Dedicated to the memory of Professor S.-S. Chern

Transcription

1 ASIAN J MATH c 2006 International Press Vol 0, No, pp 08 4, March TOPOLOGICAL STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES JUN LI, KEFENG LIU, AND JIAN ZHOU Dedicated to the memory of Professor S-S Chern Abstract We propose to use the identification of topological string partition functions as equivariant indices on framed moduli spaces of instantons to study the Gopakumar-Vafa conjecture for some local Calabi-Yau geometries Key words equivariant index, toric Calabi-Yau, moduli space AMS subject classifications 4D20; 8T30 Introduction In this work we exploit the relationship with certain equivariant genera of instanton moduli spaces to study the string partition functions of some local Calabi-Yau geometries, in particular, the Gopakumar-Vafa conjecture for them [8] Gromov-Witten invariants are in general rational numbers However as conjectured by Gopakumar and Vafa [8] using M-theory, the generating series of Gromov- Witten invariants in all degrees and all genera has a particular form, determined by some integers See [8] or Section 2 for precise formulation There have been various proposals to the proof of this conjecture [4, 0, 20] In this work we propose a new and geometric approach towards this conjecture for some interesting cases The method relates the computations of Gromov-Witten invariants to equivariant index theory and 4 dimensional gauge theory Recently there have been some progresses on the calculations of Gromov-Witten invariants, both in the physical approaches [2,, ] and mathematical treatments [29, 8] For toric local Calabi-Yau geometries, one can now express the string partition functions as sums over partitions Furthermore, one can relate them to partition functions in gauge theory according to the idea of geometric engineering (see eg [22, 2, 3, 6, 30, 7, 9]) The latter are equivariant genera of framed moduli spaces and can be computed by localization formula [2, 9] (see also Section 4) The fixed points on moduli spaces are indexed by tuples of partitions hence one gets seemingly different sums over partitions However as shown in some of the works mentioned above, one can use combinatorics to identify different expressions To prove the GV conjecture, one needs to rewrite the sums over partitions as infinite products (see eg [9] or Section 2 for explanation) The purpose of this paper is to point out that in some cases if one pushes forward the calculations from the framed moduli spaces, which are the Gieseker partial compactification of the moduli space of genuine instantons, to the Uhlenbeck partial compactification, then one can achieve this Let us briefly explain some relevant terminologies Received May 8, 2005; accepted for publication November 7, 2005 Department of Mathematics, Stanford University, Stanford, CA 94305, USA (jli@mathstanford edu) Center of Mathematical Sciences, Zhejiang University, Hangzhou 30027, China; and Department of Mathematics, University of California at Los Angeles, Los Angeles 90095, USA (liu@ cmszjueducn; liu@mathuclaedu) Department of Mathematical Sciences, Tsinghua University, Beijing 00084, China (jzhou@ mathtsinghuaeducn) 8

2 82 J LI, K LIU AND J ZHOU For a compact complex d-manifold X, one is often interested in its Hirzebruch χ y genus It is defined by: χ y (X) = d ( y) p p=0 d ( ) q dim H q (X, Λ p T X) By Hirzebruch-Riemann-Roch theorem, it can be computed as follows: χ y (X) = q=0 d X j= x j ( ye xj ), ( e xj ) where x,, x d are the formal Chern roots of T X The χ y -genus reduces to other invariants for special values of y, eg, χ (X) is the Euler number, and χ 0 (X) = d ( ) q dim H q (X, O X ) q=0 is the geometric genus of X An important generalization of the χ y -genus is the elliptic genus χ(x, y, q), which can be defined as the generating series of dimensions of some cohomology groups of series of vector bundles It can be computed by χ(x, y, q) = y d/2 It is easy to see that X j= d x j n χ y (X) = y d/2 χ(x, y, 0) ( yq n e xj )( y q n e xj ) ( q n e xj )( q n e xj ) Denote by M(N, k) the framed moduli space of torsionfree sheaves of rank N and c 2 = k on P 2 The instanton partition functions in 4D, 5D and 6D correspond to the generating series of the χ 0, χ y, and the elliptic genera, respectively, of M(N, k) Note however M(N, k) is noncompact hence the cohomology groups might be infinite dimensional, so the definition of the genera need to be modified It turns out that the weight spaces of the cohomology groups with respect to some natural torus actions are finite-dimensional, hence it makes sense to replace the dimensions of the cohomology groups by their characters In this way, one gets the equivariant versions of the χ 0, χ y and elliptic genera of M(N, k) Furthermore, one can compute them by using localization formula Since the fixed points on M(N, k) are parameterized by tuples of partitions, localization methods applied in this gauge theory setting yield sums over tuples of partitions On the string theory side, one has different looking expressions as sums over partitions However combinatorics can be used to identity these expressions [2, 3, 6, 30, 7] Hence one can relate equivariant genera of framed moduli spaces in gauge theory to the generating series of the Gromov-Witten invariants of some local toric Calabi-Yau geometries This idea appeared in physics literature in the context of geometric engineering, for example Nekrasov s conjecture [22] Our detailed treatment includes the cases with matters, hence it makes the geometric engineering of SU(N) gauge theory with matters mathematically rigorous We notice that the identification of the Gromov-Witten partition functions as equivariant genera suggests an approach to prove the Gopakumar-Vafa conjecture for

3 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 83 the relevant Calabi-Yau geometries To prove this conjecture, one needs to rewrite the sums over partition as infinite products We propose a natural geometric setting to achieve this More precisely, by exploiting the natural morphism M(N, k) M(N, k) 0 from the framed moduli spaces M(N, k) to the Uhlenbeck compactifications M(N, k) 0, one can pushforward the calculations of equivariant indices on M(N, k) to M(N, k) 0 When N =, M(, k) is the Hilbert scheme (C [k] while M(, k) 0 is the symmetric product (C (k) By standard manipulation on the symmetric products (cf eg [27, 28]) one can get infinite product expressions and prove the Gopakumar- Vafa conjecture We will discuss in detail the case of χ 0, χ y, and elliptic genera which correspond to three different local Calabi-Yau geometries We propose to deal with the N 2 cases in a similar fashion We summarize the steps involved in our approach as follows Step Compute the Gromov-Witten invariants by diagrammatic methods as sum over partitions; Step 2 Use Schur function calculus to rewrite the result in Step as sum over partitions of certain ratios; Step 3 Identify the result in Step 2 as suitable equivariant genera on M(N, k); Step 4 Identify the result in Step 3 as suitable equivariant genera on M 0 (N, k); Step 5 Obtain Gopakumar-Vafa type infinite products by series manipulations The first three steps are mostly from existing works, and the last two steps are the new features of this work Note it is a common practice in number theory that if one wants to show certain number is integral, then one tries to identify it as the dimension of certain cohomology group Also it is well-known that by taking symmetric powers, one can identify a sum expression with a product expression Our proposal is compatible with such general principles The rest of the paper is arranged as follows In Section 2 we recall the Gopakumar- Vafa Conjecture and its infinite product formulation In Section 3 we compute the equivariant elliptic genera for symmetric products, which naturally yield some infinite product expressions We recall some results on framed moduli spaces and compute their equivariant ellitpic genera in Section 4 The relationship between the equivariant genera of Hilbert schemes and symmetric products is studied in Section 5, and applied in Section 6 to compute some Gopakumar-Vafa invariants for Calabi-Yau geometries corresponding to M(, k) We give two theorems about the index expressions of the Gromov-Witten invariants of certain toric Calabi-Yau manifolds, and propose a similar approach of infinite products for Calabi-Yau geometries corresponding to M(N, k) for N > by push-froward in equivariant K-theory The detail will be worked out later The major part of this paper was done during the authors visit of Center of Mathematical Science, Zhejiang University in 2004 The authors would like to thank CMS for its hospitality The first and the second author are supported by NSF and the thrid author by NSFC During the preparation of the paper, there have been interesting progresses in the subject See [23] and [7] for the proof of the GV conjecture for toric Calabi-Yau manifolds by using combinatorics and the theory of topological vertex 2 Preliminaries In this section we recall the Gopakumar-Vafa Conjecture and its formulation in terms of infinite product

4 84 J LI, K LIU AND J ZHOU 2 Gopakumar-Vafa Conjecture Denote by F X the generating series of Gromov-Witten invariants of a Calabi-Yau 3-fold X Intuitively, one counts the number of stable maps with connected domain curves to X in any given nonzero homology classes However, because the existence of automorphisms, one has to performed the weighted count by dividing by the order of automorphism groups Hence Gromov- Witten invariants are in general rational numbers Based on M-theory considerations, Gopakumar and Vafa [8] made a remarkable conjecture on the structure of F, in particular, its integral properties More precisely, there are integers n g Σ such that () F = Σ H 2(X) {0} g 0 k k ng Σ (2 sin kλ 2g 2 Q kσ For given Σ, there are only finitely many nonzero n g Σ Let q = e λ, and regard it as an element of SU(2) represented by the diagonal matrix ( ) q 0 0 q Denote by V n the (n + )-dimensional irreducible representation of SU(2) Then the character of V n is given by: We have χ Vn (q) = qn+ q (n+) q q = q n + q n q n (2 sin λ 2 = (q + q 2) = (χ V 2χ V0 ) Recall {V n : n 0} form an integral basis of the representation ring of SU(2) Since V V n = V n+ V n, {(V V 0 V 0 ) n : n 0} also form an integral basis Therefore, given integers n g Σ there are integers N g Σ such that (2) N g Σ (qg + q g q g ) g 0 n g Σ ( )g (q 2 q 2g = g 0 22 Infinite product formulation of the Gopakumar-Vafa Conjecture The generating series of disconnected Gromov-Witten invariants are given by the string partition function: See [9] for the following: Z = exp F Proposition 2 The Gopakumar-Vafa Conjecture can be reformulated as follows: (3) Z = j ( q 2k+m+ Q Σ ) (m+)n g Σ Σ H 2(X) j k= j m=0 where j = g 2, k = j, j +,, j, j

5 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 85 Proof By (2) one has Σ H 2(X) g 0 k = Σ H 2(X) g 0 k = k ng Σ (2 sin kλ 2g 2 Q kσ ( ) g n g (q k/2 q k/ 2g Σ Q kσ k (q k 2 q k 2 Σ H 2(X) g 0 k = Σ H 2(X) g 0 k = Σ H 2(X) g 0 k = Σ H 2(X) = k N g Σ k N g Σ N g Σ Σ H 2(X) g 0 a=0 = g a=0 qk(g 2a) (q k 2 q k 2 QkΣ g a=0 qk(g 2a+) ( q k ) 2 Q kσ g k N g Σ q k(g 2a+) Q kσ (m + )q km a=0 g k N g Σ q k(g 2a+) Q kσ (m + )q km g 0 k a=0 g k (qg 2a+m+ Q Σ ) k N g Σ Σ H 2(X) g 0 a=0 + ) (m k g (m + ) log( q g 2a+m+ Q Σ ) From (3) one sees that to prove the Gopakumar-Vafa conjecture one needs an infinite product expression for the string partition function On the other hand, as will be recalled below such partition functions are usually given by a sum expression in the diagrammatic method We will show by some examples how one can convert the sum expressions to the product expressions This is achieved by relating the string partition functions to equivariant genera, first of Hilbert schemes, then of symmetric products Our motivation is very simple As noted in [9], the expression on the right-hand side of (3) looks like counting the states in the Hilbert space of a second quantized theory, a situation similar to the calculations of orbifold elliptic genera of symmetric products [5] 3 Orbifold Equivariant Elliptic Genera of Symmetric Products In this section we show that infinite product expressions arise naturally when one considers genera of symmetric products This motivates our proposal of using symmetric products to prove the Gopakumar-Vafa Conjecture For references see eg [27, 28, 4] 3 Orbifold elliptic genera Let M be a compact complex manifold and let G act on M by biholomorphic maps Furthermore, let π : E M be a holomorphic vector bundle which admits a G-action compatible with the G-action on M For g G, the Lefschetz number is defined by: L(M, E)(g) = dim M i=0 ( ) i tr(g H i (M,O(E)))

6 86 J LI, K LIU AND J ZHOU It can be computed by the holomorphic Lefschetz formula [3]: L(M, E)(g) = M g ch g E ch g Λ (N M g /M ), where M g denotes the set of points fixed by g, N M g /M denotes the normal bundle of M g in M Here and in the following it is understood that one considers each connected component of M g separately There is a natural decomposition T M M g = j N g (λ j ), where each N g (λ j ) is a holomorphic subbundle on which g acts as e 2π λ j 0 λ j < In particular, where N g (0) = T M g Define the fermionic shift by [26]: (4) F (M g ) = j (rank N λj )λ j Following [4], define E(M; q, y) g = y d 2 +F (M g i ) [ ( j n Λ yq n (T M g ) Λ y qnt M g) n (S q n(t M g ) S q nt M g )] ) [ n (Λ yq n +λj N λj Λ y q N n λj λj λj 0 Recall for complex vector bundle E, n ( S q n +λ j N λ j S q n λ j N λj )] Λ t (E) = + te + t 2 Λ 2 E +, S t (E) = + te + t 2 S 2 E + Definition 3 The orbifold elliptic genus of M/G is defined by χ(m, G; q, y) = Z(g) [g] G h Z(g) L(M g, E(M; q, y) g )(h), where G denotes the set of conjugacy classes of G, and Z(g) denotes the centralizer of g G For any h Z(g), let M (g,h) be the set of points of M fixed by both g and h By Lefschetz formula, we have L(M g, E(M; q, y) g ch h (E(M; q, y) g ) )(h) = M ch g,h h (Λ (N )) td(t M g,h ) M g,h /M g

7 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES Equivariant orbifold elliptic genera Assume now a G-manifold M admits an S -action which commutes with the G-action The equivariant orbifold elliptic genus is defined by: χ(m, G; q, y)(s) = L(M g, E(M; q, y) g )(h, s), Z(g) [g] G h Z(g) where s S This defines a character for S Denote by M (g,h,s) the points on M which is fixed by g, h, and s For our applications, it suffices to assume that M g,h,s consists of an isolated point Then by Lefschetz formula, we have L(M g, E(M; q, y) g )(h, s) = ch h,s (E(M; q, y) g ) ch h,s (Λ (N M g,h,s /M g )) 33 Equivariant elliptic genera of symmetric products Suppose X is a nonsingular projective variety admitting an action by a torus group T The diagonal action by T and the natural action by permutation group S N on the N-fold cartesian product X N commute with each other Theorem 3 Suppose the equivariant elliptic genera of X can be written as χ(x; q, y)(t,, t r ) = c(m, l, k)q m y l t k tkr r, then one has (5),l,k Q N χ(x N, S N ; q, y)(t,, t r ) N 0 = exp = N,n where ω n = exp(2π /n) Note we have Q Nn N n χ(x; (ω i n nq /n ) N, y N )(t N,, t N r ) i=0 n>0,,l,k,,t r ( Q n q m y l t k tkr r ), c(nm,l,k,,kr) N 0 Q N χ y (X N, S N )(t,, t r ) = N 0 Hence by taking q = 0 in (5) one gets: Q N χ y (X N, S N )(t,, t r ) (6) N 0 = exp m = Taking y = 0 in (6) one gets: (7) Q m χ y m(x)(t m,, t m r ) m( y m Q m ) (yq) N χ(x N, S N ; 0, y)(t,, t r ) n>0,l,k,,k r ( (yq) n y l t k ) c(0,l,k,,kr) N 0 Q N χ 0 (X N, S N )(t,, t r ) = exp( m See [27, 28, 4] for the nonequivariant version of (5)-(7) Q m m χ 0(X)(t m,, t m r ))

8 88 J LI, K LIU AND J ZHOU 34 Equivariant elliptic genera of the symmetric products: The proof Assume T X X s = v N v, where s = e 2π t acts on N v by multiplication by e 2π vt Let g = mσ l, h = h l,j,m, and n = ml Here σ l denotes an l-cycle, mσ l stands for the composition of m disjoint l-cycles For examples, the permutation (2)(34)(56) can be written as 3σ 2 See eg [5] for notations and more information on symmetric groups Then we have (8) T X n (Xn ) g,h,s = l k=0 m r=0 vn k,r,v, where N k,r,v is a copy of N v on which g acts as multiplication by e 2π k/l, and h acts as multiplication by e 2π (r/m+jk/(lm)) Lemma 32 If χ(x; q, y)(s) = c(α, β, γ)q α y β s γ, then (9) l l j=0 L((X lm ) (mσl), E(X lm ; q, y) (mσl) )(h l,j,m, s) = α,β,γ c(lα, β, γ)q mα y mβ s mγ Proof Let d v = dim N v Denote the formal Chern roots of N v by 2π x vi, i =,, d v Then we have On the other hand, = χ(x; q, y)(s) = α,β,γ d 0 X s i= (2π x 0i ) c(α, β, γ)q α y β s γ v d v i= θ(x vi z + vt, τ) θ(x i + vt, τ) L((X lm ) (mσl), E(X lm ; q, y) (mσl) )(h l,j,m, s) d 0 = (2π x 0i ) d v l m y k/l θ(x vi z kτ/l + r/m + jk/(lm) + vt, τ) θ(x vi kτ/l + r/m + jk/(lm) + vt, τ) = = X s i= d 0 X s i= X s i= = m d0 = d 0 X s i= = α,β,γ v (2π x 0i ) v (2π x 0i ) X s i= d 0 v i= k=0 r=0 d v l i= k=0 d v i= d (2π mx 0i ) (2π x 0i ) v d v i= y km/l θ(mx vi mz mkτ/l + jk/l + mvt, mτ) θ(mx vi mkτ + jk/l + mvt, mτ) θ(mx vi mz + mvt, (mτ j)/l) θ(mx vi + mvt, (mτ j)/l) v d v i= c(α, β, γ)(q m/l e 2π j/l ) α y mβ s mγ θ(mx vi mz + mvt, (mτ j)/l) θ(mx vi + mvt, (mτ j)/l) θ(x vi mz + mvt, (mτ j)/l) θ(x vi + mvt, (mτ j)/l) = χ(x, e 2π (τ j)/l, y m )(t m, t m

9 Hence we have STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 89 l = l l L((X lm ) (mσl), E(X lm ; q, y) (mσl) )(h l,j,m, s) j=0 l c(α, β, γ)(q m/l e 2π j/l ) α y mβ s mγ j=0 α,β,γ = α,β,γ c(lα, β, γ)q mα y mβ s mγ Proof of Theorem 3 χ(x n, S n ; q, y)(s)p n n 0 = n 0 = l lnl =n l N l 0 N l! = l C (l) (s), p ln l N l!l N l (N l σ l )h l =h l (N l σ l ) ( ) p l Nl l (N l σ l )h l =h l (N l σ l ) L((X ln l ) (N lσ l ), E(X ln l ; q, y) (N lσ l ) )(h l, s) L((X ln l ) (N lσ l ), E(X ln l ; q, y) (N lσ l ) )(h l, s) where C (l) (s) = N l 0 N l! ( ) p l Nl l (N l σ l )h l =h l (N l σ l ) L((X ln l ) (N lσ l ), E(X ln l ; q, y) (N lσ l ) )(h l, s)

10 90 J LI, K LIU AND J ZHOU Now C (l) (s) = N l 0 N l! ( ) p l Nl l h l = N l,j,m ((σ j l,,,),σm) m,j mn l,j,m=n l L((X ln l ) (N lσ l ), E(X ln l ; q, y) (N lσ l ) )(h l, s) = p ln l N l 0 l m j=0 = m l h l = N l,j,m ((σ j l,,,),σm) m,j mn l,j,m=n l m l N l N l! l j=0 (lm)n l,j,m Nl,j,m! l m j=0 (lm)n l,j,m Nl,j,m! ( ) L((X lm ) (mσl), E(X lm ; q, y) (mσl) Nl,j,m )(h l,j,m, s) j=0 N l,j,m 0 (p lm ) N l,j,m (lm) N l,j,m Nl,j,m! ( L((X lm ) (mσ l), E(X lm ; q, y) (mσ l) )(h l,j,m, s) l = m j=0 ) Nl,j,m ( ) p lm exp lm L((Xlm ) (mσl), E(X lm ; q, y) (mσl) )(h l,j,m, s) l p lm = exp lm L((Xlm ) (mσl), E(X lm ; q, y) (mσl) )(h l,j,m, s) m j=0 = exp p lm l L((X lm ) (mσl), E(X lm ; q, y) (mσl) )(h l,j,m, s) m l m j=0 The proof is completed by using (9) Here we have used an index calculation to prove Theorem 3 One can also prove it using the graded symmetric products generalizing the approach in [27, 4] Such a proof is closer to a second quantized theory in flavor 4 Partition Functions on Framed Moduli Spaces of Instantons In this section we recall some properties of the framed moduli spaces of instantons Our reference for this section is [2] In particular, we compute various equivariant genera on these spaces The results are expressed as sums over tuples of partitions Combined with the combinatorial results in [2, 3, 6, 30, 7], these will enable us to relate topological string theory with gauge theory 4 The framed moduli spaces Let M(N, k) denote the framed moduli space of torsion free sheaves on P 2 with rank N and c 2 = k The framing means a trivialization of the sheaf restricted to the line at infinity In particular when N = we get the Hilbert scheme (C [k] See [2] for details As proved in [2], M(N, k) is a nonsingular variety of dimension 2Nk The maximal torus T of GL N (C) together with the torus action on P 2 induces an action on M(N, k) As shown in [2], the fixed points are isolated and parameterized by N-tuples of partitions µ = (µ,, µ N ) such that i µi = k The weight decompo-

11 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 9 sition of the tangent bundle of T M(N, k) at a fixed point µ is given by (0) N α,γ= e γ e α ( t ((µ γ ) t j i) t µα i j+ (i,j) µ α 2 + t (µ α ) t j i+ t (µγ i j) (i,j) µ γ, where t, t 2 C C, and e α T 42 A cohomological property of the framed moduli space The space M(N, k) has the following remarkable property Let E be an equivariant coherent sheaf on it Even though H i (M(N, k), E) might be infinite-dimensional, the weight spaces of the induced torus action on it are finite-dimensional, hence it makes sense to define the equivariant index In other words, if H i (M(N, k), E) = V ν is the weight decomposition of H i (M(N, k), E), then for all weight ν Hence one can define dim V ν < ch H i (M(N, k), E) = (dim V ν )e ν Furthermore, one can compute the equivariant index by localization (cf [2]): Lemma 4 Let E be an equivariant coherent sheaf on M(N, k) Then χ(m(n, k), E) = 2Nk i=0 ( ) i ch H i (M(N, k), E) = µ ch ( i µ E T µ M(N, k) ) This Lemma together with (0) gives us the following formula for the equivariant elliptic genera of the framed moduli spaces: () Q k χ(m(n, k), y, p)(e,, e N, t, t k 0 = µ,,n (y N Q) (i,j) µ α (i,j) µ γ N i= µi n α,γ ( yp n e α e γ t (µγ ) t j i t (µα i j+) ( p n e α e γ t (µγ ) t j i t (µα i j+) ( y p n eα ( p n e α ( yp n e α e γ t ((µα ) t j i+) t µγ i j ( p n e α e γ t ((µα ) t j i+) t µγ i j ( y p n e α ( p n e α e γ t ((µγ ) t j i) t µα i j+ e γ t ((µγ ) t j i) t µα i j+ e γ t (µα ) t j i+ t (µγ i j) e γ t (µα ) t j i+ t (µγ i j)

12 92 J LI, K LIU AND J ZHOU By taking p = 0, one can also get: (2) Q k χ y (M(N, k))(e,, e N, t, t k 0 Q N i= µi µ,,n α,γ (i,j) µ α = (i,j) µ γ If one further takes y = 0, (3) ( ye α e γ t (µα ) t j i+) t µγ i j ( e α e γ t (µα ) t j i+) t µγ i j Q k χ 0 (M(N, k))(e,, e N, t, t = ( ye α e γ t (µγ ) t j i t (µα i j+) ( e α e γ t (µγ ) t j i t (µα i j+) k 0 Q N i= µi µ,,n α,γ (i,j) µ α ( e α e γ t (µγ ) t j i t (µα i j+) (i,j) µ γ ( e α e γ t (µα ) t j i+) t µγ i j 43 A natural bundle on the framed moduli space Recall M(N, k) can be identified with the space of equivalent classes of tuples of linear maps satisfying (B : V V ; B 2 : V V ; i : W V ; j : V W ) [B, B 2 ] + ij = 0 and a stability condition Hence one gets a vector bundle V over M(N, k) whose fibers are given by V This bundle is an equivariant bundle, and its weight decomposition at a fixed point µ is given by [2]: α V = V α e α, α V α = t i+ t j+ 2 (i,j) µ α Therefore, the weight of detv at the fixed point µ is e µα α t i t j 2 (i,j) µ α Now we take EN,k m = K 2 N,k (det V ) m, where K denotes the canonical line bundle of M(N, k) The equivariant index χ(m(n, k), EN,k m ) can be identified with the equivariant indices of Dirac operators, twisted by (det V ) m An application of the above fixed point formula gives us

13 (4) STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 93 Lemma 42 We have Q k χ(m(n, k), EN,k)(e m,, e N, t, t k 0 N Q i= µi µ,,n = N N α= e µα α t i t j 2 (i,j) µ α α,γ= (i,j) µ α (e α e γ t ((µγ ) t j i) t µα i j+ (i,j) µ γ (e α e γ t (µα ) t j i+ t (µγ i j) 2 (e α m 2 (e α e γ t (µγ ) t j i) t µα i j+ 2 e γ t (µα ) t j i+ t (µγ i j) 2 Lemma 43 When t = e βh, t 2 = e βh, e α = e βaα we have the following identity: Q k χ(m(n, k), EN,k)(e m,, e N, t, t where a i,j = a i a j, k 0 = N N (Q/4) i= µi e mβ( µα a α+hκ µ α /2) µ,,n α= N sinh β 2 (a α,γ + h(µ α i µγ j + j i)) sinh β 2 (a, α,γ + h(j i)) α,γ= i,j= Proof By Lemma 42 we have Q k χ(m(n, k), EN,k)(e m,, e N, t, t k 0 N Q i= µi µ,,n = N α= e β µα a α (i,j) µ α e βh(j i) N α,γ= 2 sinh β (i,j) µ α 2 (a α a γ + h(µ α i + (µγ ) t j i j + )) 2 sinh β (i,j) µ γ 2 (a α a γ h((µ α ) t j + µγ i i j + )) = N N (Q/4) i= µi e mβ( µα a α+hκ µ α /2) µ,,n α= N sinh β 2 (a α,γ + h(µ α i µγ j + j i)) sinh β 2 (a, α,γ + h(j i)) α,γ= i,j= m

14 94 J LI, K LIU AND J ZHOU where in the last equality we have used Lemma 44 below and the identity: l(µ) κ µ = µ + (µ 2 i 2iµ i ) = 2 (j i) i= (i,j) µ Lemma 44 [6] We have the identity: N α,γ= sinh β (i,j) R α 2 (a α,γ + h(µ α i + µt,γ j i j + )) sinh β (i,j) R γ 2 (a α,γ h(µ γ i + µt,α j i j + )) N sinh β 2 = (a α,γ + h(µ α i µγ j + j i)) sinh β 2 (a α,γ + h(j i)) α,γ= i,j= 5 Relationship between Hilbert Schemes and Symmetric Products In this section we study the equivariant genera of M(, k) which are the Hilbert schemes (C [k] These are given by sums over partitions by localization calculations By identifying them with the corresponding orbifold equivariant genera of symmetric products of C 2, one then gets infinite product expressions 5 Equivariant χ 0 of Hilbert schemes as infinite products 5 Equivariant χ 0 of Hilbert schemes By localization on (C [n] (equation (3)) we have (5) = Q n χ 0 ((C [n] )(t, t n=0 n=0 Q n µ =n e µ ( t l(e) t a(e)+ ( t l(e)+ t a(e) This is a sum over partitions 52 Equivariant χ 0 of symmetric products Denote by χ 0 ((C (n) )(t, t the orbifold equivariant geometric genera of the symmetric product (C (n) = (C n /S n Then we have (6) χ 0 ((C (n) )(t, t = exp( n Q n n( t n )( tn ) We will give three proofs Proof by direct calculations Note { H i ((C (n) C[x, y,, x n, y n ] Sn = S n (C[x, y]), i = 0,, O) = 0, i > 0,

15 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 95 where S n acts on C[x, y,, x n, y n ] by permuting (x, y ),, (x n, y n ) Now the weight decomposition of C[x, y] is p,p 2 0 t p tp2 2, corresponding to the basis {x p y p2 } Then by standard calculations for symmetric powers we have (7) Q n χ 0 ((C (n) ) = n=0 p,p 2 0 t p tp2 2 Q = exp Q n n( t n )( tn n= Proof by orbifold equivariant localization Now consider the localization on the orbifold (C (n) with respect to the natural torus action There is only one fixed point n[0] in (C (n), whose normal bundle has the following weight decomposition: n(t + t Taking into account the effect of the S n -action, one sees the contribution of this fixed point is n! ( t ) n ( t n One also has to consider the twisted sectors, associated with the conjugacy classes of S n They can be described as follows [27, 28] For every partition µ = (µ,, µ l ) of n, let g µ the product of disjoint cycles of lengths µ,, µ l respectively: l j g µ = ( µ i +, j=0 i= j µ i + 2,, i= j µ i + µ j+ ) The centralizer of g µ is denoted by Z µ It is not hard to see that Z µ = n i= i= S mi(µ) Z mi(µ) i, (semidirect product) Hence there is a surjective homomorphism Z µ n i= S m i(µ), and the order of Z µ is z µ The twisted sector corresponding to µ is given by (C (n) µ = ((C n ) gµ /Z µ For example, when µ = (n), then g µ is the n-cylce (, 2,, n), ((C n ) gµ is the diagonal in (C n, Z µ is the cyclic group generated by the n-cycle, so the twisted sector in this case is just a copy of C 2 with a trivial Z n -action There is only one fixed point situated at the origin The weight decomposition of its normal bundle inside the whole space (not the twisted sector) is n (ωnt i + ωnt i i=0

16 96 J LI, K LIU AND J ZHOU Here we have used the weight decomposition with respect to the action by T n Z n Taking into the account of the trivial action of Z n, its contribution is n i=0 ( ωi nt )( ωnt i = n ( t n )( tn n In general, it is easy to see that ((C n ) gµ can be identified with (C l, where Z µ acts through the homomorphism Z µ n i= S m i(µ) There is only one fixed point in each twisted sector, given by the origin in (C l, its normal bundle inside (C n has weight decomposition µ l j (ωµ i j t + ωµ i j t j= i=0 Taking into account of the action of Z µ, one gets the contribution from the twisted sector: z n µj µ j= i=0 ( ωi µ j t )( ωµ i j t = z n µ i= ( ti )mi(µ) ( t i mi(µ) Therefore, the total contribution is n 0 Q n µ =n z µ n i= ( ti )mi(µ) ( t i mi(µ) = n ( Q i m n 0 µ =n i= i (µ)! i( t i )( ti = ( Q i ) mi m i! i( t i i m i 0 )( ti = ( Q i ) exp i( t i i )( ti = exp i Q i i( t i )( ti Proof by general results for symmetric products We have hence by (7), χ 0 (C (t, t = ( t )( t, Q N χ 0 ((C (n) )(t, t = exp N 0 = exp Q m m( t m m )( tm ) mi(µ) Q m m χ 0(C (t m, t m

17 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES Identification of the equivariant χ 0 of Hilbert schemes and symmetric products We now show that (8) χ 0 ((C [n] )(t, t = χ 0 ((C (n) )(t, t Our first proof follow [2] First notice higher cohomology groups H i ((C [n], O)(i > 0) vanish since (C (n) is a rational singularity Secondly, note H 0 ((C [n], O) = H 0 ((C (n), O), hence (8) follows For the second proof, note π is a small resolution, we have We then apply the results in [25] π O (C = O [n] (C (n) 54 The infinite product expression for equivariant χ 0 of Hilbert schemes Let t = q and t 2 = q Putting (8) and (7) together, χ 0 ((C [n] )(q, q ) = χ 0 ((C (n) )(q, q ) = exp Q m m( q m )( q m ) = exp (qq) n n( q n ) 2 m n By the following series expansion ( x) 2 = m mx m we have (qq) n n( q n ) 2 = (Qq) n n n n m = m (Qq m ) n = log q n m n m ( m Q) m mq n(m ) Hence (9) n=0 Q n χ 0 ((C [n] )(q, q ) = m ( q m Q) m 52 Equivariant χ y genera of Hilbert schemes as infinite products 52 Equivariant χ y genera of Hilbert schemes By the N = case of (2) we have (20) Q k χ y ((C [n] )(t, t n 0 = µ = µ Q µ (i,j) µ Q µ e µ ( yt (µt j i) t µi j+ ( yt µt j i+ t (µi j) ( t (µt j i) t µi j+ ( t µt j i+ t (µi j) ( yt l(e) t a(e)+ ( yt l(e)+ t a(e) ( t l(e) t a(e)+ ( t l(e)+ t a(e)

18 98 J LI, K LIU AND J ZHOU 522 Orbifold equivariant χ y genera of symmetric products By localization formula we have χ y (C = ( yt )( yt ( t )( t This can be directly verified as follows It is easy to see that So we have C[z, z 2 ], (p, q) = (0, 0), H q (C 2, Λ p T C 2 C[z, z 2 ]dz C[z, z 2 ]dz 2, (p, q) = (, 0), ) = C[z, z 2 ]dz dz 2, (p, q) = (2, 0), 0, otherwise χ y (C (t, t = (t m tm2 m,m yt m+ t m2 2 yt m tm y 2 t m+ = χ 0 (C ( yt yt 2 + y 2 t t = ( yt )( yt ( t )( t t m2+ Therefore by standard calculations on symmetric products [27, 28] we have (2) Indeed, n 0 Q n χ y ((C n, S n )(t, t = exp n Q n χ y ((C n, S n )(t, t n 0 = l = exp l m,m 2 0 Q n n χ y n(c (t n, t n y n Q n ( y l Q l t m+ t m2 ( yl Q l t m tm2+ ( Q l y l t m tm2 ( yl+ Q l t m+ t m2+ (log( y l Qt m+ t m2 + log( yl Qt m m,m 2 0 tm2+ log( Qy l t m tm2 log( yl+ t m+ t m2+ ) = exp n Qnl y n(l ) t nm t nm2 2 ( y n t n y n t n 2 + y 2n t n t n l n = exp Q n ( y n t n )( y n t n n( y n Q n )( t n n )( tn = exp n Q n n This matches with (6) χ y n(c (t n, t n y n Q n 523 Identification of the equivariant χ y of Hilbert schemes and symmetric products Since π : (C [n] (C (n) is a semismall resolution, we have (22) χ y ((C [n] )(t, t = χ y (C 2, S n )(t, t

19 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES Equivariant χ y genera of Hilbert schemes as infinite products Again let t = q, t 2 = q By combining (22) with (2) Q n χ y (C 2, S n )(q, q ) n 0 Q n χ y ((C [n] )(q, q ) = n 0 = exp n Q n ( y n q n )( y n q n ) n( y n Q n )( q n )( q n ) Now Hence (23) n = n Q n ( y n q n )( y n q n ) n( y n Q n )( q n )( q n ) Q n ( y n q n )(q n y n ) n( y n Q n )( q n ) 2 = n Qn ( y n q n )(q n y n ) y kn Q kn (m + )q nm n k 0 = (m + ) [(Q k+ y k q m+ ) n (Q k+ y k+ q m+ n n k 0 n (Q k+ y k+ q m ) n + (Q k+ y k+2 q m+ ) n ] = (m + )[log( Q k+ y k q m+ ) log( Q k+ y k+ q m+ k 0 log( Q k+ y k+ q m ) + log( Q k+ y k+2 q m+ )] Q n χ y ((C [n] (q, q ) n = k, ( ( Q k+ y k q m+ )( Q k+ y k+2 q m+ ) m+ ) ( Q k+ y k+ q m )( Q k+ y k+ q m+2 ) 53 Equivariant elliptic genera of Hilbert schemes as infinite product 53 Equivariant elliptic genera of Hilbert schemes The N = case of () gives us Q k χ((c [k] ; y, p)(t, t k 0 = µ (y Q) µ n (24) (i,j) µ (i,j) µ ( yp n t µt j i t (µi j+) ( y p n t (µt j i) t µi j+ ( p n t µt j i t (µi j+) ( p n t (µt j i) t µi j+ ( yp n t (µt j i+) t µi j ( y p n t µt j i+ t (µi j) ( p n t (µt j i+) t µi j ( p n t µt j i+ t (µi j)

20 00 J LI, K LIU AND J ZHOU 532 Equivariant elliptic genera of symmetric products By localization formula we have (25) χ(c 2, y, p)(t, t =y n ( yp n t )( y p n t ) ( p n t )( p n t ( ypn t ( y pn t ) ( p n t ( p n t By (5) we get Q N χ((c N, S N ; p, y)(t, t N 0 = exp N,n Q Nn N n χ(c ((ω i n np /n ) N, y N )(t N, t N i=0 533 Identification of the equivariant χ y of Hilbert schemes and symmetric products Motivated by [5] we make the following conjecture: (26) Q k χ((c [k] ; p, y)(t, t = Q k χ((c k, S k ; p, y)(t, t k=0 k=0 We will refer to it as the Equivariant DMVV Conjecture 534 Equivariant elliptic genera of Hilbert schemes as infinite product Now let t = q and t 2 = q Assuming the equivariant DMVV conjecture, one then has: log Q k χ((c [k] ; p, y)(q, q ) k=0 = log n 0 Q n χ((c n, S n ; p, y)(q, q ) = l = l m m Q lm m Q lm m l l l χ(c 2, e 2π (mτ j)/l, y m )(q m, q m ) j=0 l y m j=0 ( y m e (n )2π (mτ j)/l q m )( y m e n2π (mτ j)/l q m ) ( e (n )2π (mτ j)/l q m )( e n2π (mτ j)/l q m ) n ( ym e (n )2π (mτ j)/l q m )( y m e n2π (mτ j)/l q m ) ( e (n )2π (mτ j)/l q m )( e n2π (mτ j)/l q m ) = Q lm m y m ym q m q m ym q m q m l m l l ( y m e n2π (mτ j)/l q m )( y m e n2π (mτ j)/l q m ) j=0 n ( e n2π (mτ j)/l q m )( e n2π (mτ j)/l q m ) ( ym e n2π (mτ j)/l q m )( y m e n2π (mτ j)/l q m ) ( e n2π (mτ j)/l q m )( e n2π (mτ j)/l q m )

21 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 0 Let ( yp n q)( y p n q ) ( p n q)( p n q ) n = C(a, b, c)p a q b y c a 0,b,c Z ( ypn q )( y p n q) ( p n q )( p n q) where C(a, b, c) are some integers Note the left-hand side is invariant under the changes of variables q q, or y y, ie, (27) C(a, b, c) = C(a, b, c) = C(a, b, c) Furthermore, for fixed a and c there are only finitely values of b such that C(a, b, c) 0 Hence there are integers C(a, b, c) such that C(a, b, c)p a q b y c = C(a, j, c)p a (q 2j + q 2(j ) + + q 2j )y c a 0,b,c Z a,j 0,c Z We have l = l l ( y m e n2π (mτ j)/l q m )( y m e n2π (mτ j)/l q m ) j=0 n ( e n2π (mτ j)/l q m )( e n2π (mτ j)/l q m ) ( ym e n2π (mτ j)/l q m )( y m e n2π (mτ j)/l q m ) ( e n2π (mτ j)/l q m )( e n2π (mτ j)/l q m ) l C(a, b, c)e 2π a(mτ j)/l q mb y mc j=0 a,b,c = C(la, b, c)p ma q mb y mc a,b,c = j C(la, j, c)p ma q 2km y mc a,j 0,c Z k= j

22 02 J LI, K LIU AND J ZHOU We have two ways to proceed from here log = l Q k χ((c [k] ; p, y)(q, q ) k=0 m Q lm m y m ym q m q m ym q m q m a,b,c l C(la, b, c)p ma q mb y mc = Q ln n y n q n ( y n q n y n q n + y 2n ) (m + )q mn n C(la, b, c)p na q nb y nc a,b,c = (m + ) C(la, b, c) l a,b,c Z n [(Ql p a q m+b+ y c ) n (Q l p a q m+b+2 y c ) n n (Q l p a q m+b y c ) n + (Q l p a q m+b+ y c+ ) n ] = (m + ) C(la, b, c) a,b,c l [log( Q l p a q m+b+ y c ) log( Q l p a q m+b+2 y c ) log( Q l p a q m+b y c ) + log( Q l p a q m+b+ y c+ )] therefore, (28) Q k χ((c [k] ; p, y)(q, q ) k=0 = a,b,c l ( ( Q l p a q m+b+ y c )( Q l p a q m+b+ y c+ ) (m+)c(la,b,c) ) ( Q l p a q m+b+2 y c )( Q l p a q m+b y c )

23 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 03 In the same fashion we have log = l Q k χ((c [k] ; p, y)(q, q ) k=0 n = n a,j 0,c l Q ln n y n yn q n q n yn q n q n a,j 0,c Z C(la, j, c)p na Q ln n y n q n ( y n q n y n q n + y 2n ) (m + )q mn C(la, j, c)p na = (m + ) l j k= j n a,j 0,c Z j k= j q 2kn y nc C(la, j, c) n [(Ql p a q m+2k+ y c ) n (Q l p a q m+2k+2 y c ) n (Q l p a q m+2k y c ) n + (Q l p a q m+2k+ y c+ ) n ] = (m + ) C(la, j, c) a,j 0,c l j [log( Q l p a q m+2k+ y c ) log( Q l p a q m+2k+2 y c ) k= j log( Q l p a q m+2k y c ) + log( Q l p a q m+2k+ y c+ )] j k= j q 2km y mc therefore, Q k χ((c [k] ; p, y)(q, q ) k=0 = a,j 0,c l j k= j ( Q l p a q m+2k+ y c )( Q l p a q m+2k+ y c+ ) ( Q l p a q m+2k+2 y c )( Q l p a q m+2k y c ) (m+) C(la,j,c) Now notice that = j ( Q l p a q m+2k+ y c )( Q l p a q m+2k+ y c+ ) ( Q l p a q m+2k+2 y c )( Q l p a q m+2k y c ) k= j j k= j ( Ql p a q m+2k+ y c ) j k= j ( Ql 2p a q m+2k+ y c+ ) j+ 2 k= (j+ )( Ql p a q m+2k+ y c ) j, 2 2 k= (j )( Ql p a q m+2k+ y c ) 2

24 04 J LI, K LIU AND J ZHOU hence we also have an infinite product of the type in (3): (29) Q k χ((c [k] ; p, y)(q, q ) k=0 = a,j 0,c l j k= j ( Ql p a q m+2k+ y c ) j+ 2 k= (j+ )( Ql p a q m+2k+2 y c ) 2 j k= j ( Ql p a q m+2k+ y c+ ) j 2 k= (j )( Ql p a q m+2k+ y c ) 2 (m+) C(la,j,c) 6 Rank Examples In this section we will study the local Calabi-Yau geometries corresponding to gauge theory moduli spaces M(, k), which are the Hilbert schemes (C [k] We will show that by using the results from the preceding section, one can extract the Gopakumar-Vafa invariants by identifying the string partition functions with the χ 0, χ y or elliptic genera, first of the Hilbert schemes (C [n] then of the symmetric products (C (n) 6 The 4D case In this subsection we consider equivariant χ 0 of M(, k) In this case the corresponding local Calabi-Yau geometry is the resolved conifold O( ) O( ) P The Gopakumar-Vafa invariants in this case is well-known We rederive it from our point of view for completeness In this case the web diagram for the local Calabi-Yau geometry is µ we have following formula for the closed string partition function [2,, 29]: (30) Z 0 (Q, q) = µ P ( Q) µ W µ (q)w µ t(q) = µ Q µ W µ (q)w µ (q ) We have Theorem 6 We have the identity (3) Q µ W µ (t )W µ (t = Q n χ 0 ((C [n] )(t, t where Proof Recall Hence by the standard identity: µ n=0 W µ (q) = s µ (q ρ ), p n (q ρ ) = q n, q ρ = (q 2, q 3 2, ) s µ (x)s µ (y) = exp µ n p n (x)p n (y), n

25 it follows that STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 05 Q µ W µ (t )W µ (t = exp( Q n n( t n µ n )( tn ) = The proof is completed by the results in 5 as m,m 2 0 ( Qt m tm2 Remark 6 By localization formula one can write the right-hand side of (3) n=0 Q n µ =n e µ ( t l(e) t a(e)+ ( t l(e)+ t a(e) Hence (3) is an identity that equates an infinite sum over partitions to an infinite product: n=0 Q n = exp( n µ =n e µ ( t l(e) t a(e)+ ( t l(e)+ t a(e) Q n n( t n )( tn ) = m,m 2 0 ( Qt m tm2 Now we take t = q and t 2 = q By (30), (3) and (9) we have Z 0 (Q, q) = n χ 0 ((C [n] )(q, q ) = m ( q m Q) m We have a direct proof as follows First of all (5) becomes = Q n χ 0 ((C [n] ) n=0 n=0 Q n µ =n e µ ( qh(e) )( q h(e) ) = ( Q) µ n=0 µ e µ = ( Q) µ s µ (q ρ )s µ t(q ρ ) n=0 µ q h(e)/2 q h(e)/2 q h(e)/2 e µ q h(e)/2 q h(e)/2 q h(e)/2 Then the first equality follows from the identity W µ = s µ (q ρ ), the second equality follows from the identity: s µ (x)s µ t(y) = ( + x i y j ), µ i,j where x = (x, x 2, ) and y = (y, y 2, ) By comparing with (3) we get N j d = δ d,δ j,0

26 06 J LI, K LIU AND J ZHOU 62 The 5D case This case has been discussed in Section 5 and Section 6 in [9] We now combine their result with the results in 52 to extract the Gopakumar- Vafa invariants The web diagram is The partition function given by topological vertex method is (32) Z(Q, Q m, q) = µ P ( Q) µ ν P( Q m ) ν C νt µ(0)(q)c νµt (0)(q) (33) The following result has been proved in [9, 5]: Ẑ = Z(Q, Q m, q) Z (0) (Q m, q) = µ Q µ (i,j) µ Now we take t = q and t 2 = q in (20) to get n 0 Q k χ y ((C [n] )(q, q ) = µ Q µ ( Q m q h(i,j) )( Q m q h(i,j) ) ( q h(i,j) )( q h(i,j) ) (i,j) µ ( yq h(i,j) )( yq h(i,j) ) ( q h(i,j) )( q h(i,j) ) Hence we get the following result proved in [9, 6]: After a variable change y = Q m, we have the identity Ẑ = Q k χ y (C [k] )(q, q ) k= By (23) one then gets: Ẑ = ( ( Q k+ Q k mq m+ )( Q k+ Q k+2 m q m+ ) m+ ) (34) ( Q k+ Q k+ m q m )( Q k+ Q k+, m q m+ hence (35) Z = k, k, Comparing with (3) one gets: ( ( Q k+ Q k mq m+ )( Q k Q k+ m q m+ ) m+ ) ( Q k+ Q k+ m q m )( Q k+ Q k+ m q m+ N 0 Q r Q s m N Q r Q s m N g Q r Q s m = δ r s,, = δ r,s, = 0, g > 63 The 6D case This case has been discussed in Section 5 and Section 62 in [9] We now combine their results with 53 to extract the Gopakumar-Vafa invariants in this case The web diagram is = =

27 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 07 The partition function by topological vertex method is (36) Z(Q, Q m, Q, q) = ( Q) µ ( Q ) ν ( Q m ) η C µνη (q)c µt ν t η t(q) µ,ν,η P The following expressions for the normalized topological partition function have been derived in [9]: Ẑ = Z(Q, Q m, Q, q Z(Q m, Q, q) = Q µ ( Q m q h(i,j) )( Q m q h(i,j) ) ( q h(i,j) )( q h(i,j) ) µ k= (i,j) µ ( Q k ρq m q h(i,j) )( Q k ρq m q h(i,j) )( Q k ρq m q h(i,j) )( Q k ρq m q h(i,j) ) ( Q k ρq h(i,j) ) 2 ( Q k ρq h(i,j) ) 2, where Q ρ = Q Q m The right hand side can be written in terms of the Jacobian theta-functions Now let t = q and t 2 = q in (24), then by (24) Q k χ((c [k] ; p, y)(q, q ) k= = µ n= Q µ (i,j) µ ( yq h(i,j) )( yq h(i,j) ) ( q h(i,j) )( q h(i,j) ) ( p n yq h(i,j) )( p n y q h(i,j) )( p n yq h(i,j) )( p n y q h(i,j) ) ( p n q h(i,j) ) 2 ( p n q h(i,j) ) 2 Thus one gets the following result in [9, 62]: After a variable change p = Q ρ and y = Q m one has: Ẑ = Q k χ((c [k] ; p, y)(q, q ) k= Hence if one assumes the Equivariant DMVV Conjecture, then Gopakumar-Vafa Conjecture for this case follows by (29) We give some examples below 64 Appendix: Examples of the 6D case We have C(a, b, c)p a q b y c a,b,c = + p(2(q + q ) + y(q + q ) + y (q + q )) +p 2 [6 4(y + y ) + (y 2 + y + (q + q ) + 4(q 2 + q (yq + yq + y q + y q ) 2(yq 2 + yq 2 + y q 2 + y q ] +, in particular, (37) C(0, b, c) = {, b = c = 0, 0, otherwise,

28 08 J LI, K LIU AND J ZHOU and (38), b = ±, c = ±, C(, b, c) = 2, b = ±, c = 0, 0, otherwise 64 The a = 0 case We have by (37) only one case with C(0, b, c) 0 which gives us: b,c l = l This matches with (34) ( ( Q l q b+m+ y c )( Q l q b+m+ y c+ ) (m+)c(0,b,c) ) ( Q l q b+m+2 y c )( Q l q b+m y c ) ( ( Q l q m+ y )( Q l q m+ y) ( Q l q m+( Q l q m ) ) (m+) 642 The a =, l = case We have by (38) six cases when C(, b, c) is nonzero We group them into three pairs The pair (b, c) = (±, 0) gives us: = ( ( Qpq m+2 y )( Qpq m+2 y + ) 2(m+) ) ( Qpq m+3 )( Qpq m+ ) ( ( Qpq m y )( Qpq m y + ) 2(m+) ) ( Qpq m+ )( Qpq m ) ( [( Qpq m+2 y )( Qpq m y )][( Qpq m+2 y)( Qpq m y)] [( Qpq m+3 )( Qpq m+ )( Qpq m )] ( Qpq m+ ) The pair (b, c) = (±, ) gives us: = ( ( Qpq m+( Qpq m+2 y (m+) ) ( Qpq m+3 y)( Qpq m+ y) ( ( Qpq m+ y)( Qpq m ) (m+) y) ( Qpq m )( Qpq m y ) 2(m+) ( [( Qpq m+( Qpq m )][( Qpq m+2 y ( Qpq m y (m+) )] [( Qpq m+3 y)( Qpq m+ y)( Qpq m y)] ( Qpq m+ y) The pair (b, c) = (±, ) gives us: = ( Qpq m y ( Qpq m ) ( Qpq m+ y )( Qpq m y ) ( Qpq m+2 y ( Qpq m+ ( Qpq m+3 y )( Qpq m+ y ) (m+) (m+) [( Qpq m+2 y ( Qpq m y ] [( Qpq m+( Qpq m )] [( Qpq m+3 y )( Qpq m+ y )( Qpq m y )] ( Qpq m+ y ) (m+)

29 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 09 Note the last two pairs each contains a term [( Qpq m+( Qpq m )] (m+) m 643 The a = and l = 2 case This involves the coefficients C(2, b, c), ie the coefficients of p 2 given as follows: 6 4(y + y ) + (y 2 + y + (q + q ) + 4(q 2 + q (yq + yq + y q + y q ) 2(yq 2 + yq 2 + y q 2 + y q = 2 2(y + y ) + (y 2 + y + (q + q ) + 4(q 2 + q 0 + q y(q + q ) y (q + q ) 2y(q 2 + q 0 + q 2y (q 2 + q 0 + q We have rewrite the terms of the form y c (q 2 + q as y c (q 2 + q 0 + q y c A GV type expressions can be obtained from the term 4(q 2 + q 0 + q as follows: ( ( Q 2 pq m+3 y )( Q 2 pq m+3 ) 4(m+) y) ( Q 2 pq m+4 )( Q 2 pq m+ ( ( Q 2 pq m+ y )( Q 2 pq m+ ) 4(m+) y) ( Q 2 pq m+( Q 2 pq m ) ( ( Q 2 pq m y )( Q 2 pq m ) 4(m+) y) ( Q 2 pq m )( Q 2 pq m = ( ( Q 2 pq m++2 y )( Q 2 pq m+ y )( Q 2 pq m+ 2 y ) ( Q 2 pq m++3 )( Q 2 pq m++ )( Q 2 pq m+ )( Q 2 pq m+ 3 ) ( ( Q 2 pq m++2 y)( Q 2 pq m+ y)( Q 2 pq m+ 4(m+) y) ( Q 2 pq m++ )( Q 2 pq m+ ) ) 4(m+) In the same fashion one can get a GV type expression for other terms The case of a = 2, l = can be dealt with similarly One only has to change Q 2 p to Qp 2 in the above expressions 7 Rank > Cases In this section we will study local Calabi-Yau geometries that correspond to the framed moduli spaces M(N, k) for N > We will present details for the identifications of the relevant string partition function as equivariant genera for suitable bundles on M(N, k) We also propose a method to obtain infinite product expressions 7 The rank N = 2 cases The corresponding local Calabi-Yau geometries are given by the Hirzebruch surfaces F m (m = 0,, 2) Their web diagrams are given by B B B F F 0 F F F F F F 2 F B B + F B + 2F

30 0 J LI, K LIU AND J ZHOU where B, F H 2 (F m, Z) are the homological classes of the base and the fiber respectively, and we have hence F 2 = 0, B 2 = m, BF =, (B + mf ) 2 = m Their dual graphs are The following results on the closed string partition functions can be found in [2,, 29]: Z Fm = 4 F e m κ µ 3 λ/2 W µ 4 µ 3(( )m Q B ) µ3 µ,2,3,4 W µ µ 4Q µ W µ 3 µ 2Q µ2 F e m κ µ λ/2 W µ 2 µ (( )m Q B ) µ Define the normalized partition function by: Ẑ (m) (Q B, Q F ; q) = Z F m (Q B, Q F ; q) Z (0) (Q F ; q) Theorem 7 After the following change of variables: we have the following identity: Q = Q B Q F, eβa =, a a 2 = 2a Ẑ (m) (Q B, Q F ; q) = Q k χ(m(2, k), E2,k(V m )) k= in other words, the generating series of the GW invariants on the local Calabi-Yau geometry of Hirzebruch surface F m, can be identified with the generating series of the equivariant indices of suitable bundles on the framed moduli spaces Proof It has been proved in [6, 30] that the normalized partition function of the topological string theory is given by Ẑ (m) (Q B, Q F ; q) = (( ) m Q B 2 4 Q µ,2 2 α,γ= i,j= + µ 2 F Q m µ2 ) µ F q m 2 (κ µ +κ µ sinh β 2 (a α,γ + h(µ α i µγ j + j i)) sinh β 2 (a, α,γ + h(j i)) where a,2 = a 2, = 2a, a, = a 2,2 = 0, Q F completed by Lemma 43 = e 2βa, q = e βh The proof is

31 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 72 The N > 2 cases In this case the corresponding local Calabi-Yau geometries are that of suitable ALE spaces of A N -type fibered over P We study the closed string partition functions of A N fibrations In this case the corresponding (p, q) 5-web diagrams are of the following form: Their dual graph is of the form (m =,, N + ): w m w w w N 2 wn+ w 0 w N+2 Using the topological vertex [] (for a mathematical theory see [8]), one can show that [3, 30, 7]: Ẑ (m) = ++ µ N µ,,µ N (( ) N+m 2 2N Q B ) µ [ N+m 2 ] Q (N+m 2i)( µ ++ µ i ) F i i= N i=[ N+m+ 2 ] N Q (N i)( µ ++ µ i ) i( µ i+ ++ µ R N ) F i i= q N 2 i= mκ µ i N α,γ= i,j= Q (N+m 2i)( µi+ ++ µ N ) F i sinh β 2 (a α,γ + h(µ α i µγ j + j i)) sinh β 2 (a, α,γ + h(j i)) where Q Fi = e β(ai ai+) Here Ẑ(m) is a normalized partition function introduced in [3] By splitting the terms with and without m, one gets: Ẑ (m) = µ,,µ N ( ) N 2 2N Q B [ ( ) µ ++ µ N q N 2 i= mκ µ i N [ N+m 2 ] i= α,γ= i,j= N+m 2 ] Q i F i i= Q µ ++ µ i F i N i=[ N+m+ 2 ] N i=[ N+m+ 2 ] Q (N i) F i sinh β 2 (a α,γ + h(µ α i µγ j + j i)) sinh β 2 (a α,γ + h(j i)) µ ++ µ N Q ( µi+ ++ µ N ) F i m

32 2 J LI, K LIU AND J ZHOU When t = e βh, t 2 = e βh, e α = e βaα we have the following identity by the same method as in the proof of Lemma 43: Q k χ(m(n, k), K 2 N,k (det V) m )(e,, e N, t, t k 0 = (Q/2 2N N N ) i= µi e mβ( µα a α+hκ µ α /2) µ,,n α= N sinh β 2 (a α,γ + h(µ α i µγ j + j i)) sinh β 2 (a, α,γ + h(j i)) α,γ= i,j= where a i,j = a i a j Thus we should take q = e βh, (39) and N α= Q = ( ) N Q B e β µα a α = ( ) µ ++ µ N [ N+m 2 ] Q i F i i= [ N+m 2 ] Q µ ++ µ i F i i= N i=[ N+m+ 2 ] N i=[ N+m+ 2 ] The right-hand side of the last equality can be rewritten as: N e β µα aα α= [ N+m 2 ] i= [( )e βa k ] µi N i=[ N+m+ 2 ] where k = [ N+m 2 ] Therefore, if one takes furthermore e βa k =, Q (N i) F i, Q ( µi+ ++ µ N ) F i [( )e βa k ] µi, then one has Ẑ (m) = Q k χ(m(n, k), K 2 N,k (det V) m )(e,, e N, t, t k 0 under the above specializations To summarize we have: Theorem 72 For N > 2, we have (40) Ẑ (m) = k 0 Q k χ(m(n, k), K 2 N,k (det V) m )(e,, e N, t, t with the following specialization of variables: t = e βh, t 2 = e βh, e α = e βaα, q = e βh, e βa [ N+m ] 2 =, Q Fi = e β(ai ai+), and Q = ( ) N Q B [ N+m 2 ] Q i F i i= N i=[ N+m+ 2 ] Q (N i) F i

33 STRING PARTITION FUNCTIONS AS EQUIVARIANT INDICES 3 73 Infinite product expressions for equivariant indices Now we propose a method to obtain infinite product expressions for equivariant indices In gauge theory another partial compactification has been used Denote by M reg 0 (r, n) the framed moduli space of genuine instantons on S 4 = C 2 { }, and There is a projective morphism M 0 (r, n) = n n =0M reg 0 (r, n ) (C (n n ) π : M(r, n) M 0 (r, n) which is equivariant with respect the torus actions on both spaces when r =, this is the Hilbert-Chow morphism: In particular, π : (C [n] (C (n) We now explain why we can expect a product expression by exploiting the collapsing morphism We apply the Riemann-Roch theorem in the equivariant G-theory of algebraic stacks developed by Toen [25] The rough idea is as follows We have converted the calculation of closed string partition functions to the calculation of equivariant indices of E N k on M(N, k) If we use the pushforward map π in the equivariant G-theory, we end up with a calculation on M 0 (N, k), where localization leads to a calculation on the orbifolds (C (n) Now G-theory on (C (n) is the same as the S n -equivariant K-theory of (C n Then by standard argument one can get a product expression for the partition function REFERENCES [] M Aganagic, A Klemm, M Marino, C Vafa, The topological vertex, preprint, hepth/ [2] M Aganagic, M Marino, C Vafa, All loop topological string amplitudes from Chern-Simons theory, preprint, hep-th/ [3] M Atiyah, I Singer, The index of elliptic operators III, Ann of Math, (2) 87 (968), pp [4] L Borisov, A Libgober, Elliptic genera of singular varieties, preprint, mathag/ [5] R Dijkgraaf, G Moore, E Verlinde, H Verlinde, Elliptic genera of symmetric products and second quantized strings, Comm Math Phys, 85: (997), pp [6] T Eguchi, H Kanno, Topological Strings and Nekrasovs formulas, JHEP, 2 (2003), 006, hep-th/ [7] T Eguchi, H Kanno, Geometric transitions, Chern-SImons gauge theory and Veneziano type amplitudes, Phys Lett, B585 (2004), pp [8] R Gopakuma, C Vafa, M-theory and toplogical strings-ii, hep-th/98227 [9] T Hollowood, A Iqbal, and C Vafa, Matrix model, geometric engineering and elliptic genera, preprint [0] S Hosono, M-H Saito, A Takahashi, Relative Lefschetz action and BPS state counting, mathag/00548 [] A Iqbal, All genus topological amplitudes and 5-brane webs as Feynman diagrams, preprint, hep-th/02074 [2] A Iqbal, A-K Kashani-Poor, Instanton counting and Chern-Simons theory, preprint, hepth/ [3] A Iqbal, A-K Kashani-Poor, SU(N) geometries and topological string amplitudes, preprint, hep-th/ [4] S Katz, A Klemm, C Vafa, M-theory, topological strings and spinning black holes, hepth/9908 [5] A Kerber, Representations of Permutation groups, I Lect Notes in Math, Vol 240, Springer, Berlin, 97