String Duality and Moduli Spaces

String Duality and Moduli Spaces Kefeng Liu July 28, 2007 CMS and UCLA

String Theory, as the unified theory of all fundamental forces, should be unique. But now there are Five different looking string theories. I, IIA, IIB, E 8 E 8, SO(32). Physicists: these theories should be equivalent, in a way dual to each other: their partition functions should be equivalent. All other theories should be part of string theory. The identifications of partition functions of different theories have produced many surprisingly beautiful mathematical formulas like the mirror formulas, following from the mirror principle inspired by duality between IIA and IIB string theory; the Mariño-Vafa formula, the Labastida-Mariño-Ooguri-Vafa conjecture from large N Chern-Simons string duality.

I will briefly review various results we have obtained during the past years, most of them are inspired by string duality between Chern-Simons and topological string theory (or Gromov-Witten theory). 1. Mariño-Vafa formula for the generating series of triple Hodge integrals on moduli spaces of Riemann surfaces of all genera and any number of marked points in terms of quantum dimension, or the Chern-Simons invariants of a trivial knot. (Liu-Liu-Zhou) Many corollaries: ELSV formula, Dikgraaf-Verlinde-Verlinde conjecture, new proofs of the Witten conjecture, Hodge integral identities. (Kim, Chen, Li, Liu)

2. Hopf link and the two partition Mariño-Vafa formula: Hodge integals in terms of CS invariants of Hopf link. (Liu-Liu-Zhou) 3. Topological vertex theory and applications: explicit formulas for generating series of all degree and all genera Gromov-Witten invariants for toric Calabi-Yau manifolds in terms of Chern-Simons knot invariants. (Li-Liu-Liu-Zhou). Pan Peng s proof of the Gopakumar-Vafa conjecture for toric Calabi-Yau: a remarkable integrality structure of Gromov-Witten invariants.

4. Labastida-Mariño-Ooguri-Vafa conjecture: new algebraic and integrality structure for the generating series of quantum knot invariants, inspired by string duality. Pan Peng will discuss the proof in detail. He will also discuss some history and details about large N Chern-Simons string duality. 5. Several new results about intersection numbers of the moduli spaces of Riemann surfaces: explicit formula for n-point functions; new recursion formulas for integrals of all κ and all ψ classes; proof of a conjecture of Itzkson-Zuber about divisibility of denominators of certain Hodge integrals. (Joint with Hao Xu). All of the above results are available on arxiv, or my webpage at UCLA.

In the following discussions, we will study Hodge integrals (i.e. intersection numbers of λ classes and ψ classes) on the Deligne- Mumford moduli space of stable curves M g,h. A point in M g,h consists of (C, x 1,..., x h ), a (nodal) curve and h smooth points on C. The Hodge bundle E is a rank g vector bundle over M g,h whose fiber over [(C, x 1,..., x h )] is H 0 (C, ω C ). The λ classes are Chern Classes: λ i = c i (E) H 2i (M g,h ; Q). The cotangent line T x i C of C at the i-th marked point x i gives a line bundle L i over M g,h. The ψ classes are also Chern classes: ψ i = c 1 (L i ) H 2 (M g,h ; Q).

(1). The Mariño-Vafa Formula Introduce the total Chern classes of the Hodge bundle on the moduli space of curves: Λ g (u) = u g λ 1 u g 1 + + ( 1) g λ g. Mariño-Vafa formula: the generating series over g of triple Hodge integrals Λ g (1)Λ g (τ)λ g ( τ 1) M hi=1, g,h (1 µ i ψ i ) can be expressed by close formulas of finite expression in terms of representations of symmetric groups, or Chern-Simons knot invariants. Here τ is a parameter.

Mariño-Vafa conjectured the formula from large N duality between Chern-Simons and string theory, following works of Witten, Gopakumar-Vafa, Ooguri-Vafa. Geometric side: For every partition µ = (µ 1 µ l(µ) 0), define triple Hodge integral: with G g,µ (τ) = A(τ) A(τ) = 1 µ +l(µ) Aut(µ) Introduce generating series Λ g (1)Λ g ( τ 1)Λ g (τ) M l(µ) g,l(µ) i=1 (1 µ iψ i ) l(µ) [τ(τ + 1)] l(µ) 1 i=1 µ i 1 G µ (λ; τ) = g 0 λ 2g 2+l(µ) G g,µ (τ). a=1 (µ iτ + a). (µ i 1)!,

Special case when g = 0: Λ 0 (1)Λ 0 ( τ 1)Λ 0 (τ) M l(µ) 0,l(µ) i=1 (1 µ iψ i ) = 1 M l(µ) 0,l(µ) = µ l(µ) 3 i=1 (1 µ iψ i ) for l(µ) 3, and we use this expression to extend the definition to the case l(µ) < 3. 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 points: G(λ; τ; p) = µ 1 G µ (λ; τ)p µ.

Representation side: χ µ : the character of the irreducible representation of symmetric group S µ indexed by µ with µ = j µ j, C(µ): the conjugacy class of S µ indexed by µ. Introduce: W µ (q) = q κ µ/4 1 i<j l(µ) [µ i µ j + j i] [j i] l(µ) i=1 1 µ i v=1 [v i + l(µ)] where κ µ = µ + i (µ 2 i 2iµ i), [m] = q m/2 q m/2 andq = e 1λ. The expression W µ (q) has an interpretation in terms of quantum dimension in Chern-Simons knot theory.

Define: R(λ; τ; p) = n 1 ( 1) n 1 n µ [ n n i=1 µi =µ i=1 ν i = µ i χ ν i(c(µ i )) 1(τ+ 1 e 2 )κ ν iλ/2 Wν i(λ)]p µ z µ i where µ i are sub-partitions of µ, z µ = j µ j!j µ j for a partition µ: standard for representations of symmetric groups. Theorem: Mariño-Vafa Conjecture is true: G(λ; τ; p) = R(λ; τ; p).

Remark: (1). Equivalent expression: G(λ; τ; p) = exp [G(λ; τ; p)] = µ G(λ; τ) p µ = µ 0 ν = µ χ ν (C(µ)) z µ e 1(τ+ 1 2 )κ ν λ/2 Wν (λ)p µ (2). Each G µ (λ, τ) is given by a finite and closed expression in terms of representations of symmetric groups: G µ (λ, τ) = n 1 ( 1) n 1 n n n i=1 µi =µ i=1 ν i = µ i χ ν i(c(µ i )) z µ i e 1(τ+ 1 2 )κ ν iλ/2 Wν G µ (λ, τ) gives triple Hodge integrals for moduli spaces of curves of all genera with l(µ) marked points.

(3). Mariño-Vafa formula gives explicit values of many interesting Hodge integrals up to three Hodge classes: Taking limit τ 0 we get the λ g conjecture (Faber-Pandhripande), ( ) ψ k 1 1 M ψk n 2g + n 3 2 2g 1 1 B 2g n λ g = g,n k 1,..., k n 2 2g 1 (2g)!, for k 1 + +k n = 2g 3+n, and the following identity for Hodge integrals: Mg λ 3 g 1 = M g λ g 2 λ g 1 λ g = 1 2(2g 2)! B 2g are Bernoulli numbers. And other identities. B 2g 2 2g 2 B 2g 2g,

Taking limit τ, we get the ELSV formula. Zhou) (Liu-Liu- Kim-Liu gave a new proof of the Dikgraaf-Verlinde-Verlinde conjecture which implies the Witten conjecture by using asymptotic method. Lin Chen, Yi Li and myself have given another proof of the Witten conjecture from ELSV formula. Yi Li has derived more Hodge integral identities from the MV formula.

The idea to prove the Mariño-Vafa formula is to prove that both G and R satisfy the Cut-and-Join equation: Theorem : Both R and G satisfy the following differential equation: F τ = 1 ( ( )) F F F 1λ (i + j)p i p j +ijp i+j + 2 F 2 p i+j p i p j p i p j i,j=1 This is equivalent to linear systems of ODE. They have the same initial value at τ = 0: The solution is unique! G(λ; τ; p) = R(λ; τ; p).

Cut-and-Join operator, denoted by (CJ), in variables p j on the right hand side gives a nice match of Combinatorics and Geometry. The proof of cut-and-join equation for R is a direct computation in combinatorics. The first proof of the cut-and-join equation for G used functorial localization formula. The second proof is to prove a convolution formula for G.

2. Two Partition Mariño-Vafa Formula. Let µ +, µ be any two partitions. Introduce Hodge integrals: G µ +,µ (λ; τ) = B(τ) λ 2g 2 with µ + i µ + i g 0 Λ g (1)Λ g (τ)λ g ( τ 1) ( ) ( M g,l(µ + )+l(µ ) l(µ + ) 1 1 l(µ i=1 ψ ) τ i j=1 B(τ) = ( 1λ) l(µ+ )+l(µ ) z µ + z µ µ i τ µ j ψ j+l(µ + ) [τ(τ + 1)] l(µ+ )+l(µ ) 1 ) l(µ + ) i=1 µ + i 1 a=1 ( µ + i τ + a ) µ + i! l(µ ) i=1 µ i 1 a=1 ( µ i 1 τ + a ) µ i!.

They naturally arise in open string theory. Introduce notations: Geometry side: G (λ; p +, p ; τ) = exp (µ +,µ ) P 2 G µ +,µ (λ, τ)p + µ +p µ p ± µ ± are two sets of formal variables associated to the two partitions., Representation side: R (λ; p +, p ; τ) = ν ± = µ ± 0 χ ν +(C(µ + )) χ ν (C(µ )) 1(κν e +τ+κ ν τ 1 )λ/2 Wν +,ν p + µ +p µ. z µ + z µ

Here W µ,ν = q l(ν)/2 W µ s ν (E µ (t)) = ( 1) µ + ν q κ µ+κν+ µ + ν 2 ρ q ρ s µ/ρ (1, q,... )s ν/ρ (1, q,... ) in terms of skew Schur functions s: Chern-Simons invariant of Hopf link. Theorem: We have the equality: G (λ; p +, p ; τ) = R (λ; p +, p ; τ). Idea of Proof: Both sides satisfy the same equation: τ H = 1 2 (CJ)+ H 1 2τ 2(CJ) H,

where (CJ) ± cut-and-join operator: differential with respect to p ±, and the same initial value at τ = 1: Ooguri-Vafa formula. G (λ; p +, p ; 1) = R (λ; p +, p ; 1), The cut-and-join equation can be written in a linear matrix form, follows from the convolution formula, which naturally arises from localization technique on moduli.

(3). Mathematical Theory of Topological Vertex Topological vertex theory, as developed by Aganagic-Klemm- Marino-Vafa from string duality and geometric engineering, gives complete answers for all genera and all degrees in the toric Calabi-Yau cases in terms of Chern-Simons knot invariants! We developed the mathematical theory of topological vertex by using localization technique. This formula gives closed formula for the generating series of the Hodge integrals involving three partitions in terms of Chern-Simons knot invariants. The corresponding cut-and-join equation has the form: τ F = (CJ) 1 F + 1 τ 2(CJ)2 F + 1 (τ + 1) 2(CJ)3 F

where (CJ) denotes the cut-and-join operator with respect to the three groups of infinite numbers of variables associated to the three partitions. We derived the convolution formulas both in combinatorics and in geometry. Then we proved the identity of initial values at τ = 1. We introduced the new notion of formal toric Calabi-Yau manifolds to work out the gluing of Calabi-Yau and the topological vertices. We then derived all of the basic properties of topological vertex, like the fundamental gluing formula.

By using gluing formula of the topological vertex, we can derive closed formulas for generating series of GW invariants, all genera and all degrees, open or closed, for all toric Calabi-Yau, in terms Chern-Simons invariants, by simply looking at The moment map graph of the toric Calabi-Yau. Each vertex of the moment map graph contributes a closed expression to the generating series of the GW invariants in terms of explicit combinatorial Chern-Simons knot invariants. Let us look at an example to see the computational power of topological vertex.

Let N g,d denote the GW invariants of a toric Calabi-Yau, total space of canonical bundle on a toric surface S. It is the Euler number of the obstruction bundle on the moduli space M g (S, d) of stable maps of degree d H 2 (S, Z) from genus g curve into the surface S: N g,d = [M g (S,d)] v e(v g,d) with V g,d a vector bundle induced by the canonical bundle K S. At point (Σ; f) M g (S, d), its fiber is H 1 (Σ, f K S ). Write F g (t) = d N g,d e d t.

Example: Topological vertex formula of GW generating series in terms of Chern-Simons invariants. For the total space of canonical bundle O( 3) on P 2 : exp ( g=0 λ 2g 2 F g (t)) = ν 1,ν 2,ν 3 W ν1,ν 2 W ν2,ν 3 W ν3,ν 1 ( 1) ν 1 + ν 2 + ν 3 q 1 2 3i=1 κ νi e t( ν 1 + ν 2 + ν 3 ). Here q = e 1λ, and W µ,ν are from the Chern-Simons knot invariants of Hopf link. Sum over three partitions ν 1, ν 2, ν 3. Three vertices of moment map graph of P 2 three W µ,ν s, explicit in Schur functions. For general (formal) toric Calabi-Yau, the expressions are just similar: closed formulas.

Applications of Topological Vertex. Recall the interesting: Gopakumar-Vafa conjecture: There exists expression: g=0 such that n g d λ 2g 2 F g (t) = n g d k=1 g,d 1 d (2 sin dλ 2 )2g 2 e kd t, are integers, called instanton numbers. By using the explicit knot invariant expressions from topological vertex in terms of the Schur functions, we have the following applications:

(1). First motivated by the Nekrasov s work, by comparing with Atiyah-Bott localization formulas on instanton moduli we have proved (Li-Liu-Zhou): Theorem: For conifold and the toric Calabi-Yau from the canonical line bundle of the Hirzebruch surfaces, we can identify the n g d as equivariant indices of twisted Dirac operators on moduli spaces of anti-self-dual connections on C 2. A complicated change of variables like mirror transformation is performed. R. Waelder s recent work on equiavriant elliptic genera of the Hilbert schemes of C 2 identifies GW invariants to coefficients of elliptic genera, therefore modular forms.

(2). The proof of the flop invariance of the GW invariants of (toric) Calabi-Yau by Konishi and Minabe, with previous works of Li-Ruan and Chien-Hao Liu-Yau. (3). There are a lot of efforts and several new conjectures to understand the GV conjecture. But the following result is still by now the only complete result about Gopakumar-Vafa conjecture, proved by Pan Peng: Theorem: (Pan Peng) The Gopakumar-Vafa conjecture is true for all (formal) local toric Calabi-Yau for all degree and all genera. More applications expected from the computational power of the topological vertex.

4. The Labastida-Mariño-Ooguri-Vafa Conjecture. Based on Chern-Simons/Topolofical String (CS/TS) duality, Labastida- Mariño-Ooguri-Vafa conjectured the following highly nontrivial structure. Generating function of quantum group invariants of link L Z(L) = 1 + A 1,,A L W A 1,,A L (L) L α=1 s A α(x α ) CS/TS duality = generating function of GW invariants?

Schur function s A (x) is determined by the following formula s A (x) = µ χ A (C µ ) z µ l(µ) α=1 x µ α j χ A is the character of symmetric group. C µ is the conjugate class labeled by µ. l(µ) is the length of the partition µ. We write F = log Z, then F = d=1 A 1,A 2,...,A L 1 d f A 1,...,A L (q d, t d ) L α=1 s A α(x d ). Define: M AB (q) = λ C λ n! χ A(C λ )χ B (C λ ) l(λ) j=1 ( q λ j /2 q λ j/2 ) q 1/2 q 1/2.

Theorem: (Liu-Peng)(LMOV Conjecture). F has the following structures: f A 1,..,A L (q, t) = (q 1 2 q 1 2) L 2 and P B 1,..,B L (q, t) = g 0 B 1,,B L P B 1,..,B L (q, t) L α=1 M A α B α(q); N (B 1,,B L ),g,q (q1/2 q 1/2 ) 2g t Q where N (C 1,...,C L ),g,q Q Z 2 are integers. Tomorrow Pan Peng will discuss the motivation, origin and the ideas of the proof of LMOV conjetcure.

Remarks: (1) From the definition of quantum group invariants of links, q = 1 is a pole of f A 1,..,A L (q, t), possibly of very high order. However, LMOV conjecture tells us it is at most 1. (2) Even assume the algebraic structure of f A 1,..,A L (q, t), the integrality of N (C 1,...,C L ),g,q are still mysterious mathematically. It is only safe to say they are rational numbers. (3) In physics, the integer coefficient N (C 1,...,C L ),g,q can be interpreted in terms of a generalization of Gromov-Witten invariants which involve Riemann surfaces with boundaries in the same spirit as the Gopakumar-Vafa invariants.

We have seen close connections between knot invariants and Gromov-Witten invariants. There may be a more interesting and grand duality picture between Chern-Simons invariants for real three dimensional manifolds and Gromov-Witten invariants for complex three dimensional toric Calabi-Yau. General correspondence between the geometry of real dimension 3 and complex dimension 3?!. In any case String Duality has already inspired exciting duality and unification among various mathematical subjects.

5. New results of Intersection Numbers on Moduli Spaces Natural cohomology classes, the κ classes, were originally defined by Mumford, Morita, Miller and Arbarello-Cornalba. It is known that the κ (or λ) and ψ classes generate the tautological cohomology ring of the moduli spaces, and most of the known cohomology classes are tautological. The following intersection numbers τ d1 τ dn j 1 κ b j j g := ψ d 1 1 M ψd n n g,n are called the higher Weil-Petersson volumes j 1 κ b j j.

We call the following generating function F (x 1,..., x n ) = g=0 dj =3g 3+n τ d1 τ dn g n j=1 x d j j the n-point function. The Witten conjecture states that this function satisfies KdV hierarchy and the Virasoro constraints, infinite number of differential equations. The Dijkgraaf-Verlinde-Verlinde conjecture, proved by Kim-Liu, shows that Witten conjecture is equivalent to the following simple recursion formula of the ψ integrals.

Theorem (Kim-Liu):(DVV Coonjecture) We have the identity: 1 2 a+b=n 2 σ n σ k g = (2k + 1) σ n+k 1 k S k S σ a σ b l a,b σ l g 1 + 1 2 S=X Y, a+b=n 2, g 1 +g 2 =g σ a k X l k σ l g + σ k g1 σ b l Y σ l g2. Here the notation S = {k 1,, k n } = X Y ; σ n = (2n + 1)!!ψ n and n j=1 σ kj g = M g,n n j=1 σ kj. We have obtained rather explicit expression for n point function.

Theorem (Liu-Xu): Let G(x 1,..., x n ) = exp and n 2. Then we have G(x 1,..., x n ) = r,s 0 ( nj=1 x 3 j 24 ) F (x 1,..., x n ) (2r + n 3)!!P r (x 1,..., x n ) (x 1,..., x n ) s 4 s (2r + 2s + n 1)!! where P r and are homogeneous symmetric polynomials = ( n j=1 x j ) 3 n j=1 x 3 j, 3 P r = = 1 2 n j=1 x j 1 2 n j=1 x j n=i J( i I x i ) 2 ( i J n=i J( i I x i ) 2 ( i J x i ) 2 G(x I )G(x J ) x i ) 2 r r =0 3r+n 3 G r (x I )G r r (x J ).

It is rather hard to find the formula, but it is easy to prove it. Special case of n = 2 was obtained by Dijkgraaf. n = 3 was obtained by Zagier. The case of From this we can derive many new identities and recursion formulas about ψ integrals. We only give one example: Let d j 1 and n j=1 (d j 1) = g 1, then 2g j=0 ( 1) j τ 2g j τ j n i=1 τ di g = (2g + n 1)! 4 g (2g + 1)! n j=1 (2d j 1)!!.

Let m, t, a 1,..., a n N, m = n i=1 a i, and s := (s 1, s 2,... ) be a family of independent formal variables. ( m) := t i 1 m := im i, m := m i, i 1 i 1 ( m i t i s m := i 1 ), ( m s m i i, m! := a 1,..., a n ) := i 1 i 1 m i!, ( m ) i. a 1 (i),..., a n (i) Let b N, we denote a formal monomial of κ classes by κ(b) := κ b i i. We have very general recursion formulas for the general Hodge integrals for the κ and ψ classes. i 1

Theorem (Liu-Xu): Let b N and d j 0. (2d 1 + 1)!! κ(b) = + 1 2 n j=2 L+L =b n j=1 α L ( b L τ dj g ) (2( L + d 1 + d j ) 1)!! κ(l )τ (2d j 1)!! L +d1 +d j 1 L+L =b r+s= L +d 1 2 α ( L r+s= L +d 1 2 + 1 2 L+e+f =b I J={2,...,n} α ( b) L (2r+1)!!(2s+1)!! κ(l )τ r τ s L κ(e)τ r i 1 b ) (2r + 1)!!(2s + 1)!! L, e, f i I τ di g κ(f)τ s i J τ di g g. i 1,j τ di g 1 These tautological constants α L can be determined recursively τ di g

from the following formula namely L+L =b ( 1) L α L L!L!(2 L + 1)!! = 0, b 0, α b = b! L+L =b L 0 with the initial value α 0 = 1. ( 1) L 1 α L L!L!(2 L + 1)!!, b 0, Let s := (s 1, s 2,... ) and t := (t 0, t 1, t 2,... ), we introduce the following generating function G(s, t) := g m,n κ m 1 1 κm 2 2 τ n 0 0 τ n 1 1 g sm m! i=0 t n i i n i!,

where s m = i 1 s m i i. We proved that, for any fixed values of s, G(s, t) is a τ-function for a more general KdV hierarchy. When b = (l, 0, 0,... ), this theorem recovers Mirzakhani s famous recursion formula of Weil-Petersson volumes for moduli spaces of bordered Riemann surfaces, which only involves κ 1. The above theorem also provides an effective algorithm to compute higher Weil-Petersson volumes recursively. In fact we can generalize almost all pure ψ intersections to identities of higher Weil-Petersson volumes.

Proof of a conjecture of Itzkson-Zuber Let denom(r) denotes the denominator of a rational number r in reduced form (coprime numerator and denominator, positive denominator). We define D g,n = lcm denom and for g 2, D g = lcm { denom n j=1 ( τ dj g M g κ(b) n j=1 ) where lcm denotes least common multiple. d j = 3g 3 + n b = 3g 3 } Since denominators of intersection numbers on M g,n all come from orbifold quotient singularities, the divisibility properties of D g,n and D g should reflect overall behavior of singularities.

Hao Xu and I have proved the following interesting properties of D g,n and D g. Proposition: We have D g,n D g,n+1, D g,n D g and D g = D g,3g 3. Theorem: For 1 < g g, the order of any automorphism group of a Riemann surface of genus g divides D g,3. The following corollary is a conjecture raised by Itzykson and Zuber in 1992. Corollary: For 1 < g g, the order of any automorphism group of an algebraic curve of genus g divides D g.

Thank You Very Much!