Multiplicative expression for the coefficient in fermionic 3-3 relation

Recently, a family of fermionic relations were discovered corresponding to Pachner move 3-3 and parameterized by complex-valued 2-cocycles, where the weight of a pentachoron (4-simplex) is a Grassmann-Gaussian exponent. Here, the proportionality coefficient between Berezin integrals in the l.h.s. and r.h.s. of such relations is written in a form multiplicative over simplices.


Introduction
This paper continues the series of papers [5], [6] and [7]. The reader is referred especially to [7] for definitions and facts that are only briefly mentioned here. Also, the reader is referred to [1] for a concise exposition of Grassmann-Berezin calculus of anticommuting variables (or to [2] for a more modern and detailed exposition), and to [8] for a pedagogical introduction to Pachner moves.
In paper [6], a large family was discovered of Grassmann-Gaussian relations corresponding to Pachner move 3-3, with pentachoron (4-simplex) weights depending on a single Grassmann variable attached to each 3-face. In paper [7], a full parameterization was given for (a Zariski open set of) such relations, in terms of a 2-cocycle given on both l.h.s. and r.h.s. of the Pachner move. Many questions still remain, however, to be solved before we arrive at a full-fledged four-dimensional topological quantum field theory on piecewise-linear manifolds.
In the present paper, we solve one of such questions, and show that the answer is remarkably nontrivial. It consists in finding the coefficient called 'const' in [7, formula (53)] (as well as [6, formula (6)]) in a form that would make possible further construction of a manifold invariant. Namely, the coefficient should be represented as a ratio, const = c r /c l (compare relation (1) below), of two expressions belonging to the two sides of the move, and each of these must be multiplicative -have the form of a product over simplices belonging to the corresponding side. This was the case in an earlier paper [5], see formula (1) and Theorem 1 there, also reproduced in [6,Section 6], although the 3-3 relations in these papers must be regarded as degenerate from the viewpoint of the present paper. This was also the case in [3, formula (38)] and [4, formula (12)], where different but similar relations were considered.

PL manifold invariants and Pachner moves
In order to construct invariants of piecewise linear (PL) manifolds, it makes sense to construct algebraic relations corresponding to Pachner moves, see, for instance, [8,Section 1]. Pachner's theorem states that a triangulation of a PL manifold can be transformed into any other triangulation using a finite sequence of these moves [9], so there is a hope to pass then from such relations to some quantities characterizing the whole manifold.
In the four-dimensional case, the Pachner moves are 3-3, 2-4 and 1-5. The first of them is usually regarded as 'central', and we will be dealing with it in this paper. Here we describe this move and fix notations for the involved vertices and simplices.
Let there be a cluster of three pentachora (4-simplices) 12345, 12346 and 12356 situated around the 2-face 123. Move 3-3 transforms it into the cluster of three other pentachora, 12456, 13456 and 23456, situated around the 2-face 456. The inner 3-faces (tetrahedra) are 1234, 1235 and 1236 in the l.h.s., and 1456, 2456 and 3456 in the r.h.s. The boundary of both sides consists of nine tetrahedra.
Note that we have listed in the previous paragraph exactly all simplices in which the l.h.s. of move 3-3 differs from its r.h.s. And the boundary of both sides is, of course, the same, it consists of nine tetrahedra.
1.3 The results of this paper, and how they are explained The results are explicit formulas for everything in (1): Grassmann weights W ijklm and coefficients c l and c r -in terms of a 2-cocycle ω, in accordance with [7]. As all formulas are algebraic, the author might have presented just these formulas, saying: and now the validity of (1) can be checked using computer algebra. The formulas look, however, rather intricate, so we follow another way, focusing on the actual author's reasonings.

Explicit formulas for matrix elements
In this Section, as well as in the next Sections 3 and 4, we work within a single pentachoron u = 12345. The changes to be made for other u are quite simple and will be explained later.

Convention 2.
We also write the simplices by their vertices, e.g., s = ijk or, as we have written above, u = 12345. The vertices are thus given by their numbers, and in writing so, we assume by default that the vertices are ordered : i < j < k, etc. If, however, we need a triangle whose order of vertices in unknown or unessential, we write s as {ijk}, as in Lemma 1 below.

Edge operators
Our Grassmann-Gaussian pentachoron weight is is the column of Grassmann variables corresponding to the 3-faces t ⊂ u, and F -a 5 × 5 antisymmetric matrix.
We are going to recall the construction of matrix F from [7]. Moreover, we will write down some specific explicit expressions for the entries of F that do not appear in [7]. On the other hand, we will skip some details for which the reader is referred to [7].
Our starting point is a 2-cocycle ω: it takes complex values ω s = ω ijk on triangles s = ijk ⊂ u such that Then, there are edge operators d b for the ten edges b = ij ⊂ u that make the bridge between ω and matrix F . Edge operators have the following properties: • they belong to the 10-dimensional space of operators • more specifically, the sum (4) for a given d b runs only over such three tetrahedra t that t ⊃ b, • each of them annihilates the pentachoron weight: • they are antisymmetric with respect to changing the edge orientation: • they obey the following linear relations for each vertex i ∈ u: • and there is one more linear relation: where ν is any 1-cocycle such that ω makes its coboundary: • they form a maximal (5-dimensional) isotropic subspace in the (10dimensional) space of all operators of the form (4), where the scalar product is, by definition, the anticommutator:

Partial scalar products of edge operators
Due to the form (4), we have t-components of edge operators, and the (vanishing) scalar product of two edge operators is a sum over tetrahedra: where d b 1 , d b 2 t -we call it the partial scalar product of d b 1 and d b 2 with respect to tetrahedron t -is by definition the same as Lemma 1. Choose a tetrahedron t ⊂ u and a triangle {ijk} ⊂ t (see Convention 2 for this notation). Then the partial scalar product d ij , d ik t remains the same under any permutation of i, j, k.
Proof. Let us prove, for instance, that Setting i = 3 in (6) and taking its t-component, we have (keeping in mind also (5)): We want to take the scalar product of (9) with d 12 . As 1234 is the only tetrahedron common for the edges 12 and 34, and all edge operators are orthogonal to each other, we get So, the mentioned scalar product, together with (10), gives (8) at once.
For a tetrahedron t ⊂ u, construct the expression Here tetrahedron t is considered as oriented, s is any of its 2-faces with the induced orientation, and b 1 , b 2 ⊂ s are two edges sharing the same initial vertex (thus also oriented). Then, the expression (11) does not depend on a specific choice of s, b 1 and b 2 , and thus pertains solely to t.
Proof. Let us prove, for instance, that (the minus sign accounts for opposite orientations of 123 and 124). A small exercise shows that the following linear relation is a consequence of (7): Multiplying this scalarly by d 12 and using once again orthogonality (10), we get (12).
Lemma 3. Expression (11) also remains the same for all tetrahedra t forming the boundary of pentachoron u, if these tetrahedra are oriented consistently (as parts of the boundary ∂u).
Proof. It is enough to consider the situation where s is the common 2-face of two tetrahedra t, t ′ ⊂ u, and show that Indeed, as the orientation of s as part of ∂t is different from its orientation as part of ∂t ′ , there are two values ω s differing in sign, and (13) will yield at once that (11) is the same for t and t ′ .
To prove (13), we note that t and t ′ are the only tetrahedra containing both b 1 and b 2 , so Convention 3. We normalize edge operators in such way that quantity (11) becomes unity.
Here is the matrix of scalar products d a , d b 1234 calculated according to Convention 3. The rows (resp. columns) correspond to edge a (resp. b) taking values in lexicographic order: 12, 13, 14, 23, 24, 34: Remark 1. To calculate diagonal elements in (14) is an easy exercise using linear relations similar to (9).
Remark 2. As for tetrahedron 1235, we must not only replace '4' by '5' in (14), but also change all signs -due to its different orientation! Similarly, analogues of matrix (14) for other tetrahedra can be calculated.

Superisotropic operators and matrix F
Superisotropic operators are such operators of the form (4) that annihilate the weight W u and whose each t-component is isotropic, i.e., either γ t = 0 or β t = 0. The rows of matrix F correspond to superisotropic operators in the following sense: every component of the column where x is given by (2) and p, similarly, by is superisotropic. We recall [7, Subsection 4.2] how superisotropic operators proportional to entries of (15) are constructed in terms of edge operators. They all are linear combinations written as First, we choose and fix one of two square roots of each ω s : Second, we define "initial" α ij as Example 1. As the 2-faces s ⊂ 12345 containing edge 12 are 123, 124 and 125, and the only 2-face not intersecting with 12 is 345, such "initial" α 12 is Finally, the operator proportional to the i-th entry in (15), and thus corresponding to the tetrahedron t not containing the vertex i, is obtained by the following change of signs: We want to identify the entries in column (15) with the operators given by (16), (17) and (18). Such identifications are determined to within a renormaliza- This renormalization leads to multiplying matrix F from both sides by the diagonal matrix diag(λ −1 2345 , . . . , λ −1 1234 ). To fix the mentioned arbitrariness, we choose a distinguished edge a in every tetrahedron t and assume that the restriction of d a onto t has a unit coefficient before ∂ t : d a | t = ∂ t + γx t .
As ∂ t , x t = 1, this implies Convention 4. In this paper, the distinguished edge a in any tetrahedron t will always be the lexicographically first one, for example, a = 12 in tetrahedron t = 1234.
We now denote g (t) the superisotropic operator defined according to (16), (17) and (18). If such operator contains a summand γx t ′ , then γ = g (t) , d a t ′ , and if it contains β∂ t , then β = 2 g (t) , d a t d a , d a t . Hence, the matrix element F tt ′ = γ/β (because the coefficient at ∂ t must be set to unity, according to (15)), i.e., The scalar products are calculated according to (14) and Remark 2.
Here is a typical matrix element:

Divisors of matrix elements
The central part of the present work consisted in finding a nice description for poles and zeros of matrix elements F tt ′ of the typical form (20). The point is, of course, that the quantities ω ijk = q 2 ijk make a cocycle, so there are dependencies for all tetrahedra ijkl.

Variables a ij and their relation to "initial" α ij
Recall that we are working within one pentachoron 12345. It has ten 2-faces, as well as ten edges. This fact, together with the accumulated experience (compare [7, formula (50)]), suggests the idea to introduce a 1-chain a ik such that ω ijk is written as a product of its three values, namely: Given all ω ijk , the a ij are found from the system of equations which become linear after taking logarithms and are easily solved. Interestingly, the result is, up to an overall factor, our old alphas from formula (17): . The cocycle relations are now written (instead of (23)) as a kl a jl a jk − a kl a il a ik + a jl a il a ij − a jk a ik a ij = 0. (26) Remark 4. We do not permute the indices of a ij in this paper, but if needed, the natural idea is to assume that a ij = −a ji .

Matrix elements in terms of a ij
Matrix elements F tt ′ can now be calculated in terms of a ij . To be exact, here is what we do: set α ij = a ij /p according to (25); the value of p is not of great importance because it will soon cancel out. Then apply formula (19) with g (t) expressed using (16)     Of course, even if it is not immediately obvious from (27) and (28). We will shed some light on this by studying the poles and zeros of these expressions.

The variety of zeros of the main factor in the denominator of a matrix element as function of six variables
The main factor in the denominator of (27) is and its pleasing feature is that is depends on the six variables a ij belonging to just one tetrahedron 2345. There is just one dependence between these a ij : Remark 6. The goal of this paper consists in finding the expressions (56) and (57) below, for example, by guess. It looks hardly possible to guess these expressions based on nothing, but studying divisors on M ′ proves to be enough for achieving this goal, so, we content ourself with M ′ . Nevertheless, studying divisors on the whole M might be also of interest, because, for instance, (33) and (34) lie exactly in M \ M ′ .
In M ′ , we introduce the following subvarieties of codimension 1, denoted as D with indices because we think of them as Weil divisors: • D u : this is the subvariety given by the old formulas (36), but now ten of them: 1 ≤ i ≤ j ≤ 5, • (D u ) K : choose now subset K ⊂ {1, 2, 3, 4, 5}, and define (D u ) K by the same formulas (36) except that we change the signs of those a ij whose exactly one subscript i or j is in K. We write also (D u ) 1 , (D u ) 12 , etc. instead of (D u ) {1} , (D u ) {1,2} , etc., • D − t : for a tetrahedron t = ijkl ⊂ u, let b = ij be the distinguished edge. Then D − t is given by the following equations (compare to (32)!): • D + t , similarly: Lemma 6. For a tetrahedron t = ijkl and its distinguished edge ij, the sum D − t + D + t gives, on M ′ , exactly the zero divisor of σ t a ij = a jk a ik − a jl a il (compare with the first factor in the numerator of either (27) or (28)!), where we denoted Proof. Due to the cocycle relation (3),  (ii) The zero divisor of (27), restricted to M ′ , is (D u ) 12 + D + 2345 + D + 1345 (the last two are defined in (39)).
Proof. First, note that the component (32) of the divisor of function (30) cancels out with the first factor in the numerator of (27), that is, and what remains of the zero divisor of (41) after this canceling is D + 2345 , according to Lemma 6.
For item (i), this means that, on the pole variety of (27), all the expressions ψ s /ω s are the same for s ⊂ 2345. And analyzing (28) similarly (and taking into account (29)), we arrive at the conclusion that the same are also ψ s /ω s for s ⊂ 1345. It is not hard to deduce now (through a small calculation) that ψ s /ω s are the same for the whole pentachoron 12345, including s = 123, 124 and 125. So, item (i) is proved.
For item (ii), we first notice that the main factor in the numerator of F 12 (resp. F 21 ) is the same (up to an overall sign) as the main factor in the denominator of F 21 (resp. F 12 ) except that the sign is changed of all a ij with i = 1 (resp. i = 2). For F 12 , this means that D + 1345 appears as a component of zero divisor, in analogy with (32), while the first paragraph of this proof means that D + 2345 is also there. The rest, namely (D u ) 12 appears in full analogy with D u in the previous paragraph. So, item (ii) is also proved.
Remark 7. Theorem 1 speaks about a specific matrix element and divisors. It applies, however, to all similar objects, with obvious changes.

Function ϕ 12345
On our subvariety M ′ ⊂ M (37), we can express all a ij in terms of q ijk according to (25), where the factor p never vanishes and can be ignored as long as we are considering the zero or pole varieties of expressions homogeneous in variables a ij .
Remark 8. And all functions of a ij or q ijk in this paper are homogeneous.
Remark 9. Also, the fact that p is multivalued makes no obstacle on our way. has the divisor (zeros with sign plus, poles with sign minus) Proof. The formulas for divisors of the first five functions make a simple variations on the theme of Lemma4, where, of course, Convention 5 must be also taken into account. Then, (48) follows by adding/subtracting relevant divisors.
Motivated by Lemma 6, we divide the expression (47) by Theorem 2. The divisor of the so obtained expression considered as a function on M ′ , is Proof. This follows from (48) and Lemma 6.
The symmetry of divisor (50) suggests the following theorem. Proof. Direct calculation.
Function ϕ 12345 is thus an interesting highly symmetric function of variables q ijk belonging to the pentachoron u = 12345 and obeying the restrictions (23).

The poles and zeros of the coefficient in 3-3 relation, and its explicit form
We now pass from the single pentachoron 12345 to Pachner move 3-3, where six pentachora are involved. The l.h.s. and r.h.s. of move 3-3 are triangulated manifolds with boundary. We can orient the pentachora in both these manifolds consistently, and also so that these orientations induce the same orientation on the boundary ∂(l.h.s.) = ∂(r.h.s.). For one such orientation (of two), the signs in the following table show whether this consistent orientation of a pentachoron coincides with the orientation given by the natural order of its vertices: We do now all calculations in terms of variables q ijk and not a ij . This is due to the following important remark.
Remark 11. Variables a ij depend on a pentachoron (i.e., two a ij for the same edge ij, but calculated within two different pentachora containing this edge, are different), while q ijk do not.

Matrix elements for all six pentachora involved in move 3-3
In Section 2, we explained how to calculate matrix F elements for pentachoron 12345. For any pentachoron ijklm (recall that i < · · · < m, according to Convention 2), the obvious substitution 1 → i, . . . , 5 → m must be made. Besides this, the sign of matrix element must be changed for the pentachora marked with minus sign in table (51), as we are going to explain in (the proof of) Lemma 9, where we study the way how our normalization of edge operators, given by Convention 3, propagates from one pentachoron to another.
Lemma 9. Expression (11) can be normalized to unity for a whole oriented triangulated manifold.
Proof. Let tetrahedron t be the common 3-face of two pentachora, t = u 1 ∩ u 2 . Let a ⊂ t be its edge, and d We see now that, on passing to a neighboring pentachoron, first, the orientation of t changes (and this affects the orientation of s in(11)!), and secondpartial scalar products of edge operators also change their signs because of (52). Hence, the quantity (11) remains the same, as before in Lemmas 1, 2 and 3. This means that it pertains to the whole triangulated manifold, if it is orientable and connected. Hence, we can normalize all edge operators globally so that quantity (11) stays always equal to unity.
And it is clear from (19) that, indeed, changing the sign of partial scalar products implies changing the sign of matrix elements.

Components
in the r.h.s.
Remark 12. Two tetrahedra in the subscripts of a matrix element in (53) or (54) clearly determine the relevant pentachoron.
Our goal is now to guess the form of c l and c r . As we can then check the correctness of our guess with a direct computer calculation, informal reasoning will be quite enough for us at this moment.
So, we analyze poles and zeros of L, R, and other similar Grassmann polynomial coefficients, in order to invent such c l and c r that will compensate these poles and zeros. First, we do so assuming that no one of values q ijk vanish, that is, within the 'global analogue' of set M ′ (37). The poles of at least one component in the l.h.s. are relevant, while the zeros must be common for all components; similarly for r.h.s. We see this way that the poles are situated on divisors D u (see Subsection 3.4) for all pentachora in the relevant side of Pachner move, while the zeros are situated on divisors D + t of all inner tetrahedra, again in the relevant side of Pachner move.

Fitting the divisors, and the formulas for c l and c r
The above analysis of poles and zeros of triple Berezin integrals in (1), when confronted with the divisor (50) of function ϕ 12345 , suggests that square roots of such functions may be the key ingredient of our c l and c r . So, we introduce, in analogy with ϕ 12345 , quantities ϕ u for each pentachoron u (simply making relevant subscript substitutions).
Now we look at what may happen where some q ijk do vanish. Motivated by the products of q ijk factored out in the denominators of expressions like (20), we introduce the quantities ̺ t = q ijk q ijl for tetrahedra t = ijkl. These ̺ t are expected to compensate the poles appearing where the mentioned denominators vanish.