Intersection Theory on Moduli Space of Curves and Their Connection to Integrable Systems

Intersection Theory on Moduli Space of Curves and their connection to Integrable Systems

ICTS Bangalore

Chitrabhanu Chaudhuri

IISER Pune

17 July, 2018 Genus = 2 Genus =1

The genus of the surface is the number of handles that we attach to the sphere. For example the 2-sphere has genus 0, where as the torus has genus 1.

Moduli of Riemann Surfaces

A Riemann surface is a 1 dimensional complex manifold. We are interested in compact Riemann surfaces. Any such surface is topologically, a sphere with g handles attached.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 2 / 20 The genus of the surface is the number of handles that we attach to the sphere. For example the 2-sphere has genus 0, where as the torus has genus 1.

Moduli of Riemann Surfaces

A Riemann surface is a 1 dimensional complex manifold. We are interested in compact Riemann surfaces. Any such surface is topologically, a sphere with g handles attached.

Genus = 2 Genus =1

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 2 / 20 Moduli of Riemann Surfaces

A Riemann surface is a 1 dimensional complex manifold. We are interested in compact Riemann surfaces. Any such surface is topologically, a sphere with g handles attached.

Genus = 2 Genus =1

The genus of the surface is the number of handles that we attach to the sphere. For example the 2-sphere has genus 0, where as the torus has genus 1.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 2 / 20 It can be realized as a quotient of a complex manifold of dimension 3g − 3 by the action of a discrete group. Such a space is called a complex orbifold. This space is not compact.

We shall also consider the moduli spaces of genus g, compact, Riemann surfaces with n marked points, (C; x1,..., xn). This space is denoted by Mg,n. The points x1,..., xn are all distinct.

The moduli space, Mg , is a space parametrizing isomorphism classes of compact Riemann Surfaces of genus g.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 3 / 20 We shall also consider the moduli spaces of genus g, compact, Riemann surfaces with n marked points, (C; x1,..., xn). This space is denoted by Mg,n. The points x1,..., xn are all distinct.

The moduli space, Mg , is a space parametrizing isomorphism classes of compact Riemann Surfaces of genus g.

It can be realized as a quotient of a complex manifold of dimension 3g − 3 by the action of a discrete group. Such a space is called a complex orbifold. This space is not compact.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 3 / 20 The moduli space, Mg , is a space parametrizing isomorphism classes of compact Riemann Surfaces of genus g.

It can be realized as a quotient of a complex manifold of dimension 3g − 3 by the action of a discrete group. Such a space is called a complex orbifold. This space is not compact.

We shall also consider the moduli spaces of genus g, compact, Riemann surfaces with n marked points, (C; x1,..., xn). This space is denoted by Mg,n. The points x1,..., xn are all distinct.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 3 / 20 Similarly there is a compactiﬁcation of Mg,n denoted by Mg,n. This space parametrizes pinched and marked Riemann surfaces which are degenerations of Riemann surfaces of genus g with n marked points.

Such a surface is called stable, in particular

χ(C − {x1,..., xn} − {singular points}) < 0. (1)

Mg,n is a compact complex orbifold of dimension 3g − 3 + n.

There is a map π : Mg,n+1 → Mg,n obtained by forgetting the last marked point and deleting any unstable component.

There is a compactiﬁcation of the moduli space Mg which we denote by Mg . This space parametrizes “pinched” Riemann surfaces. The pinchings are called singularities.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 4 / 20 Such a surface is called stable, in particular

χ(C − {x1,..., xn} − {singular points}) < 0. (1)

Mg,n is a compact complex orbifold of dimension 3g − 3 + n.

There is a map π : Mg,n+1 → Mg,n obtained by forgetting the last marked point and deleting any unstable component.

There is a compactiﬁcation of the moduli space Mg which we denote by Mg . This space parametrizes “pinched” Riemann surfaces. The pinchings are called singularities.

Similarly there is a compactiﬁcation of Mg,n denoted by Mg,n. This space parametrizes pinched and marked Riemann surfaces which are degenerations of Riemann surfaces of genus g with n marked points.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 4 / 20 Mg,n is a compact complex orbifold of dimension 3g − 3 + n.

There is a map π : Mg,n+1 → Mg,n obtained by forgetting the last marked point and deleting any unstable component.

There is a compactiﬁcation of the moduli space Mg which we denote by Mg . This space parametrizes “pinched” Riemann surfaces. The pinchings are called singularities.

Such a surface is called stable, in particular

χ(C − {x1,..., xn} − {singular points}) < 0. (1)

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 4 / 20 There is a map π : Mg,n+1 → Mg,n obtained by forgetting the last marked point and deleting any unstable component.

There is a compactiﬁcation of the moduli space Mg which we denote by Mg . This space parametrizes “pinched” Riemann surfaces. The pinchings are called singularities.

Such a surface is called stable, in particular

χ(C − {x1,..., xn} − {singular points}) < 0. (1)

Mg,n is a compact complex orbifold of dimension 3g − 3 + n.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 4 / 20 There is a compactiﬁcation of the moduli space Mg which we denote by Mg . This space parametrizes “pinched” Riemann surfaces. The pinchings are called singularities.

Such a surface is called stable, in particular

χ(C − {x1,..., xn} − {singular points}) < 0. (1)

Mg,n is a compact complex orbifold of dimension 3g − 3 + n.

There is a map π : Mg,n+1 → Mg,n obtained by forgetting the last marked point and deleting any unstable component.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 4 / 20 Degenerations of marked Riemann Surfaces

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 5 / 20 Let Li be the line bundle on Mg,n whose ﬁber at (C; x1,..., xn) is the cotangent line at xi . Deﬁne

2 ψi = c1(Li ) ∈ H (Mg,n, C). (2)

Let H be the vector bundle on Mg,n whose ﬁber at (C; x1,..., xn) is the 0 vector space H (C, ΩC ). This is a rank g vector bundle. Deﬁne

2i λi = ci (H) ∈ H (Mg,n, C). (3)

Psi classes and the Hodge bundle

There are several line bundles and vector bundles which naturally occur on Mg,n.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 6 / 20 Let H be the vector bundle on Mg,n whose ﬁber at (C; x1,..., xn) is the 0 vector space H (C, ΩC ). This is a rank g vector bundle. Deﬁne

2i λi = ci (H) ∈ H (Mg,n, C). (3)

Psi classes and the Hodge bundle

There are several line bundles and vector bundles which naturally occur on Mg,n.

Let Li be the line bundle on Mg,n whose ﬁber at (C; x1,..., xn) is the cotangent line at xi . Deﬁne

2 ψi = c1(Li ) ∈ H (Mg,n, C). (2)

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 6 / 20 Psi classes and the Hodge bundle

There are several line bundles and vector bundles which naturally occur on Mg,n.

Let Li be the line bundle on Mg,n whose ﬁber at (C; x1,..., xn) is the cotangent line at xi . Deﬁne

2 ψi = c1(Li ) ∈ H (Mg,n, C). (2)

Let H be the vector bundle on Mg,n whose ﬁber at (C; x1,..., xn) is the 0 vector space H (C, ΩC ). This is a rank g vector bundle. Deﬁne

2i λi = ci (H) ∈ H (Mg,n, C). (3)

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 6 / 20 If d1 + ... + dn = 3g − 3 + n, deﬁne Z d1 dn hτd1 ··· τdn i = ψ1 ··· ψn . Mg,n otherwise 0.

For example M0,3 is a point hence Z 3 hτ0τ0τ0i = hτ0 i = 1 = 1. M0,3 where as if n 6= 3 n hτ0 i = 0.

Intersection numbers

There are certain cohmology classes on Mg,n which are called tautological classes. It turns out that the intersections of all tautological classes can be obtained just by knowing the intersections of the ψ classes.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 7 / 20 For example M0,3 is a point hence Z 3 hτ0τ0τ0i = hτ0 i = 1 = 1. M0,3 where as if n 6= 3 n hτ0 i = 0.

Intersection numbers

If d1 + ... + dn = 3g − 3 + n, deﬁne Z d1 dn hτd1 ··· τdn i = ψ1 ··· ψn . Mg,n otherwise 0.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 7 / 20 Intersection numbers

If d1 + ... + dn = 3g − 3 + n, deﬁne Z d1 dn hτd1 ··· τdn i = ψ1 ··· ψn . Mg,n otherwise 0.

For example M0,3 is a point hence Z 3 hτ0τ0τ0i = hτ0 i = 1 = 1. M0,3 where as if n 6= 3 n hτ0 i = 0.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 7 / 20 Some of these intersection numbers were already known. For example Z 1 hτ1i1,1 = ψ1 = . M1,1 24

It was also known but harder to show that if g > 0 Z 3g−2 1 hτ3g−2ig,1 = ψ1 = g Mg,1 24 g!

and if d1 + ··· + dn = n − 3 (n − 3)! hτd1 ··· τdn i0,n = . d1! ··· dn!

Sometimes we write hτd1 ··· τdn i = hτd1 ··· τdn ig,n, where, as before d1 + ··· + dn = 3g − 3 + n.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 8 / 20 It was also known but harder to show that if g > 0 Z 3g−2 1 hτ3g−2ig,1 = ψ1 = g Mg,1 24 g!

and if d1 + ··· + dn = n − 3 (n − 3)! hτd1 ··· τdn i0,n = . d1! ··· dn!

Sometimes we write hτd1 ··· τdn i = hτd1 ··· τdn ig,n, where, as before d1 + ··· + dn = 3g − 3 + n.

Some of these intersection numbers were already known. For example Z 1 hτ1i1,1 = ψ1 = . M1,1 24

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 8 / 20 and if d1 + ··· + dn = n − 3 (n − 3)! hτd1 ··· τdn i0,n = . d1! ··· dn!

Sometimes we write hτd1 ··· τdn i = hτd1 ··· τdn ig,n, where, as before d1 + ··· + dn = 3g − 3 + n.

Some of these intersection numbers were already known. For example Z 1 hτ1i1,1 = ψ1 = . M1,1 24

It was also known but harder to show that if g > 0 Z 3g−2 1 hτ3g−2ig,1 = ψ1 = g Mg,1 24 g!

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 8 / 20 Sometimes we write hτd1 ··· τdn i = hτd1 ··· τdn ig,n, where, as before d1 + ··· + dn = 3g − 3 + n.

Some of these intersection numbers were already known. For example Z 1 hτ1i1,1 = ψ1 = . M1,1 24

It was also known but harder to show that if g > 0 Z 3g−2 1 hτ3g−2ig,1 = ψ1 = g Mg,1 24 g! and if d1 + ··· + dn = n − 3 (n − 3)! hτd1 ··· τdn i0,n = . d1! ··· dn!

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 8 / 20 The ﬁrst few terms can be calculated easily and we list a few of them 1 1 1 F (t , t , t ,...) = t3 + t3t + t 0 1 2 6 0 6 0 1 24 1 1 1 1 1 + t3t2 + t4t + t2 + t t + ... 6 0 1 24 0 2 48 1 24 0 2

The Witten conjecture gives an elegant way of recursively ﬁnding the coeﬃcients of this generating function and hence for computing all the intersections of the ψ classes.

We form the generating function

∞ X X 1 F (t , t , t ,...) = hτ ··· τ it ··· t . 0 1 2 n! d1 dn d1 dn n=1 d1,...,dn

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 9 / 20 The Witten conjecture gives an elegant way of recursively ﬁnding the coeﬃcients of this generating function and hence for computing all the intersections of the ψ classes.

We form the generating function

∞ X X 1 F (t , t , t ,...) = hτ ··· τ it ··· t . 0 1 2 n! d1 dn d1 dn n=1 d1,...,dn

The ﬁrst few terms can be calculated easily and we list a few of them 1 1 1 F (t , t , t ,...) = t3 + t3t + t 0 1 2 6 0 6 0 1 24 1 1 1 1 1 + t3t2 + t4t + t2 + t t + ... 6 0 1 24 0 2 48 1 24 0 2

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 9 / 20 We form the generating function

∞ X X 1 F (t , t , t ,...) = hτ ··· τ it ··· t . 0 1 2 n! d1 dn d1 dn n=1 d1,...,dn

The ﬁrst few terms can be calculated easily and we list a few of them 1 1 1 F (t , t , t ,...) = t3 + t3t + t 0 1 2 6 0 6 0 1 24 1 1 1 1 1 + t3t2 + t4t + t2 + t t + ... 6 0 1 24 0 2 48 1 24 0 2

The Witten conjecture gives an elegant way of recursively ﬁnding the coeﬃcients of this generating function and hence for computing all the intersections of the ψ classes.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 9 / 20 The Witten conjecture says that ∂U ∂ = Ri (U, U˙ , U¨ ,...); ∂ti ∂t0

where the polynomials Ri are deﬁned recursively by

3 ∂Rn+1 1 ˙ ∂ 1 ∂ R0 = U, = U + 2U + 3 Rn. ∂t0 2n + 1 ∂t0 4 ∂t0

This system of equations is called the KdV (Korteweg-de Vries) hierarchy, and the polynomials Ri are called the Gelfand-Dikii polynomials.

Witten conjecture Let 2 2 ∂ ˙ ∂ ¨ ∂ U = 2 F , U = U, U = 2 U ... ∂t0 ∂t0 ∂t0

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 10 / 20 This system of equations is called the KdV (Korteweg-de Vries) hierarchy, and the polynomials Ri are called the Gelfand-Dikii polynomials.

Witten conjecture Let 2 2 ∂ ˙ ∂ ¨ ∂ U = 2 F , U = U, U = 2 U ... ∂t0 ∂t0 ∂t0

The Witten conjecture says that ∂U ∂ = Ri (U, U˙ , U¨ ,...); ∂ti ∂t0

where the polynomials Ri are deﬁned recursively by

3 ∂Rn+1 1 ˙ ∂ 1 ∂ R0 = U, = U + 2U + 3 Rn. ∂t0 2n + 1 ∂t0 4 ∂t0

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 10 / 20 Witten conjecture Let 2 2 ∂ ˙ ∂ ¨ ∂ U = 2 F , U = U, U = 2 U ... ∂t0 ∂t0 ∂t0

The Witten conjecture says that ∂U ∂ = Ri (U, U˙ , U¨ ,...); ∂ti ∂t0

where the polynomials Ri are deﬁned recursively by

3 ∂Rn+1 1 ˙ ∂ 1 ∂ R0 = U, = U + 2U + 3 Rn. ∂t0 2n + 1 ∂t0 4 ∂t0

This system of equations is called the KdV (Korteweg-de Vries) hierarchy, and the polynomials Ri are called the Gelfand-Dikii polynomials.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 10 / 20 The ﬁrst equation in the KdV hierarchy, that is, ∂U ∂ = R1 ∂t1 ∂t0 is called the KdV equation.

The ﬁrst few Gelfand-Dikii polynomials are

R0 =U, 1 1 R = U2 + U¨ , 1 2 12 5 5 1 .... R = U3 + UU¨ + U . 2 6 12 48

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 11 / 20 The ﬁrst few Gelfand-Dikii polynomials are

R0 =U, 1 1 R = U2 + U¨ , 1 2 12 5 5 1 .... R = U3 + UU¨ + U . 2 6 12 48

The ﬁrst equation in the KdV hierarchy, that is, ∂U ∂ = R1 ∂t1 ∂t0 is called the KdV equation.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 11 / 20 In this paper Witten also proved that F satisﬁes the “string equation”:

2 ∞ ∂ t0 X ∂F F = + ti+1 . ∂t0 2 ∂ti i=0 In terms of intersection numbers the string equation reads X hτ0τd1 ··· τdn i = hτd1 ··· τdi −1 ··· τdn i.

{i | di >0}

It can be shown that the string equation and the KdV equation generate the KdV hierarchy.

Witten conjecture appeared ﬁrst in his 1991 paper “Two-dimensional Gravity and Intersection Theory on Moduli Space”.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 12 / 20 In terms of intersection numbers the string equation reads X hτ0τd1 ··· τdn i = hτd1 ··· τdi −1 ··· τdn i.

{i | di >0}

It can be shown that the string equation and the KdV equation generate the KdV hierarchy.

Witten conjecture appeared ﬁrst in his 1991 paper “Two-dimensional Gravity and Intersection Theory on Moduli Space”.

In this paper Witten also proved that F satisﬁes the “string equation”:

2 ∞ ∂ t0 X ∂F F = + ti+1 . ∂t0 2 ∂ti i=0

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 12 / 20 It can be shown that the string equation and the KdV equation generate the KdV hierarchy.

Witten conjecture appeared ﬁrst in his 1991 paper “Two-dimensional Gravity and Intersection Theory on Moduli Space”.

In this paper Witten also proved that F satisﬁes the “string equation”:

2 ∞ ∂ t0 X ∂F F = + ti+1 . ∂t0 2 ∂ti i=0 In terms of intersection numbers the string equation reads X hτ0τd1 ··· τdn i = hτd1 ··· τdi −1 ··· τdn i.

{i | di >0}

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 12 / 20 Witten conjecture appeared ﬁrst in his 1991 paper “Two-dimensional Gravity and Intersection Theory on Moduli Space”.

In this paper Witten also proved that F satisﬁes the “string equation”:

2 ∞ ∂ t0 X ∂F F = + ti+1 . ∂t0 2 ∂ti i=0 In terms of intersection numbers the string equation reads X hτ0τd1 ··· τdn i = hτd1 ··· τdi −1 ··· τdn i.

{i | di >0}

It can be shown that the string equation and the KdV equation generate the KdV hierarchy.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 12 / 20 The ﬁrst proof of this theorem was given By Kontsevic (1992) using a certain cellular decomposition of Mg,n and matrix integrals.

There is a particularly beautiful proof of the theorem by Mirzakhani (2007). She uses symplectic and hyperbolic geometry.

Here we shall discuss a proof by Kazarian and Lando (2007), which uses Hurwitz numbers and the ELSV formula.

Witten conjecture, now a theorem due to Kontsevic and many others, thus can be simply stated as: Theorem (Witten, Kontsevic) ∂ ∂U 1 ∂3U U = U + 3 (KdV equation). ∂t1 ∂t0 12 ∂t0

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 13 / 20 There is a particularly beautiful proof of the theorem by Mirzakhani (2007). She uses symplectic and hyperbolic geometry.

Here we shall discuss a proof by Kazarian and Lando (2007), which uses Hurwitz numbers and the ELSV formula.

Witten conjecture, now a theorem due to Kontsevic and many others, thus can be simply stated as: Theorem (Witten, Kontsevic) ∂ ∂U 1 ∂3U U = U + 3 (KdV equation). ∂t1 ∂t0 12 ∂t0

The ﬁrst proof of this theorem was given By Kontsevic (1992) using a certain cellular decomposition of Mg,n and matrix integrals.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 13 / 20 Here we shall discuss a proof by Kazarian and Lando (2007), which uses Hurwitz numbers and the ELSV formula.

Witten conjecture, now a theorem due to Kontsevic and many others, thus can be simply stated as: Theorem (Witten, Kontsevic) ∂ ∂U 1 ∂3U U = U + 3 (KdV equation). ∂t1 ∂t0 12 ∂t0

The ﬁrst proof of this theorem was given By Kontsevic (1992) using a certain cellular decomposition of Mg,n and matrix integrals.

There is a particularly beautiful proof of the theorem by Mirzakhani (2007). She uses symplectic and hyperbolic geometry.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 13 / 20 Witten conjecture, now a theorem due to Kontsevic and many others, thus can be simply stated as: Theorem (Witten, Kontsevic) ∂ ∂U 1 ∂3U U = U + 3 (KdV equation). ∂t1 ∂t0 12 ∂t0

The ﬁrst proof of this theorem was given By Kontsevic (1992) using a certain cellular decomposition of Mg,n and matrix integrals.

There is a particularly beautiful proof of the theorem by Mirzakhani (2007). She uses symplectic and hyperbolic geometry.

Here we shall discuss a proof by Kazarian and Lando (2007), which uses Hurwitz numbers and the ELSV formula.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 13 / 20 If genus of S is g then the number of branch points m other than ∞ is given by the Riemann-Hurwitz formula

m = 2g − 2 + n + d.

The complex structure on S is determined by the topological type of the covering.

Hurwitz numbers

Let (b1,..., bn) be an ordered partition of d. Consider connected branched 1 covers S of the Riemann Sphere CP with the following branching data:

∞ has n pre-images with ramiﬁcation indices b1,..., bn, for any other branch point the branching is simple, that is the ramiﬁcation indices above it are 2, 1, 1,..., 1.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 14 / 20 The complex structure on S is determined by the topological type of the covering.

Hurwitz numbers

Let (b1,..., bn) be an ordered partition of d. Consider connected branched 1 covers S of the Riemann Sphere CP with the following branching data:

∞ has n pre-images with ramiﬁcation indices b1,..., bn, for any other branch point the branching is simple, that is the ramiﬁcation indices above it are 2, 1, 1,..., 1.

If genus of S is g then the number of branch points m other than ∞ is given by the Riemann-Hurwitz formula

m = 2g − 2 + n + d.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 14 / 20 Hurwitz numbers

Let (b1,..., bn) be an ordered partition of d. Consider connected branched 1 covers S of the Riemann Sphere CP with the following branching data:

∞ has n pre-images with ramiﬁcation indices b1,..., bn, for any other branch point the branching is simple, that is the ramiﬁcation indices above it are 2, 1, 1,..., 1.

If genus of S is g then the number of branch points m other than ∞ is given by the Riemann-Hurwitz formula

m = 2g − 2 + n + d.

The complex structure on S is determined by the topological type of the covering.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 14 / 20 1

1 2 m Genus g surface

d sheeted branched covering n

8 Riemann Sphere

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 15 / 20 Up to a combinatorial factor hg,b1,...,bn counts the number of ways of factoring a transitive d-permutation with conjugacy class (b1,..., bn) into m transpositions.

The Hurwitz numbers are interesting in their own right and can be 1 thought of as certain Gromov-Witten invariants of CP .

Hurwitz numbers

1 If we fix the images of the ramification points on CP , then there are only finitely many distinct covers upto isomorphism. Denote that number by

hg,b1,...,bn . These numbers are called Hurwitz numbers.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 16 / 20 The Hurwitz numbers are interesting in their own right and can be 1 thought of as certain Gromov-Witten invariants of CP .

Hurwitz numbers

1 If we fix the images of the ramification points on CP , then there are only finitely many distinct covers upto isomorphism. Denote that number by

hg,b1,...,bn . These numbers are called Hurwitz numbers.

Up to a combinatorial factor hg,b1,...,bn counts the number of ways of factoring a transitive d-permutation with conjugacy class (b1,..., bn) into m transpositions.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 16 / 20 Hurwitz numbers

1 If we fix the images of the ramification points on CP , then there are only finitely many distinct covers upto isomorphism. Denote that number by

hg,b1,...,bn . These numbers are called Hurwitz numbers.

Up to a combinatorial factor hg,b1,...,bn counts the number of ways of factoring a transitive d-permutation with conjugacy class (b1,..., bn) into m transpositions.

The Hurwitz numbers are interesting in their own right and can be 1 thought of as certain Gromov-Witten invariants of CP .

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 16 / 20 The numerator in the integral is just the total Chern class of the dual of the Hodge bundle that is c(H∨).

The integral is understood as expanding the denominator as a power series and only picking up monomials of the correct degree that is 3g − 3 + n in the entire product.

The ELSV formula

Ekedahl-Lando-Shapiro-Vainshtein (ELSV) formula gives an amazing relation between the Hurwitz numbers and some intersection numbers on Mg,n n bi Z Y bi 1 − λ1 + · · · ± λg hg,b1,...,bn = m! . bi ! (1 − b1ψ1) ··· (1 − bnψn) i=1 Mg,n

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 17 / 20 The integral is understood as expanding the denominator as a power series and only picking up monomials of the correct degree that is 3g − 3 + n in the entire product.

The ELSV formula

The numerator in the integral is just the total Chern class of the dual of the Hodge bundle that is c(H∨).

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 17 / 20 The ELSV formula

The numerator in the integral is just the total Chern class of the dual of the Hodge bundle that is c(H∨).

The integral is understood as expanding the denominator as a power series and only picking up monomials of the correct degree that is 3g − 3 + n in the entire product.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 17 / 20 Okounkov (2000) showed that eH satisﬁes the Kadomtsev-Petviashvili (KP) hierarchy, which in particular shows that H satisﬁes the KP equation.

∂2H ∂2H 1 ∂2H 2 1 ∂4H 2 = − 2 − 4 . ∂s2 ∂s1∂s3 2 ∂s1 12 ∂s1

Kazarian and Lando use this equation to deduce the KdV equation for U.

Consider the generating function for Hurwitz numbers

∞ m X X x sb ··· sb H(x, s , s ,...) = h 1 n . 1 2 g,b1,...,bn m! n! g=0 b1,...,bn

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 18 / 20 Kazarian and Lando use this equation to deduce the KdV equation for U.

Consider the generating function for Hurwitz numbers

∞ m X X x sb ··· sb H(x, s , s ,...) = h 1 n . 1 2 g,b1,...,bn m! n! g=0 b1,...,bn

Okounkov (2000) showed that eH satisﬁes the Kadomtsev-Petviashvili (KP) hierarchy, which in particular shows that H satisﬁes the KP equation.

∂2H ∂2H 1 ∂2H 2 1 ∂4H 2 = − 2 − 4 . ∂s2 ∂s1∂s3 2 ∂s1 12 ∂s1

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 18 / 20 Consider the generating function for Hurwitz numbers

∞ m X X x sb ··· sb H(x, s , s ,...) = h 1 n . 1 2 g,b1,...,bn m! n! g=0 b1,...,bn

Okounkov (2000) showed that eH satisﬁes the Kadomtsev-Petviashvili (KP) hierarchy, which in particular shows that H satisﬁes the KP equation.

∂2H ∂2H 1 ∂2H 2 1 ∂4H 2 = − 2 − 4 . ∂s2 ∂s1∂s3 2 ∂s1 12 ∂s1

Kazarian and Lando use this equation to deduce the KdV equation for U.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 18 / 20 By a simple combinatorial technique K-L eliminate the λ classes from the ELSV formula and obtain the following explicit formula

d +1 d +1 n ! X1 Xn 1 Y (−1)di −b1+1 hτd ··· τd i = ··· hg,b ,...b . 1 n bi −1 1 n m! (di − b1 + 1)!b b1=1 bn=1 i=1 i

This gives a simple relation between the generating functions U and H after some clever change of variables. Finally KdV equation for U is deduced from the KP equation for H.

The method of proof is the following:

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 19 / 20 This gives a simple relation between the generating functions U and H after some clever change of variables. Finally KdV equation for U is deduced from the KP equation for H.

The method of proof is the following:

By a simple combinatorial technique K-L eliminate the λ classes from the ELSV formula and obtain the following explicit formula

d +1 d +1 n ! X1 Xn 1 Y (−1)di −b1+1 hτd ··· τd i = ··· hg,b ,...b . 1 n bi −1 1 n m! (di − b1 + 1)!b b1=1 bn=1 i=1 i

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 19 / 20 Finally KdV equation for U is deduced from the KP equation for H.

The method of proof is the following:

By a simple combinatorial technique K-L eliminate the λ classes from the ELSV formula and obtain the following explicit formula

d +1 d +1 n ! X1 Xn 1 Y (−1)di −b1+1 hτd ··· τd i = ··· hg,b ,...b . 1 n bi −1 1 n m! (di − b1 + 1)!b b1=1 bn=1 i=1 i

This gives a simple relation between the generating functions U and H after some clever change of variables.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 19 / 20 The method of proof is the following:

By a simple combinatorial technique K-L eliminate the λ classes from the ELSV formula and obtain the following explicit formula

d +1 d +1 n ! X1 Xn 1 Y (−1)di −b1+1 hτd ··· τd i = ··· hg,b ,...b . 1 n bi −1 1 n m! (di − b1 + 1)!b b1=1 bn=1 i=1 i

This gives a simple relation between the generating functions U and H after some clever change of variables. Finally KdV equation for U is deduced from the KP equation for H.

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 19 / 20 References

1 P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Etudes Sci. Publ. Math.(1969)

2 E. Witten, Two-dimensional gravity and intersection theory on moduli spaces, Surveys in Diﬀerential Geometry (1991)

3 M. Kontsevich, Intersection theory on the moduli space of curves and the Airy function, Comm. Math. Phys.(1992)

4 A. Okounkov, Toda equations for Hurtwitz numbers, Math. Res. Lett.(2000)

5 T. Ekedahl, S. K. Lando, M. Shapiro, A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. math.(2001)

6 M. Mirzakhani, WeilPetersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc.(2007)

7 M.E. Kazarian, S.K. Lando, An algebro-geometric proof of Witten’s conjecture, J. Amer. Math. Soc.(2007)

C. Chaudhuri (IISER Pune) Witten Conjecture 17 July, 2018 20 / 20