Tautological Ring of Moduli Spaces of Curves

Transcription

1 Talk at Colloquium of University of California at Davis April 12, 2010, 4:10pm

2 Outline of the presentation Basics of moduli spaces of curves and its intersection theory, especially the integrals of ψ classes. We will present our work (with Prof. Kefeng Liu) on n-point functions, higher Weil-Petersson volumes, Faber intersection number conjecture, explicit tautological relations in the moduli spaces of curves. The starting point of our work is the Witten-Kontsevich theorem relating intersection theory and integrable systems.

3 Riemann surfaces A Riemann surfaces is a 2-dimensional complex manifolds (or smooth algebraic curve over C).

4 Riemann surfaces A Riemann surfaces is a 2-dimensional complex manifolds (or smooth algebraic curve over C). Let n 1. The Fermat curve {(x, y, z) P 2 x n + y n = z n }. is a Riemann surface with genus (n 1)(n 2)/2.

5 Stable curves A stable curve with n marked points is a connected and compact curve with nodes with n smooth points labeled by {1,..., n} and satisfy: {(x, y) C 2 xy = 0} (i) each genus 0 component has at least 3 node-branches or marked points; (ii) each genus 1 component has at least one node-branch or marked point.

6 Stable curves A stable curve with n marked points is a connected and compact curve with nodes with n smooth points labeled by {1,..., n} and satisfy: {(x, y) C 2 xy = 0} (i) each genus 0 component has at least 3 node-branches or marked points; (ii) each genus 1 component has at least one node-branch or marked point. a stable curve in M 3,2 unstable curve

7 Fine moduli spaces of curves does not exist Consider contravariant functors Scheme Sets Hom(, S) : B Hom(B, S) (S is a scheme) F V : B (family of k-dim subspaces of V) F Mg : B (family of genus g curves over B)

8 Fine moduli spaces of curves does not exist Consider contravariant functors Scheme Sets Hom(, S) : B Hom(B, S) (S is a scheme) F V : B (family of k-dim subspaces of V) F Mg : B (family of genus g curves over B) Fine moduli: A functor F is representable, i.e. there exists a scheme S, such that F is isomorphic to Hom(, S).

9 Fine moduli spaces of curves does not exist Consider contravariant functors Scheme Sets Hom(, S) : B Hom(B, S) (S is a scheme) F V : B (family of k-dim subspaces of V) F Mg : B (family of genus g curves over B) Fine moduli: A functor F is representable, i.e. there exists a scheme S, such that F is isomorphic to Hom(, S). The grassmannian Grass(k,n) is the fine moduli space of F V. Fine moduli space of curves over C fail to exist due to automorphisms.

10 Coarse moduli spaces of curves There exists a scheme M g called coarse moduli spaces of curves.

11 Coarse moduli spaces of curves There exists a scheme M g called coarse moduli spaces of curves. M g is in fact quasi-projective.

12 Coarse moduli spaces of curves There exists a scheme M g called coarse moduli spaces of curves. M g is in fact quasi-projective. 1. F Mg (SpecC) = Hom(SpecC, M g ) (isomorphism classes of genus g curves points of M g )

13 Coarse moduli spaces of curves There exists a scheme M g called coarse moduli spaces of curves. M g is in fact quasi-projective. 1. F Mg (SpecC) = Hom(SpecC, M g ) (isomorphism classes of genus g curves points of M g ) 2. F Mg φ Hom(, M g )

14 Coarse moduli spaces of curves There exists a scheme M g called coarse moduli spaces of curves. M g is in fact quasi-projective. 1. F Mg (SpecC) = Hom(SpecC, M g ) (isomorphism classes of genus g curves points of M g ) 2. F Mg ψ φ Hom(, M g ) Hom(, N)

15 Coarse moduli spaces of curves There exists a scheme M g called coarse moduli spaces of curves. M g is in fact quasi-projective. 1. F Mg (SpecC) = Hom(SpecC, M g ) (isomorphism classes of genus g curves points of M g ) 2. F Mg ψ φ η Hom(, M g ) Hom(, N)

16 Compactification of moduli spaces of curves Constructions of M g and M g The quotient of Teichmüller spaces by the action of the mapping class group. Geometric invariant theory (Mumford, Gieseker, etc).

17 Compactification of moduli spaces of curves Constructions of M g and M g The quotient of Teichmüller spaces by the action of the mapping class group. Geometric invariant theory (Mumford, Gieseker, etc). The coarse moduli space M g is a dense open subset of its compactification M g, which is projective. It s more popular to treat M g as a Deligne-Mumford stack.

18 Compactification of moduli spaces of curves Constructions of M g and M g The quotient of Teichmüller spaces by the action of the mapping class group. Geometric invariant theory (Mumford, Gieseker, etc). The coarse moduli space M g is a dense open subset of its compactification M g, which is projective. It s more popular to treat M g as a Deligne-Mumford stack. The description of M g indicates that it is natural to consider curves with marked points.

19 Moduli spaces of curves with marked points There are two type of boundary morphisms: M g1,n 1+1 M g2,n 2+1 M g1+g 2,n 1+n 2

20 Moduli spaces of curves with marked points There are two type of boundary morphisms: M g1,n 1+1 M g2,n 2+1 M g1+g 2,n 1+n 2 M g,n+2 M g 1,n

21 Dual graph of nodal curves The following is a nodal curve of genus 3 with two components and two marked points. a curve in M 3,2

22 Dual graph of nodal curves The following is a nodal curve of genus 3 with two components and two marked points. Its associated dual graph is: a curve in M 3,2 Dual graph also denotes the class of the corresponding strata in moduli space.

23 Degeneration in M 0,4 M 0,4 is not complete.

24 Degeneration in M 0,4 M 0,4 is not complete.

25 Degeneration in M 0,4 M 0,4 is not complete.

26 The strata of M 0,4 stratum of nonsingular curves In fact, we have M 0,4 = P 1. boundary strata

27 Intersection theory (Bézout theorem) Let Y, Z be generic distinct curves in P 2, having degree d and e. Then the number of intersection points of Y, Z is de.

28 Intersection theory on moduli spaces of curves Mumford defined the Chow ring A (M g,n ) on moduli spaces of curves in He emphasized the tautological subring: R (M g,n ) A (M g,n ), which contains all geometrically natural classes.

29 Intersection theory on moduli spaces of curves Mumford defined the Chow ring A (M g,n ) on moduli spaces of curves in He emphasized the tautological subring: R (M g,n ) A (M g,n ), which contains all geometrically natural classes. Looijenga, Boggi-Pikaart proved that M g,n is a global quotient of a smooth projective varieties M g,n with action by a finite group G p : M g,n M g,n.

30 Intersection theory on moduli spaces of curves Mumford defined the Chow ring A (M g,n ) on moduli spaces of curves in He emphasized the tautological subring: R (M g,n ) A (M g,n ), which contains all geometrically natural classes. Looijenga, Boggi-Pikaart proved that M g,n is a global quotient of a smooth projective varieties M g,n with action by a finite group G p : M g,n M g,n. For α, β A (M g,n ) Q, define α β = 1 #G (p [p α p β]).

31 Tautological classes on M g,n ψ i = c 1 (L i ), where L i is the line bundle whose fiber over each pointed stable curve is the cotangent line at the ith marked point. L i T i C M g,n [C]

32 Tautological classes on M g,n ψ i = c 1 (L i ), where L i is the line bundle whose fiber over each pointed stable curve is the cotangent line at the ith marked point. L i T i C M g,n [C] λ i = c i (E) the ith Chern class of the Hodge bundle E (with fibre H 0 (C, ω C )). E H 0 (C, ω C ) M g,n [C]

33 Tautological classes on M g,n ψ i = c 1 (L i ), where L i is the line bundle whose fiber over each pointed stable curve is the cotangent line at the ith marked point. L i T i C M g,n [C] λ i = c i (E) the ith Chern class of the Hodge bundle E (with fibre H 0 (C, ω C )). E H 0 (C, ω C ) M g,n [C] κ i = π (ψ i+1 n+1 ), (π : M g,n+1 M g,n is the forgetful morphism).

34 Hodge integrals We call the following integrals the Hodge integrals on moduli spaces of curves τ d1 τ dn κ a1 κ am λ k1 1 λkg g ψ d1 1 ψdn n κ a1 κ am λ k1 1 λkg g. M g,n

35 Hodge integrals We call the following integrals the Hodge integrals on moduli spaces of curves τ d1 τ dn κ a1 κ am λ k1 1 λkg g ψ d1 1 ψdn n κ a1 κ am λ k1 1 λkg g. M g,n Based on Mumford s Chern character formula ch 2m 1 (E) = B n 2m κ 2m 1 ψ 2m 1 i + 1 (2m)! 2 i=1 ξ ( 2m 2 l ξ i=0 ) ψ1( ψ i 2m 2 i 2 ) Faber s algorithm reduces the calculation of general Hodge integrals to those with pure ψ classes.

36 Intersection of pure ψ classes The following integrals are called descendent integrals: τ d1 τ dn g := ψ d1 1 ψdn n, M g,n where d d n = 3g 3 + n.

37 Intersection of pure ψ classes The following integrals are called descendent integrals: τ d1 τ dn g := ψ d1 1 ψdn n, M g,n where d d n = 3g 3 + n. Mathematician: They are intersection numbers over varieties! Physicist: They are correlation functions of 2-D gravity!

38 Numerical properties of intersection numbers We know from Okounkov s work that g=0 dj =3g 3+n converges for all positive real numbers x j. τ d1 τ dn g n j=1 x d j j

39 Numerical properties of intersection numbers We know from Okounkov s work that g=0 dj =3g 3+n converges for all positive real numbers x j. τ d1 τ dn g n j=1 For d 1 < d 2, we have the multinomial value property: x d j j τ d1 τ d2 τ dn g τ d1+1τ d2 1 τ dn g. We have confirmed its validity for all g 20.

40 Moduli of elliptic curves M 1,1 We may identify M 1,1 with the compactification of PSL(2, Z)\H. The unique singular curve at The orbifold structure of M 1,1 arises from a finite quotient of the Riemann sphere CP 1.

41 Compute M 1,1 ψ 1 (Arithmetic) The sections of Ω n over M 1,1 correspondes exactly to entire modular forms of weight n. The Ramanujan tau function = q (1 q n ) 24, where q = e 2πiτ, τ H n=1 is a cusp form of weight 12 with a simple zero at q = 0 (τ = i ).

42 Compute M 1,1 ψ 1 (Arithmetic) The sections of Ω n over M 1,1 correspondes exactly to entire modular forms of weight n. The Ramanujan tau function = q (1 q n ) 24, where q = e 2πiτ, τ H n=1 is a cusp form of weight 12 with a simple zero at q = 0 (τ = i ). Since generic elliptic curve has an involution, degω 12 = deg([ ]) = 1 2 hence we have M 1,1 ψ 1 = degω = 1 24.

43 KdV hierarchy The KdV hierarchy is the following hierarchy of differential equations: U = R n+1, n 1 t n t 0 where R n are differential polynomials in U, U, Ü,... defined recursively by R n+1 R 1 = U, = 1 ( U R n + 2U R n ) R n. t 0 2n + 1 t 0 t 0 4 t 3 0

44 KdV hierarchy The KdV hierarchy is the following hierarchy of differential equations: U = R n+1, n 1 t n t 0 where R n are differential polynomials in U, U, Ü,... defined recursively by R n+1 R 1 = U, = 1 ( U R n + 2U R n ) R n. t 0 2n + 1 t 0 t 0 4 It is easy to compute R n recursively R 2 = 1 2 U U t0 2, R 3 = 1 6 U3 + U 3 U 12 t ( U ) t U t0 4, t 3 0

45 1-soliton solution of KdV hierarchy This is the traveling wave solution u(x, t) = 1 2 c sech2 ( 1 2 c(x ct) + x0 ) of the KdV equation u t + 6uu x + u xxx = 0

46 Scott Russell Aqueduct On the Scott Russell Aqueduct, 12 July 1995 at Edinburgh, Scotland

47 Witten s conjecture The Witten-Kontsevich theorem states that the generating function for ψ class intersection numbers F (t 0, t 1,...) = g n=(n 0,n 1, ) i=0 τ n i i g i=0 t n i i n i! is a τ-function for the KdV hierarchy, i.e. U 2 F / t0 2 equations in the KdV hierarchy. obeys all

48 A diagram

49 The n-point function Definition The following generating function F (x 1,..., x n ) = is called the n-point function. g=0 dj =3g 3+n τ d1 τ dn g n j=1 x d j j

50 The n-point function Definition The following generating function F (x 1,..., x n ) = is called the n-point function. g=0 dj =3g 3+n τ d1 τ dn g n j=1 x d j j Note that n-point functions encoded all information of intersection numbers on moduli spaces of curves. Note its difference with Witten s free energy F (t 0, t 1,... ) = g=0 dj =3g 3+n t d1 t dn τ d1 τ dn g. n!

51 Recursive formula of n-point functions Let n 2. G(x 1,..., x n ) = r,s 0 (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 j 3 3 P r = 1 2 n j=1 x j = 1 2 n j=1 x j n=i J( i I n=i J( i I, x i ) 2 ( i J x i ) 2 ( i J x i ) 2 G(x I )G(x J ) x i ) 2 3r+n 3 r G r (x I )G r r (x J ). r =0

52 New effective recursion formula n (2g + n 1)(2g + n 2) τ dj g j=1 = 2d n τ0 4 τ d1+1 τ dj g 1 2g + n 1 n τ0 3 τ dj g j=2 j=1 + (2d 1 + 3) τ d1+1τ0 2 τ di g τ0 2 τ di g g {2,...,n}=I J {2,...,n}=I J i I i J (2g + n 1) τ d1 τ 0 τ di g τ0 2 τ di g g. When d j 1, all non-zero intesection numbers on RHS have genera strictly less than g. i I i J

53 New effective recursion formula n (2g + n 1)(2g + n 2) τ dj g j=1 = 2d n τ0 4 τ d1+1 τ dj g 1 2g + n 1 n τ0 3 τ dj g j=2 j=1 + (2d 1 + 3) τ d1+1τ0 2 τ di g τ0 2 τ di g g {2,...,n}=I J {2,...,n}=I J i I i J (2g + n 1) τ d1 τ 0 τ di g τ0 2 τ di g g. When d j 1, all non-zero intesection numbers on RHS have genera strictly less than g. This formula of us is used in Stephani Yang s Macaulay2 package KaLaPs. i I i J

54 Why integrals of ψ classes Intersection theory on moduli spaces of curves has several applications and connections: Tautological ring of moduli spaces of curves (Faber s conjecture) Gromov-Witten theory Landau-Ginzburg theory (FJRW invariants) Hurwitz numbers

55 Weil-Petersson volumes V g,n = κ 3g 3+n 1. M g,n Mathematician: They are integrals of Weil-Petersson metric! Physicist: They are multiloop amplitudes of Polyakov string!

56 Mirzakhani s recursion formula of Weil-Petersson volumes Mirzakhani proved a beautiful recursion formula for the Weil-Petersson volume of the moduli space M g,n (L) of genus g hyperbolic surfaces with n geodesic boundary components of specified length L = (L 1,..., L n ). Vol g,n (L) = 1 2L L1 2L 1 0 g 1+g 2=g n=i J L xyh(t, x + y) Vol g1,n 1 (x, L I )Vol g2,n 2 (y, L J )dxdydt L 1 n j=2 xyh(t, x + y)vol g 1,n+1 (x, y, L 2,..., L n )dxdydt 0 L1 where the kernel function 0 H(x, y) = 0 x ( H(x, L 1 + L j ) + H(x, L 1 L j ) ) Vol g,n 1 (x, L 2,..., ˆL j,..., L n )dxdt, e + 1 (x+y)/2 1 + e. (x y)/2

57 Mulase-Safnuk differential form of Mirzakhani s recursion n (2d 1 + 1)!! τ dj κ a 1 g = n a j=2 b=0 a j=1 a! (a b)! b=0 r+s=b+d a (2(b + d 1 + d j ) 1)!! β b κ a b (2d j 1)!! 1 τ b+d1+d j 1 a! (a b)! (2r + 1)!!(2s + 1)!!β b κ a b b=0 c+c =a b I r+s=b+d 1 2 J={2,...,n} τ di g i 1,j 1 τ r τ s τ di g 1 i 1 a! c!c! (2r + 1)!!(2s + 1)!!β b κ c 1τ r τ di g κ c 1 τ s τ di g g, i I i J

58 Mulase-Safnuk differential form of Mirzakhani s recursion n (2d 1 + 1)!! τ dj κ a 1 g = n a j=2 b=0 a j=1 a! (a b)! b=0 r+s=b+d a (2(b + d 1 + d j ) 1)!! β b κ a b (2d j 1)!! 1 τ b+d1+d j 1 a! (a b)! (2r + 1)!!(2s + 1)!!β b κ a b b=0 c+c =a b I r+s=b+d 1 2 J={2,...,n} τ di g i 1,j 1 τ r τ s τ di g 1 i 1 a! c!c! (2r + 1)!!(2s + 1)!!β b κ c 1τ r τ di g κ c 1 τ s τ di g g, i I i J β b = (2 2b+1 4) ζ(2b) (2π 2 ) b = ( 1)b 1 2 b (2 2b 2) B 2b (2b)!.

59 Recursion formula of Higher Weil-Petersson Volumes m = (m 1, m 2,... ) N, define m := i 1 i m i, m := i 1 m i We proved the following recursion formula of Higher WP Volumes. (2d 1 + 1)!! κ(b)τ d1 τ dn g n ( ) b (2( L + d1 + d j ) 1)!! = α L κ(l )τ L +d1+d L (2d j 1)!! j 1 τ di g j=2 L+L =b i 1,j + 1 ( ) b n α L (2r + 1)!!(2s + 1)!! κ(l )τ r τ s τ di g 1 2 L L+L =b r+s= L +d 1 2 i=2 + 1 ( ) b α L (2r + 1)!!(2s + 1)!! 2 L, e, f L+e+f=b I r+s= L +d 1 2 J={2,...,n} κ(e)τ r τ di g κ(f)τ s τ di g g. i I i J

60 Tautological rings Denote by M g the moduli space of Riemann surfaces of genus g 2. The tautological ring R (M g ) is defined to be the Q-subalgebra of the Chow ring A (M g ) generated by the tautological classes κ i and λ i.

61 Tautological rings Denote by M g the moduli space of Riemann surfaces of genus g 2. The tautological ring R (M g ) is defined to be the Q-subalgebra of the Chow ring A (M g ) generated by the tautological classes κ i and λ i. R (M g ) has the following properties: i) (Mumford) R (M g ) is in fact generated by the g 2 classes κ 1,..., κ g 2 ; ii) (Looijenga) R j (M g ) = 0 for j > g 2 and dim R g 2 (M g ) 1 (Faber showed that actually dim R g 2 (M g ) = 1).

62 Tautological rings Denote by M g the moduli space of Riemann surfaces of genus g 2. The tautological ring R (M g ) is defined to be the Q-subalgebra of the Chow ring A (M g ) generated by the tautological classes κ i and λ i. R (M g ) has the following properties: i) (Mumford) R (M g ) is in fact generated by the g 2 classes κ 1,..., κ g 2 ; ii) (Looijenga) R j (M g ) = 0 for j > g 2 and dim R g 2 (M g ) 1 (Faber showed that actually dim R g 2 (M g ) = 1). R (M g ) = C[κ 1,..., κ g 2 ], I where I is the ideal of tautological relations.

63 Faber s conjecture Around 1993, Faber proposed a series of remarkable conjectures about the structure of the tautological ring R (M g ). It is a major conjecture in the subject of moduli spaces of curves. Roughly speaking, Faber s conjecture asserts that R (M g ) behaves like the cohomology ring of a (g 2)-dimensional complex projective manifold.

64 Faber s conjecture Around 1993, Faber proposed a series of remarkable conjectures about the structure of the tautological ring R (M g ). It is a major conjecture in the subject of moduli spaces of curves. Roughly speaking, Faber s conjecture asserts that R (M g ) behaves like the cohomology ring of a (g 2)-dimensional complex projective manifold. i) (Perfect pairing conjecture) When an isomorphism R g 2 (M g ) = Q is fixed, the following natural pairing is perfect R k (M g ) R g 2 k (M g ) R g 2 (M g ) = Q; Faber s perfect paring conjecture is still open to this day. ii) The [g/3] classes κ 1,..., κ [g/3] generate the ring, with no relations in degrees [g/3]; (proved by Morita and Ionel)

65 Faber intersection number conjecture An important part (the only quantitative part) of Faber s conjecture is the famous Faber intersection number conjecture. (2g 3 + n)! 2 2g 1 (2g 1)! n j=1 (2d j 1)!! = τ 2g n j=1 τ dj +2g i j n τ dj g j=1 τ di g + 1 2g 2 ( 1) j τ 2g 2 j τ j 2 n=i J j=0 n τ di g 1 i=1 2g 2 ( 1) j τ j τ di g τ 2g 2 j τ di g g, j=0 i I i J

66 λ g λ g 1 theorem Faber intersection number conjecture is equivalent to ψ d (2g 3 + n)! B 2g ψdn n λ g λ g 1 = M g,n 2 2g 1 (2g)! n j=1 (2d j 1)!!,

67 λ g λ g 1 theorem Faber intersection number conjecture is equivalent to ψ d (2g 3 + n)! B 2g ψdn n λ g λ g 1 = M g,n 2 2g 1 (2g)! n j=1 (2d j 1)!!, This was proved by Getzler and Pandharipande (1998) conditional to Givental s work on Virasoro conjecture for P n. Recently Teleman announced a proof of the Virasoro conjecture for all manifolds with semi-simple quantum cohomology using the Mumford conjecture.

68 Tautological relations in R (M g ) Faber intersection number conjecture is equivalent to the following tautological relation in R (M g ) π (ψ d ψn dn+1 ) = (2g 3 + n)!(2g 1)!! κ σ = (2g 1)! n σ S n j=1 (2d j + 1)!! κ g 2, where π : M g,n M g is the forgetful morphism.

69 Tautological relations in R (M g ) Faber intersection number conjecture is equivalent to the following tautological relation in R (M g ) π (ψ d ψn dn+1 ) = (2g 3 + n)!(2g 1)!! κ σ = (2g 1)! n σ S n j=1 (2d j + 1)!! κ g 2, where π : M g,n M g is the forgetful morphism. In 2006, Goulden, Jackson and Vakil give an enlightening proof of this relation for up to three points. Their remarkable proof relied on relative virtual localization in Gromov-Witten theory and some tour de force combinatorial computations.

70 Importance The Faber intersection number determines the ring structure of R (M g ) if Faber s perfect pairing conjecture is true is a perfect pairing. R k (M g ) R g 2 k (M g ) R g 2 (M g ) = Q

71 Importance The Faber intersection number determines the ring structure of R (M g ) if Faber s perfect pairing conjecture is true is a perfect pairing. R k (M g ) R g 2 k (M g ) R g 2 (M g ) = Q Counterexamples of analogues of Faber s perfect pairing conjecture on partially compactified moduli spaces of curves have recently been found by R. Cavalieri and S. Yang.

72 Proof of Faber intersection number conjecture Now we describe our proof. Since one and two-point functions in genus 0 are F 0 (x) = 1 x 2, F 0(x, y) = 1 x + y = ( 1) k x k y k+1, k=0 it is consistent to define the virtual intersection numbers τ 2 0 = 1, τ k τ 1 k 0 = ( 1) k, k 0.

73 Relations with n-point functions i) 2g 2 1 ( 1) j τ 2g 2 j τ j 2 j=0 n τ di g 1 = [F g 1 (y, y, x 1,..., x n )] y 2g 2 i=1

74 Relations with n-point functions i) 2g 2 1 ( 1) j τ 2g 2 j τ j 2 j=0 n τ di g 1 = [F g 1 (y, y, x 1,..., x n )] y 2g 2 i=1 ii) 1 2 n=i J = 1 2 2g 2 n ( 1) j τ j τ di g τ 2g 2 j τ di g g + τ dj τ 2g g j=0 n=i J j Z i I n j=1 τ dj +2g 1 i J τ di g i j j=1 ( 1) j τ j τ di g τ 2g 2 j τ di g g g = g =0 n=i J i I i J F g (y, x I )F g g ( y, x J ) y 2g 2 n i=1 x d i i

75 Explicit tautological relation in R g 2 (M g ) Let g 3 and m = g 2. Then the following relation κ(m) = 1 ( m 1) L+L =m L 2 A g,l ( m L where A g,l are some explicitly known constants. ) κ(l + δ L ), (1)

76 Explicit tautological relation in R g 2 (M g ) Let g 3 and m = g 2. Then the following relation κ(m) = 1 ( m 1) L+L =m L 2 A g,l ( m L where A g,l are some explicitly known constants. ) κ(l + δ L ), (1) Note that in the right-hand side, L + δ L < m, so it is indeed an effective recursion relation.

77 Another tautological relation in R (M g ) Let m g 2. Then m F g (m) = (g 1) L+L =m L 0 C L F g (L ), where F g (0) = 1 and C L are some explicitly known constants.

78 Another tautological relation in R (M g ) Let m g 2. Then m F g (m) = (g 1) L+L =m L 0 C L F g (L ), where F g (0) = 1 and C L are some explicitly known constants. So these virtual numbers F g (m) when m < g 2 are useful.

79 Gromov-Witten invariants If γ a1,..., γ an H (X, Q), the Gromov-Witten invariants are defined by τ d1 (γ a1 )... τ dn (γ an ) X g,β = Ψ d1 1 Ψdn n ev (γ a1 γ an ). [M g,n(x,β)] virt Mathematician: They are virtual enumerative numbers of curves! Physicist: They are path integrals of the topological field theory!

80 Universal relations of Gromov-Witten invariants String equation τ 0,0 τ k1,a 1... τ kn,a n X g,β = Dilaton equation n τ k1,a 1... τ ki 1,a i... τ kn,a n X g,β. τ 1,0 τ k1,a 1... τ kn,a n X g,β = (2g 2 + n) τ k1,α 1... τ kn,α n X g,β. Divisor equation i=1 τ 0 (ω)τ k1,a 1... τ kn,a n X g,β = ( ω β ) τ k1,a 1... τ kn,a n X g,β n + τ k1,a 1... τ ki 1(ω γ a )... τ kn,a n X g,β, where ω H 2 (X, Q). i=1

81 Topological recursion relations We may pull back tautological relations on M g,n via the forgetful map π : M g,n+1 (X, β) M g,n to get universal equations for Gromov-Witten invariants by the splitting axiom and cotangent line comparison equations.

82 Topological recursion relations We may pull back tautological relations on M g,n via the forgetful map π : M g,n+1 (X, β) M g,n to get universal equations for Gromov-Witten invariants by the splitting axiom and cotangent line comparison equations. On M 1,1, the relation implies the genus 1 TRR τ k (x) 1 = τ k 1 (x)γ α 0 γ α τ k 1(x)γ α γ α 0.

83 WDVV equation From the simple fact that the three boundary divisors of M 0,4 = P 1 are equal, there is the well-known WDVV equation τ k1,a 1 τ k2,a 2 γ α 0 γ α τ k3,a 3 τ k4,a 4 0 = τ k1,a 1 τ k3,a 3 γ α 0 γ α τ k2,a 2 τ k4,a 4 0 which is the associativity condition of the quantum cohomology ring.

84 Vanishing Identities of Gromov-Witten Invariants Our proof of Faber intersection number conjecture (X = pt) motivates us to formulate the following conjecture Conjecture Let x i, y i H (X ) and k 2g 3 + r + s. Then g r s ( 1) j τ j (γ a ) τ pi (x i ) g τ k j (γ a ) τ qi (y i ) g g = 0. g =0 j Z i=1 Note that j runs over all integers. where we adopt Gathmann s convention τ 2 (pt) X 0,0 = 1 and τ m (γ 1 )τ 1 m (γ 2 ) X 0,0 = ( 1) max(m, 1 m) i=1 X γ 1 γ 2, m Z. The conjecture has recently been proved recently by Xiaobo Liu and R. Pandharipande.

85 Virasoro conjecture Let r = dim X and {γ a } a basis in H (X ). i) R b a γ b = c 1 (X ) γ a, t a k = ta k δ a0δ k1 ii) [x] k i = e k+1 i (x, x + 1,..., x + k) iii) If γ a H pa,qa (X ), b a = p a + (1 r)/2. L k = k+1( [b a +m] k i (R i ) b a t m a b,m+k i m=0 i=0 + 2 ( 1)m+1 [b a m 1] k i (R i ) ab a,m b,k m i 1 ) (Rk+1 ) ab t0t a 0 b + δ k0 48 X ( (3 r)cr (X ) 2c 1 (X )c r 1 (X ) ). Virasoro conjecture (Eguchi, Hori and Xiong and also by Katz) L k (exp F ) = 0, k 1.

86 Eynard-Orantin s recursion Originated from random matrix theory, Eynard and Orantin developed a theory of symplectic invariants of curves. Bouchard-Klemm-Mariño-Pasquetti (BKMP) further conjectures that Eynard-Orantin s recursion can be used to compute the Gromov-Witten invariants of toric Calabi-Yau 3-folds through mirror symmetry. Bouchard-Mariño conjectured a recursive formula for simple Hurwitz numbers based on BKMP. Bouchard-Mariño conjecture has been proved recently by Borot, Eynard, Mulase and Safnuk.

87 ADE singularities A n : W = x n+1, n 1 D n+2 : W = x n+1 + xy 2, n 2 E 6 : W = x 3 + y 4 E 7 : W = x 3 + xy 3 E 8 : W = x 3 + y 5 Mathematician: They are ADE singularities! Physicist: They are potential functions of Landau-Ginzburg theory!

88 Landau-Ginzburg model In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy with a degenerate critical point is called a Landau-Ginzburg theory.

89 Landau-Ginzburg model In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy with a degenerate critical point is called a Landau-Ginzburg theory. (Greene-Vafa-Warner) Landau-Ginzburg theory is related by a renormalization group flow to sigma models on Calabi-Yau manifolds. (Witten) Landau-Ginzburg theory and sigma model on Calabi-Yau manifolds are different phases of the same theory, similar to the relation between the gas and the liquid phases of a fluid.

90 Landau-Ginzburg model In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy with a degenerate critical point is called a Landau-Ginzburg theory. (Greene-Vafa-Warner) Landau-Ginzburg theory is related by a renormalization group flow to sigma models on Calabi-Yau manifolds. (Witten) Landau-Ginzburg theory and sigma model on Calabi-Yau manifolds are different phases of the same theory, similar to the relation between the gas and the liquid phases of a fluid. Examples include: The quintic Calabi-Yau 3-fold W = x x x x x 5 5 Orbifold Calabi-Yau in weighted projective spaces. Simple, unimodal and bimodal singularities.

91 Landau-Ginzburg model The Landau-Ginzburg potential is a quasi-homogeneous polynomial W : C N C with unique weights and an isolated singularity at the origin.

92 Landau-Ginzburg model The Landau-Ginzburg potential is a quasi-homogeneous polynomial W : C N C with unique weights and an isolated singularity at the origin. There are weights (or charges) q 1,..., q N such that W (λ q1 x 1,..., λ q N x N ) = λw (x 1,..., x N ) for all λ C with central charge ĉ W = (1 2q i ).

93 Landau-Ginzburg B-model The Landau-Ginzburg B-model is the Milnor ring (or Chiral ring): A local Artin ring. D W := C[x 1,..., x N ] ( W x 1,..., W x N )

94 Landau-Ginzburg B-model The Landau-Ginzburg B-model is the Milnor ring (or Chiral ring): A local Artin ring. D W := C[x 1,..., x N ] ( W x 1,..., W The unique highest degree element is det( 2 W x i x j ) x N )

95 Landau-Ginzburg A-model For any quasi-homogeneous, non-degenerate singularity W and an admissible group G of diagonal symmetry of W. Following a suggestion of Witten, Fan, Jarvis and Ruan constructed i) FJRW state space (a Frobenius algebra) H W,G = g G H g, where H g = H mid (Fixg, W g, C) G ii) Moduli of stable W -orbicurves and its virual cycles.

96 Landau-Ginzburg A-model For any quasi-homogeneous, non-degenerate singularity W and an admissible group G of diagonal symmetry of W. Following a suggestion of Witten, Fan, Jarvis and Ruan constructed i) FJRW state space (a Frobenius algebra) H W,G = g G H g, where H g = H mid (Fixg, W g, C) G ii) Moduli of stable W -orbicurves and its virual cycles. The associated cohomological field theory defines Gromov-Witten type invariants for W.

97 Intersection theory and singularities Table: Simple and elliptic singularities W ĉ W moduli of W curves integrable hierarchy A 1 0 M g,n KdV A r 1, r 2 D n, n 4 E E E P 8 = x 3 + y 3 + z 3 1 X 9 = x 4 + y 4 1 J 10 = x 3 + y 6 1 r 2 r M 1/r g,n n 2 n 1 Gelfand-Dickey Drinfeld-Sokolov Kac-Wakimoto Don t know yet

98 Thank you!