession I C ( M ) = ± Q (f ) det ζ j 2 j 3 Q (e) det ζ i 1 i 2 · Q (t) det ζ k 3 k 4 · τ , (29) wher e the pr o ducts ar e define d in t he very same way as explaine d after formula (2 7) , taken for given subset C of c omp onents of b oundary dϕ as explai n e d after for- mula (5) , is an invariant of manifold M with the t riangulate d one-c omp onent b ound- ary ∂ M . In other w or ds, I C ( M ) do es not change under mov es 2 ↔ 3 and 1 ↔ 4 within M , not affecting the fixed triangula tion of ∂ M . Pr o of. A move 2 ↔ 3 changes only minor f 3 in (28 ). The fa ctor by which minor f 3 is multiplied is deter mined in essentially the sa me wa y as in the pro o f of Theorem 3 ; the a dditional tetra hedra (with resp ect to the siuation where there w er e just t wo tetrahedra in the l.h.s. and three in the r.h.s.) do no t affect this fa c tor. As for the mov e 1 → 4, it can be considere d a s a co mpo sition o f mov es 0 → 2 and 2 → 3, wher e 0 → 2 means that we take an inner 2-face let it b e face 1 23, and glue in its place tw o o pp o sitely or iented tetra hedra, s ay 12 34, in such w ay that they ar e g lued to ea ch other by their r esp ective faces 1 24, 13 4 and 23 4. Thus, old face 1 23 is r eplaced with a “triangula r pillow” with the new vertex 4 inside. 9 which is quite common when the sub j ect is related to torsions, see Section 6 MA TRIX SOLUTION TO PENT A GON WITH ANTICOMMUTING V ARIABLES 11 W e thus consider this mov e 0 → 2. One po ssibility o f changing the minors in (28) under this move is a s follows: • extend minor f 2 by blo ck r ow co rresp onding to dϕ 124 and blo ck column corres p o nding to dz 4 , • extend minor f 3 by blo ck rows cor resp onding to dψ (1) 1 , 1234 , dψ (1) 2 , 1234 and dψ (2) 1 , 1234 , where sup ers cript (1) indicates one tetrahedro n and (2) — the other, and blo ck columns co r resp onding to dϕ 134 , dϕ 234 and one of the t wo dϕ 123 , • extend minor f 4 by blo ck row corres po nding to dχ 4 and blo ck column cor- resp onding to dψ (2) 2 , 1234 . Standard arg umen t using blo ck tria ngularity shows that the three resp ective mi- nors a re thus mult iplie d by D ϕ 124 D z 4 , D ψ (1) 1 , 1234 ∧D ψ (1) 2 , 1234 ∧D ψ (2) 1 , 1234 D ϕ 134 ∧D ϕ 234 ∧D ϕ 123 , and D χ 4 D ψ (2) 2 , 1234 , where D means the exterio r pro duct of different ia ls of n comp onents of the resp ective quantit y . The first of these quantities is co mputed us ing (11), the s econd — (23), and the last — (13) together with (9) and (10). The result is that τ is m ultiplied (up to a sign) by det ζ 14 det ζ 34 det ζ 23 . One can see that, miraculo us ly , this ag rees with how the pro ducts in formula (29) change.  Of course, in the case of no inner vertices, the inv ar iants (29) are nothing but co efficients at the pro ducts of an tico mm uting v ariables in (27), thus we hav e proved that these co efficients ar e top olo gical in v ar iants — pr ovided a triangulation with no inner p oints 10 exists. 7. Discussion W e hav e constructed the fir st ever solution of p entagon eq ua tion with anticom- m uting v ariables and incor po rating, in an ess ent ia l wa y , the noncommutativ e matrix m ultiplica tion; this can b e seen in formula (23) fro m which the tetra he dr on weigh t is ma de accor ding to (1 9). W e also show ed in Section 6, on the example o f as simple algebraic co mplexes a s we co uld inv ent, how the obtained inv ar iants a r e rela ted to the torsion of acyclic complex e s. This also showed the g o o d be havior of our inv a ri- ants with resp ect to Pachner mov es 1 ↔ 4 (while the pentagon equation dealt only with mov es 2 ↔ 3). W e plan to wr ite another , a nd longer , a rticle, containing int er esting calcula- tions, esp ecia lly for “twisted” complexes (like those in [8, 9], but for manifolds with bo undary), a nd other material such as the generaliza tion of o ur complex (5) for the case of b ounda ry having any num b er of co mpo nents. Moreov e r , it turns out that complex (5) admits a rather str aightforw a rd gen- eralization o nto four -dimensional manifolds — this will b e the theme o f s e pa rate resear ch. The ex istence of inv ariants lik e (29), with a factor, multiplicativ e in some v alues belo nging to simplexes of tr iangulation, multiplied by a Reidemeister-type torsion, alwa ys comes a s a miracle. The po int is that we first co nstruct a complex like (5) (already guided b y so me no t very for mal ideas), and complex (5) b elongs to a fi xe d triangulation of a manifold M . It alw ays turns o ut, how ever, tha t the torsion of such a complex behaves bea utifully under a ll types of Pachner mov es changing the triangulation. T o co nclude, we rema rk that coming from a “ na ¨ ıve” state-sum inv aria nt like (4) or (27) that turns in many c ases int o zero, to inv ariants inv olving tors io n ca n be considered as a s ort of r e no rmalization pro cedur e. This pro cedure int r o duces 10 and of course not conta i ning edges starting and ending at the same vertex 12 S. I. BEL’KO V, I. G. K ORE P ANO V new v ariables dz i and dχ i , and, in physics, s uch v ariables may corre s po nd to new ph y sical entities. This raise s an interesting question of poss ible relations b etw een acyclic complexes and renorma lization. Ac kno wl edgements. One of the a uthors (I.K.) thanks Ir ina Aref ’ev a and all the organize rs for the great p o ssibility of making a rep ort at the conference SFT’09, and for their warm hospitality . The idea o f using for mu la s like (16), (17) and (18 ) was suggested to I.K . by Rinat Kashaev [3]. W e would like to expre ss him o ur gr atitude for this sug gestion. The work of I.K. w a s supp orted in pa r t by the Russian F oundation for B asic Research (Grant No. 07- 01-00 081a ). References [1] F.A. Berezin, Introduction to s up eranalysis. Mathematical Phy si cs and Applied Mathematics, v ol. 9, D. Reidel Publishing Compan y , Dordrech t, 1987. [2] S.I. Bel’ko v, I.G. Kor epano v, E.V. Martyushe v, A si m ple top ological quantu m field theory for manifolds with tri angulated b oundary , ar X iv:0907.3787v1 (2009). [3] R.M. Kashaev, priv ate communication (2006). [4] I.G. Korepanov, In v ar iant s of PL manifolds from metrized simplicial complexes. Thr ee- dimensional case, J. Nonlin. M ath. Ph ys., vol. 8 (2001), no. 2, 196–210. [5] I.G. Korepanov, Geometric torsions and inv arian ts of manifolds with a triangulated b oundary , Theor. Math. Phys., vol. 158 (2009), 82–95. [6] I.G. Korepanov, Geometric torsions and an At i ya h- s t yle top ological field theory , Theor. Math. Ph ys. , vol. 158 (2009), 344–354. [7] W.B.R. Li c korish, Simplicial mov es on complexes and manifolds, Geom. T op ol. M onographs 2 (1999), 299–320. [8] E.V. Martyushev, Euclidean simpl ices and inv ar iant s of three-manifolds: a mo dification of the inv ariant f or lens spaces, P r o ceedings of the Chelya bi nsk Scientific Cente r 19 (2003), No. 2, 1–5. [9] E.V. Mar tyushev, Eucli dean geometric inv arian ts of links in 3-sphere, Pro ceedings of the Chely abinsk Scien tific Cen ter 26 (2004), No. 4, 1–5. [10] E. Moise, Affine structures in 3-manif olds, V, Ann. of Math., 5 6 (1952), 96–114. [11] U . Pac hner, PL homeomorphic manif ol ds are equiv alent by elementary shellings, Europ. J. Comb i natorics, 12 (1991), 129–145. [12] V .G. T uraev, In tro duction to combinat ori al torsions, Boston: Birkh¨ auser, 2000.