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Nuclear Physics B ••• (••••) «

www.elsevier.com/locate/nuclphysb

A reciprocity formula from abelian BF and Turaev-Viro

theories

P. Mathieu, F. Thuillier *

LAPTH, Université Savoie Mont Blanc, CNRS, 9, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France

Received 19 April 2016; accepted 9 May 2016

Editor: Hubert Saleur

Abstract

In this article we show that the use of Deligne-Beilinson cohomology in the context of the U(1) BF theory on a closed 3-manifold M yields a discrete Zn BF theory whose partition function is an abelian TV invariant of M. By comparing the expectation values of the U(1) and Zn holonomies in both BF theories we obtain a reciprocity formula.

© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

The impact of Deligne-Beilinson cohomology in the context of Quantum Field Theory was carefully investigated in [1]. In a previous article [2] a study of the U(1) BF theory within the Deligne-Beilinson cohomology [3,4] framework was initiated, following what was done in the U(1) Chern-Simons (CS) theory case [5-9]. In this first article the partition function of the BF theory was computed and compared with the absolute square of the Chern-Simons partition function thus highlighting significant differences from the non-abelian case. In this same article an abelian Turaev-Viro (TV) invariant, whose construction is based on a generalisation of V. Tu-raev and O. Viro [10] approach as proposed by B. Balsam and A. Kirillov [11], was exhibited and it was shown that up to a normalisation this abelian TV invariant coincides with the U(1) BF partition function.

* Corresponding author.

E-mail address: frank.thuillier@lapth.cnrs.fr (F. Thuillier).

http://dx.doi.org/10.1016/j.nuclphysb.2016.05.007

0550-3213/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

2 P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-•••

In the second section of this article we complete the study of the U(1) BF theory on a closed 3-manifold M by computing expectation values of U(1) holonomies, still in the Deligne-Beilinson (DB) cohomology framework. In section 3 we show that the Turaev-Viro invariant can be seen as the partition function of a discrete ZN BF theory whose observables are ZN holonomies. Some gauge fixing procedures are also discussed in this section together with the usefulness of a Heegaard splitting of M. Finally by taking for N the quantized coupling constant of the original U(1) BF theory a relationship between expectation values of the BF and TV theories is made explicit in section 4. This yields a reciprocity formula which is comparable with Deloup-Turaev one [12], this last formula being related to the U(1) Chern-Simons theory and the Reshetekhin-Turaev surgery formula [13,8,9,2].

The use of DB cohomology proves to be very effective in the U(1) BF theory since unlike the non-abelian SU(2) case we find that: 1) the discretisation of the original U(1) BF theory is a consequence of the construction and not an input; 2) no regularisation of the expectation values is required in the discrete abelian case because all sums occurring are finite whereas a Quantum Group has to be introduced by hand in the non-abelian case to get well-defined expressions [14, 15].

The results obtained in this article can be gathered into:

Proposition. For a smooth, closed, connected and oriented three-manifold M endowed with dual cellular decompositions C and C* we have:

(1) In the U(1) BF theory the expectation values of the U(1)-holonomies along two cycles y1 and y2 are:

((y1,y2))bfn = ff] e-^(y^y0^)

x e-2in(NQ(K1,K2)+Q(K 1,T 1)+Q(K2,T2)) , (1.1)

K 2 €71

where yi = Y0 + Yf + yt and yi = Y0 + Yf + yt is a decomposition of these cycles into their trivial, free and torsion part, f1 and f2 denote the free homology classes and t 1 and t2 the torsion homology classes of y1 and y2, and Q is the linking form on torsion.

(2) There is an abelian TV theory whose observables are ZN -holonomies, the expectation values of which are defined by:

«Z1,Z2»7VN = NFT7-T E E e'Z1+m'Z2), (1.2)

msZN lsZN

where z1 and z2 are two cycles of C and C* respectively represented by z1 e ZE and z2 e ZF, and with F, E and V the number of faces, edges and vertices of C.

(3) The TV and BF observables expectation values satisfy:

((z1,z2))7VN = - ((z1,z2))BFN >

N P1- ■ ■ Pn

which provides a reciprocity formula.

All along this article M is a smooth, closed, connected and oriented three-manifold. We use

= to denote equality in R/Z, that is to say modulo integers, as well as Einstein summation convention.

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) «

2. Abelian BF theory

2.1. Reminders on Deligne-Beilinson cohomology

integral periods by Qp (M)• The Pontrjagin dual of Qp (M) is QPp(M)* = Hom (QPp(M),

We denote by Zp(M) the set of singular p-cycles in M and by Hp(M) (resp. Hp(M)) the corresponding homology (resp. cohomology) group. The space of smooth p-forms on M is denoted by Qp(M), the subset of closed p-forms by Qp(M) and the one of closed p-forms with

and the set of de Rham p-currents in M is the (topological) dual of QP p (M). In particular, every p-chain c in M defines a de Rham (3 - p)-current denoted by jc. Poincare duality states that H2(M) ~ H1(M). Hence when referring to the class of a 1-cycle in M we indifferently refer to its homology class or to the cohomology class of its Poincare dual. We use the canonical decomposition of the abelian group H2(M) into its free and torsion part according to: Zb1 © Zpi © ••• © Zpn, where b1 is the first Betti number of M, and pi |pi+1 e Z for i = 1, ■ •• , n — 1.

As in the U(1) Chern-Simons theory, the space of fields of the U(1) BF theory is built from the first Deligne-Beilinson cohomology group of M, HD(M, Z), or its Pontrjagin dual HXD(M, Z)* = Hom (H1 (M, Z), R/Z). This means in particular that we deal with classes of U( 1)-connections rather than with connections.

The spaces H1(M, Z) and HXD(M, Z)* are Z-modules. They can be embedded into the exact sequence:

Q1(M) 1 2

0 —> , —— h1 (M, Z) —> H2(M, Z) -— 0, (2.3)

QPp(M) D

for the former and:

0 —> QPp(M)* —> h1 (M, Z)* —— H2(M, Z) —— 0, (2.4)

for the latter. The space Qp (M) is thus the global gauge group of smooth U( 1)-connections on M.

The configuration space of the U(1) BF theory is the product HXD(M, Z) x HXD(M, Z), or at the level of distributions HlD(M, Z)* x HlD(M, Z)*.

Let us list some important properties of ^ (M, Z) and H 1 (M, Z)*.

(1) Regular DB classes. Relating the two previous exact sequences by the mean of the canonical injection Q1(M)/Qp(M) — Qp(M) we deduce that there is a canonical injection:

h1 (M, Z) — h1 (M, Z)* . (2.5)

Hence we can identify smooth DB classes as regular elements of ^ (M, Z) — H 1 (M, Z)* just like smooth functions are identified with regular distributions. Moreover there is a canonical injection of de Rham 1-currents into Qp(M) . For instance the de Rham currents of two surfaces with the same boundary define the same DB class. However, since there is no real possible confusion, we will use the same notation for a current and its image in Qp(M) .

(2) Bundle structure. Exact sequence (2.3) (resp. (2.4)) tells us that ^ (M, Z) (resp. ^(M, Z)*) is a bundle over the discrete set H2(M, Z) whose fibres are affine spaces with associated vector space Q1(M)/Qp(M) (resp. Qp(M) ). Hence for any A e HXD(M, Z) and any

4 P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-•••

M e we write A + m the DB class obtained from A by the translation m. The

fibre over the zero class of H2(M) is called the trivial fibre. A fibre over a purely free class of H2(M) is called a free fibre. A fibre over a purely torsion class of H2(M) is called a torsion fibre.

(3) DB product. There is a commutative product:

*: HlD (M, Z) x Hi (M, Z) —> , . (2.6)

Composing this product with integration over M provides a R/Z-valued symmetric linear pairing in H D (M, Z):

j o * : HD (M, Z) x HD (M, Z) —> R/Z. (2.7)

The DB product and the pairing can be straightforwardly extended to HD (M, Z) x HXD (M, Z)*.

(4) Holonomy. There is a pairing:

: H1 (M, Z) x Zi (M) R/Z, (2.8)

which defines integration of DB classes along cycles in M. From this pairing we deduce the inclusion:

Zi(M) c H1 (M, Z)* , (2.9)

which means that we can associate to a 1-cycle y a unique DB class nY e ^ (M, Z)* defined by:

VA e H1 (M, Z), j) A = j A + nY . (2.10)

This pairing yields the holonomy of a DB class A according to:

e2i^YA = e2infMA*VY_ (2.11)

(5) Régularisation. For the same reason as the product of distributions is ill-defined, some regularisation procedure has to be chosen to extend product (2.6) and pairing (2.7) to ^ (M, Z)* x H 1 (M, Z)*. For DB classes of 1-cycles of M we adopt the so-called zero régularisation convention which is defined by:

= 0. (2.12)

(6) Origins. The zero fibre admits as canonical origin the zero DB class which is the class of the zero U(i) connection. This choice of origin allows to identify this fibre with the translation group Q}(M)/&\(M) (or &Z(M)*). On any other fibre of ^ (M, Z) and HlD (M, Z)* there is no such canonical choice.

Even if there is no specific origin on free fibres, injection (2.9) suggests the following in ^ (M, Z)*: let qa (a = 1, ■ ■ ■ , bD) be a set of once for all chosen i-cycles of M which generate

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-••• 5

F1(M); then the DB class of each 1-cycle ^maga is taken as origin of the fibre over m = (m1, ■ •• , mbl) e F2(M) ~ F1(M). Such origins will be referred as free origins and denoted by Am. Note that Am = m"nga so that zero regularisation also applies to Am.

Although torsion 1-cycles could also be chosen as origin on torsion fibres in H1 (M, Z)*, there exists specific origins on these fibres for both H^ (M, Z) and H^ (M, Z)*. Indeed it can be shown [8] that on each torsion fibre there exists a DB class, AcT, such that:

Z , f ,c Z

J AcTl * Acx2 = -Q(T 1, r2) and J Acr = 0, (2.13)

for any m e Q1(M)/Q1Z(M) (or Q2Z(M)*), with Q : T1(M) x T1(M) —> R/Z the linking form of M. With some abuse such particular origins are called canonical torsion origins. Since for any representative t of a torsion class r there is pT e Z and a 2-chain £T of M such that pTt = d£T then we have:

vt = AT + ]£r/p (2.14)

and thus:

I nT*nT = f Acr *ACr + 21 Acr + j (2.15)

M M M M

Let us recall that even if £'t is another 2-chain such that prt = d£'T we have j£r/p = j£>z/p in (M)*. Using relations (2.12) and (2.13) we find that:

f j£r j£r Z Z f j£r A ,j£'r

I -*- = Q(r, r) = I -A d-, (2.16)

p p p p

MM which shows the consistency of the construction.

A generic DB class A e H 1 (M, Z)* is decomposed according to:

A = Am + AcK + a, (2.17)

with a e nZ(M)*.

(7) Zero modes. The set Q1(M)/QZ(M) of translations in H 1 (M, Z) can be embedded on its turn into an exact sequence:

Q0(M) Q1(M) Q1(M) 0 0 ,( ) / ) 0, (2.18) qZz(m) qZz(m) Q1(M)

where Ql0(M) denotes the space of closed 1-forms on M (see details in [7,8]). This implies that we can (non-canonically) write:

( QM) - () x ( QM ) . (2.19)

\ttZ(M)) \n10(M)J \qZ(m))

Elements of q0(M)/QZ(M) are called zero modes. It is obvious that:

'q1(MA / R\b1

0 ' - ' ' (2.20)

i2lZ(M)j \Z

6 P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-•••

We refer to this quotient as the space of zero modes. It can be shown [7,8] that for any zero-mode a0:

V« e OZ(M)*, ja0 = 0. (2.21)

By combining the second equation of (2.13) together with decomposition (2.17) and property (2.21) we find that:

VA e H1(M, Z)*, fa0 *A = j a0 *Ai = ^ma^a0 . (2.22)

M M a = 1 Sa

Let us consider a set of smooth closed 1-forms pa (a = 1, ■ ■ ■ , b1) such that:

Pb = Sab , (2.23)

the sa's being the 1-cycles defining the free origins Aim. The images in o0(M)/oZ(M) of these 1-forms form a basis {pa }a=i,...,b1 for zero-modes according to:

a0 = 0aPa, (2.24)

with 0a e R/Z. The components 0a depends on the zero mode a0 and not on the basis {Pa}a=1,...,b1. We have:

/b1 „ b1

Aim *(0bPb) = J2 m"0b / Pb = Ema°a = m' (2.25)

M a,b=1 Sa a=1

By definition the closed 1-forms pa are the Poincare dual of some closed surfaces S° in M which generate F2(M) = H2(M). This means that instead of pa we could use the de Rham currents jS0 of these surfaces, relation (2.23) then becoming the intersection number of S^ and sa. This means that the splitting of (O1 (M)/&Z(M)) straightforwardly extends to OZ(M)*. The decomposition of a e &Z(M)* according to this splitting is written:

a = a0 + a± . (2.26)

In fact it is more rigorous to say that a0 + a^ biunivocally span O~Z(M)* when a0 runs trough Ol0(M)/OlZ(M) and ax through Ol(M)/Ol0(M) (or its distributional version). Finally any A e H 1 (M, Z)* is decomposed as:

A = Ai + AcK + a0 + ax . (2.27)

2.2. Abelian BF action, measure, partition function and observables

Locally, i.e. in any open set diffeomorphic to R3, A * B = A a dB, which is the Lagrangian usually considered in the U (1) BF theory. This suggests to set:

BFn(A,B) = j bfn(A,B) = N j A* B , (2.28)

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-••• 7

as generalised U(1) BF action with coupling constant N, where (A, B) e H^ x H^. From pairing (2.7) we deduce that BFN (A, B) is well defined if and only if N e !

At the quantum level we assume that our gauge fields live in a configuration space H c (H which contains H¿ and Z1 x Z1 so that (A, B) e H2 = H x H. In particular H has an affine bundle structure over H2(M) whose translation group T c (M) . We also assume that the set of free and torsion origins previously discussed, A^ and AcT, has been set on H. We provide H2 with the (formal) measure d^N defined by:

V (A,B) e H2, d^N(A,B) = DADB e2inBFN(AB), (2.29)

where D stands for the (formal) Lebesgue measure on H. The measure d^N satisfies the fundamental property:

d^N (A + Ao,B + Bo) = d^N (A, B) e2in{BFN(Ao,B)+BFN(A,Bo)+BFN(Ao,m , (2.30)

for any fixed (A0, B0) e H2. This means that, unlike D, the measure d^N is not invariant by translation. However it has an invariance associated with zero modes. Consider ji and j2 the de Rham currents of two closed surfaces Si and S2 in M. Then ji and j2 are zero modes and for all (m, n) e Z2 and all (A, B) e H2, properties (2.25) and (2.30) imply that:

d^N(^A + mj ,B + nj2^ = d^N(A,B). (2.31)

The BF partition function for a given coupling constant N is defined as:

Zbfn = ¡DA DB e2inBFN(AB), (2.32)

Nn J h2

Nn = j Da Dpe2inBFN(a^). (2.33)

The observables for this theory are the U(\) holonomies, also called Wilson loops, that is to say:

,Trí , ~ 2in A 2in B /(,

W (A,y1,B,y2) = e Jn e . (2.34)

The expectation values are computed through the formula:

«K1,K2»BFN = «W(A,K1,B,K2)»

i / * n\ 2in é, A 2in B

dßN(A,B)e Jn e ^ . (2.35)

We use the notation (( ■, ■ ))bfn to emphasise the fact that we are working with a particular normalisation: usually Nn is chosen so that (0,0)bfn = 1 while here ((0, 0))bfn = Zbfn. It can be checked that the expectation value ((y1,Y2))bfn is N nilpotent, that is to say:

J((NK1,Y2))bfn = ((0,Y2))bfn

1«Y!,NY2»bfn = ((Y1, 0))bfn .

8 P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-•••

2.3. Computation of expectation values

Consider y1 = y0 + y{ + yT and y2 = y20 + yf + y2T where the superscript 0 refers to the homologically trivial part of the loop, f to its non-trivial free part and t to its non-trivial torsion part.

If y1 is N times a generator of the free part of the homology, then thanks to property (2.36):

((Y1 = Y10 + Yf + Y1T,Y2» ^fn = ( (y° + Yl,Y2))BFN (2.37)

and the same for y^ . If not, given any closed surface £ and any integer m, we can write, together with the measure invariance:

,R //wm R „2inmL N

^W^A + mJ-^,Y1,B, Y2jjj = {{W (A, y1,B, Y2)» e "myY1 (2 38)

= {{W(A,Y1,B,Y2))) . Thus, if {{W (A, y1,B,y2)>> = 0, we must have for any m:

e2inm$n N = 1, (2.39)

which means that the intersection number of y{ and £ is 0 modulo N for all closed surface £.

This contradicts the hypothesis that yf is non-trivially free. Hence, {{W (A, y1,B,y2)>> must be 0. The same reasoning apply to B thus yielding:

{{Y1,Y2>>BFN = CC ((Y10 + YT,Y20 + Yl))BFN , (2.40)

where f1 and f2 denoting the homology class of y[ and y^ , and with:

fN = j1 iff= 0modulo N, (2.41)

f 10 otherwise.

We thus consider now:

a _\\ 1 f 2in 4 0| tA 2in 4 0 tB

{{y0 + YT,Y20 + YT))BFn = J^j d^N(A,B)e X+^e W . (2.42)

Dualizing the loops of integration with Deligne classes no associated to the trivial part y0 and nT associated to the torsion part yT we can write:

((y1° + y1T,y20 + y2T\\ = -L f d^N (A, B) e2in SM{Mn0i+n1)+B*(n°+n2)}, (2.43) \\ I IBFN NN J

and using decomposition (2.14) the right-hand side of the previous expression reads:

-L f d,N(A: B,/"ia*('!+at 1+££)). (2.44)

By performing in this last expression the change of variables:

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) «

A A-( ni +

N p2N I expression (2.44) takes the form:

B b _( n1 +

-_^w+yt^+yi1) r 2in r iAA +BA 1

--- dßN(Ä,B)e Jm\' ti + M .

(2.45)

(2.46)

Using decomposition (2.27) we get: (a = Am + ACK J + ao + ax

B = Aim + AK 2 + ßo + ßx

(2.47)

^2 + AK2 ' ' '

Contributions Am *Am2 cancel by zero régularisation whereas contributions ACKJ *jo, jo *ACr , ACK 2 *ao, ao * ACT ^ AK 1 * jx, jx * ACT 2, AK 2 *ax, ax * ACT ^ ao * jx, ax * jo et ao * jo cancel thanks to properties (2.13) and (2.21). Thus, the only non trivial contributions are:

N j Aii * AK2 + Aii *AcK 1 + Aii *ßo + Aii *ao M

N j Ai *ßx + Ai2 *«x + AK1 * AK2 + ax *ßx

Ami * AT1 + AK1 *At 1

Am2 * AT2 + AK2 * At2

We factorize out:

j Va0e2inNMAmi Vß0e2inNMAmi

and use relation (2.25) to obtain:

(2.48)

(2.49)

£ j Va0e2inN fMAm1 *a0 = £ j j dd'e2' m1eF1 m1eF1 ya=1K/Z

= E Sm1,0.

Similarly we have:

£ j Vp0e2inN MBm2*fk> = £ Sm2,0 . m2 6F1 m2eF1

Our expectation value thus takes the form:

2inNmleb i apb

(2.5o)

(2.5i)

10 P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-•••

((Kl0 + Yit,Y20 + Yi)L =-T7--X (2.52)

BFN Nn

J2 f Va±. VPe1* f^NACKl *Ai2 +a±*P±+AKl 1 +AK 2 *AC 2} .

K1,K 2 €71

In the same spirit, by factorizing out the zero modes contribution in the expression of Nn, we obtain:

Nn = J Va±Vp±e2inNfMa^ , and since fM ACK1 * ACK = -Q (k 1, k2) we finally have:

«Y!,Y2»bFn = We-^W+y^+YÎ) x (2.53)

x e-2in{NQ(K1,K2)+Q(ri,ki)+Q(T2,K2)},

K1,K 2€7i

which is the announced result. Note that we recover that [2]:

«0, 0))BFn = Zbfn = £ e-2inNQ(K lK2). (2.54)

K i,K 2€Ti

Furthermore, if M has no torsion the linking form Q is trivial and hence [5]:

((Y, y))bf« = ((Y))csk . (2.55)

3. Towards an Abelian TV theory

3.1. Reminders on cellular decompositions

We provide M with an oriented cellular decomposition C = (P, F, E, V) where P is the set of 3-cells (polyhedra), F the set of 2-cells (faces), E the set of 1-cells (edges) and V the set of 0-cells (vertices). These sets are given by:

P = . ,P , F = (Sa)a=i,.. .,f , E = (ei)i=h.. .,E , V = (xa)a=h.. .,v . (3.56)

As M is closed, we can consider a dual oriented decomposition C* = (P*, F*, E*, V*) of C given by:

V* = • ,p , E* = (e°)a=i,. ,f , F* = (S0!=1, . .,E , P* = (Pa)-=i,• ,v , (3.57)

in such a way that:

P„ © xv = s; , Sa O eb = Sba , ei O Sj = Sj , xa © P% = 8%, (3.58)

with © denoting the intersection number in M. The decompositions C and C* are naturally endowed with the structure of abelian graded groups.

Let us list some important construction and properties that these dual decompositions yield.

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) «

(1) Boundary operator. We provide C and C* with boundary operators 9 and 9* such that:

9 Pu = dbuSb

dSa = die

dei = d^x1 i i

dxa = 0

9* Pa = d*aSj

d*Sl = 9' be

d*ea = 9*Vx 9*xu = 0

(3.59)

a o 9 = 0 = 9*o 9*,

all matrix elements of 9 and 9* being integers. By introducing the matrix notation: /

0 0 0 0\

(9(3)au)a,u 0 0 0

0 (9(2)la)l,a 0 0

0 0 (9(1)ai ) a,i

we have:

(9*3) = 9ÎDLv'(

d(2) = 9(2))fxE' (9*1) = 9(3),

(3.60)

(3.61)

(3.62)

The boundary operators 9 and 9* turn C and C* into differential groups [16] thus yielding homology groups H,(C) and H.(C*). We will always assume that the decomposition C is good, which means that:

H.(C) ~ H,(M). (3.63)

By construction the dual decomposition C* is good too.

(2) Cochains and differentials. Relations (3.58) lead to the following correspondences:

Pa ^ Pa e Hom (V, Z) = C£ / Pa(xp) =

Si ^ Si e Hom(E, Z) = Cl / Si(ej) = 81

c 1 1 , (3.64)

ea ^ea e Hom (F, Z) = C£ / ea (Sb) = 8ab

x* ^X* e Hom (P, Z) = C£ / XPv) = 8*

and once (C*)* has been canonically identified with C the following additional correspondences can be done:

Pu ^ Pu e Hom (V*, Z) = C0c* / Pu(xV) = 8U Sa ^ Sa e Hom (£*, Z) = CC* / Sa (eb) = 8ha

ei ^ êi e Hom (F *, Z) = CC * / (SJ) = 8j xa ^Xa e Hom (P*, Z) = CC* / xa(Pp) =

(3.65)

A cochain of C and C* is then a linear combination of these fundamental cochains.

We write C£ (resp. C£*) the graded group of cochains of C (resp. C*). We turn C£ (resp. C£*) into a differential group by endowing it with the endomorphism d : C£ ^ C£ (resp. d*: C£* ^ C£*) defined by:

NUPHB:13737

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) <

VU e CÇ, d o U = U o 9

(resp. Vv e Cç*, d * o V = V o 9*) '

Since the decomposition C is good the cohomology groups of (CÇ,, d) and (C^* ,d*) with the ones of M. With respect to expression (3.61) we have the matrix relations:

(d(0) = 9(1)Lv ' (d(1) = ^FxE ' ^(2) = 9(3))

d*m = dL = d(

(2) = d(3)

/ FXP \

d(1)=dh)=9

(2)Lf' ('

d*2) = d(0) = d

(3.66) coincide

(3.67)

(3.68)

(3) Cap and cup. The symmetric non-degenerate pairings defined by:

P^xA = Plx(xv) = P„ O xv = 8

„b vb

{Sa, ebj = Sa (eb) = Sa O eb = 8b (êi, S^ = êi (s^ = ei O Sj = 8j (xa,P?) = Xa (PP) = xa O PP = 8Pa yield the following cap products:

(3.69)

M - P| = P| M - Sa = Sa M - ei = ei M — xa = xa

M — Pa = Pa

m — Si = s!'

M — ea = ea M — x1 = x1

(3.70)

These relations are nothing but Poincare duality. For instance, for a 1-chain c = c'ei then its Poincare dual is just c = c'ei e C^». Note that we start with a chain in C and end with a cochain in C*.

The cup products associated to the previous cap products are:

P - x^ (M)

= Pi(M — xv) = x V(M — Pi) = Pi(xv) = 8l

(3.71)

(Sa - eb) (M) = Sa (M ^ eb) = eb(M - Sa) = Sa (eb) = sb .

{ex - S^ (M) = ei (M - Sj) = Sj(M - ei) = e^ (Sj) = sj (xa - P%)(M) = X a(M - P%) = P%(M - Xa) = Xa(P%) = 8%

(4) Labellings and gaugings. The previous construction extends to ZN-valued cochains of C and C* the differential groups of which are denoted C^* and CN *. In the context of Turaev-Viro theory [10,11,2] elements of C^1 (resp. cN*1) are called ZN labellings of C (resp. C*) whereas elements of cN,( (resp. cN*0) are called ZN gaugings of C (resp. C*).

By construction, the differential of a ZN gauging is a ZN labelling.

Let us consider l e CN1 and m e CN,1 such that

l = li S' and m = maSa,

(3.72)

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) 13

a N 2 N2

with li , ma e ZN. The 2-cochains dl e CN and d* m e Cc*' are defined by equation which gives:

dl= (iid^ea = (di)aea and d*m = (mad*a) ei = (d*m)iei. (3.73)

Note that (dl)a e ZN since 9a e Z and ZN is a Z-module. Thanks to the ring structure of ZN we can extend the cup products (3.70) to ZN-valued cochains. In particular we have:

(m - d/) (M) = ma (dl)b (Sa - eb) (M) = ma (dl)a . (3.74)

As C^'1 = Hom (E, ZN) ~ ZN and CN*. 1 = Hom (E*, ZN) ~ ZN we introduce the canonical bijections:

l = liS1 e CN, 1 l = (li)i=h..., e e Zen, m = miSi e C^1 —> m = (ma)a=1,... ,f e ZN , so that we have:

dî e C^2 dl = ((dl)a)a=1,... ,f e ZN,

(3.75)

d*m e CN2 d*m = ((d*m)^ j ^ e ZN-

(3.76)

Poincare duality implies that a chain has the same components than its Poincare dual regardless of the fact that these components are taken in Z or ZN. For instance the Poincare dual of c = c' ei is the 2-cochain c = c'£' since M ^ (c'£') = c'(M ^ ei) = ciei.

Using correspondences (3.75) and (3.76), we can rewrite equation (3.74) as:

(m - dfj (M) = m ■ dl = (d*m - fj (M), (3.77)

where the ■ denotes the Euclidean scalar product.

(5) Holonomy. If z = z' ei is a 1-cycle of C then for any l = I' S' e CCN,1 we have:

l(z) = (liS^ (z1ej) = liZ = l ■ z = (1 - Z)(M), (3.78)

with z = (z')'=1,... ,e e ZE and z = z'S' the Poincare dual of z. In the same way if z* = z*ea is a cycle of C* then for any cochain m = maSa e C^1 we have:

m(z*) = (maSa) (z*eb) = maz* = m ■ z* = (m - z*)(M), (3.79)

with z* = (z*)a=i,... ,f e ZF and z = z*ea the Poincare dual of z*.

For any l e CN'1 and m e CN1 the cochains l/N and m/N are R/Z-valued. Their holonomies are:

e2i*U.z) = eir'(z) = el-z and e2inm{z*) = e2rm(z*) = e2Nm-z*, (3.80)

where z is a cycle of C* and z* a cycle of C*. In particular this means that l/N and m/N are ZN-connections on C* and C* respectively.

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3.2. Abelian TV partition function and observables

Let us assume that M is provided with a cellular decomposition C as described in (3.1). In [2] we presented an abelian version of the TV invariant whose expression in C is:

^ = nU E (FKN. (3.81)

îeCN-1 \SeF

where £lS = dl(S). Using correspondences (3.75) and (3.76) we can write TN as:

Tn = ^E SdN] - (3.82)

and by transforming Kronecker symbols into complex exponentials we obtain:

t*=w+v-1 EE « * =w+v-r EE « * <"di),M) ■ (3.83)

meZ* leZ* msC*1 leC*1

We can also write TN in terms of the ZN connections of C and C* as:

t*^^^ t, «***<<m^<*))(M)■ (3-84)

- ,-r-N,1 Î,- ^N,1 I l €Cq

Remembering that the cup product of cochains is the equivalent of the wedge product of forms and that locally A * B = A a dB, we can notice the similarity of expression (3.84) with the BF partition function (2.32).

Under the form (3.84) the invariant TN appears like a discretisation of the abelian BF partition function for the action ^m) ^ d{N). The fields appearing in this action are ZN connections of C and C* and the coupling constant is N like in BF. We hence refer to (3.84) as the ZN TV theory.

After these remarks it seems natural to consider as relevant observables of the ZN TV theory the ZN -holonomies:

2in i rw 2in rw

e~lzi and eNmz2, (3.85)

with z1 e ZE representing a cycle of C and z2 e ZF a cycle of C*. The expectation values of these observables with respect to TN are obviously defined as:

1 X—\ X—\ 2in ^ ^ 2in i „ 2in,

((Z1,Z2))TVN = NF+V-T E N^N"Z1 eNm'z2 - (3.86) msZN lsZN

or in terms of the ZN connections of C and C* as:

«zi,Z2))TVN = NF+V-T E E "{N((mHN))(M)+(N)(Zl)+(m)(Z2)} • (3.87)

This last expression has to be compared with the expectation value of holonomies in the usual U(1) BF theory but also with expression (2.35). We can introduce the Poincare duals z1 and z2 of z1 and z2 thus getting:

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) 15

«Z1,Z2>>tVn = N?+V-r £ £ H(n(mMN)+(Nm. (3.88)

- ,-/~N,1 tr /-N,1 meC^* l

In the U(1) BF theory this corresponds to write holonomies with the use of the de Rham currents of z1 and z2, and in the DB framework of section 2 to relation (2.11).

3.3. Gauge fixing procedures

Once the TV invariant (3.81) as been written under the form of the partition function (3.84) we can wonder whether a gauge fixing procedure could be used instead of the normalisation factor 1/NV-1. Before discussing this let us make a remark concerning the expression of the TV partition function. By construction [10] it depends on M and not on the chosen cellular decomposition of M. Hence instead of using the cellular decomposition C we can use the dual one C*. This means that we have:

_L_ y S[N] = —^ y 8™ . (3.89)

NV-1 di nV 1 d*m

irtnN,1 - „N,1

i mi eC^ *

Even by noticing that V* = P this equality does not seem trivial. However if we use the exponential form of the Kronecker symbol 8[N] to rewrite this relation we obtain:

1 v^ v^ 2Nf(m^di)(M) 1 nNrii^d *m)(M)

T. = T. <3.90)

m€C^* l€Cc m€C^* i€Cq

and since V* = P, F* = E and (l ^ d(M) = (m ^ dPj (M) we have just to compare

1/NF +v-1 with 1/Ne+p-1. It turns out that the Euler characteristic of M is zero which implies that V - E + F - P = 0 and hence that E + P - 1 = F + V - 1 thus showing that (3.90) and hence (3.89) hold true.

In [15] a geometrical gauge fixing procedure is proposed in the non-abelian context. To apply this procedure to our abelian case we consider an oriented spanning tree T in C rooted at a vertex x0 of C. Such a graph always exists thanks to M connectedness, reaches any vertex of C and does not contain any cycle. The orientation of T is defined by going from the root x0, which is the only vertex with no incoming edge, to any vertex of C. This orientation induces a canonical orientation of the edges of T so that for any e e T we write de = t(e) - s(e), where t(e) (resp. s(e)) denotes the target (resp. the source) of e with respect to its canonical orientation.

The gauge fixing procedure is then to restrict the sum over labellings which defines the TV invariant (3.81) of M to the labellings l e C^1 which satisfy:

Ve e T, l (e) = 0. (3.91)

For such a gauge fixed labelling l the gauge transformed labelling l+dp, with p e C^0 satisfies:

Ve e T, (l + d/Pj(e) = p(de) = p(t(e) - s(e)) = p(t(e)) - p(s(e)). (3.92)

On the one hand every vertex of C belongs to T and on the other hand x0 is the root of T hence we have:

(l + dp)(e) = 0 ^ p(x) = p(x0), (3.93)

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16 P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-•••

which means that f is a constant gauging. Hence the geometrical gauge fixing selects one and only one representative in each cohomology class of ZN cocycles of C^'1. This coincides with the result of [2] where the partition function was normalised by the quotient of the set of gaugings by the set of constant gaugings thus yielding the normalisation factor 1 /NV -1. The construction is independent of the root x0.

The "spanning tree" gauge fixing we just described can be seen as a homotopic gauge in the following sense: consider a neighbourhood of the tree T. This defines a contractible open set of M with origin x0, the contraction being done along the edges of T until we reach x0. The first and second homology and cohomology groups of this open set are trivial. This means that the restriction of a closed labelling of C is necessarily trivial, that is to say a gauging, hence gauge condition (3.91). What is remarkable is that the gauge fixing constraint applied in this open set is enough to gauge fix all the closed labellings on C. Actually we can shrink the original decomposition along T thus getting a new cellular decomposition with only one vertex, x0, and which provides the same TV invariant as the original decomposition. This reduced decomposition of M has only cyclic edges based at x0. If we denote by CT the reduced decomposition of C with respect to T we have:

_!y S[N] = y S[N]. (3.94)

NV-1 L^, dl ¿-^ dl y '

- N'1 /<=<rN,1

l sCN'1 l eC

Let us note that this gauge fixing procedure is quite unusual since it is not a constraint on labellings - i.e. fields - of C but rather a change of cellular decomposition for M. This is why it was referred to as a geometrical gauge fixing.

With the previous geometrical gauge fixing we do not really need expression (3.83) of the TV partition function. By considering the TV action (l ^ d(M) appearing in (3.83) we can think about some other gauge fixing procedures inspired by what is usually done in Gauge Field Theory. The first example that comes to mind is that of the covariant (or Lorentz) gauge which in our discrete context takes the form:

d*hl = 0 [N], (3.95)

where hl is the Hodge dual of l = li S', Hodge duality being defined in CN1 by:

hS' = j e CN2. (3.96)

We want to compare expression (3.90) with the supposedly gauge fixed one

s[N ]s[Nh]-,

dl d*hl

'C1;. (3.97)

or rather expression (3.83) with:

1 T J2 J2 J2 e2M(m~dî)(M)+(l~dhî){M)}. (3.98)

^n* l e Cn ^ e Cn *

where dh = hd *h. Unfortunately we are faced with several difficulties. First of all the covariant gauge fixing procedure is usually done on differential forms which are by definition real valued. In other words from Hodge decomposition theorem we know that in the cohomology class of

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-••• 17

a real 1-cocycle r there is a unique co-closed representative, that is to say a real 1-cocycle l such that d *hl = 0. However real cohomology forgets about torsion hence it is hopeless to try to impose (3.95) on torsion cocycles. Even if M has no torsion a Z-valued cohomology class does not necessarily have a Z-valued co-closed representative, and dealing with ZN-valued cocycles does not improve the situation.

Let us assume for a moment that M is torsionless and such that each ZN-valued cohomology class have a representative which fulfils (3.95). The cellular decomposition introduces possible degeneracies in this gauge fixing procedure. Indeed if a closed labelling l e CN,1 fulfils (3.95) then the gauge transformed closed labelling l + dp fulfils it too if and only if:

d o dhp = Ap = Np, (3.99)

for some p e CN, 0. However gaugings p such that

dp = Nrn, (3.100)

has to be excluded since they do not change ZN labellings. Solving the diophantine system (3.99) while excluding solutions of (3.100) can be a tedious task. Fortunately there is a loophole if a Heegaard splitting H U* H of M is used. A cellular decomposition C of the Riemann surface £ = dH compatible with the diffeomorphism * - which means that *(C) is also a decomposition of £ - canonically induces a cellular decomposition C* of M. Remarkably a dual decomposition C* of C* contains only two vertices. Taking into account the remarks made at the beginning of this subsection we find that:

<v 1 ^ v^ *mW) ,„.nn

^N = NE+T 2. 2. eN v ; . (3.101)

- ,-r-N.l iV /-N,1 m€C_i I €Cr

Trying to replace the normalisation factor 1/N in (3.101) by using the covariant gauge fixing (3.95) leads us to consider:

1 v^ v^ i^d*m)(M)\

2 E J2 J2 J2 eN[(l~d*m)(M)+(k~dhm)(M)^ (3.102)

N -,-^N,1 i „N,1 ^N,3

m€C_i l eCr XeCr

with dh = hdh and P = 2. The degeneracy of the gauge constraint is now much easier to study as 3(3)(Pi + P2) = d(3)M = 0 and hence d(3)P2 = -d(3)P1. Thus the matrix of 9(3) has the simple form:

( £1 —£1 \

62 —62

(3.103)

\£F Sf/

with €i = 0, ±1 for i = 1, ■ •• , F, and the matrix representing A* = hd(2)hd*0) is:

A* = = (-'„ -;) = n( - -/). (3.104)

where n = J2i is the number of common faces of P1 and P2, or equivalently the number of edges joining the two points of the dual cellular decomposition C*. Equation (3.99) now reads:

18 P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-•••

Д., = NP = (»_ -n)^) = n(2). ,3.105)

There are gcd (N, n) = к non-trivial - i.e. such that d., = d(3), = Nrn - solutions of this system. These are the degeneracies of the covariant gauge fixing and hence we have:

Tn = _j_ £ £ £ eN{(^т)(м)+(^т)(:м)}, ,3.106)

C. C<P C<P

that is to say:

TN = - V 8lNl S™ • (3.107)

N k ' ' d*m d*m y '

An example where the covariant gauge fixing procedure just described can be applied is provided by Heegaard splittings H иф H with ф = Ids. Such a splitting defines a manifold M such that H1(M) = Zg, with g the genus of S. The case of S1 x S2 presented in subsection 5.2 is of this kind. In any event, as natural as it seems to be the covariant gauge fixing procedure turns out to be much less effective than the geometrical one in the context of U(-) TV theory.

As ZN holonomies are gauge invariant, once the partition function has been properly gauge fixed, expectation values of these holonomies can be computed in the chosen gauge.

4. Reciprocity formula

We now show the main result of this paper:

«Z1,Z2»7-vn = N 1 «Z1,Z2»bfn • (4.108)

" P1 •••Pn "

Since any cellular cycle is a cycle in M whereas the converse is not true it seems natural to start from the TV theory. So let z1 = z0 + z{ + z[ e C and z2 = z0 + zf + z2 e C* be two cellular cycles. They yield the following expectation value:

1 X—^ X—^ 2in ^ Л1 2in i „ 2in,

«Z1,Z2»rVN = N^+7-T E Ee N md1 e N - e N

'tvn nf+v 1

meZN leZN

. N N (4.109)

= V e N1-Z1 S[N]

NV-1 Z^ e °d\+Z2 . leZN

The sum over m yields the constraint:

d l + z2 = —N u, (4.110)

for some u e ZF. The minus sign appearing in the right hand side is only here for later convenience. Constraint (4.110) implies that the cycle z2 of C* can be seen - through Poincare duality - as a ZN-coboundary. Moreover since z2 is a cycle this same constraint also implies that:

du = 0, (4.111)

which states that u represents a 2-cocycle. Hence we deduce that equation (4.110) does not admit

any solution if z2 is not 0 modulo N. The same reasoning applies to z1 when factorizing out l instead of m. Therefore we have

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-••• 19

«zi,z2= ^ff((z0 + + ))rVw , (4.112)

as with BF.

Consider two cycles z\ = z0 + z[ e C and z'2 = z0 + z2 e C* of order p1 and p2 respectively, and hence without any free part. Then there exists a 2-chain S1 e C such that:

Pizl = 9Si, (4.113)

or equivalently for the vector a1 e ZF representing the Poincare dual of Si such that:

piz'j = d a 1. (4.114)

The quantity:

a 1 ■ dl, (4.115)

represents the intersection number of S1 with the boundary whose Poincare dual is represented by dl. Constraint (4.110) then yields:

a 1 ■ dl = —a 1 ■ (Nu + z2) = -Na 1 ■ u - a 1 ■ z2. (4.116)

Due to the symmetry property of the intersection we have:

a 1 ■ dl = d a 1 ■ l = p1 z1 ■ l, (4.117)

which gives:

, N a 1 ■ u a 1 ■ z2

z1 ■ l =--1---1—2 . (4.118)

The construction above allows to associate an element u e ZF to an element l e ZE. We need to determine the degeneracy of this pairing when trying to use it in order to relate the sum over m and l in (4.109) with the sum over u and v in (2.53). For this purpose we set:

S = {l e ZE | 3u e ZF | dl + z2 = — Nu} , (4.119)

S' = {u e ZF | 3l e ZE | dl + z2 = — Nu} . (4.120)

Thus, the set of summation in our computation are S/NZ and S'I Imd. We now use the following lemma that is proven in appendix:

Lemma.

(S/NZ)

S' Imd ~ , ' , ) . (4.121)

' (Ker d/NZ)

Thus our degeneracy factor is | Kerd/NZ|. Since H 1(M) ~ H2(M) is free, we have b1 independent directions in this set (b1 being the first Betti number) to which we can add an ambiguity dx = X0 — X, with x0 corresponding to an arbitrary vertex. Hence there remains V — 1 possibilities for x in order to get a non-zero dx. All the elements of | Kerd/NZ| having coefficients in ZN we get:

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20 P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-•••

| Ker d/NZ|= Nb1+V—1' (4.122)

and therefore:

M ^ = ^V—rE « N *C-

1 X—^ i z

- > «N i

v-i Z^ «

_ M-z'

Nv- ' « * Is Z

—2in[ No i-u , g N \ Pi + Pi ^

= Nbi £ « M P1 PW (4.123)

us y/Im d

2in o 1-z2 __/-»• oi u

= «—N ^ £ «—2in -Pr

us Sr/ Im d

= Nbi «— N^Z 4) J2 «—2Mk(z1 '"),

S'/ Im d

where z'j, z'2 and u are the cycles associated respectively to z'j, z2 and u. Since Ik (z'vu) = Q (ni, u), where ni is the cohomology class of z'j, we can write:

((4z2))TVw = Nb1 e— ,z2) £ e—2inQ(n1'u)

(4.124)

us S'/Im d

= Nb1 e— Mz1 e—2'ng(n1'u)S—Nu—n2,0 .

Since Q is a non-degenerate quadratic form on T1 (M) we can use it to dualize the Kronecker symbol, thus getting:

{U z )) = Nb1 e — 2^(214) y^ e—2inQ(n1'u)e—2inQ(Nu+n2,v)

W 1' 2//TVN P1 ¿T

u'VeT1

(4.125)

= N 1 e — ^^(z^) V^ e—2in{NQ(u'V)+Q(n1,u)+Q(n2'V)}

P1 •••Pn u^ .

Hence we have shown that:

«z1'z2»TVN = - «z1'z2»BFN ' (4.126)

P1 •••Pn

which is the reciprocity formula we were looking for. If 3i e [1 ; n] | gcd (N, pi) f n1 or n2 then both sides of the equality vanish. Indeed, in this case, equation (4.110) has no solution. The proportionality factor appearing in (4.126) is closely related to the one appearing in the Deloup-Turaev reciprocity formula [12], which in turn emerges as a Reshetekhin-Turaev surgery formula in the context of the U(1) Chern-Simons theory [8].

Strictly speaking formula (4.126) has a sense of reading - from the left to the right - since as already noticed not every cycle in M is a cycle of the cellular decomposition, whereas the converse is always true. Let us note that in equations (4.125) it is the linking form Q that was

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-••• 21

used to exponentiate S—Nu—n2,0 not just because it is non degenerate but also because it goes to homology classes and is defined by the linking number which itself appears in equation (4.118).

5. Examples

For the following examples, we exploit the fact that a Heegaard splitting can lead to a good cellular decomposition. However this is far than being the only possibility. Our examples are lens spaces and thus admit a genus 1 decomposition that we consider here. To reconstruct the manifold with the diagrams given below, we first identify the left and right edge of each rectangle, to generate two solid cylinders, whose opposite faces are bounded by the upper and lower edge. Those faces are then identified for each solid cylinder, giving two solid tori. Finally, we identify the boundary of these two solid tori via the gluing rule h. The orientations can be complicated when considering a cycle passing through the (common) boundary of the solid cylinders. The doted lines appearing in the drawings below are not considered as elements of the cellular decomposition but are drawn only for the convenience of the representation. By convention z1 denotes a 1-cycle of C and z2 a 1-cycle of C*.

5.1. S3

A A -►-

-►--

ei ei

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) <

Here, F = 3 (the faces S2 and S3 which do not appear on the diagram are the sections of the left and right solid cylinders whose boundaries are e1 and e2), E = 2 and V = 1. The operator d(1) being the transpose of the matrix giving the components of dSj in the basis ej, we get:

(5.127)

With z1 = e1 = 3S2 and z2 = e2 such that Ik (z1,z2) = S2 O e2 = 1, we obtain:

((z1,z2>>TVN = eN = eT^1-^ = N0 ((Y1,Y2>>bfn .

(5.128)

The general case arises by taking z1 = n1e1 and z2 = n2e2 so that tk (z1;z2) = n1 n2. Note that no gauge fixing is required in this example.

5.2. S1 x S2

T w 1 e1 —w— e2 —w • e3 I

S1 S2 S1

! e4 J ie4 e5 i ke5 !

0 0 0

! e1 e2 - e3 ; —h ■

1 0 0 1

T W i e1 —w— e2 -W T e3 |

S1 S2 S1

! e4 i ke4 e5 A ^ e5 !

0 0 0

! e1 * h e2 -fc— e3 | -fci é

Here, F = 4 (the faces S3 and S4 which do not appear on the diagram are the sections of the left and right solid cylinders whose boundaries are e1 + e2 + e3), E = 5 and V = 3. The operator d(1) being the transpose of the matrix giving the components of dSi in the basis ej, we get:

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) <

d(1) =

0 0 0 1 -1

0 0 0 -1 1

1 1 1 0 0

1 1 1 0 0

(5.129)

Computing the TV partition function, we obtain:

NF+v-1=6

1=6 E E

lsZN=5 msZN=4

= N = — zbfn-

(5.130)

Computing now the expectation value of z1 = e1 + e2 + e3 = 9S3 and z2 = 0, we obtain:

«Z1,Z2))rv- = -6 E E e^^ = - = n1 «K1,K2))bf- - (5.131)

lsZ- msZ-

With z1 = e1 + e2 + e3 = 9S3 and z2 = e1 — e2 + e3 + e4 trivial such that tk (z1;z2) = S3 O (e1 — e2 + e3 + e4) = S3 O e3 = 1, we obtain:

«Z1.Z2))tyn = -e- = Neir^1^ = _ «K1,K2»bfn -

(5.132)

With z1 = e4 non-trivial and z2 = 0, we obtain:

<<Z1,Z2))TVn = 0 = — (<K1,K2))BFN >

(5.133)

as expected. It can be checked that the covariant gauge fixing procedure can be applied in this example.

5.3. RP3 = L (2, 1)

Here, F = 4 (the faces S3 and S4 which do not appear on the diagram are the sections of the left and right solid cylinders whose boundaries are e1 + e2 and e3 + e4), E = 4 and V = 2. The operator d(1) being the transpose of the matrix giving the components of dSi in the basis ej, we get:

d(i) =

f 1 -1 -1 1

-1 1 1 -1

1 1 0 0

0 0 1 1

(5.134)

For z1 = e\ + e4 (with 2zi = 9(Si + S3 + S4)) and z2 = e3 + e4 trivial such that tk (zi,z2) = 1 tk(2zi,Z2) = 2(Si + S3 + S4) O (e3 + e4) = 1, we obtain:

«zi,Z2»rVv = e-N (i - = e-T^1^ (i - 42]) = — «Yi,y2))bfn , (5.i35)

the cohomology class ni associated to zi being i and n2 associated to z2 being 0.

With zi = ei + e4 and z2 = e3 torsion such that 1tk(2z1,z2) = 2(Si + S3 + S4) O e3 = 1, we obtain:

^z2))TVN = -e-N (l - SN^) = -e-V««.»^ 1 - ¿f) = Nr ((Yi,Yi)) bfn

(5.136)

the cohomology class ni associated to zi being i and n2 associated to z2 being i.

The presence of &N in the above expressions comes from the fact that for even N, gcd (N, p = 2) = 2 { n1, n2 implying that (4.110) has no solution. It can be checked at the level of the partition function that in this case the covariant gauge fixing does not apply properly since it produces a gcd (N, 4) factor.

6. Conclusion

In this article we showed how the use of Deligne-Beilinson cohomology allows to prove that the U(\) BF theory can be turned into a discrete ZN BF theory without resorting to the usual guessworks of the non-abelian case. For instance all the sums occurring in the discrete theory are finite thanks to the emergence of ZN as "gauge" group whereas in the non-abelian

P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-••• 25

case a Quantum Group is introduced as a way to regularise the infinite sums the non-abelian discrete BF theory yields. In addition it is only under this regularisation assumption that the non-abelian BF theory is related with a TV invariant whereas in the U(1) this relation is proven too. However it has to be stressed out that although the discrete BF action on M is (i ^ d*m^ (M) the action of the corresponding U(1) BF theory is NOT fMB A dA but fM A * B. It's only on S3 that fMB a dA becomes a possible expression for the U(1) BF action since the set of U( 1)-connections on S3 can be identified with ^1(S3) and the gauge group with dQ°(S3) (see exact sequence (2.3)).

Finally, all we have done in this article can be extended to connected, closed, smooth and oriented manifold of dimension m = 41 + 3 with configuration space of the BF theory being HD+1(M) x H2D+1(M) instead of HXD(M) x HlD(M) and the one of TV being cN,21+1 x cN*,21+1 instead of C^1 x cN*1 for some cellular dual decompositions C and C* of M.

7. Appendix: Proof of the lemma

We now want to prove the following:

Lemma.

.. (S/NZ)

S' Imd ~ , ' , ) . (7.137)

' (Ker d/NZ)

Proof. Let's consider:

y : S'/Imd ^ (S/NZ)/(Kerd/NZ)

_ = , (7.138) u ^ l

where we use bars to emphasise the fact that we work with classes in the appropriate quotient sets.

First we check that y is well-defined, that is to say

u = v ^ y(u) = y(v). (7.139)

Indeed,

u = v ^ u — v = 0 ^ u — v = 0, (7.140)

which means by definition that:

u — v e Imd ^ 3a e ZE | u — v = da. (7.141)

u e S' ^ 3l e ZE | dl + z2 = — Nu, (7.142)

v e S' ^ 3m e ZE | dm + z2 = — Nv, (7.143)

(the minus sign in the right-hand side being purely conventional) thus

26 P. Mathieu, F. Thuillier /Nuclear Physics B ••• (••••) •••-•••

N (u - v) = Nda =-d(l - m) or d (l - m + N a) = 0, (7.144)

and so

l = m - N a + f , (7.145)

with f e Ker d, so

l = m + § , (7.146)

and hence

I = m ^ p(U) = y(v). (7.147)

Then, we note that, by construction, p is necessarily surjective. Thus we only need to prove that it is injective. For that we consider I = m, that is to say

l - m = l - m = 0 ^ l - m = f , (7.148)

with f e Ker d IZN so

3a e ZE 11 - m = f - N a. (7.149)

l e S ^ 3u e ZF | dl + z2 = -Nu, (7.150)

m e S ^ 3v e ZF | d m + z2 = -N v, (7.151)

-N u = d l + z2 = d (m - N a + f ) + z2 = (d m + z2 ) - Nda = N (v - d a) , (7.152)

u = v - da ^ u = v. (7.153)

Hence, y is bijective. □

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