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A Large k Asymptotics of Witten’s Invariant of Seifert Manifolds

로잔스키(L. Rozansky)의 논문 "A Large k Asymptotics of Witten’s Invariant of Seifert Manifolds"의 요약입니다.

로잔스키는 Witten의 SU(2) 불변량을 계산하는 방법을 개발한 바 있습니다. 이 방법은 3차원 다양체 M 위의 Chern-Simons 액션에 대한 경로 적분을 이용하여 M의 topological 불변량인 Z(M, k)를 계산합니다.

로잔스키는 Witten-Reshetikhin-Turaev의 방법을 사용하여 SU(2) 불변량을 계산하고, 이 방법은 flat connection A(i)에 대한 Chern-Simons 액션 S(i)의 classical exponential exp[ikS(i)]과 quantum correction ∆(i)n를 포함하는 asymptotic series를 제공합니다.

로잔스키는 1-loop factor ∆0를 계산하고, Reidemeister-Ray-Singer torsion τR(A)와 spectral flow Ii를 이용하여 Z(M, k)를 구현합니다. 로잔스키는 또한 flat connection A(i)의 Chern-Simons 인변량 S(i)과 quantum correction S(i)n을 계산하는 방법을 제공합니다.

로잔스키의 결과는 Chern-Simons 이론에서 topological 불변량의 계산에 중요한 기여를 할 것으로 예상됩니다.

A Large k Asymptotics of Witten’s Invariant of Seifert Manifolds

arXiv:hep-th/9303099v4 18 Oct 1993UTTG-06-93hep-th/9303099A Large k Asymptotics of Witten’s Invariant of Seifert ManifoldsL. Rozansky1Theory Group, Department of Physics, University of Texas at AustinAustin, TX 78712-1081, U.S.A.AbstractWe calculate a large k asymptotic expansion of the exact surgery formulafor Witten’s SU(2) invariant of Seifert manifolds.

The contributions of all ﬂatconnections are identiﬁed. An agreement with the 1-loop formula is checked.A contribution of the irreducible connections appears to contain only a ﬁnitenumber of terms in the asymptotic series.

A 2-loop correction to the contributionof the trivial connection is found to be proportional to Casson’s invariant.1Work supported by NSF Grant 9009850 and R. A. Welch Foundation.

Contents1Introduction22Calculation of Witten’s Invariant52.1Loop Expansion . .

. .

. .52.2Surgery Calculus.

. .

. .92.3SU(2) Formulas and Poisson Resummation .

. .

. .123A Large k Limit of the Invariants of 3-Fibered Seifert Manifolds153.1Stationary Phase Points.

. .

. .153.2The Integrals.

. .

. .203.3Framing Corrections.

. .

. .244One-Loop Approximation Formulas254.1Irreducible Flat Connections .

. .

. .254.2General Reducible Flat Connections .

. .

. .284.3Special Reducible Connections .

. .

.295A Large k Limit of the Invariants of General Seifert Manifolds325.1A Bernstein-Gelfand-Gelfand Resolution and Verlinde Numbers. .

. .

.325.2A Contribution of Irreducible Flat Connections. .

. .

. .375.3A Contribution of Reducible Flat Connections .

. .

. .416Discussion441

1IntroductionA Chern-Simons action is an “almost” gauge invariant function of a gauge connection on a3-dimensional manifold M:SCS = 12πǫµνρTrZM(Aµ∂νAρ + 23AµAνAρ)d3x(1.1)(a trace is taken in the fundamental representation of the gauge group G). A quantum ﬁeldtheory built upon this action is topological.

This means that a partition function presentedby a path integral over the gauge equivalence classes of connectionsZ(M, k) =ZDAµeikSCS[Aµ](1.2)does not depend on the metric of the manifold M and is therefore its topological invariant.An exact calculation of this invariant was carried out by Witten in his paper [1] on Jonespolynomial. The calculation requires a construction of M by a surgery on a link in S3 (orin some other simple manifold, say S1 × S2).

Reshetikhin and Turaev proved in [2] thatWitten’s procedure really leads to a topological invariant. Their proof does not use pathintegral (1.2) which is not a rigorously deﬁned object for mathematicians yet.A Chern-Simons action enters the exponential of the path integral (1.2) with an arbitraryinteger factor k. Its inverse k−1 plays a role of the Planck constant ¯h, which appears inquantum theories and sets a scale of quantum eﬀects.

A stationary phase approximationfor the integral (1.2) in the limit of k →∞expresses a partition function Z(M, k) as anasymptotic series in k−1. Physicists call this series a “loop expansion”, because the terms oforder k1−n come from the n-loop Feynman diagrams.The loop expansion of Z(M, k) has been studied in [3], [4] and [5], as well as in papers[6] and [7] which were aimed at producing Vassiliev’s knot invariants.

Feynman rules wereformulated, however the actual calculation of loop corrections for particular manifolds wentonly up to the 1-loop order. The 1-loop correction was found in [1],[8],[9] to contain suchinvariants of the manifold M as the Reidemeister-Ray-Singer torsion, spectral ﬂow anddimensions of cohomologies.2

Thus there are two diﬀerent methods of calculating Z(M, k): a “surgery calculus” ofWitten-Reshetikhin-Turaev and a loop expansion which is a standard method of quantumﬁeld theory. Both methods should give the same value of Z(M, k) if the path integral (1.2)has the properties that physicists expect it to have.

D. Freed and R. Gompf suggested tocheck this by computing an exact value of Z(M, k) for large values of k through the surgerycalculus and then comparing it to the quantum ﬁeld theory 1-loop approximation. Theycarried out their program in [8] for some lens spaces and Brieskorn spheres.

A computercalculation showed a close correspondence between the values of exact and 1-loop partitionfunctions. In a subsequent paper [9], L. Jeﬀrey used a Poisson resummation trick to deriveanalyticly a large k expansion of an exact surgery formula for lens spaces and mapping tori.She also observed a correspondence between the surgical and 1-loop expressions (at leastup to some minor factors, which we will discuss in the next section).

Similar results wereobtained in [10].In this paper we carry out a large k expansion of an exact surgery formula for Seifertmanifold SU(2) invariants. We identify the contributions of all ﬂat connections and showthat they correspond to the slightly modiﬁed 1-loop approximation formula of [8] and [9].In contrast to the lens spaces, Seifert manifolds have irreducible ﬂat connections.

We ﬁndrather surprisingly that although the reducible connections contribute to all orders in loopexpansion, the contribution of irreducible connections appers to be ﬁnite loop exact. Wealso ﬁnd that a 2-loop correction to the contribution of the trivial connection is proportionalto Casson’s invariant.In section 2 we review the basic features of loop expansion and surgery calculus.

Section3 describes an application of the Poisson resummation to the surgery formula for Seifertmanifolds with 3 ﬁbers. In section 4 the asymptotic expansion of the surgery formula forthose manifolds is compared to the 1-loop formula.

In section 5 a Poisson resummation isapplied to a general n-ﬁbered Seifert manifold and the contributions of both irreducible andreducible ﬂat connections are calculated.3

Summary of ResultsFor reader’s convenience we summarize brieﬂy the main results of the calculations in Sec-tion 3. We present the SU(2) Witten’s invariant of a 3-ﬁbered Seifert manifold X( p1q1 , p2q2 , p3q3 )as a sum over ﬂat connections in the spirit of eqs.

(2.3) and (2.4).The contribution of an irreducible connection is 2-loop exact:12 exph2πiK P3i=1ripi ˜n2i −qisiλ2ie2πiλ+i 3π4 sign( HP )sign(P)×Q3i=11√|pi|2i sin 2πripi ˜ni + siλe−iπ2K φ. (1.3)HereP = p1p2p3, H = p1p2q3 + p1q2p3 + q1p2p3, λ = 0, 12,˜ni = ni + qiλ, ni ∈Z, ps −qr = 1, s, r ∈Z,(1.4)s(q, p) is a Dedekind sum.

For more details see subsection (4.1). The phaseφ = 3signHP+3Xi=1 12s(qi, pi) −qipi!

(1.5)is the 2-loop correction. As we will see, this phase appears in the contributions of all ﬂatconnections.The contribution of a reducible connection contains an asymptotic series of loop correc-tions:−exp 2πiKhP3i=1ripin2i + HP c20iei π2 sign( HP )√2K|H| sign(P)e−iπ2K φ×P∞j=01j!

(8πiK)−j PHj∂(2j)cQ3i=1 2i sin2π rini+cpi2i sin 2πcc=c0(1.6)here c0 =PHP3i=1nipi , for notations see section 4.2.A logarithm of this series has to becalculated in order to put (1.6) into the form (2.4).A reducible ﬂat connection for which c0 = 0, 1/2 (see “point on a face” in subsection 4.3),contributes−exp 2πiKhP3i=1ripin2i + HP c20iei π2 sign( HP )√2K|H| e2πic0sign(P)e−iπ2K φ4

×ei π4 sign( HP )rK8 HP Q3i=1 2i sinrini+c0pi+ P∞j=01j! (8πiK)−j PHj ∂(2j)ǫQ3i=1 2i sin2π rini+c0+ǫpi2i sin 2πǫ−Q3i=1 2i sin2π rini+c0pi4πiǫǫ=0.

(1.7)The contribution of the trivial connection is−ei π2 sign( HP )q8K|H|sign(P)e−iπ2K φ∞Xj=01j! π2iKPHj ∂(2j)ǫQ3i=1 2i sin ǫpi2i sin ǫǫ=0.

(1.8)A remarkable feature of this formula is that its full 2-loop term is proportional to Casson’sinvariant (4.14).The formulas analogous to eqs. (1.6), (1.7) and (1.8) for the n-ﬁbered Seifert manifoldare eqs.

(5.48), (5.52) and (5.54).2Calculation of Witten’s Invariant2.1Loop ExpansionWe start with a brief description of a stationary phase approximation to the path inte-gral (1.2). The stationary phase points are the extrema of the action (1.1).

SinceδSδAµ= 12πǫµνρFνρ,(2.1)these extrema are ﬂat connections, i.e. connections with Fµν = 0.

The gauge equivalenceclasses of ﬂat connections are in one-to-one correspondence with the homomorphismsπ1(M) A→G, A : x 7→g(x) ∈G(2.2)(G is a gauge group) up to a conjugacy, that is, the homomorphisms g(x) and h−1g(x)h areconsidered equivalent.Each stationary phase point A(i) contributes a classical exponential exp[ikSi] times anasymptotic series in k:Z(M, k) =XieikSi ∞Xn=0k−n∆(i)n!. (2.3)5

Another form of presenting the same expansion isZ(M, k) =Xi∆(i)0 exp ik"Si +∞Xn=2k−nS(i)n#. (2.4)Here Si are Chern-Simons invariants of the ﬂat connections A(i)µ , and S(i)nare the n-loopquantum corrections coming from the n-loop 1-particle irreducible Feynman diagrams.

Aset of Feynman rules for their calculation has been developed in [3], however the actualcalculations have been carried out only up to the 1-loop order.Generally in quantum ﬁeld theory a 1-loop factor is an inverse square root of a determi-nant of the second order variations of the classical action taken at the stationary phase point.However a gauge invariance of the action (1.1) requires a gauge ﬁxing and an introduction ofthe Faddeev-Popov ghost determinant (see [1] for details). So for the Chern-Simons theory∆0 =| det(k∆)|[det(−ikL−)]1/2.

(2.5)Here ∆is a covariant Laplacian∆= DµDµ, Dµ = ∂µ + Aµ(2.6)acting on the Lie algebra valued functions, while L−isL−=∗dA−dA∗dA∗0(2.7)acting on the Lie algebra valued 1-forms and 3-forms. A diﬀerential dA is built upon acovariant derivative Dµ, d2A = 0 for ﬂat connections.According to [1], the absolute value of the ratio (2.5) is a square root of the Reidemeister-Ray-Singer torsion τR(A).

A detailed expression for the phase of that ratio has been workedout in [8]. The 1-loop formula for Z(M, k) presented there is a sum over the ﬂat connectionsA(i):Z(M, k) = e−i π4 (dim G)(1+b1) Xie2πiKS(i)CSτ 1/2R e−i π2 IiC0C1,(2.8)here K = k + cv, cv is a dual Coxeter number or, equivalently, a quadratic Casimir invariantof the adjoint representation, b1 is the ﬁrst Betti number and Ii is a spectral ﬂow.

The6

factors C0 and C1 reﬂect the presence of the 0-form (i.e. 3-form) and 1-form zero modes inthe operators ∆and L−of eq.(2.5).

These factors have to be slightly modiﬁed from theiroriginal values in [8].The zero modes are related to the elements of the cohomology spaces H0(M, dA(i)) andH1(M, dA(i)). For each element of H0 there is a zero mode of ∆and a zero mode of L−.For each element of H1 there is another zero mode of L−.

It is also known that H0 canbe identiﬁed with a tangent space of the symmetry group Hi of the connection A(i)µ . Thegroup Hi consists of the gauge transformations that do not change A(i)µ .

Equivalently, Hi isa subgroup of G whose elements commute with the image of the homomorphism (2.2). Asfor H1, its elements represent inﬁnitesimal deformations of the connection A(i)µ which do notviolate the ﬂatness condition.

This picture is reminiscent of the string theory. There thezero modes of the ghosts c and b were identiﬁed with the elements of tangent spaces of thesymmetry group and moduli of the complex structure.

However in our case generally notall the elements of H1 can be extended to ﬁnite deformations of ﬂat connections. In otherwords, dim H1 ≤dim Xi(M), where Xi is a connected component of the moduli space ofﬂat connections.Let us ﬁrst assume that dim H1 = dim Xi.

If operators ∆, L−have zero modes, thenthe Reidemeister torsion can still be obtained from eq. (2.5) if the zero modes and zeroeigenvalues are removed from there.L.

Jeﬀrey noted in [9] that τ 1/2Rthus deﬁned is anelement of ΛmaxH0 ⊗(ΛmaxH1)∗. She suggested to take a canonical element v ∈(ΛmaxH0)∗derived from the basic inner product on H0 which is a Lie algebra of Hi.

A pairing of vand τ 1/2Rproduces a volume form on the moduli space Xi. A sum over the ﬂat connectionsin eq.

(2.8) then includes a natural integration over Xi. However, according to [9], thisprocedure does not quite agree with the leading term in the 1/k expansion of Z(S3, k).We propose a slightly diﬀerent prescription.

We take any element v ∈(ΛmaxH0)∗andbalance the integral over Xi, deﬁned by pairing of v and τ 1/2R , with a factor of 1/Vol (Hi),volume of Hi being deﬁned by the same element v2. We show in Appendix why this factor2 The factor 1/Vol (Hi) appeared in slightly diﬀerent circumstances in [11].

It also appeared in [12] and7

should appear after the removal of the zero modes from eq. (2.5) by considering a simpleﬁnite dimensional version of a gauge invariant path integral.

We also demonstrate in theend of subsection 2.3 how our prescription ﬁts the value of Z(S3, k).There is another consequence of dropping the zero modes from the determinants ineq. (2.5).Each nonzero mode of the operator ∆carries a factor of k in eq.

(2.5) andeach nonzero mode of L−carries there a factor3 of (−ik)−1/2. By dropping the modes, weloose these factors.

Therefore dropping an element of H0 produces an extra factor (ik)−1/2while dropping an element of H1 creates a factor (−ik)1/2. ThusC0 =1Vol (Hi)(ik)−(dim H0)/2.

(2.9)We could also assume thatC1 = (−ik)(dim H1)/2(2.10)However the 1-form zero modes of L−that can not be extended to ﬁnite deformations ofthe ﬂat connection, should not be simply dropped from eq. (2.5).

A nonzero mode of L−contributes to eq. (2.5) through a gaussian integralZ +∞−∞exp(iπkλx2)dx ∼(−ik)−1/2(2.11)A zero mode 1-form that hits obstruction contributes through the integralZ +∞−∞exp(iπkλx4)dx ∼(−ik)−1/4(2.12)Therefore a corrected version of the formula for C1 isC1 = (−ik)dim H12(−ik)−dim H1−dim Xi4= (−ik)dim H1+dim Xi4(2.13)[13] where the Alexander polynomial was produced from a Chern-Simons theory based on a supergroupU(1|1).

It was shown there that Vol (U(1|1)) = 0, so that the ﬂat connections for which Hi = U(1|1), gaveinﬁnite contributions to the partition function. These inﬁnities helped to explain the nonmultiplicativity ofthe Alexander polynomial which distinguished it from the family of the SU(N) Jones polynomials.3Actually a partition function (1.2) has also a factor k# of all modes of ∆−12 # of all modes of L−hidden in theintegration measure DAµ.8

and the 1-loop formula (2.8) takes the formZ(M, k)=Xiexp2πiKS(i)CSexp −iπ4"(1 + b1) dim G + 2Ii + dim H0 + dim H1 + dim Xi2#×1Vol (Hi)τ 1/2R k−dim H02+ dim H1+dim Xi4. (2.14)2.2Surgery CalculusHere we brieﬂy present Witten’s recipe of an exact calculation of the partition function (1.2).Witten used the fact that the Hilbert space of the Chern-Simons quantum ﬁeld theory isisomorphic to the space of conformal blocks of the level k 2-dimensional WZW model basedon the same group G. More speciﬁcally, a Chern-Simons Hilbert space corresponding to a2-dimensional torus is equivalent to the space of aﬃne characters of G (see, e.g.

[14]).Consider a path integral (1.2) calculated over a solid torus with a Wilson line carryingrepresentation VΛ of G going inside it. Λ denotes the shifted highest weight, i.e.

the highestweight of VΛ is Λ −ρ, ρ being half the sum of positive roots of G: ρ = 12Pλi∈∆+ λi. Aninclusion of the Wilson line means that the integrand of eq.

(1.2) is multiplied by a traceof a holonomy trVΛPexp (H Aµdxµ). Such integral is a function of the boundary conditionsimposed on Aµ on the boundary of the solid torus.

Therefore it is an element |Λ⟩of theHilbert space of T 2. Witten claimed that this element corresponds to the aﬃne characterof level k built upon VΛ and that all such elements corresponding to the integrable aﬃnerepresentations form an orthonormal basis in that Hilbert space.The group SL(2, Z) acting as modular transformations on T 2, generates canonical trans-formations in the phase space of the classical Chern-Simons theory.

Therefore SL(2, Z) canbe unitarily represented in the Hilbert space. This representation is determined by the actionof the matrices S and TS =0−110, T =1101(2.15)9

on the aﬃne characters. An action of a general unimodular matrixM(p,q) =prqs∈SL(2, Z)(2.16)is determined by its presentation as a productM(p,q) = T atS .

. .

T a1S. (2.17)The integer numbers ai form a continued fraction expansion of p/q:pq = at −1at−1 −1...−1a1.

(2.18)For more details on this construction see e.g. [8] and [9].A lens space L(p, q) can be constructed by gluing the boundaries of two solid tori.

Theboundaries are identiﬁed trivially after a matrix SM(p,−q) acts on one of them, i.e. thatmatrix determines how the boundaries are glued together.

Before the gluing, each solidtorus produces an aﬃne character Vρ growing out of the trivial representation, as a stateon its boundary. Hence according to the postulates of a quantum ﬁeld theory, a partitionfunction (1.2) is equal to a matrix elementZ(L(p, q), k) = ( ˜S ˜M(p,−q))ρρ(2.19)here tilde denotes a representation of the SL(2, Z) matrices in the space of aﬃne characters.Note that the lens space depends only on the numbers p and q. Diﬀerent choices ofthe entries r and s of the matrix (2.16) correspond to diﬀerent framings of the same lensspace.

A framing is a choice of three vector ﬁelds which form a basis in the tangent spaceat each point of the manifold. A phase of a partition function Z depends on a choice offraming.

Formula (2.14) gives a 1-loop approximation to Z(M, k) in the standard framing.The surgical formulas should also be reduced to the standard framing in order to yield atrue invariant of the manifold. We will discuss this reduction in the end of this subsectionand in subsection 3.3.10

Consider now a manifold S2 × S1. A Seifert manifold X( p1q1 , .

. .

, pnqn ) is constructed bycutting out the tubular neighborhoods of n strands going parallel to S1 and then gluingthem back after performing the M(pi,qi) transformations on their boundaries4. These trans-formations change the states on the surfaces of the solid tori from |ρ⟩into|Λ′i⟩=XΛ|Λ⟩M(pi,qi)Λρ(2.20)Therefore an invariant of a Seifert manifold is given by a multiple sumZ(X, k) =XΛ1,...,ΛnM(p1,q1)Λ1ρ.

. .

M(pn,qn)ΛnρNΛ1...Λn(2.21)Here NΛ1...Λn is a Verlinde number, which is equal to the invariant of the manifold S2 × S1containing n Wilson lines carrying representations VΛi along S1. This number is also “almost”equal to the number of times that a trivial representation appears in a decomposition of atensor productNni=1 VΛi.

NΛ1...Λn will be equal to that number if VΛi are representations ofthe quantum algebra Gq. An expression for NΛ1...Λn in the case of n = 3 and G = SU(2) ispresented in subsection 3.1, a general case will be considered in subsection 5.1.As in the case of a lens space, the phase of a partition function (2.21) also depends onthe choice of numbers ri, si.

Witten found in [1] that a change of framing by one unit isaccompanied by a change in that phase by πc12, c being a central charge of the level k WZWmodel.To get a partition function Z(X, k) in the standard framing, the r.h.s. of eq.

(2.21) shouldbe multiplied by a phase factor determined in [8] to be equal toexp iπc12−3σ +nXi=1tiXj=1a(i)j. (2.22)Here a(i)jform a continued fraction expansion of pi/qi, whileσ = −sign nXi=1qipi!+nXi=1sign piqi!+nXi=1ti−1Xj=1sign(a(i)j )(2.23)4It is not hard to see that X( pq ) and X( p1q1 , p2q2 ) are the lens spaces.11

2.3SU(2) Formulas and Poisson ResummationExplicit formulas for the SL(2, Z) representation in the space of aﬃne SU(2) characters oflevel k were derived in [9]. There are k + 1 integrable gSU(2) representations with spins0 ≤j ≤k2.

We use the shifted highest weight α = 2j + 1 instead of j and K = k + 2 insteadof K, so that 0 < α < K. The weight ρ is equal to 1 for SU(2).The formulas for ˜Sαβ and ˜Tαβ are well known:˜Sαβ =s2K sin παβK ,˜Tαβ = e−iπ4 eiπ2K α2δαβ(2.24)A substitution of these expressions in the r.h.s. of eq.

(2.17) turns it into a multiple ﬁnitegaussian sum.A summation over the intermediate indices goes from 1 to K −1.Anapplication of the Poisson resummation formula in [9] converted that sum into anothergaussian sum with a summation interval independent of K:˜M(p,q)αβ= i sign(q)q2K|q|e−iπ4 Φ(M(p,q)) Xµ=±1q−1Xn=0µ exp iπ2Kqhpα2 −2µα(β + 2Kn) + s(β + 2Kn)2i(2.25)Here Φ(M) is a Rademacher phi function deﬁned as followsΦprqs=p+sq −12s(s, q)ifq ̸= 0rsifq = 0,(2.26)a function s(s, q) being a Dedekind sum:s(m, n) = 14nn−1Xj=1cot πjn cot πmjn . (2.27)We illustrate the use of the Poisson formulaXn∈Zf(n) =Xm∈ZZ +∞−∞e2πimxf(x)dx(2.28)by explicitly deriving an expression for the matrix element( ˜S ˜T p ˜S)αβ = −e−iπ4 p2KK−1Xγ=1Xµ1,µ2=±1µ1µ2 exp iπ2Khpγ2 + 2γ(µ1α + µ2β)i(2.29)12

This excercise will prepare us for the calculations that we will perform in section 3.We have to extend the summation range of γ to Z in order to be able to use eq. (2.28).The summand in eq.

(2.29) is even and periodic with a period of 2K. We ﬁrst double therange of summation: PK−1γ=1 →12PK−1γ=−K.

Then a formulaN−1Xn=0f(n) = N limǫ→0 ǫ1/2 Xn∈Ze−πǫn2f(n), if f(n) = f(n + N),(2.30)and a Poisson resummation allow us to transform eq. (2.29) into( ˜S ˜T p ˜S)αβ = −e−iπ4 p2K limǫ→0(Kǫ1/2)Xγ∈ZXµ1,2=±1µ1µ2e−πǫγ2 exp iπ2Khpγ2 + 2γ(µ1α + µ2β)i= −e−iπ4 p2K limǫ→0(Kǫ1/2)Xn∈ZZ +∞−∞dγXµ1,2=±1µ1µ2e−πǫγ2 exp iπ2Khpγ2 + 2γ(µ1α + µ2β + 2Kn)i(2.31)Note that a change from a sum to an integral over γ has been essentially accomplishedthrough a substitutionβ →β + 2Kn(2.32)and a subsequent summation over n. This is a trick that works for a general expression (2.17).Just one substitution like (2.32) for an initial or ﬁnal index converts all the intermediate sumsin eq.

(2.17) into gaussian integrals. This is, in fact, the origin of the expression (β + 2Kn)in eq.

(2.25).From this point we can proceed in two ways. A straightforward way is to integrate ther.h.s.

of eq. (2.31) over γ.

Then, after neglecting some irrelevant terms we get a formula( ˜S ˜T p ˜S)αβ = −e−iπ4 psiK2p limǫ→0Xn∈ZXµ1,2=±1µ1µ2ǫ1/2e−4πǫ K2p2 n2 exp"−iπ2Kp(µ1α + µ2β + 2Kn)2#(2.33)The second exponential here is periodic in n with a period p. Therefore a second applicationof eq. (2.30), this time backwards, leads to the ﬁnal expression( ˜S ˜T p ˜S)αβ = −e−iπ4 psi8Kpp−1Xn=0Xµ1,2=±1µ1µ2 exp"−iπ2Kp(µ1α + µ2β + 2Kn)2#(2.34)An equivalent way to treat the r.h.s.

of eq. (2.31) is to notice that since an integralover γ is gaussian, a stationary phase approximation is exact.

An array of stationary phase13

points γst(n) = −2Kn/p and their contributions exhibit the same symmetry properties asa summand of eq. (2.29).

Therefore an inverse use of eq. (2.30) shows that we can drop afactor (Kǫ1/2) from the r.h.s.

of eq. (2.31) and restrict the sum there to those values of nfor which0 ≤γst(n) ≤K(2.35)The formula that we get this way is slightly diﬀerent in its form from eq.

(2.34). For p – oddwe get( ˜S ˜T p ˜S)αβ=−e−iπ4 psi2KpXµ1,2=±1µ1µ2"12 exp −iπ2Kp(µ1α + µ2β)2!+p−12Xn=1exp −iπ2Kp(µ1α + µ2β + 2Kn)2!(2.36)The 1/2 factor in front of the ﬁrst exponential here is due to the fact that the stationarypoint γst(0) = 0 is on the boundary of the interval (2.35).

There is another such stationarypoint γst(p/2) for p – even.In the next section we will apply eqs. (2.28) and (2.30) to formula (2.21).

Meanwhile weuse eq. (2.34) together with eq.

(2.19) in order to get an expression for Z(L(p, −1), k) andcheck a factor of 1/Vol(H) in the 1-loop approximation formula (2.14). In the large k limitfor odd pZ(L(p, −1), k) = ( ˜S ˜T p ˜S)11 ≈e−iπ4 p√2π iKp!3/2+ 4si2Kpp−12Xn=1e−2πiKn2/p sin2 2πnp(2.37)The ﬁrst term in the square brackets is a contribution of the trivial connection, while theremaining sum goes over nontrivial maps π1(L(p, −1)) = Zp →SU(2).

The SU(2) subgroupH commuting with the image of π1 is SU(2) and U(1) respectively.According to [8], a square root of the Reidemeister torsion of the trivial connection isp−3/2 and that of a nontrivial one is 4p−1 sin2(2πn/p). Therefore if eq.

(2.14) is correct, thenVol (SU(2)) =1π√2, Vol (U(1)) =√2. (2.38)14

The same value of Vol (SU(2)) is predicted by a large k limit of Z(S3, k), which is equal to√2πk−3/2. The group SU(2) is a 3-dimensional sphere and U(1) is its big circle.

We getboth volumes (2.38) if we assume that the radius of that sphere is 1/(√2π). This value isnot unnatural.

The path integral measure in quantum theories contains an implicit factor(2π¯h)−1/2 coming with each of the 1-dimensional integrals comprising the path integral. Forthe Chern-Simons theory ¯h = π/k.

The radius of SU(2) was calculated in the “reduced”measure, so its actual value is 1.3A Large k Limit of the Invariants of 3-Fibered SeifertManifolds3.1Stationary Phase PointsLet us try to apply the Poisson formula (2.28) to the partiton function (2.21) for the case ofn = 3 in order to put it in the form (2.3) or (2.4).We start by giving an explicit expression for Nα1,α2,α3 inside the fundamental cube0 < α1, α2, α3 < K(3.1)According to its deﬁnition, Nα1,α2,α3 = 1 iﬀα1+α2+α3 is odd and the following 4 inequalitiesare satisﬁed:α1 + α2 −α3>0α1 −α2 + α3>0−α1 + α2 + α3>0α1 + α2 + α3<2K(3.2)Otherwise, Nα1,α2,α3 = 0. We can drop a restriction on the parity of α1+α2+α3 if we change15

the formula (2.21) intoZ X(p1q1, p2q2, p3q3), k!=X0<α1,α2,α3

1).To simplify this picture we draw its section by a plane α3 = const in Fig. 2 (region 1+).At this stage we could use a discrete stationary phase approximation method describedin Appendix A of [8], in order to get the 1-loop approximation of Z.

The stationary phasepoints inside the tetrahedron would correspond to the irreducible ﬂat connections.Theconditional stationary phase points on the faces of the tetrahedron would correspond to thereducible ﬂat connections.We use a diﬀerent appoach close to the one in subsection 2.3 in order to get the full 1/kexpansion of Z(X, k). We want to extend the sum in eq.

(3.3) from the fundamental cubeto the whole 3-dimensional space. We do this in two steps by using the symmetries of thematrices ˜M(p,q)α,βunder the aﬃne Weyl transformations.

These transformations include theordinary Weyl reﬂections as well as the shifts by the root lattice multiplied by K:˜M(p,q)−α,β = −˜M(p,q)α,β ,˜M(p,q)α+2K,β = ˜M(p,q)α,β(3.4)The easiest way to see these symmetries is to use eq. (2.17) and expressions (2.24) for ˜Sαβand ˜Tαβ.The ﬁrst of eqs.

(3.4) enables us to extend the sum (3.3) to a bigger cube:P0<α1,α2,α3

The translational invariance of ˜Mαβ together witheq. (2.30) brings us to another formula for Z:Z = limǫ→0(2Kǫ1/2)38Xα1,α2,α3∈ZXλ=0, 1212e2πiλ(1−α1−α2−α3)−πǫ(α21+α22+α23) ˜M(p1,q1)α11˜M(p2,q2)α21˜M(p3,q3)α31˜Nα1,α2,α3(3.5)if we require ˜Nα1,α2,α3 to be a periodic function of its indices with the period of 2K (seeFig.

3).16

A Poisson formula (2.28) transforms the sum over αi into an integral:Xα1,α2,α3∈Z=Zdα1dα2dα3Xn1,n2,n3∈Z3Yi=1δ(αi−ni) =Zdα1dα2dα3Xm1,m2,m3∈Zexp 2πi3Xi=1αimi! (3.6)On the other hand,e2πiαm ˜Mαβ = ˜Mα,β+2Kqm,(3.7)so the sum over mi and the exponential in the r.h.s.

of eq. (3.6) can be absorbed by extendingthe sum over n in eq.

(2.25) to all integer numbers. FinallyZ(X, k)=Z1Z2,(3.8)Z1=i3sign(q1q2q3) exp"−iπ43Xi=1Φ(M(pi,qi))#,(3.9)Z2=limǫ→0(2Kǫ1/2)3812Xλ=1, 12e2πiλZ˜Nα1α2α33Yi=1dαiq2K|qi|Xµi=±1Xni∈Zµie−πǫα2i× expiπ2Kqihpiα2i −2αi(2K(ni + qiλ) + µi) + si(2Kni + µi)2i.

(3.10)The integral in eq. (3.10) is gaussian, but the function ˜Nα1α2α3 carves a rather complicatedregion out of the 3-dimensional α-space.

A slice of that region for α3 = const is depicted inFig. 3.

Fortunately, this region can be represented as a linear combination of positive strips(double wedges in 3-dimensional space)α1 −α3 + 2Kl < α2 < α1 + α3 + 2Kl, l ∈Z(3.11)and negative stripsα3 −α1 + 2Kl < α2 < −α3 −α1 + 2Kl, l ∈Z(3.12)Each strip (double wedge) is in turn a diﬀerence between two half-planes (half-spaces).Overall we have a superposition of half-spaces3Xi=1νiαi + 2Kl > 0, ν1 = −1, ν2, ν3 = ±1(3.13)These half-spaces are related to the Bernstein-Gelfand -Gelfand resolution of aﬃne modules.We will use this relation in subsection 5.1.17

A sign of the contribution coming from a half-space (3.13) is determined by the productν2ν3. Therefore we can changeR ˜Nα1α2α3 in eq.

(3.10) for−Xn∈ZXν2,3=±1ν1ν2ν3ZP3i=1 νiαi+2Kl>0(3.14)The stationary points of the phase in eq. (3.10) areα(st)i= 2K ˜nipi, here ˜ni = ni + qiλ(3.15)There are also conditional stationary points on the boundary planes3Xi=1νiαi + 2Kl = 0(3.16)They areα(cst)i= 2Kpiνi(ni −qic0),(3.17)herec0 = HP3Xj=1njpj+ l, P = p1p2p3, H = P3Xj=1qipi= p1p2q3 + p1q2p3 + q1p2p3.

(3.18)|H| is the order of homology group of the Seifert manifold Xp1q1 , p2q2 , p3q3. The points (3.17)form a 2-dimensional lattice on the plane (3.16).

Note that α(cst)iare not changed under asimultaneous shiftni −→ni + qim, m ∈Z(3.19)Consider now an integral from eq. (3.10) with a substitution (3.14):ZP3i=1 νiαi+2Kl>03Yi=1dαiq2K|qi|expiπ2Kqihpiα2i −2αi(2K(ni + qiλ) + µi) + si(2Kni + µi)2i(3.20)We dropped a regularization factor exp(−πǫP3i=1 α2i ), while keeping in mind that it willsuppress a contribution of the stationary points (3.15) and (3.17) by its value at thosepoints.

If a point (3.15) does not belong to the half-space of (3.20), then the integral is equalto a contribution of the conditional point (3.17). If, however, a point (3.15) is within thehalf-space, then we use an obvious relationZP3i=1 νiαi+2Kl>0 =ZR3 −ZP3i=1 νiαi+2Kl<0(3.21)18

The ﬁrst integral in the r.h.s. of this equation is purely gaussian, it is determined by thepoint (3.15).

The second integral is again determined by a conditional point (3.17). We willcalculate both integrals in the next subsection.

Here we just note that as it follows fromeq. (3.21), a contribution of a point (3.15) to the integral (3.20) is either zero or a quantitywhich does not depend on the half-space to which it belongs.

Therefore if a point (3.15)belongs to an array of tetrahedra whose slice is depicted in Fig. 3, then its contribution tothe whole expression (3.10) is equal to the ﬁrst integral in the r.h.s.

of eq. (3.21).

If thepoint does not belong to the array, then its contribution is zero.The overall picture is this: we have two lattices (3.15) and (3.17). Z2 is equal to thesum of the contributions of the points of these lattices.

The lattices and the contributions oftheir points exhibit the same symmetry under the aﬃne Weyl group transformations, as thesummand of eq. (3.5).

Therefore by using the inverse eq. (2.30) in exactly the same way aswe did in deriving eq.

(2.36), we drop the factor (2Kǫ1/2)3/8 from the integral in eq. (3.10).At the same time we restrict the sum over ni to those stationary points (3.15) which belongto the fundamental tetrahedron (3.2) and to those conditional stationary points (3.17) whichlie on its faces.The stationary points that belong to the intersection of the planes (3.16) require a specialcare.

Their contribution to the integral over the region carved by ˜Nα1α2α3 is proportionalto the number of planes to which they belong. However the reduction to the fundamentaltetrahedron should also account for the fact that these points are invariant under the actionof a subgroup W ′ of the aﬃne Weyl group.

Therefore the total contribution of such pointsis equal to their contribution to the integral (3.10) times a factor# of planes# of elements in W ′(3.22)This factor is similar to the factor 1/2 in eq. (2.36).

It is equal to 22 = 1 for the points onthe edges of the tetrahedron and to 48 = 12 for the points on the vertices of the tetrahedron.We can “unfold” the surface of the tetrahedron and require the points (3.17) to belongto the intersection of the plane α1 + α2 + α3 = 0 with the cube −2K < α1 < 0, −2K <α2 < 0, 0 < α3 < 2K. This intersection is an equilateral triangle (see Fig.

4) consisting of 419

smaller triangles that can be mapped by Weyl reﬂections onto the faces of the fundamentaltetrahedron.There is yet another way to view the fundamental set of conditional stationary phasepoints. As we have noted, the triplets ni related by a transformation (3.19) deﬁne the samepoint through eqs.

(3.17). A transformationni −→ni + mpi, nj −→nj −mpj, i ̸= j(3.23)does not change c0 and shifts α(cst)iby 2Km and α(cst)jby −2Km, thus leaving them withinthe same equivalence class of aﬃne Weyl transformations.

We can describe a fundamentalregion of the conditional stationary phase points as a factor of a lattice of all integer tripletsni over a lattice generated by three vectors⃗v1 = (q1, q2, q3), ⃗v2 = (p1, −p2, 0), ⃗v3 = (0, p2, −p3)(3.24)The number of triplets ni inside that factor is equal to the volume of a parallelepiped formedby the vectors ⃗vi# of conditional points =q1q2q3p1−p200p2−p3= |H|(3.25)We should be interested only in approximately half of the triplets, because the volume ofthe prism built upon a triangle of Fig. 4 is twice as small as that of the parallelepiped whichis built upon the whole parallelogram.

Thus the number of the conditional stationary phasepoints within the fundamental domain is approximately equal to half the rank of the homol-ogy group. This result is not surprising since we intend to identify the conditional stationaryphase points with reducible SU(2) ﬂat connections.

The number of these connections is alsoapproximately equal to |H|/2.3.2The IntegralsStationary Phase PointsWe start with the simplest case of a contribution of the point (3.15) which is inside the20

fundamental tetrahedron (3.2). As we saw in the previous subsection, it is equal to thegaussian integral taken over the whole α-space:Z2=12Xλ=0, 12e2πiλ3Yi=1Xµi=±1µiZ +∞−∞dαiq2K|qi|expiπ2Kqihpiα2i −2αi(2K˜ni + µi) + si(2Kni + µi)2i=Z312Xλ=0, 12e2πiλ3Yi=11q|pi|2i sin 2π ripi˜ni + siλ!exp 2πiK"ripi˜n2i −qisiλ2#.

(3.26)Here Z3 is a factor that will be present in all the subsequent expressions for the contributionsto Z2:Z3 =3Yi=1eiπ4 sign(piqi) exp iπ2Kripi! (3.27)Conditional Stationary Phase PointsConsider now a contribution of the pointsα(cst)i= 2Kpi(ni −qic0), c0 = PH3Xi=1nipi(3.28)which belong to the planeα1 + α2 + α3 = 0(3.29)We start with an integral (3.26) that we take over the regionα1 + α2 + α3 < 0(3.30)We can return to the integral over the whole α-space if we introduce an extra factorθ(−α1 −α2 −α3) =Z +∞0dxZ +∞−∞dc exp [2πic(α1 + α2 + α3 + x)](3.31)The invariance of α(cst)iunder the transformation (3.19) implies that we should make asubstitution (3.19) in the integral (3.26) and take a sum over all integer m. As a result, thefull contribution of the point α(cst)ito Z2 isZ2=12Xλ=0, 12e2πiλ Xm∈ZZ +∞0dxZ +∞−∞dc e2πicx3Yi=1Xµi=±1µiZ +∞−∞dαiq2K|q|× expiπ2Kqihpiα2i −2αi (2K(ni + qi(λ + m −c)) + µi) + si (2K(ni + qim) + µi)2i=Z3e−iπ4 sign(H/P )q2K|H|exp 2πiK" 3Xi=1ripin2i + HP c20#Xµ1,2,3=±1( 3Yi=1µi exp"2πiµirini + c0pi#)×12Xλ=0, 12e2πiλ Xm∈ZI(m),(3.32)21

hereI(m) =Z +∞0dx exp2πix(c0 + m + λ) + iπ2KPH x +3Xi=1µipi!2(3.33)We calculate the integral (3.33) in the spirit of remarks preceding eq. (3.21).

If PH (c0 + m +λ) < 0, then the stationary point of the phase in eq. (3.33) lies outside the integration region,and the dominant contribution comes from the boundary point x = 0:I(m)=∞Xj=01j!

iπ2KPHj Z ∞0dx x +3Xi=1µipi!2jexp[2πix(c0 + m + λ)]=∞Xj=01j! (8πiK)−j PHj∂(2j)ǫ"Z ∞0dx exp 2πix(c0 + m + λ + ǫ) + 2πiǫ3Xi=1µipi!#ǫ=0.

(3.34)If however PH (c0 + m + λ) > 0, then we use a relation similar to eq. (3.21):Z ∞0dx =Z +∞−∞dx −Z 0−∞dx(3.35)The integralR +∞−∞dx is fully determined by the stationary phase point and therefore has beenaccounted for in eq.

(3.26). The integral −R 0−∞dx is dominated by the boundary point x = 0and leads to the same expression (3.34).

Thus eq. (3.34) is valid if c0 + m + λ ̸= 0.A Poisson formula (2.28) allows us to convert a sum over m into a “discretization” of theintegral over x:12Xλ=0, 12e2πiλ Xm∈ZI(m)=∞Xj=01j!

(8πiK)−j PHj∂(2j)ǫ12Xλ=0, 12e2πiλ+ǫP3i=1µipi ∞Xx=0e2πix(c0+λ+ǫ)ǫ=0=−∞Xj=01j! (8πiK)−j PH∂(2j)ǫe2πiǫ P3i=1µipi2i sin 2π(c0 + ǫ)ǫ=0(3.36)By substituting this expression into eq.

(3.32) and taking a sum over µi we get the ﬁnalexpression for the contribution of the stationary point (3.28) to Z2:Z2=−Z3e−i π4 sign( HP )q2K|H|exp 2πiK" 3Xi=1ripin2i + HP c20#×∞Xj=01j! (8πiK)−j PHj ∂(2j)cQ3i=1 2i sin2π rini+cpi2i sin 2πcc=c0.

(3.37)22

A Stationary Phase Point on the BoundaryA stationary phase point (3.15) presents a special case when it belongs to the boundary ofone of the planes (3.13). Suppose, that α(cst)isatisfy conditions (3.29).

Comparing eqs. (3.15)and (3.28), we see that this may happen ifc0 + λ = 0(3.38)or, in other words, c0 = 0, −12.We can proceed with the same analysis as for an ordinary conditional stationary phasepoint up to eq.

(3.34).The integral I(0) requires a separate calculation.According toeq. (3.33),I(0)=Z ∞0dx expiπ2KPH x +3Xi=1µipi!2=sK2HPei π4 sign(H/P ) −Z P3i=1µipi0dx exp iπ2KPH x2=sK2HPei π4 sign(H/P ) −∞Xj=01j!

(8πiK)−j PHj ∂(2j)ǫe2πiP3i=1µipi −12πiǫǫ=0. (3.39)The remaining part of eq.

(3.36) is equal to−∞Xj=01j! (8πiK)−j PHe2πic0∂(2j)ǫe2πiǫ P3i=1µipi12i sin 2πǫ −14πiǫǫ=0 .

(3.40)Adding I(0) with an extra factor 12e2πic0 to this expression and substituting it into eq. (3.32)brings us to the formulaZ2=Z3e−i π4 sign( HP )q2K|H|exp 2πiK" 3Xi=1ripin2i + HP c20#e2πic0×ei π4 sign( HP )sK8HP3Yi=12i sin 2πrini + c0pi!(3.41)−∞Xj=01j!

(8πiK)−j PHj∂(2j)ǫQ3i=1 2i sin2π rini+c0+ǫpi2i sin(2πǫ)−Q3i=1 2i sin2π rini+c0pi4πiǫǫ=0.Trivial ConnectionIn the case of n1 = n2 = n3 = c0 = 0 the formula (3.41) requires an extra 1/2 factor23

coming from the ratio (3.22). After some simpliﬁcations it becomesZ2 = −Z3e−i π4 sign( HP )q8K|H|∞Xj=01j!

π2iKPHj ∂(2j)ǫQ3i=1 2i sinǫpi2i sin ǫǫ=0. (3.42)3.3Framing CorrectionsWe have to reduce our formulas for Z to the standard framing in order to obtain a truetopological invariant that is independent, for example, of the choice of ri and si in thematrices M(pi,qi).

We add an extra phase factor (2.22) to Z. According to [9],tiXj=1a(i)j −3ti−1Xj=1signa(i)j= Φ(M(pi,qi)).

(3.43)Also note thatΦ(M(pi,qi)) −3sign(piqi) = Φ(SM(pi,qi))(3.44)The central charge of the SU(2) WZW model isc = 3(K −2)K.(3.45)Therefore the full framing correction isZf=exp iπ4" 3Xi=1Φ(M(pi,qi)) −3sign(piqi)+ 3signHP#× exp −iπ2K" 3Xi=1Φ(SM(pi,qi)) + 3signHP#(3.46)so that a corrected version of the product of phase factors Z1Z2 isZ1Z3Zf = ei 3π4 sign( HP )3Yi=1sign(pi) exp −iπ2K"3signHP+3Xi=1 12s(qi, pi) −qipi!#. (3.47)We used the following property of the Dedekind sum in order to derive this formula:s(m∗, n) = s(m, n), if mm∗= 1(mod n).(3.48)Eq.

(3.47) together with eqs. (3.27), (3.37), (3.41) and (3.42) leads to the ﬁnal formulas (1.3)–(1.8).24

4One-Loop Approximation Formulas4.1Irreducible Flat ConnectionsIrreducible ﬂat connections on a Seifert manifold Xp1q1 , . .

. , pnqnare the ones for which thesubgroup H commuting with the image of the homomorphism (2.2) does not have contin-uous parameters.

In the case of G = SU(2) this simply means that the image of (2.2) isnoncommutative.The fundamental group π1 of the Seifert manifold is generated by the elements x1, . .

. , xn, hsatisfying relationsxpii hqi = 1, hxi = xih,nYi=1xi = 1.

(4.1)The elements xi go around the solid tori that make up the manifold, while h goes along theS1 cycle of the “mother-manifold” S1 × S2 (see [15] and [8] for details).Suppose that the image of h does not belong to the center of SU(2). Since h commuteswith all the elements of π1, then the whole image of π1 belongs to U(1) ⊂SU(2).

ThereforeH ⊃U(1), so this is a reducible case. An irreducible connection is produced only if theimage of h belongs to the center of SU(2):h A7→e2πiλ100−1, λ = 0, 12.

(4.2)The images of the elements xi belong to the conjugation classes of diagonal matrices whosephases we denote as ui:xiA7→g−1ie2πiui00e−2πiuigi(4.3)The ﬁrst of relations (4.1) determines the possible values of these phases:ui = ˜nipi,(4.4)here the numbers ˜ni are deﬁned in eq. (3.15).

The phases ui determine the map (2.2) uniquelyup to an overall conjugation if the number of the solid tori is n = 3. If n > 3, then each25

particular choice of the phases ui corresponds to a connected component of the 2(n −3)dimensional moduli space of these maps, which is a moduli space of ﬂat connections on aSeifert manifold. Such connected component is isomorphic to the space of ﬂat connections onan n-holed sphere if the holonomies around the holes are ﬁxed by eq.

(4.3). We will discussthis subject further in subsection 5.2.

Here we specialize to the case n = 3 and present theexpressions for the manifold invariants entering eq. (2.14).

A Chern-Simons invariant of anirreducible ﬂat connection on a Seifert manifold was computed in [17]:SCS =3Xi=11pi(rin2i −2λni −qiλ2) =3Xi=1 ripi˜n2i −qisiλ2! (mod 1).

(4.5)According to [16], a square root of the corresponding Reidemeister torsion isτ 1/2X=3Yi=12√pi| sin(2πφi)|,(4.6)hereφi = rini −λpi= ripi˜ni + siλ (mod 1)(4.7)are the phases of the conjugation classes of the holonomies along the central ﬁbers of thesolid tori that make up the Seifert manifold.The spectral ﬂow was calculated in [15]:IA = −3 + 8SCS +3Xi=12pipi−1Xl=1cot πrilpi!cot πlpi!sin2"2πlpi(rini −λ)#.(4.8)L. Jeﬀrey presented in her paper [9] a proof by D. Zagier of an equation(−i)sign sin 2πrnpsin 2πnp!= exp iπ28rn2p−2pp−1Xl=1cot πlp cot πqlp sin2 2πnlp,(4.9)here n, p, q, r ∈Z, qr = −1 (mod p).

A slight modiﬁcation of that proof shows that eq. (4.9)works also for half-integer n if we multiply its l.h.s.

by an extra factor of e2πin. Then anapplication of eq.

(4.9) with a substitutionq = ri, r = qi, n = rini −λ(4.10)to the r.h.s. of eq.

(4.8) leads to a formula for the exponential of the spectral ﬂow:exp−iπ2 IA= −3Yi=1e2πiλsign sin 2π˜nipisin 2πφi!= −e2πiλ3Yi=1sign (sin 2πφi) ,(4.11)26

here we used the fact that 0 <˜nipi <12 because 0 < α(st)i< K. Apparently the role ofthe factor (−i)IA is to remove the absolute value from the sines in the square root of theReidemeister torsion (4.6). A similar eﬀect was observed for lens spaces in [9].For an irreducible connection on a 3-ﬁbered Seifert manifold Xp1q1 , p2q2 , p3q3, dim H0 =dim H1 = b1 = 0 and dim SU(2) = 3, while H is a center of SU(2) which consists of 2elements, Vol(H) = 2.Therefore according to eq.

(2.14), the 1-loop contribution of anirreducible ﬂat connection should be equal to−12e−i 3π4 e2πiλ3Yi=12q|pi|sin 2π ripi˜ni + siλ!exp 2πiK ripi˜n2i −qisiλ2!. (4.12)If we compare this expression to eq.

(1.3), then we see that5 the exact contribution diﬀersfrom eq. (4.12) only by a phase factorexp −iπ2K"3signHP+3Xi=1 12s(qi, pi) −qipi!#.

(4.13)It comes from the overall phase factor Z1Z3Zf and can be interpreted as a 2-loop correctionaccording to eq. (2.4).

Note that this factor is the same for all the stationary phase contri-butions. It does not “feel” the background gauge ﬁeld and seems to be of “gravitational”origin.

A similar 2-loop phase factor has been found in [10] and [9] for the lens spaces L(p, q)to be equal to exphiπ2K 12s(q, p)i. S. Garoufalidis noted in [10], that this phase is proportionalto Casson’s invariant extended by K. Walker to rational homology spheres. According to[18], this invariant is equal to s(q, p) for the lens space L(p, q).

C. Lescop computed theCasson-Walker invariant for n-ﬁbered Seifert manifolds in [20]:λCW X(p1q1, . .

. , pnqn)!= 112PH 2 −n +nXi=1p−2i!−112"3signHP+nXi=1 12s(qi, pi) −qipi!#.

(4.14)We see that the phase of (4.13) is indeed proportional to the second term in the r.h.s. ofeq.

(4.14), however the ﬁrst term (which is dominating in the limit of large pi) is missing. Wewill see that the missing part appears in the total 2-loop correction to the trivial connection,which includes some terms of the asymptotic series together with the phase (4.13).5in assumption of p1, p2, p3, H > 027

4.2General Reducible Flat ConnectionsAs we noted in the previous subsection, the image of π1 under the homomorphism (2.2)belongs to the U(1) subgroup of SU(2) for the reducible ﬂat connections. This generallyhappens when the image of h does not belong to the center of SU(2):h A7→e2πic000e−2πic0, c0 ̸= 0, 12.

(4.15)Since h commutes with all xi, their images should also be diagonal.The ﬁrst of equa-tions (4.1) again determines the phases:xiA7→e2πiui00e−2πiui, ui = ni −qic0pi,(4.16)while c0 is determined by the second equation in (3.28). The phases of the holonomies goingalong the ﬁbers of the solid tori areφi = rini + c0pi(4.17)A Chern-Simons action and a square root of the Reidemester torsion are known to be6SCS =3Xi=1ripin2i + HP c20, τ 1/2X= |H|−1/2Q3i=1 2 sin(2πφi)2 sin(2πc0) .

(4.18)Reducibility of connection means that this time dim H0 = 1, H = U(1), Vol(H) = 1/√2.All these formulas are compatible with the leading term in 1/k expansion of the condi-tional stationary phase contribution (1.6) at least up to a phase factor. Indeed, we see thatthe 1-loop part of eq.

(1.6) (assuming that p1, p2, p3, H > 0) is equal toi√2KHQ3i=1 2 sin(2πφi)2 sin(2πc0)exp 2πiK" 3Xi=1ripin2i + HP c0#(4.19)6see e.g. [13] where these quantities were calculated by using a U(1|1) Chern-Simons-Witten theory28

4.3Special Reducible ConnectionsThe special reducible ﬂat connections are those for which one or more sines in eq. (4.18) areequal to zero.

This amounts to a condition that a stationary phase point α(st) deﬁned byeq. (3.15) belongs to a face, an edge or a vertex of the fundamental tetrahedron (3.2).Point on a FaceSuppose that a condition (3.38) is satisﬁed for some value of λ.

This means that theelement h ∈π1 is mapped , according to eq. (4.15), to e2πiλ1001, so that the condi-tional stationary phase point on a face of the tetrahedron is, in fact, unconditional.

Theapproximation (4.19) breaks down, the reason being that eq. (1.7) should be used insteadof eq.

(1.6). Then the leading contribution to a partition function is equal to one-half ofeq.

(4.12):14e−i 3π4 e2πiλ3Yi=12q|pi|sin 2π ripi˜ni + siλ!exp 2πiK ripi˜n2i −qisiλ2!. (4.20)Let us reconcile this expression with eq.

(2.14). The fact that a denominator of eq.

(4.18)is zero for c0 = 0, 12 indicates a presence of 1-form zero modes in the operator L−. Indeed,the ﬁrst two conditions (4.1) ﬁx the images of xi in SU(2) only up to arbitrary conjugationsbecause the image of h again belongs to the center of SU(2).

The last condition (4.1) saysthat the points x1, x2x1 and x3x2x1 = 1 form a “curved” triangle inside SU(2). The sizeof the sides of this triangle is ﬁxed by the ﬁrst condition (4.1), but their orientation isconstrained only by a condition that the triangle is closed.

Such a triangle is a rigid objectand can be rotated into a predetermined position by an overall conjugation. This is why theirreducible connections on a 3-ﬁbered Seifert manifold have no moduli.The rigidity of the triangle is considerably decreased if all three of its vertices belong tothe same big circle (see Fig.

4), i.e. if the images of all xi belong to the same U(1) subgroupof SU(2), as it happens for a stationary phase point on a face.

In this case, say, a middlevertex can be inﬁnitesimally shifted in the plane perpendicular to the line of triangle, withthe sizes of the sides of triangle changing only to the second order in the shift. Thus the29

operator L−has two zero modes, dim H1 = 2, however there are still no moduli, because anobstruction prevents an extension of those modes to a 1-parameter family of ﬂat connections.The two zero modes form a 2-dimensional representation of U(1) which is a symme-try group of the reducible ﬂat connections. This means that the modes are gauge equiva-lent.

However a procedure of dropping the zero modes of ∆and L−from the determinantsof eq. (2.5) amounts to neglecting the global U(1) gauge transformations (see Appendix).The integration in path integral (1.2) includes the directions along both zero modes.

Theexponent corresponding to these directions has no quadratic terms, the qubic terms areprohibited by the U(1) symmetry. Therefore the dominating term in the exponent is gen-erally of the fourth order in coordinates along the zero modes.

Each direction contributesan integral (2.12) which results in the formula (2.13) for th factor C1. Since in our casedim H0 = 1, dim H1 = 2, dim X = 0 we see that eq.

(2.14) predicts an overall power of Kto be equal to zero in agreement with the surgery asymptotics (4.20).Point on an EdgeSuppose that in addition to the previous conditions, one of the phases φi is equal to zeroor 12. This means that α(st)iequals either 0 or K, so that a stationary phase point belongsto an edge of the tetrahedron (3.2).Let, for example, φ1 = α(st)1= u1 = 0.

Then the image of x1 in SU(2) is the identitymatrix. The triangle x1, x2x1, x3x2x1 = 1 is even more degenerate, becasue its ﬁrst sidehas shrunk.

The rigidity of the construction is restored, the zero modes of L−disappeared,dim H1 = 0. Since dim H0 = 1, then according to eq.

(2.14) we expect a contribution to beproportional to k−1/2. Indeed, the approximation (4.20) breaks down and the ﬁrst subleadingterm in eq.

(1.7) contributesi e2πiλ√2KH1p1 3Yi=22 sin(2πφi)!exp 2πiK 3Xi=1ripin2i + HP λ2!. (4.21)This expression is very similar to eq.

(4.19).We easily recognize the same constructionblocks assuming that nowτ 1/2R=1√H1p13Yi=22 sin(2πφi) . (4.22)30

Point on a VertexThis is the case when the image of π1 is a subgroup of the center of SU(2), that is,c0, φ1, φ2, φ3 = 0, 12. Let us take a particular case of a trivial connection, for which c0 =φ1 = φ2 = φ3 = 0.

The Chern-Simons invariant is zero, the square root of the Reidemeistertorsion is known to be (see, e.g. [8])τ 1/2X= H−dim G2= H−3/2(4.23)The group H is the whole SU(2), its volume in the proper normalization is 1/(√2π) (seeeq.

(2.38)). We also know that dim H0 = 3, dim H1 = 0.

As for the phase factor in eq. (2.14),it is shown in [8] that for the trivial connectionexp −iπ4h2Ia + 3(1 + b1) + dim H0 + dim H1i= 1.

(4.24)Therefore, according to eq. (2.14), the 1-loop contribution of the trivial connection shouldbeZ =√2π(KH)−3/2.

(4.25)We get the same expression from the surgery calculus if we take the term with j = 1 ineq. (1.8).2-Loop CorrectionLet us use eq.

(1.8) to calculate the next subleading correction to the formula (4.25)7. Inother words, we are looking for a 2-loop term S2 as deﬁned by eq.

(2.4) (note, however, thatwe are using now K = k + 2 instead of k as an expansion parameter). One obvious source ofS2 is the 2-loop phase −iπ2φ (see eqs.

(1.5) or (4.13)). The other source is the j = 2 term inthe asymptotic series of (1.8).

Actually, we have to take a logarithm of that series to bringit to the form (2.4). At the 2-loop level of approximation this amounts to dividing the j = 2term by the leading j = 1 term.

Since∂(4)ǫQ3i=1 2i sinǫpi2i sin ǫǫ=0= 16P" 3Xi=1p−2i−1#,(4.26)7I am indebted to D. Freed for turning my attention to this calculation.31

then the whole 2-loop correction S2 isS2 = iπ2" PH 3Xi=1p−2i−1!−3signHP−3Xi=1 12s(qi, pi) −qipi!#= 6πλCW,(4.27)here λCW is a Casson-Walker invariant (see eq. (4.14)).5A Large k Limit of the Invariants of General SeifertManifolds5.1A Bernstein-Gelfand-Gelfand Resolution and Verlinde Num-bersA calculation of Witten’s invariant for a general Seifert manifold can proceed along the samelines as that for the 3-ﬁbered one, described in section 3.We will try again to convertthe sums in eq.

(2.21) into the integrals over the n-dimensional half-spaces.We need arepresentation for Verlinde numbers Nα1,...,αn similar to that of subsection 3.1. We will do thiswith the help of the Bernstein-Gelfand-Gelfand resolution, which presents a representationspace of a Lie group G as a cohomology over a complex of certain vector spaces (see e.g.

areview [21] and references therein).We introduce the following notation.∆with various subscripts will denote the setsof weights of G coming with multiplicities.In other words, the elements of ∆are pairs(v, m), where v is a weight and m ∈Z is its multiplicity. The weights form an abelian group.Consider its group algebra A with the coeﬃcients in Z.

There is a one-to-one correspondencebetween the sets ∆and the elements of A:∆←→X(v,m)∈∆mv. (5.1)We deﬁne the sums and products of the sets ∆which parallel the operations in A. Thesum ∆1 + ∆2 consisits of weights belonging to either of the sets ∆1, ∆2 and coming withthe multiplicities which are sums of their multiplicities in ∆1 and ∆2.

To build a product32

∆1 ◦∆2 we take all the pairs of weights v1 ∈∆1, v2 ∈∆2. Their sums v1 + v2 appear in theproduct ∆1 ◦∆2 with multiplicities m1m2.

If the same weight v appears more than once asa sum v1 + v2, then we add all its multiplicities in order to account for the similar terms. Asum Pv∈∆F(v) is a shorthand for P(v,m)∈∆mF(v), in the same way a product Qv∈∆F(v) isequivalent toQ(v,m)∈∆F m(v).Consider a representation space VΛ of a Lie group G with the shifted highest weight Λ(we remind that a highest weight of VΛ is Λ −ρ, ρ = 12Pλi∈∆+ λi, ∆+ is a set of positiveroots of G. A Weyl formula for the character of VΛ as a function of the element x of a Cartansubalgebra, is a ratioχΛ(x) ≡Xv∈∆Λeiv·x =Pw∈W(−1)|w|ei[w(Λ)−ρ]·xQλi∈∆+ (1 −e−iλi·x).

(5.2)Here ∆Λ is a set of weights of VΛ, W is a Weyl group and |w| is a number of elementaryWeyl reﬂections (mod 2) whose product is equal to w.An individual term in the formula (5.2) can be presented as a sum(−1)|w|ei[w(Λ)−ρ]·xQλi∈∆+ (1 −e−iλi·x) = (−1)|w|Xv∈∆(1)w (Λ)eiv·x,(5.3)here a set ∆(1)Λ contains all the weights of the formv = Λ −ρ −Xiniλi, ni ≥0(5.4)with multiplicity 1 (in fact, many weights may ultimately have a bigger multiplicity, becausegenerally the positive roots λi are not linearly independent).It follows from eqs. (5.2)and (5.3) that∆Λ =Xw∈W(−1)|w|∆w(Λ).

(5.5)The inﬁnite dimensional modules of the Bernstein-Gelfand-Gelfand resolution consist of vec-tors with the weights and multiplicities of ∆(1)w(Λ).The “classical” Verlinde numbers appear in the decomposition of the tensor productVΛ1 ⊗VΛ2 =XΛ3NΛ3Λ1Λ2Vλ3. (5.6)33

A product of characters decomposes asχΛ1(x)χΛ2(x) =XΛ3∈∆Λ1,Λ2χΛ3(x),(5.7)here ∆Λ1,Λ2 is a set of shifted highest weights of all representations appearing in the decom-position (5.6) and coming with the multiplicities NΛ3Λ1Λ2. Note that we raised a third index ofVerlinde numbers.

The indices are raised and lowered by the metric NΛ1Λ2 which is equal to1 if VΛ1 and VΛ2 are conjugate representations, and is zero otherwise. For G = SU(2) thereis no distinction between the upper and lower indices since Nα1α2 = δα1α2.If we use the r.h.s of eq.

(5.2) for χΛ1 and χΛ3, and the middle expression of eq. (5.2) forχΛ2, then we see thatXw∈WXv∈∆Λ2(−1)|w|ei[w(Λ1)+v]·x =Xw∈WXΛ3∈∆Λ1,Λ2(−1)|w|eiw(Λ3)·x.

(5.8)Let us denote by ∆W (Λ) a set containing all the weights w(Λ), w ∈W with multiplicities(−1)|w|. Then it is easy to translate eq.

(5.8) into a statement about sets:∆W (Λ1) ◦∆Λ2 =XΛ3∈∆Λ1,Λ2∆W (Λ3). (5.9)Finally applying eq.

(5.5) to ∆Λ2 we getXw∈W(−1)|w|∆W (Λ1) ◦∆(1)w(Λ2) =XΛ3∈∆Λ1,Λ2∆W (Λ3),(5.10)or equivalently,Xw1,2∈W(−1)|w1|+|w2|∆(1)w1(Λ1)+w2(Λ2) =XΛ3∈∆Λ1,Λ2∆W (Λ3). (5.11)It is easy to generalize this relation to a tensor product of n −1 vector spaces:Xw1,...,wn−1∈W(−1)|w1|+···+|wn−1|∆(n−2)w1(Λ1)+···+wn−1(Λn−1) =XΛn∈∆Λ1,...,Λn−1∆W (Λn),(5.12)here ∆Λ1,...,Λn−1 is a set of weights Λn taken with multiplicities NΛnΛ1...Λn−1, while ∆(n)Λcontainsall the weightsv = Λ −nρ −XinXj=1ni,jλi, ni,j ≥0(5.13)34

coming with multiplicities 1 (before the counting of similar terms). There aren −1m + n −1ways in which a number m can be represented as a sum of n nonnegative numbers.

Thereforewe can say that ∆(n)Λconsists of the weightsv = Λ −nρ −Xiniλi,(5.14)coming with multiplicities Qin −1ni + n −1.We need an analog of eq. (5.12) for the case of aﬃne Lie algebra (or a quantum group Gq).The aﬃne Weyl group ˜W is a semidirect product of W and an abelian group T of translationsby the elements of the root lattice multiplied by K. We can not simply substitute ˜W forW in eq.

(5.12), because the previous reasoning does not quite apply to the case of aﬃnealgebras (e.g. eq.

(5.7) is no longer valid). Still it turns out that a simple modiﬁcation ofeq.

(5.12) makes it work for aﬃne algebras or quantum groups:Xt∈TXw1,...,wn−1∈W(−1)|w1|+···+|wn−1|∆(n−2)t(w1(Λ1)+···+wn−1(Λn−1)) =XΛn∈∆Λ1,...,Λn−1∆˜W (Λn),(5.15)here ∆˜W (Λ) is a set of weights w(Λ), w ∈˜W coming with multiplicities (−1)|w|, |w| countsonly the number of reﬂections8.8 The same equation can be derived directly from Verlinde’s formulaNΛ1,...,Λn =XΛ∈∆ nYi=1SΛΛi!/Sn−2ρΛ(∆being a set of integrable highest weights), by expanding its denominator in a geometric series similar tothat of eq. (5.3) and performing a Poisson resummation on Λ.

In fact, Λ plays a role very similar to c ineq. (5.26).A generalized Verlinde’s formulaN (g)Λ1,...,Λn =XΛ∈∆ nYi=1SΛΛi!/Sn−2+2gρΛ(5.16)for the number of conformal blocks on a g-handled Riemann surface Σg with n primary ﬁelds VΛi, allowsus to generalize the results of our calculations to the case of a Seifert manifold constructed by a surgeryon circles in S1 × Σg.

Eq. (5.16) suggests that the presence of handles can be accounted for by a simple35

The l.h.s. of eq.

(5.17) consists of all the weightsv =n−1Xi=1wi(Λi) −(n −2)ρ −Xiniλi + KXimiλi, ni, mi ∈Z, ni ≥0(5.17)coming with multiplicities(−1)Pn−1i=1 |wi| Yin −3ni + n −3. (5.18)For a given set of numbers mi and Weyl reﬂections wi, these vectors form a half-space similarto that of eq.

(3.13). The r.h.s.

of eq. (5.15) consists of the highest weights Λn of integrablerepresentations of aﬃne Lie algebra coming with multiplicities NΛnΛ1...Λn−1 together with alltheir images w(Λn), w ∈˜W, whose multiplicities have an extra factor (−1)|w|.

In otherwords, the r.h.s. of eq.

(5.15) consists of all the weights Λn coming with the multiplicities˜NΛnΛ1...Λn−1 which are Verlinde numbers NΛnΛ1...Λn−1 extended to all the weights of G by the aﬃneWeyl group: the extended numbers ˜NΛnΛ1...Λn−1 are invariant under the shifts of T and theyare antisymmetric under the Weyl reﬂections. Since the matrices ˜M of eq.

(2.21) exhibitthe same properties under the action of ˜W, we can use the extended Verlinde numbers asdeﬁned by eqs. (5.15)-(5.18) in order to extend the sums in eq.

(2.21) from the integrablehighest weights to the whole weight lattice of G and to transform it into a sum over the“half-spaces” (5.17),(5.18).The Case of SU(2)Let us study speciﬁcally the case of G = SU(2). We remind that the variables α play therole of shifted highest weights, ρ = 1 and the only positive root is equal to 2.

Thus accordingto eq. (5.15), we can drop Verlinde numbers Nα1···αn from eq.

(2.21) if we take the sum thereover all the n-dimensional vectors (α1, . .

. , αn) satisfying an equationnXi=1νiαi = (n −2) + 2m + 2Kl(5.19)substitution n →n + 2g in eqs.

(5.17)-(5.20) and (5.24). Thus only a multiplicity factor Nn(x) is aﬀected(it is substituted by Nn+2g), while other quantities, such as Chern-Simons action of ﬂat connections, remainunchanged, so that eq.

(4.5) is still valid in agreement with [17]. We hope to discuss this subject further ina forthcoming paper.36

and coming with multiplicitiesn −3m + n −3nYi=1νi. (5.20)Hereν1, .

. .

, νn−1 = ±1, νn = −1, m, l ∈Z, m ≥0. (5.21)The multiplicity appears as a new factor in eq.

(2.21) taking the place of Verlinde numbers.We can make a substitutionm = 12(x −n + 2),(5.22)so that eq. (5.19) transforms intonXi=1νiαi = x + 2Kl(5.23)and the multiplicity factor isNn(x) = −Qni=1 νi2n−3(n −3)!Y−n+4≤i≤n−4i+n even(x −i).

(5.24)The substitution (5.22) requires that x −n is even and x ≥n −2. In fact we may demandonly that x ≥0, because the ﬁxed parity together with the last factor of eq.

(5.24) eliminatesall possible extra values of x.5.2A Contribution of Irreducible Flat ConnectionsWe have to calculate a sumXx>0x−n evenXαi: Pni=1 νiαi=x+2KlNn(x)nYi=1˜M(pi,qi)αi1. (5.25)Similarly to subsection 3.1 we turn a sum over αi into an integral by extending the sum overn in eq.

(2.25) to all integer numbers. We take care of a condition (5.23) by adding a factorZ +∞−∞dc exp 2πic x + 2Kl −3Xi=1νiαi!

(5.26)37

to eq. (5.25).

Another familiar factor12Xλ=0, 12exp 2πiλ nXi=1αi + n! (5.27)guarantees together with the factor (5.26) that x −n is even.

Finally since the factor (5.26)makes x an integer, we can take an integral over x rather than a sum:Zν1,...,νn;l=Zf12Xλ=0, 12e2πiλnZ ∞0dx Nn(x)Z +∞−∞dc exp 2πic(x + 2Kl)×nYi=1Xni∈ZXµi=±1µiZ +∞−∞dαi i sign(qi)q2K|qi|e−i π4 Φ(M(pi,qi))× expiπ2Kqihpiα2i + 2αi(2K(˜ni + νiqic) + µi) + si(2Kni + µi)2i, (5.28)here Zf is a framing correction given by eq. (3.46) with a substitutionP3i=1 −→Pni=1.We ﬁx the numbers ni in order to study a contribution of a particular point (3.15).

Afterintegrating over αi and c, a partition function becomes a product of two factors Z1 and Z2:Zν1,...,νn;l = Z1Z2(5.29)Z1=ei 3π4 sign( HP ) nYi=1sign pi!exp −iπ2K"3signHP+nXi=1 12s(qi, pi) −qipi!#(5.30)Z2=e−i π4 sign( HP )q2K|H|12Xλ=0, 12e2πiλnXµ1,...,µn=±1" nYi=1µi exp 2πiK ripi˜n2i −siqiλ2!× exp 2πiµi ripi˜ni + siλ!#I(ν1, . .

. , νn; µ1, .

. .

, µn; l),(5.31)hereI(ν1, . .

. , νn; µ1, .

. .

, µn; l)=Z ∞x0dx Nn(x −x0) expiπ2KPH x −nXi=1νiµipi!2(5.32)x0=2Kl −nXi=1νiα(st)i,(5.33)and we remind that α(st)iare deﬁned by eq. (3.15).

We made a change of variables x −→x−x0in deriving eq. (5.32).The integral (5.32) is gaussian apart from a polynomial factor Nn(x −x0).

This integralis similar to the integral (3.33) and should be treated in a similar way. Here we are concerned38

with a contribution of a stationary phase point x = 0, which contributes to the integral (5.32)if x0 < 0.Consider a speciﬁc point α(st)iof eq. (3.15).

As we know from subsection 3.1, we shouldlimit our attention to the points belonging to the fundamental cube0 ≤α(st)i< K.(5.34)We have to determine a set S of arrays (ν1, . .

. νn−1; l) for which a stationary phase pointα(st)icontributes to the integral (5.32).

Then we should substitute a sumN(tot)n(x) =XSNn(x +nXi=1νiα(st)i−2Kl)(5.35)instead of Nn(x −x0) in eq. (5.32) and extend the integral over x toR +∞−∞in order to get afull contribution of the point α(st)ito the partition function (5.29).If we compose the set S of all arrays (ν1, .

. .

νn−1; l) for which2Kl −nXi=1νiα(st)i≤0,(5.36)then the sum (5.35) will contain inﬁnitely many terms. However most of these terms willcancel each other.

Therefore we propose another procedure that will express N(tot)nas a ﬁnitesum. Suppose for simplicity that α(st)n̸= 0.

Consider a line in the α-spaceαi(t) = tα(st)i, i = 1, . .

. , n −1; αn(t) = α(st)n .

(5.37)Obviously, αi(1) = α(st)i. Remember now that N(tot)nis a Verlinde number.

Therefore N(tot)n=0 for αi(0). This means that all the terms in eq.

(5.35) cancel each other and we may dropthem altogether. As t starts to grow, suppose that for some value t∗2Kl −nXi=1νiαi(t∗) = 0.

(5.38)If for t > t∗the l.h.s. of eq.

(5.38) is negative, then by passing t = t∗we gained a contributionof the array (ν1, . .

. , νn−1; l) to eq.

(5.35) and the corresponding term should be added there.If, however, for t > t∗the l.h.s. of eq.

(5.38) is positive, then we lost the contribution of thearray (ν1, . .

. , νn−1; l), and the corresponding term should be subtracted.

As a result,N(tot)n(x) = −XS′sign n−1Xi=1νiαi!Nn x +nXi=1νiα(st)i−2Kl!,(5.39)39

here a set S′ consists of all arrays (ν1, . .

. , νn−1; l) such that for some t∗∈[0, 1] eq.

(5.38) issatisﬁedSince α(st)iare proportional to K, the function N(tot)n(x) is a polynomial in K and x:N(tot)n(x) =n−3Xj=0CjKn−3−jxj. (5.40)A Verlinde number Nα(st)1...α(st)n= N(tot)n(0) = C0 is a number of the WZW conformal blocks ofn primary ﬁelds Vα(st)ion a sphere9.

This number to the leading power in K is proportionalto the volume of the moduli space of ﬂat connections on a sphere with n punctures, theholonomies around which are ﬁxed by eq. (4.3).

This moduli space coincides with a connectedcomponent of the moduli space of ﬂat connections on the Seifert manifold, for which themap (2.2) is determined by eq. (4.3).

Therfore the coeﬃcient C0 is equal to the volume ofthat component of the moduli space (calculated with the proper measure).Let us substitute eq. (5.40) into the integral (5.32) modiﬁed in order to get the contribu-tion of the stationary phase point α(st)i:I =n−3Xj=0CjKn−3−jZ +∞−∞dx x +nXi=1νiµipi!jexp iπ2KPH x2.

(5.41)The dominant contribution comes from the term with j = 0. It is proportional to Kn−3:I ≈ei π4 sign( HP )vuut2K|H||P|C0Kn−3,(5.42)so that the whole 1-loop partition function contribution coming form the point (3.15) is (forp1, p2, p3, H > 0)Z≈12ei 3π4 exp −iπ2K"3 +nXi=1 12s(qi, pi) −qipi!#Kn−3C0×Xλ=0, 12e2πiλnnYi=1exp 2πiK ripi˜n2i −siqiλ2!

2i√pisin 2π ripi˜ni + siλ!. (5.43)Comparing this expression with eq.

(2.14) we note that dim H1 = 2(n −3), hence thefactor Kn−3. Also an integral over the moduli space (which is included in the sum over ﬂatconnections in eq.

(2.14)) produces its volume C0.9The numbers α(st)iare not necessarily integer, so in fact, we should take the closest integer numbers.This does not change a conclusion that C0 is the volume of the moduli space40

In contrast to the results of subsection 4.1, the contribution of the irreducible ﬂat con-nection on an n-ﬁbered Seifert manifold (n ≥4) contains higher loop corrections comingfrom the sum in eq. (5.41).

However the number of these corrections is ﬁnite. The highestorder correction is of the order of K0, so the number of loop corrections is equal to half thedimension of the moduli space.5.3A Contribution of Reducible Flat ConnectionsA contribution of a conditional stationary phase point (3.17) is easier to calculate, thanthat of an unconditional one α(st), because the former involves an integral only over onehalf-space (5.23), to which boundary it belongs.

We will use an expression for Z2 which isslightly diﬀerent from that of eq. (5.28) and is a generalization of eq.

(3.32). We express themultiplicity factor Nn(x) of eq.

(5.24) as a derivative:Nn(x) = −Qni=1 νi(n −3)! ∂(n−3)aa12(x+n−4)a=1 .

(5.44)We also shift the integration range of x from x ≥0 to x ≥4 −n. The contribution of theextra values of x is killed by the zeros of Nn(x) (recall that x is actually an even integer).After a shift in the integration variable x →x + n −4 we get the following expression for Z2(the other factor Z1 is deﬁned by eq.

(5.30)):Z2=−" nYi=1e−i π4 sign(piqi) exp −iπ2Kripi!# Xλ=0, 12e2πiλn×Xm∈ZZ ∞0dx"1(n −3)!∂(n−3)aax2a=1# Z +∞−∞dc e−2πic(x+4−n+2Kl)nYi=1Xµi=±1µiZ +∞−∞dαiq2K|qi|× expiπ2Kqihpiα2i −2νiαi(2Kni + 2Kqi(λ + m −c) + µi) + si(2K(ni + qim) + µi)2i=−e−i π4 sign( HP )q2K|H|exp 2πiK" nXi=1ripin2i + HP c20#×Xµ1,...,µn=±1" nYi=1µi exp 2πiµirini + c0pi!# 12Xλ=0, 12Xm∈ZI(m, λ),(5.45)here c0 is deﬁned by eq. (3.18), whileI(m, λ)=e2πic0(n−4)(n −3)!Z ∞0dxh∂(n−3)aax2a=1i41

× exp−2πix(c0 + m + λ) + iπ2KPH x −n + 4 −nXi=1µipi!2=e2πic0(n−4)(n −3)!∞Xj=01j! (8πiK)−j PHj∂(2j)ǫ∂(n−3)a(e2πiǫPni=1µipi +n−4×Z ∞0dx ax2 exp[−2πix(c0 + m + λ + ǫ)]a=1ǫ=0 .

(5.46)General Reducible ConnectionA sum over m and λ converts an integral over x in eq. (5.46) into a sum over even x:12Xλ=0, 12Xm∈ZI(m)=e2πic0(n−4)(n −3)!∞Xj=01j!

(8πiK)−j PHj×∂(2j)ǫ∂(n−3)ae2πiǫPni=1µipi +n−4Xx≥0x−evenax2 e−2πix(c0+ǫ)a=1ǫ=0=∞Xj=01j! (8πiK)−j PHj∂(2j)ǫe2πiǫ Pni=1µipi[2i sin 2π(c0 + ǫ)]n−2ǫ=0.

(5.47)Therefore the total contribution of a general reducible connection isZ=−ei π2 sign( HP )q2K|H|sign(P) exp −iπ2K"3signHP+nXi=1 12s(qi, pi) −qipi!#× exp 2πiK nXi=1ripin2i + HP c20! ∞Xj=01j!

(8πiK)−j PHj×∂(2j)cQni=1 2i sin2π rini+cpi[2i sin(2πc)]n−2c=c0. (5.48)Special Reducible ConnectionIf c0 + m0 + λ0 = 0 for some values λ0 = 0, 12,m0 ∈Z, then a calculation of I(m0, λ0)has to be performed separately:I(m0, λ0)=e2πic0n(n −3)!Z ∞0dxh∂(n−3)aax2a=1iexpiπ2KPH x −n + 4 −nXi=1µipi!2=e2πic0n(n −3)!

Z ∞0−Z n−4+Pni=1µipi0!dx"∂(n−3)aa12x+n−4+Pni=1µipia=1#exp iπ2KPH x2=e2πic0n(n −3)! (Z ∞0dx"∂(n−3)aa12x+n−4+Pni=1µipia=1#exp iπ2KPH x2(5.49)42

−∞Xj=01j! (8πiK)−j PHj∂(2j)ǫ∂(n−3)ae2πiǫn−4+Pni=1µipi−a12n−4+Pni=1µipi12 log a −2πiǫ ǫ=0a=1.A remaining part of the sum (5.47) is12Xλ,mλ+m+c0̸=0I(m, λ)=e2πinc0∞Xj=01j!

(8πiK)−j PHj∂(2j)ǫe2πiǫ Pni=1µipi(5.50)×"1(2i sin 2πǫ)n−2 + e2πiǫ(n−4)(n −3)! ∂(n−3)a112 log a −2πiǫa=1#)ǫ=0,so that the whole expression for the contribution of a special reducible connection isZ=−ei π2 sign( HP )q2K|H|e2πinc0sign(P) exp −iπ2K"3signHP+nXi=1 12s(qi, pi) −qipi!#× exp 2πiK nXi=1ripin2i + HP c20!

12Xµ1,...,µn=±1" nYi=1µi exp 2πiµirini + c0pi!#×1(n −3)!Z ∞0dx"∂(n−3)aa12x+n−4+Pni=1µipia=1#exp iπ2KPH x2+∞Xj=0(8πiK)−j PHj(5.51)×∂(2j)ǫQni=1 2i sin2π rini+c0+ǫpi[2i sin(2πǫ)]n−2+1(n −3)!∂(n−3)aan−42Qni=1 sin2πrini+c0+ i2 log api12 log a −2πiǫa=1ǫ=0.It is not hard to see that the term1(n −3)!∂(n−3)aan−42Qni=1 sin2πrini+c0+ i2 log api12 log a −2πiǫa=1(5.52)contains only the negative powers of ǫ in its Laurent series expansion. Therefore the onlypurpose of this term is to cancel the negative powers of ǫ in the expansion of the termQni=1 2i sin2π rini+c0+ǫpi[2i sin(2πǫ)]n−2(5.53)so that the whole expression has a smooth limit of ǫ →0Trivial ConnectionWhen all ni = 0, eq.

(5.52) can be simpliﬁed. In particular, the integral over x and theterm (5.52) are both equal to zero after taking a sum over µi.

After adding a factor (3.22),43

a total contribution of the trivial connection isZ=−ei π2 sign( HP )2q2K|H|e2πinc0sign(P) exp −iπ2K"3signHP+nXi=1 12s(qi, pi) −qipi!#×∞Xj=01j! π2iKPHj∂(2j)ǫQni=1 2i sinǫpi(2i sin ǫ)n−2ǫ=0.

(5.54)As we have noted in the end of subsection 4.3, a ratio of the j = 2 and j = 1 termscontributes together with the phase φ of eq. (1.5) to the 2-loop correction S2 as deﬁned byeq.

(2.4). A simple calculation similar to that of eq.

(4.26) shows again that S2 is proportionalto Casson’s invariant (4.14):S2 = 6πλCW. (5.55)6DiscussionIn this paper we calculated a full asymptotic large k expansion of the exact surgery formulafor Witten’s invariant of Seifert manifolds.

We found a complete agreement between ourresults and the 1-loop quantum ﬁeld theory predictions thus extending the results of the pa-pers [8],[9] on this subject. To achieve this agreement we had to modify slightly the previous1-loop formulas for the case of reducible ﬂat connections and for the case of obstructions inextending the elements of H1 to the moduli of ﬂat connections.It seems that the method of Poisson resummation used in our calculations can be appliedto Witten’s invariants of graph manifolds, i.e.

manifolds constructed by “plumbing” theSeifert manifolds (the solid tori parallel to the ﬁbers are cut out of Seifert manifolds and thecorresponding 2-dimensional boundaries are glued together after the modular transforma-tions are performed). This method can also be applied to the invariants built upon simpleLie groups other than SU(2).

We showed that the applicability of the Poisson resummationis based on Bernstein-Gelfand-Gelfand resolution.A rather surprising result of our calculations is the ﬁnite loop exactness of the contribu-tions of irreducible ﬂat connections. This exactness is somewhat reminiscent of the formulas44

of paper [11] in which Witten applied a localization principle to the 2-dimensional gaugetheory. The order of the highest loop corrections is equal to half the dimension of the mod-uli space of ﬂat connections.

This may suggest that these corrections are related to someintersection numbers in the moduli space.The contributions of all ﬂat connections have a speciﬁc 2-loop phase correction φ (seeeq. (1.5)).

This phase is the same for all ﬂat connections of a given manifold. In case of a3-ﬁbered Seifert manifold, φ is the only 2-loop correction for the contribution of irreducibleﬂat connections.

The phase φ looks similar to Casson’s invariant, however certain terms aremissing there. It seems, however, that φ is a manifold invariant in its own right.

It wouldbe interesting to understand its topological nature.The “missing terms” appear when the full 2-loop correction to the contribution of thetrivial connection is calculated.The trivial connection is reducible and its contributioncontains an asymptotic series in K−1. The whole 2-loop correction is a combination of φand the second term in this series, and it turns to be proportional to Casson’s invariant.

Wechecked this observation for n-ﬁbered Seifert manifold and found a full agreement with theformula (4.14).It is worth noting that the change in Casson’s invariant under a surgery on a knotdepends on the second derivative of the Alexander polynomial of that knot (see e.g. [18]and references therein).

This derivative is a second order Vassiliev invariant of the knot andcomes as a 2-loop correction to any (i.e., say, either Alexander or Jones) knot polynomial ifthe latter is calculated through Feynman diagrams. Thus a change in the 2-loop correctionunder a surgery on a knot depends on a 2-loop invariant of that knot.

We could go a stepbackwards and observe a similar relationship between the self-linking number of a framedknot and the order of homology group (or its logarithm) of the manifold10 (see e.g. [22]).Both objects can be interpreted as 1-loop corrections.

It would be interesting to derive aformula (if it exists) expressing the change in the n-loop correction to the contribution ofthe trivial connection under a surgery on a knot through Vassiliev invariants of this knot up10 I am thankful to N. Reshetikhin for discussing the results of his research on this subject with me.45

to order n.Let us try to conjecture the formula relating Vassiliev invariants of a knot to loop cor-rections of a trivial connection, basing on our formula (1.8). The higher loop corrections tothe contributions of the reducible connections (see eqs.

(1.6) and (5.48)) are proportional tothe “derivatives” of the U(1) Reidemeister torsion. This looks rather strange in view of thefact that the ﬂat U(1) connections on Seifert manifolds do not have any moduli along whichthey could be changed.

We therefore propose a diﬀerent interpretation of these formulas.Note that an equation∞Xj=01j! (8πiK)−j PHj∂(2j)cf(c)c=c0 = ei π4 sign( HP )s2K HPZ +∞−∞dc f(c) exp−2πiK HP (c −c0)2(6.1)can transform the derivatives in eqs.

(1.6) and (5.46) into an integral over c. Eq. (6.1) canbe derived either by expanding f(c) in Taylor series at c = c0 and checking it for every termseparately, or by noting that the l.h.s.

of eq. (6.1) is an exponential of the 1-dimensionalLaplacian, which can be expressed through the heat kernel.

In particular, a contribution ofthe trivial connection (1.8) can be cast in a form−ei 34 πsign( HP )2q|P|sign(P)e−iπ2K φZ +∞−∞dβ e−iπ2KHP β2Q3i=1 2i sin πβpi2i sin πβK. (6.2)Here we made a substitution β = 2Kc/π.The formula (6.2) can be given a following interpretation.

A Seifert manifold Xp1q1 , p2q2 , p3q3can be constructed by a surgery on a link consisting of 3 “ﬁber” loops linked to a “base”loop (see, e.g. [8],[20]).

A surgery on the ﬁber loops produces a connected sum of three lensspaces. We apply a Reshetikhin-Turaev formula related to the ﬁnal surgery on the base loopin order to ﬁnd Witten’s invariant of X p1q1 , p2q2 , p3q3.

The Jones polynomial of the base loopcan be expressed as a sum over ﬂat connections in the connected sum of lens spaces. A con-tribution of the trivial connection turns out to be proportional to the integrand of (6.2) upto a factor sin(πβ/K).

Therefore expression (6.2) is actually a Reshetikhin-Turaev formulain which only a trivial connection part of the Jones polynomial is taken and a sum over anintegrable weightPK−1β=1 is substituted by an integral 12R +∞−∞dβ. We conjecture that these46

two changes in the Reshetikhin-Turaev formula produce a trivial connection contribution toWitten’s invariant of the manifold constructed by a simple T pS surgery on a knot belongingto some other manifold (in case of a general surgery (2.16), only the n = 0 term should beretained in the sum of eq. (2.25)).Consider now a logarithm of the trivial connection part of Jones polynomial of a knot.According to [6], the coeﬃcients in its expansion in powers of 1/K are Vassiliev’s invariantsof the knot.

A 1-loop piece in this expansion, which is proportional to πβ2/K, comes fromthe self-linking number of the knot. This number can be fractional if the original manifoldis nontrivial (for example, it is equal to H/P in (6.2)).

We conjecture that in the otherterms appearing in the logarithm, the power of β is less or equal to the negative power of K.Therefore, if we split oﬀthe self-linking exponential factor and expand the remaining part ofthe Jones polynomial in 1/K, then the gaussian integral in the modiﬁed Reshetikhin-Turaevformula will produce the 1/K expansion of the trivial connection contribution to Witten’sinvariant of the new manifold. Each term in this expansion will be expressed through a ﬁnitenumber of Vassiliev invariants of the knot.

We hope to present this calculation in more detailsin a forthcoming paper. Here we just want to mention that its results seem to agree [19]with Walker’s formula [18] for Casson invariant (if we assume that Casson invariant is indeedproportional to a 2-loop correction).

This is partly due to the fact that the second derivativeof Alexander polynomial is also a 2-loop correction to Jones polynomial, as established byD. Bar-Natan in his paper [6].AcknowledgementsI want to thank Profs.

D. Auckly, R. Gompf, C. Gordon, N. Reshetikhin and H. Saleur formany valuable discussions. I am especially indebted to Profs.

D. Freed and A. Vaintrob fortheir help, consultations and encouragement.47

AppendixWe present a simple ﬁnite dimensional example which illustrates the appearance of the factor1/Vol(H) in the gauge invariant theory. Consider a 2-dimensional integralIgauge =Z +∞−∞dX1Z +∞−∞dX2 exp2πik f(qX21 + X22)(6.3)for some function f(r).

The integrand of this integral is obviously invariant under the U(1)rotation around the origin. Let us treat it as a gauge symmetry.

Then a “physical” quantitywould be an integral Igauge divided by the volume of the gauge group Vol(U(1)) = 2π. Afull machinery of Faddeev-Popov gauge ﬁxing will lead to a well known expression for the“physical” integral:Iphys =IgaugeVol(U(1)) =Z ∞0dr r exp(2πikf(r)).

(6.4)In order to parallel the discussion of subsection 2.1 we use a stationary phase approximation.Suppose that the function f(r) has a critical point at r = r0, i.e. f ′(r0) = 0.

We want totake an integral (6.3) over the vicinity of that point. We take a representative point on thegauge orbit:X1 = r0, X2 = 0.

(6.5)This is our “backgroud gauge ﬁeld”. A simplest choice for the gauge ﬁxing condition toimpose on the ﬂuctuations (x1, x2) around the background (6.5), would be x2 = 0.

Howeverwe make a diﬀerent choice:kxivi(X0, X1) = 0,(6.6)here vi is a vector ﬁeld representing the inﬁnitesimal gauge transformation:vi(X1, X2) = ǫijXj. (6.7)The gauge ﬁxing (6.6) closely resembles a background covariant gauge ﬁxing Dµaµ of theChern-Simons theory used in [1].

Indeed, for an inﬁnitesimal gauge transformation φ, theanalog of vi is Dµφ and eq. (6.6) is similar to a conditionZ(Dµφ) aµ d3x = 0(6.8)48

for any function φ, which is equivalent to Dµaµ = 0.By substituting eqs. (6.7) and (6.5) into eq.

(6.6) we get an explicit form of the covariantgauge ﬁxing condition:kx2r0 = 0. (6.9)A Faddeev-Popov ghost determinant is a variation of the gauge ﬁxing condition with respectto the gauge transformation:∆gh = kr20.

(6.10)We supplement a quadratic term iπkf ′′(r0)x21 in the exponent of eq. (6.3) with a gauge ﬁxingterm 2πikr0yx2.

An integral over y produces a δ-function of the condition (6.9). Thereforean operator corresponding to a quadratic form in the exponent of eq.

(6.3) isL−= ikf ′′(r0)0000r00r00. (6.11)The 1-loop “ﬁeld-theoretic” prediction of the physical integral Iphys isIphys =det ∆ghqdet(−L−)e2πikf(r0) = ei pi4r0qkf ′′(r0)e2πikf(r0)(6.12)in full agreement with the stationary phase approximation of the integral in the r.h.s ofeq.

(6.4).Let us see now what happens in the special case when r0 = 0. A background point (6.5)lies at the origin and is invariant under the action of U(1).

In other words, a “backgroundﬁeld” has a U(1) symmetry, so it is similar to a reducible gauge connection of the Chern-Simons theory. A ghost determinant (6.10) has a zero mode.

The operator L−is diﬀerentfrom its ordinary form (6.11):L−= ikf ′′(0)000f ′′(0)0000(6.13)49

and it also has one zero mode. The prescription for the Reidemeister torsion (2.5) in a similarsituation was to drop the zero modes.

Since no modes are left for the ghosts, we have∆gh = 1,(6.14)whiledet(−L−) = −[kf ′′(0)](6.15)A nondegenerate part of the operator (6.13) is the same as if we were calculating the station-ary phase approximation of the integral (6.3) at the origin without remembering the U(1)symmetry. The same is suggested by the ghost determinant (6.14).

In other words, we seethat by dropping the zero modes of ghost determinant and covariant gauge ﬁxing we “forgot”about the U(1) symmetry. Therefore if we substitute expressions (6.13) and (6.14) in themiddle part of eq.

(6.12), then we will get the whole integral Igauge rather than its physi-cal “gauge ﬁxed” counterpart Iphys. So in order to get Iphys from the determinants (6.13)and (6.14) we have to add a factor 1/Vol(U(1)) “by hands” as we did in eq.

(2.9).References[1] E. Witten, Commun.Math.Phys. 121 (1989) 351.

[2] N. Reshetikhin, V. Turaev, Invent.Math. 103 (1991) 547.

[3] S. Axelrod, I. Singer, Chern-Simons Perturbation Theory, Proceedings of XXth Con-ference on Diﬀerential Geometric Methods in Physics, Baruch College, C.U.N.Y., NY,NY.

[4] L. Alvarez-Gaume, J. Labastida, A. Ramallo, Nucl.Phys. B334 (1990) 103.

[5] E. Guadagnini, M. Martellini, M. Mintchev, Phys.Lett. 277B (1989) 111.

[6] D. Bar-Natan, Perturbative Chern-Simons Theory, Princeton Preprint, August 23, 1990. [7] M. Kontsevich, Vassiliev’s Knot Invariants, Advances in Soviet Mathematics.50

[8] D. Freed, R. Gompf, Commun.Math.Phys. 141 (1991) 79.

[9] L. Jeﬀrey, Commun.Math.Phys. 147 (1992) 563.

[10] S. Garoufalidis, Relations among 3-Manifold Invariants, University of Chicago preprint,1991. [11] E. Witten, Two Dimensional Gauge Theories Revisited, preprint IASSNS-HEP-92/15.

[12] L. Rozansky, H. Saleur, Nucl.Phys. B389 (1993) 365.

[13] L. Rozansky, H. Saleur, Reidemeister torsion, the Alexander Polynomial and the U(1, 1)Chern-Simons Theory, preprint YCTP-P35-1992, hep-th/9209073. [14] S. Elitzur, G. Moore, A. Schwimmer, N. Seiberg, Nucl.Phys.

B326 (1989) 108. [15] R. Fintushel, R. Stern, Proc.Lond.Math.Soc.

61 (1990) 109. [16] D. Freed, J. reine angew.

Math. 429 (1992) 75.

[17] D. Auckly, Topological Methods to Compute Chern-Simons Invariants, to be publishedin Math. Proc.

Camb. Phil.

Soc. [18] K. Walker, An extension of Casson’s Invariant to Rational Homology Spheres, preprint.

[19] K. Walker, private communication. [20] C. Lescop, C.R.Acad.Sci.Paris 310 (1990) 727.

[21] P. Bouwknegt, J. McCarthy, K. Pilch, Progr.Theor.Phys.Suppl. 102 (1990) 67.

[22] J. Mattes, M. Polyak, N. Reshetikhin, On Invariants of 3-Manifolds Derived fromAbelian Groups, preprint.51

Figure CaptionsFig.1 A fundamental tetrahedronFig.2 A section of the fundamental tetrahedron and its Weyl reﬂections by a plane α3 =constFig.3 A section of the full array of terahedra by a plane α3 = constFig.4 A fundamental triangle for reducible connectionsFig.5 The zero modes of deformations of a degenerate triangle52

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A Large k Asymptotics of Witten’s Invariant of Seifert Manifolds

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