On the BCOV Conjecture

Transcription

1 Department of Mathematics University of California, Irvine December 14, 2007

2 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called the mirror of X, such that the quantum field theory over both manifolds are the same. At this moment, it is not possible to verify Mirror Symmetry in physics. However, mathematically, we can verify some conjectures predicted by Mirror Symmetry.

3 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called the mirror of X, such that the quantum field theory over both manifolds are the same. At this moment, it is not possible to verify Mirror Symmetry in physics. However, mathematically, we can verify some conjectures predicted by Mirror Symmetry.

4 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called the mirror of X, such that the quantum field theory over both manifolds are the same. At this moment, it is not possible to verify Mirror Symmetry in physics. However, mathematically, we can verify some conjectures predicted by Mirror Symmetry.

5 The objects to study Introduction Mirror Symmetry The objects to study We want to study the following manifold: A simply connected compact Kähler manifold with zero first Chern class. Vacuum, Yau s theorem

6 The objects to study Introduction Mirror Symmetry The objects to study We want to study the following manifold: A simply connected compact Kähler manifold with zero first Chern class. Vacuum, Yau s theorem

7 The objects to study Introduction Mirror Symmetry The objects to study We want to study the following manifold: A simply connected compact Kähler manifold with zero first Chern class. Vacuum, Yau s theorem

8 Mirror Symmetry The objects to study On a complex manifold, one can define local holomorphic functions without ambiguity.

9 Mirror Symmetry The objects to study Note that the dimensions of a complex manifold is 2, 4, 6,. Thus if the dimension is high, we can t visualize the manifold. Thus the following theorem of Yau is extremely important in understanding the so-called Calabi-Yau manifold: Theorem (Yau) There exists a unique Ricci flat Kähler metric ω such that ω is in the same cohomology class of ω 0.

10 Mirror Symmetry The objects to study Note that the dimensions of a complex manifold is 2, 4, 6,. Thus if the dimension is high, we can t visualize the manifold. Thus the following theorem of Yau is extremely important in understanding the so-called Calabi-Yau manifold: Theorem (Yau) There exists a unique Ricci flat Kähler metric ω such that ω is in the same cohomology class of ω 0.

11 Mirror Symmetry The objects to study Note that the dimensions of a complex manifold is 2, 4, 6,. Thus if the dimension is high, we can t visualize the manifold. Thus the following theorem of Yau is extremely important in understanding the so-called Calabi-Yau manifold: Theorem (Yau) There exists a unique Ricci flat Kähler metric ω such that ω is in the same cohomology class of ω 0.

12 The Weil-Petersson metric Variation of Hodge structure How to take deriviatives of CY manifolds? The derivative of CY manifolds are elements in cohomology class! Kodaira-Spencer theory.

13 The Weil-Petersson metric Variation of Hodge structure How to take deriviatives of CY manifolds? The derivative of CY manifolds are elements in cohomology class! Kodaira-Spencer theory.

14 The Weil-Petersson metric Variation of Hodge structure How to take deriviatives of CY manifolds? The derivative of CY manifolds are elements in cohomology class! Kodaira-Spencer theory.

15 The Weil-Petersson metric Variation of Hodge structure On a CY manifold, there is a holomorphic non-zero (n, 0) form. The following result can be used as the definition of the Weil-Petersson metric: Theorem (Tian) Let M be a CY moduli and let Ω be a local holomorphic section of (n, 0) forms. Let t M which is represented by a CY manifold X t. Then 1 ω WP = log Ω Ω. 2π X t

16 The Weil-Petersson metric Variation of Hodge structure Recall that for any compact complex manifold, we can define the cohomology groups H p,q. By Hodge theorem, they are made from harmonic forms. In particular, holomorphic (n, 0) forms are harmonic forms. They are in the group H n,0.

17 The Weil-Petersson metric Variation of Hodge structure Recall that for any compact complex manifold, we can define the cohomology groups H p,q. By Hodge theorem, they are made from harmonic forms. In particular, holomorphic (n, 0) forms are harmonic forms. They are in the group H n,0.

18 The Weil-Petersson metric Variation of Hodge structure If we move the CY manifolds, we move the Hodge flags. This is called the variation of Hodge structure.

19 The Weil-Petersson metric Variation of Hodge structure We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric.

20 The curvature of the Weil-Petersson metric is given by R ijkl = g ij g kl + g il g kj terms related to Yukawa coupling So it is neither positive nor negative.

21 The curvature of the Weil-Petersson metric is given by R ijkl = g ij g kl + g il g kj terms related to Yukawa coupling So it is neither positive nor negative.

22 The Hodge metric Introduction Definition (L-97) Let ω WP be the Kähler form of the WP metric. Let m = dim M and let ω H = (m + 3)ω WP + Ric (ω WP ). We call the metric ω H the Hodge metric (ω H is necessarily positive definite).

23 Theorem 1 The Hodge metric is a Kähler metric, and 2ω WP ω H ; 2 The bisectional curvatures of the Hodge metric are nonpositive; 3 The holomorphic sectional curvatures of the Hodge metric are bounded from above by a negative constant (with the bound being (( m + 1) 2 + 1) 1 ); 4 The Ricci curvature of the Hodge metric is bounded above by a negative constant.

24 Theorem 1 The Hodge metric is a Kähler metric, and 2ω WP ω H ; 2 The bisectional curvatures of the Hodge metric are nonpositive; 3 The holomorphic sectional curvatures of the Hodge metric are bounded from above by a negative constant (with the bound being (( m + 1) 2 + 1) 1 ); 4 The Ricci curvature of the Hodge metric is bounded above by a negative constant.

25 Theorem 1 The Hodge metric is a Kähler metric, and 2ω WP ω H ; 2 The bisectional curvatures of the Hodge metric are nonpositive; 3 The holomorphic sectional curvatures of the Hodge metric are bounded from above by a negative constant (with the bound being (( m + 1) 2 + 1) 1 ); 4 The Ricci curvature of the Hodge metric is bounded above by a negative constant.

26 Theorem 1 The Hodge metric is a Kähler metric, and 2ω WP ω H ; 2 The bisectional curvatures of the Hodge metric are nonpositive; 3 The holomorphic sectional curvatures of the Hodge metric are bounded from above by a negative constant (with the bound being (( m + 1) 2 + 1) 1 ); 4 The Ricci curvature of the Hodge metric is bounded above by a negative constant.

27 By Mirror Symmetry, we know that for a compact CY 3 fold X, there is the mirror pair X such that the quantum field theories on both manifolds are identical. We introduce the so-called BCOV conjecture. The BOCV conjecture is related to the quintic mirror pair.

28 By Mirror Symmetry, we know that for a compact CY 3 fold X, there is the mirror pair X such that the quantum field theories on both manifolds are identical. We introduce the so-called BCOV conjecture. The BOCV conjecture is related to the quintic mirror pair.

29 By Mirror Symmetry, we know that for a compact CY 3 fold X, there is the mirror pair X such that the quantum field theories on both manifolds are identical. We introduce the so-called BCOV conjecture. The BOCV conjecture is related to the quintic mirror pair.

30 We consider the quintics in CP 4 Z Z 5 4 5λZ 0 Z 4 = 0. For general complex number λ, it is a smooth hypersurface. In fact, it is a CY 3-fold. The CY manifold and its mirror pair is the most studied.

31 We consider the quintics in CP 4 Z Z 5 4 5λZ 0 Z 4 = 0. For general complex number λ, it is a smooth hypersurface. In fact, it is a CY 3-fold. The CY manifold and its mirror pair is the most studied.

32 We consider the quintics in CP 4 Z Z 5 4 5λZ 0 Z 4 = 0. For general complex number λ, it is a smooth hypersurface. In fact, it is a CY 3-fold. The CY manifold and its mirror pair is the most studied.

33 What is the BCOV Conjecture? BCOV refers to M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa. Very important paper Bershadsky, M. and Cecotti, S. and Ooguri, H. and Vafa, C. Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes Comm. Math. Phys, 165(2): , 1994.

34 What is the BCOV Conjecture? BCOV refers to M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa. Very important paper Bershadsky, M. and Cecotti, S. and Ooguri, H. and Vafa, C. Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes Comm. Math. Phys, 165(2): , 1994.

35 The BCOV Conjecture is related to the g = 1 Mirror symmetry. Based on physics observation, they conjecture a relation between two (presumably unrelated) Calabi-Yau threefolds.

36 For the general quintics Z Z 5 4 5λZ 0 Z 4 = 0, its mirror pair was explicitly constructed by Greene and Plesser.

37 For the general quintics Z Z 5 4 5λZ 0 Z 4 = 0, its mirror pair was explicitly constructed by Greene and Plesser.

38 Definition The mirror map is the holomorphic map from a neighborhood of P 1 to a neighborhood of 0 defined by the following formula q := (5ψ) 5 exp 5 y 0 (ψ) n=1 (5n)! (n!) 5 5n j=n+1 1 j 1 (5ψ) 5n, where ψ 1, and y 0 (ψ) := n=0 (5n)! (n!) 5, ψ > 1. (5ψ) 5n The inverse of the mirror map is denoted by ψ(q).

39 Define the multi-valued function F top 1,B (ψ) as ( ) 62 F top ψ 1,B (ψ) := 3 (ψ 5 1) 1 dψ 6 q y 0 (ψ) dq, and F top top 1,A (q) := F1,B (ψ(q)).

40 Conjecture (A) Let n g (d) be the genus-g degree-d instanton number of a quintic in CP 4 for g = 0, 1. Then the following identity holds: q d top log F dq 1,A (q) = nd qnd n 1 (d) 1 q nd n,d=1 d=1 n 0 (d) 2d q d 12(1 q d ).

41 Conjecture (A) was proved by Aleksey Zinger. Aleksey Zinger The Reduced Genus-One Gromov-Witten Invariants of Calabi-Yau Hypersurfaces ArXiv: v2, 2007.

42 Let X be a Riemannian manifold. Let = p,q be the Laplacian operator. In the Euclidean space, the Laplace operator can be defined as = 2 x x 2 n On manifolds, we can define the operator in the similar way.

43 Let X be a Riemannian manifold. Let = p,q be the Laplacian operator. In the Euclidean space, the Laplace operator can be defined as = 2 x x 2 n On manifolds, we can define the operator in the similar way.

44 Let X be a Riemannian manifold. Let = p,q be the Laplacian operator. In the Euclidean space, the Laplace operator can be defined as = 2 x x 2 n On manifolds, we can define the operator in the similar way.

45 Riemann ζ function Introduction Define ζ(z) = z n z + For Re(z) > 1, the above series converges absolutely. Thus ζ(z) defines a holomorphic function on Re(z) > 1.

46 Riemann ζ function Introduction Define ζ(z) = z n z + For Re(z) > 1, the above series converges absolutely. Thus ζ(z) defines a holomorphic function on Re(z) > 1.

47 ζ function regularization We have the identity 0 t s e tn dt = Γ(s + 1) n s+1.

48 The same method is true for the ζ function of eigenvalues. We can define ζ(z) = 1 λ z. λ i 0 i

49 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. Not well-defined? ζ function regularization (for example: Riemann ζ function)

50 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. Not well-defined? ζ function regularization (for example: Riemann ζ function)

51 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. Not well-defined? ζ function regularization (for example: Riemann ζ function)

52 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. Not well-defined? ζ function regularization (for example: Riemann ζ function)

53 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. Not well-defined? ζ function regularization (for example: Riemann ζ function)

54 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. Not well-defined? ζ function regularization (for example: Riemann ζ function)

55 Setup of Conjecture B Bershadsky-Ceccotti-Ooguri-Vafa defined T def = p,q (det p,q ) ( 1)p+q pq. Why define such a strange quantity? Answer: Riemann-Roch-Grothendieck theorem

56 Setup of Conjecture B Bershadsky-Ceccotti-Ooguri-Vafa defined T def = p,q (det p,q ) ( 1)p+q pq. Why define such a strange quantity? Answer: Riemann-Roch-Grothendieck theorem

57 Setup of Conjecture B Bershadsky-Ceccotti-Ooguri-Vafa defined T def = p,q (det p,q ) ( 1)p+q pq. Why define such a strange quantity? Answer: Riemann-Roch-Grothendieck theorem

58 Conjecture (B) Let be the Hermitian metric on the line bundle (π K W/CP 1) 62 (T (CP 1 )) 3 CP 1 \D induced from the L 2 -metric on π K W/CP 1 and from the Weil-Petersson metric on T (CP 1 ). Then the following identity holds: 1 τ BCOV (W ψ ) = Const. F top 1,B (ψ)3 ( ) 62 ( Ωψ y 0 (ψ) q d dq ) 3 where Ω is the local holomorphic section of the (3, 0) forms. 2 3,

59 Conjecture B was proved by Fang-L-Yoshikawa. H. Fang, Z. Lu, and K-I, Yoshikawa Asymptotic behavior of the BCOV torsion of Calabi-Yau moduli ArXiv: math/ , 2006.

60 Combining Conjecture A and B, we verified the Mirror Symmetry prediction to the case g = 1. For higher genus, the B-side of the conjectures have not been set up. On the other hand, the g = 0 Mirror Symmetry Conjecture was proved by Lian-Liu-Yau and Givental.

61 Combining Conjecture A and B, we verified the Mirror Symmetry prediction to the case g = 1. For higher genus, the B-side of the conjectures have not been set up. On the other hand, the g = 0 Mirror Symmetry Conjecture was proved by Lian-Liu-Yau and Givental.

62 The DDVV Conjecture and the Böttcher-Wenzel Conjecture The DDVV conjecture is also called the normal scalar curvature conjecture. Its matrix form is as follows Conjecture Let A 1,, A m be n n symmetric matrices. Then we have ( A A m 2 ) 2 i,j [A i, A j ] 2.

63 Conjecture Let A, B be n n matrices. Then 2 [A, B] 2 ( A 2 + B 2 ) 2.

64 Both conjectures have been proved now.

65 Infinite dimensional case We can make the following conjecture: Conjecture () Let A, B be the bounded trace class operators in a separable Hilbert space. Then we have where the normal is defined as 2 [A, B] 2 ( A 2 + B 2 ) 2, A = Tr(A A).

66 Thank you! Introduction Q.E.D.