On the BCOV Conjecture


 Irma Carter
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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 socalled CalabiYau 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 socalled CalabiYau 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 socalled CalabiYau 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 WeilPetersson metric Variation of Hodge structure How to take deriviatives of CY manifolds? The derivative of CY manifolds are elements in cohomology class! KodairaSpencer theory.
13 The WeilPetersson metric Variation of Hodge structure How to take deriviatives of CY manifolds? The derivative of CY manifolds are elements in cohomology class! KodairaSpencer theory.
14 The WeilPetersson metric Variation of Hodge structure How to take deriviatives of CY manifolds? The derivative of CY manifolds are elements in cohomology class! KodairaSpencer theory.
15 The WeilPetersson metric Variation of Hodge structure On a CY manifold, there is a holomorphic nonzero (n, 0) form. The following result can be used as the definition of the WeilPetersson 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 WeilPetersson 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 WeilPetersson 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 WeilPetersson 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 WeilPetersson metric Variation of Hodge structure We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric.
20 The curvature of the WeilPetersson 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 WeilPetersson 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 (L97) 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 socalled 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 socalled 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 socalled 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 3fold. 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 3fold. 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 3fold. 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. KodairaSpencer 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. KodairaSpencer 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) CalabiYau 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 multivalued 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 genusg degreed 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 GenusOne GromovWitten Invariants of CalabiYau 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 welldefined? 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 welldefined? 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 welldefined? 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 welldefined? 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 welldefined? 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 welldefined? ζ function regularization (for example: Riemann ζ function)
55 Setup of Conjecture B BershadskyCeccottiOoguriVafa defined T def = p,q (det p,q ) ( 1)p+q pq. Why define such a strange quantity? Answer: RiemannRochGrothendieck theorem
56 Setup of Conjecture B BershadskyCeccottiOoguriVafa defined T def = p,q (det p,q ) ( 1)p+q pq. Why define such a strange quantity? Answer: RiemannRochGrothendieck theorem
57 Setup of Conjecture B BershadskyCeccottiOoguriVafa defined T def = p,q (det p,q ) ( 1)p+q pq. Why define such a strange quantity? Answer: RiemannRochGrothendieck 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 WeilPetersson 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 FangLYoshikawa. H. Fang, Z. Lu, and KI, Yoshikawa Asymptotic behavior of the BCOV torsion of CalabiYau 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 Bside of the conjectures have not been set up. On the other hand, the g = 0 Mirror Symmetry Conjecture was proved by LianLiuYau and Givental.
61 Combining Conjecture A and B, we verified the Mirror Symmetry prediction to the case g = 1. For higher genus, the Bside of the conjectures have not been set up. On the other hand, the g = 0 Mirror Symmetry Conjecture was proved by LianLiuYau and Givental.
62 The DDVV Conjecture and the BöttcherWenzel 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.
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