Localization and Duality

0-0 Localization and Duality Jian Zhou Tsinghua University ICCM 2004 Hong Kong December 17-22, 2004

Duality Duality is an important notion that arises in string theory. In physics duality means the equivalence of different physical theories. Different theories may involve different mathematical objects. Duality then suggests some connections between different branches of mathematics. 1

These connections are often difficult to find by traditional mathematical methods. To give a sample of the branches of mathematics involved, we list the rough correspondence between some physical theories and mathematical theories: 2

In dimension 2, 2D quantum gravity Deligne-Mumford moduli spaces 2D Yang-Mills theory Moduli spaces of flat connections on Riemann surfaces Wess-Zumino-Witten models Representation theory of Kac-Moody algebras 3

In dimension 3, Chern-Simons theory link invariants Jones (HOMFLY) polynomials Witten 3-manifold invariants Reshetikhin-Turaev quantum group invariants 4

In dimension 4, Topological Yang-Mills theory Donaldson theory Physical Seiberg-Witten theory Mathematical Seiberg-Witten theory Moduli spaces of noncommutative instantons Moduli spaces of torsion free sheaves 5

In dimension 6, on a Calabi-Yau 3-fold, Type IIA topological string theory Gromov-Witten theory Type IIB topological string theory Variation of Hodge structure In dimension 7, M-theory G 2 -manifolds In dimension 8, F-theory Spin(7)-manifolds 6

Localization Localization: when there is symmetry, one can express global invariants by local contributions. For example, let π : E M be a holomorphic G-bundle, then one has for g G, dim M i=0 ( 1) i tr g H i (M, O(E)) = M g ch g E M g ch g Λ 1 (N M g /M), where M g is the set of points fixed by g, N M g /M is the normal bundle of M g in M. 7

String duality suggests some surprising connections between the following mathematical objects: 1. Hodge integrals on Delgine-Mumford moduli spaces 2. Representations of symmetric groups, quantum groups, and Kac-Moody algebras. 3. Partitions and symmetric functions. 4. Relative moduli spaces of stable maps 5. Gromov-Witten invariants of open Calabi-Yau 3-folds 6. Integrable hierarchies (KP and Toda hierarchies) 7. Hilbert schemes of points on C 2 8. Equivariant indices of framed moduli spaces of torsion free sheaves 8

We will briefly describe their mathematical proofs. Localization plays an important role in both the physical derivations and the mathematical proofs. We would like to give some general remarks on the relationship between the relevant physical and mathematical theories. 9

In a quantum field theory, one is interested in computing the partition function. This is defined as a Feynman integral over an infinite-dimensional space. Such integrals are in general not mathematically defined. However, the intrinsic symmetry of the theory often makes it possible to formally apply localization techniques to reduce to an integral on a finite-dimensional moduli space of classical solutions. 10

This is where one can develop a corresponding mathematical theory: One can often make mathematical definitions and give mathematically rigorous proofs of physical predictions. For example, in string theory one actually starts with integrals on the space of all maps from a Riemann surface to a Calabi-Yau 3-fold. In mathematics one starts with holomorphic stable maps and studies integrals on their moduli spaces. However, the mathematical theory is often at cost of technical difficulty, and often does not provide much insight. Therefore, mathematicians still have much to learn from physicists. 11

The plan for the rest of the talk is as follows. Hodge integrals Mariño-Vafa formula and its generalizations Relationship with representation theory Relationship with link invariants Relationship with integrable hierarchies Applications to Gromov-Witten theory Relationship with Hilbert schemes Relationship with 4D Yang-Mills theory Concluding remarks 12

Hodge Integrals Deligne-Mumford moduli spaces M g,n : These parameterize the stable curves (C, x 1,..., x n ). Here C is a (nodal) curve of genus g. The Hodge bundle E has fiber H 0 (C, ω C ), where ω C is the dualizing sheaf. This bundle has rank g, its Chern classes λ 1 = c 1 (E),..., λ g = c g (E) are called the Hodge classes. The points x 1,..., x n are regular points on the curve. The line bundle L i has fiber T x i C. Define ψ i = c 1 (L i ), i = 1,..., n. 13

The Hodge integrals are integrals of the form: ψ i 1 1 ψ i n n λ j 1 1 λ j g g, Mg,n i 1,..., i n, j 1,..., j g 0. By Witten Conjecture/Kontsevich Theorem, a suitable generating series of ψ i 1 1 ψ i n n, Mg,n i 1,..., i n 0, is a τ-function of the KdV hierarchy. 14

Mariño-Vafa Formula and Its generalizations Mariño and Vafa derived a remarkable formula for certain Hodge integrals using a sequence of physical dualities: Combinatorial expressions Witten Chern-Simons link invariants Witten Open string theory on T S 3 Gopakumar-Vafa Closed string theory on O P 1( 1) O P 1( 1) Katz-Liu One partition Hodge integrals 15

This formula has been proved by Chiu-Chu Melissa Liu, Kefeng Liu, and Z. Details of the proof are presented in Melissa s talk. A key technical tool is localization on the relative moduli spaces of stable maps developed by Jun Li. A generalization to the two-partition Hodge integrals is conjectured by Z. and also proved by Chiu-Chu Melissa Liu, Kefeng Liu, and Z. The two-partition Hodge integral formula plays an important role in the mathematical calculations of Gromov-Witten invariants of toric local Calabi-Yau geometries. 16

There is also a three-partition generalization which plays an important role in the mathematical theory of the topological vertex developed by Jun Li, Chiu-Chu Melissa Liu, Kefeng Liu, and Z. The physical theory of the topological vertex was proposed by Aganagic, Klemm, Mariño, and Vafa. 17

Geometric side of Mariño-Vafa formula Define Λ g (u) = u g λ 1 u g 1 + + ( 1) g λ g. For every partition µ = (µ 1,..., µ l(µ), define: µ +l(µ) 1 G g,µ (τ) = [τ(τ + 1)] l(µ) 1 Aut(µ) l(µ) i=1 µi 1 a=1 (µ iτ + a) (µ i 1)! and generating series Λ Mg,l(µ) g (1)Λ g ( τ 1)Λ g (τ) l(µ) i=1 (1 µ, iψ i ) G µ (λ; τ) = g 0 λ 2g 2+l(µ) G g,µ (τ). 18

Introduce formal variables p = (p 1, p 2,..., p n,...), and define for any partition µ: p µ = p µ1 p µl(µ) Generating series for all genera and all possible marked point: G(λ; τ; p) = G µ (λ; τ)p µ. µ 1 Define G(λ; τ; p) = exp G(λ; τ; p). 19

Representation side of the Mariño-Vafa formula For a partition µ, let µ = j µ j; χ µ : the character of the irreducible representation of symmetric group S µ indexed by µ; C(µ): the conjugacy class of S µ indexed by µ; z µ = j µ j!j µ j ; κ µ = j µ j(µ j 2j + 1). W µ (q) = 1 i<j l(µ) [µ i µ j +j i] [j i] l(µ) i=1 where [x] = q x/2 q x/2, q = e 1λ ; µi i v= i+1 qv/2 µi v=1 [v i+l(µ)], R(λ; τ; p) = µ χ ν (C(µ)) z µ e 1(τ+ 1 2 )κ νλ/2 W ν (q)p µ. 20

The Mariño-Vafa formula: G(λ; τ; p) = R(λ; τ; p). One can recover from this formula most of the Hode integral identities obtained by Faber and Pandharipande (Liu-Liu-Z.). One can in principle compute almost all Hodge integrals, including the ones in Witten Conjecture/Konsevich Theorem from this formula (Wen-Xuan Lu). 21

Key ideas in our proof: Both sides satisfy the cut-and-join equation : 1λ τ C = ( 2 ) (i + j)p i p j + ijp i+j C 2 p i+j p i p j and have the same initial values: G(λ; 1; p) = R(λ; 1; p). Both localization on the relative moduli spaces and some combinatorics of symmetric functions are required. Details are presented in Chiu-Chu Melissa Liu s talk. 22

For our purpose we use the algrebro-geometric relative moduli spaces developed by Jun Li. There is also symplectic geometric relative moduli spaces developed by Anming Li and Yongbin Ruan, and Ionel and Parker. In a suitable limit the Mariño-Vafa formula yields the ELSV formula that relates Hodge integrals to Hurwitz numbers and the Burnside formula. The cut-and-join equation for Hurwitz numbers was established by Goulden and Jackson by combinatorial method, and by Anming Li, Guosong Zhao and Quan Zheng by symplectic method. 23

Relationship with representation theory The right-hand side of the Mariño-Vafa formula R(λ; τ; p) = µ χ ν (C(µ)) z µ e 1(τ+ 1 2 )κ νλ/2 W ν (q)p µ is related to representations of various algebraic objects. These include symmetric groups, quantum groups and Kac-Moody algebras. 24

Representation of symmetric groups: Denote by s µ the Schur functions and by p µ the Newton symmetric functions. Then one has s ν = µ χ ν (C(µ)) z µ p µ, W µ (q) = s µ (q 1/2, q 3/2,... ), κ ν = ν ( ν 1) χ ν(c(2, 1,..., 1)) dim R ν. 25

Representations of quantum groups: For large enough N, let dim q R µ denote the quantum dimension of the irreducible module of SU(N) q indexed by µ, then one has dim q R µ = W µ = 1 i<j l(µ) [µ i µ j + j i] [j i] l(µ) i=1 µi i v= i+1 [v] e t µi v=1 [v i + l(µ)], where [x] y = y 1/2 q x/2 y 1 q x/2. Then W µ is the leading term of W µ in the power series of e t/2. One can also think of W µ (λ) as the quantum dimension of the Hecke algebra of type A. 26

Representation of Kac-Moody algebras Let ŝu(n) denote the affine Kac-Moody of algebra of su(n). For a fixed level k, the integrable highest weight representations are indexed by partitions µ with µ 1 k. Denote by χ µ (τ) their characters. By the modular property, one has functions S µν (τ) such that χ µ ( 1 τ ) = ν 1 k S µν (τ)χ ν (τ). One has W µ = S µ0 S 00. In general, the leading term of W µν = S µν S 00 plays an important role in the two- and three-partition generalizations of the Mariño-Vafa formula. κ µ is related to the transform under τ τ + 1. 27

Relationship with link invariants By Witten s Chern-Simons theoretical approach to link invariants, one finds that the HOMFLY polynomial of the unknot colored by µ is given by W µ. Similarly, the HOMFLY polynomial of the Hopf link with two components colored by µ and ν respectively is given by W µν. Mathematically, these can be proved by skein theory (Morton-Lukac formula). One can also establish these facts by representation theory of quantum groups and Hecke algebras (Xiao-Song Lin and Hao Zheng). 28

Relationship with integrable hierarchies One can realize R(λ; τ; p) as the vacuum expectation value of a suitable operator on the fermionic Fock space. By the theory of integrable hierarchies developed by the Kyoto school, one then finds by the Mariño-Vafa formula that G(λ; τ; p) is a τ-function of the KP hierarchy (Z.). Similarly, the generating function of the two-partition Hodge integrals we consider is a τ-function of the 2-Toda hierarchy (Z.). According to Aganagic-Dijkgraaf-Klemm-Vafa, the generating function of the three-partition Hodge integrals we consider is a τ-function of the 3-component KP hierarchy. 29

Applications to Gromov-Witten theory The two- and three-partition generalizations of the Mariño-Vafa formula can be used to compute Gromov-Witten invariants of open Calab-Yau spaces. For example, the canonical line bundle of P 1 P 1 is an open Calabi-Yau 3-fold. Denote by M g,0 (P 1 P 1, (m, n)) the moduli space of stable maps of genus g to P 1 P 1, of degree (m, n). Denote by K g,(m,n) the bundle whose fiber at a map f : C P 1 P 1 is H 1 (C, f κ P 1 P 1). 30

Define and K g,(m,n) = [M g,0 (P 1 P 1,(m,n))] vir ] e(k g,(m,n) ), F (λ; t 1, t 2 ) = K g,(m,n) λ 2g 2 t m 1 t n 2, (m,n) (0,0) Z(λ; t 1, t 2 ) = exp F (λ; t 1, t 2 ). In physics literature, F is often referred to as the free energy, and Z is often referred to as the partition function. Mathematically, F is the counting of stable maps with connected domain curves, Z is the counting of stable maps with possibly disconnected domain curves. 31

Physicists have developed a diagrammatic method to compute F and Z by duality with Chern-Simons theory. This is first discovered by Aganagic, Mariño and Vafa. Iqbal reformulated their results as a Feynman rule. Iqbal s Feynman rule has been proved by Z. using the two-partition Hodge integral formula and a combinatorial technique called the chemistry of graphs. 32

For example, the web diagram of κ P1 P 1 is µ 3 µ 4 µ 2 µ 1 Then one has : Z(λ) = µ 1,...,4 W µ 1,µ 2W µ 2,µ 3W µ 3,µ 4W µ 4,µ 1t µ 1 + µ 3 1 t µ2 + µ 4 2. 33

The mathematical proof makes use of the natural torus action on P 1 P 1 and the induced actions on M g,0 (P 1 P 1, (m, n)) and K g,(m,n) By localization, one can write Z as a sum over fixed point components of M g,0 (P 1 P 1, (m, n)), which are essentially products of Deligne-Mumford moduli spaces. The local contributions can be formulated as the two-partition Hodge integrals, hence one can apply our two-partition generalization of the Mariño-Vafa formula. The difficulty is that there are many fixed point components to consider. The trick of chemistry of graphs takes care of that. 34

Aganagic, Klemm, Mariño, and Vafa generalize the above method to the theory of topological vertex. This theory can be used to compute some open string partition functions, or mathematically, relative Gromov-Witten theory. Furthermore, it can be used to compute open Calabi-Yau spaces corresponding to diagrams of the form: 35

36

Jun Li, Chiu-Chu Melissa Liu, Kefeng Liu and Z. have developed a mathematical theory of the topological vertex. This theory reduces the calculations of the Gromov-Witten invariants of the formal toric Calabi-Yau spaces described by the diagrams in terms of three-partition Hodge integrals. By our three-partition Hodge integral formula we obtain a combinatorial expression. This expression looks different from the one given by Aganagic et al. Klemm wrote a program to check it for all cases where each partition has weight 6. 37

Relationship with Hilbert schemes The Hilbert scheme (C 2 ) [n] of n points on C 2 parameterizes ideals I C[x, y] such that dim C[x, y]/i = n. It is a smooth variety of dimension 2n. In recent years, there have been many interesting work done on Hilbert schemes of surfaces, e.g. Göttsche, Nakajima, Haiman, Lehn, Vesserot, Weiping Li, Zhenbo Qin, Weiqiang Wang, etc. Since Hilbert schemes also involve partitions and symmetric functions, it is natural to guess that they are related to Hodge integrals and Gromov-Witten invariants. 38

Relationship with quantum cohomology of Hilbert schemes Consider the rank n bundle E n on (C 2 ) [n] whose fiber at I is C[x, y]/i, and let D = c 1 (E n ). The natural 2-torus action on C 2 induces an action on (C 2 ) [n]. Denote by M D the quantum multiplication by D in equivariant cohomology of (C 2 ) [n], defined by localization contributions on the moduli spaces of genus 0 stable maps to (C 2 ) [n]. 39

In a recent work of Okounkov and Pandharipande, it is shown that M D is related to the Calegero-Sutherland operator : H CS = 1 ( ) 2 z i + θ(θ 1) 1 2 z i z i z j 2. i i<j The function φ(z) = i<j (z i z j ) θ is an eigenfunction of H CS. 40

A calculation shows that the operator φh CS φ 1 equals 1 θ k 2 p k + 1 (klp 2 k+l + θ(k + l)p k p l 2 p k 2 p k p l k k,l on the space of symmetric functions. p k+l When θ = 1, which corresponds to the diagonal action of S 1 on C 2, one gets φh CS φ 1 = 1 (klp 2 ) k+l + (k + l)p k p l. 2 p k p l k,l This is the cut-and-join operator! p k+l ). 41

Relationship with equivariant genera of Hilbert schemes The Hirzebruch χ y genus of a compact complex d-fold X is defined by: χ y (X) = d ( y) p p=0 d ( 1) q dim H q (X, Λ p T X). q=0 The χ y -genus specializes to other invariants. E.g., χ 1 (X) is the Euler number, and χ 0 (X) = d ( 1) q dim H q (X, O X ) q=0 is the geometric genus of X. 42

The elliptic genus χ(x, y, q) generalizes the χ y -genus. It is the Riemann-Roch number of It is easy to see that Ell(T X; q, y) = y dim X/2 n 1 (Λ yq n 1(T X) Λ y 1 q n(t X) S q n (T X) S q n(t X)). χ y (X) = y d/2 χ(x, y, 0). When a group acts on X, one can also consider the equivariant versions of the above genera. Equivariant genera can also be computed by localization. 43

Fixed points of the torus action on Hilbert schemes are parameterized by partitions. It follows that equivariant genera of Hilbert schemes can be written as sums over partitions. Sums over partitions also appear in the diagrammatic computations of topological string partition functions. This suggests the relationship between them. It is actually part of the physical theory of geometric engineering that reproduces gauge theory from string theory. Again one can give mathematical proofs of such dualities by localization and combinatorics. 44

For example, for the resolved conifold described by the following diagram: One has Z 0 (Q, q) = µ ( Q) µ W µ (q)w µ t(q) = Q n χ 0 ((C 2 ) [n] )(q, q 1 ). n=0 45

For the geometry described by the following diagram (the vertical edges are glued with each other): Hollowood, Iqbal and Vafa found Z(Q, Q m, q) = ( Q) µ ( Q m ) ν C ν t µ(0)(q)c νµ t (0)(q) µ,ν = Z 0 (Q, q) Q k χ Qm (C [k] )(q, q 1 ). k=0 46

They also found for the geometry described by the following diagram (the vertical edges are glued with each other, so do the horizontal ones): Z(Q, Q m, Q 1, q) = µ,ν,η( Q) µ ( Q 1 ) ν ( Q m ) η C µνη C µ t ν t η t = Z 0 (Q, q) Q k χ(c [k] ; Q 1 Q m, Q m )(q, q 1 ). k=0 47

In a recent joint work with Jun Li and Kefeng Liu, we use the Hilbert-Chow morphism π : (C 2 ) [n] (C 2 ) (n) = (C 2 ) n /S n to push the calculations to the symmetric products. This produces the following infinite product expressions required by the Gopakumar-Vafa conjecture for the partition functions of the relevant Calabi-Yau geometries: m,a,j,c,l where 1 m 1 (1 qm Q) m, ( (1 Q k+1 Q k+1 m qm )(1 Q k+1 Q k+1 k,m 0 (1 Q k+1 Q k m qm+1 )(1 Q k+1 Q k+2 ( m qm+2 ) m q m+1 ) ) m+1, (1 Q l p a q b+k+2 y c )(1 Q l p a q b+k y c ) (1 Q l p a q b+k+1 y c 1 )(1 Q l p a q b+k+1 y c+1 )) (m+1) C(la,j,c), χ(c 2 ; p, y)(q, q 1 ) = a,j,c C(a, j, c)p a (q 2j +q 2(j 1) + +q 2j )y c. 48

Relationship with 4D Yang-Mills theory Another kind of moduli spaces on which one can apply localization and use combinatorics to relate to the string partition functions are the framed moduli spaces torsion-free sheaves on P 2. Such spaces have been extensively studied by Nakajima and Yoshioka. They provide important examples of geometric engineering. The relationship among such spaces, Seiberg-Witten prepotential function, and the topological string partition functions of A n -fibered spaces, was conjectured by Nekrasov. 49

This relationship has been clarified by the works of Nekrasov, Iqbal-Kashani-Poor, Eguchi-Kanno, Z., Li-Liu-Z. 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. When N = 1, M(1, k) is the Hilbert scheme (C 2 ) [k]. 50

It is isomorphic to the space of B 1, B 2 End(V ), i Hom(W, V ) and j Hom(V, W ), satisfying [B 1, B 2 ] + ij = 0 and a stability condition, modulo the action of GL(V ) = GL N (C) given by g (B 1, B 2, i, j) = (gb 1 g 1, gb 2 g 1, gi, jg 1 ). This is an ADHM construction. Here dim C V = k, dim C W = N. In physics literature, they correspond to the moduli space of instatons in noncommutative gauge theory. 51

We recall some of their properties: 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). The fixed points are isolated and parametrized by an N-tuple of partitions µ = (µ 1,, µ N ) such that i µi = k. Even though M(N, k) is noncompact, one can still apply the localization techniques on it. The rank N canonical bundle V on M(N, k) whose fibers are given by V is naturally an equivariant bundle. 52

(Li-Liu-Z.) After suitable identification of variables, one has Ẑ Fm = Q k χ(m(2, k), κ 1 2 M(N,k) (det V ) m )(q, q 1 ), k=0 where LHS=instanton part of the string partition function of the open Calabi-Yau manifold κ Fm (m=0, 1,2); F m is the Hirzebruch surface, e.g. F 0 = P 1 P 1 ; RHS = generating series of the equivariant index of natural bundles on M(N, k). There are similar results for N > 2. 53

Concluding Remarks In conclusion, we have seen that the following contributions of three great Chinese scientists are related: Chern: Chern-Weil theory, Chern-Simons theory Yang: Yang-Mills theory, Yang-Baxter theory Yau: Calabi-Yau spaces, SYZ Conjecture Our main technical tool (localization) can be traced back to Professor Chern s work. I dedicate this talk to his memory. 54

Thank you very much! 55