Gauss-Bonnet Theorem on Moduli Spaces. 陆志勤 Zhiqin Lu, UC Irvine

Gauss-Bonnet Theorem on Moduli Spaces 陆志勤 Zhiqin Lu, UC Irvine 台大数学科学中心 July 28, 2009 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 1/57

Background Calabi-Yau 3-folds are one of the most important objects in mathematics and string theory They are defined as the Kähler manifolds with zero first Chern class Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 2/57

Background Calabi-Yau 3-folds are one of the most important objects in mathematics and string theory They are defined as the Kähler manifolds with zero first Chern class By the work of Yau, we know that within any fixed Kähler class, there is a unique Ricci flat metric Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 2/57

A K3 surface is a Calabi-Yau 2-fold Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 3/57

A K3 surface is a Calabi-Yau 2-fold Currently it is not possible to write down the Ricci flat metric explicitly even for K3 surfaces Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 3/57

A K3 surface is a Calabi-Yau 2-fold Currently it is not possible to write down the Ricci flat metric explicitly even for K3 surfaces Indirect way (or the standard way): Linerization moduli space of Calabi-Yau manifold, Calabi-Yau moduli 模空间的理论之所以会成功是因为它是某些东西的线性化! Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 3/57

Why moduli space (CY moduli)? Moduli space gives a way to compare CY manifolds that are close to each other So it gives a platform of linearization Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 4/57

有两类模空间是适合微分几何的研究的 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 5/57

有两类模空间是适合微分几何的研究的 1 一类是黎曼面的模空间 ( 它的万有覆盖就是 Teichmüller 空间 ) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 5/57

有两类模空间是适合微分几何的研究的 1 一类是黎曼面的模空间 ( 它的万有覆盖就是 Teichmüller 空间 ) 2 另一类就是卡拉比 - 丘成桐空间的模空间 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 5/57

有两类模空间是适合微分几何的研究的 1 一类是黎曼面的模空间 ( 它的万有覆盖就是 Teichmüller 空间 ) 2 另一类就是卡拉比 - 丘成桐空间的模空间因为这两类空间都是光滑的 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 5/57

By a theorem of Tian-Todorov, the Calabi-Yau moduli is smooth Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 6/57

By a theorem of Tian-Todorov, the Calabi-Yau moduli is smooth We shall study the Differential Geometry of CY moduli Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 6/57

By a theorem of Tian-Todorov, the Calabi-Yau moduli is smooth We shall study the Differential Geometry of CY moduli with respect to the so-called Weil-Petersson metric Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 6/57

Weil-Petersson metric The metric is an L 2 metric, defined by the following Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 7/57

Weil-Petersson metric The metric is an L 2 metric, defined by the following Definition Let Z be a polarized Calabi-Yau manifold with the Ricci flat Kähler metric µ whose Kähler form defines the polarization Let X, Y H 1 (Z, T (1,0) Z) Define the L 2 inner product by (X, Y) = 1 X, Y µ n n! For a Calabi-Yau manifold, via the Kodaira-Spencer map: T Z M H 1 (Z, T (1,0) Z), which is an isomorphism, the above inner product defines a metric on the smooth part of M The metric happens to be Kählerian, and is called the Weil-Petersson metric of M Z Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 7/57

A very quick review of Kodaira-Spencer theory A complex structure J is a real operator such that J 2 = I J : T C Z T C Z Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 8/57

A very quick review of Kodaira-Spencer theory A complex structure J is a real operator J : T C Z T C Z such that J 2 = I For fixed frames, we have ( 1I J = 1I ) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 8/57

A very quick review of Kodaira-Spencer theory A complex structure J is a real operator J : T C Z T C Z such that J 2 = I For fixed frames, we have ( 1I J = 1I ) A variation of the complex structure is a real matrix A such that (J + εa) 2 = I Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 8/57

A very quick review of Kodaira-Spencer theory-2 Or we have Thus we have AJ + JA = 0 ( A = Ā1 A 1 ) for some A 1 : T 1,0 Z T 0,1 Z, or equivalently A 1 can be represented by φ Λ 0,1 (T 1,0 Z) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 9/57

A very quick review of Kodaira-Spencer theory-2 Or we have Thus we have AJ + JA = 0 ( A = Ā1 A 1 ) for some A 1 : T 1,0 Z T 0,1 Z, or equivalently A 1 can be represented by φ Λ 0,1 (T 1,0 Z) Roughly speaking, the Kodaira-Spencer map is defined as ε φ Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 9/57

The differential geometry of the Weil-Petersson metric on Calabi-Yau moduli is called the Weil-Petersson geometry Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 10/57

The differential geometry of the Weil-Petersson metric on Calabi-Yau moduli is called the Weil-Petersson geometry The Aim: We want to be able to tell the properties of the moduli space through geometric analysis Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 10/57

Local Weil-Petersson geometry Good properties Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 11/57

Local Weil-Petersson geometry Good properties Kählerian Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 11/57

Local Weil-Petersson geometry Good properties Kählerian Formula of Strominger and 王金龙 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 11/57

Local Weil-Petersson geometry Good properties Kählerian Formula of Strominger and 王金龙很多其他的性质 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 11/57

Local Weil-Petersson geometry Good properties Kählerian Formula of Strominger and 王金龙很多其他的性质 Bad property Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 11/57

Local Weil-Petersson geometry Good properties Kählerian Formula of Strominger and 王金龙很多其他的性质 Bad property The curvature is neither positive nor negative Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 11/57

Rebuilding the local geometry The following result was proved in my thesis (1997) Theorem Let M be the moduli space of a CY 3-fold with dimension m Then We define the the following metric ω H = (m + 3)ω WP + Ric(ω WP ) which I called it Hodge metric The curvature has good properties (non-positive bisectional curvature, negative Ricci and holomorphic sectional curvature, etc) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 12/57

We are not able to prove that the sectional curvature is non-positive However, we don t expect that, otherwise the CY moduli may be too restrictive Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 13/57

Hodge theory helps us! 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 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 14/57

If we deform the CY manifolds, we deform the Hodge flags This is called the variation of Hodge structure Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 15/57

We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric Local theory Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 16/57

We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric Local theory Semi-global theory Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 16/57

We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric Local theory Semi-global theory The BCOV conjecture (settled by Zinger-07 方浩 - 陆 -Yoshikawa-06) Incompleteness of the Weil-Petersson metric ( 王 -96 for 1d, L-Wang-09 in preparation) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 16/57

The BCOV Conjecture BCOV=Bershadsky-Cecotti-Ooguri-Vafa Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 17/57

The BCOV Conjecture BCOV=Bershadsky-Cecotti-Ooguri-Vafa 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 1 j (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) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 17/57

Define the multi-valued function F top 1,B (ψ) as ( ) 62 ψ F top 1,B (ψ) := 3 (ψ 5 1) 1 dψ 6 q y 0 (ψ) dq, and F top 1,A (q) := Ftop 1,B (ψ(q)) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 18/57

Conjecture (A) 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 log Ftop 1,A dq (q) = 50 12 2nd qnd n 1 (d) 1 q nd n,d=1 d=1 n 0 (d) 2d q d 12(1 q d ) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 19/57

Conjecture (A) was proved by Aleksey Zinger Aleksey Zinger The Reduced Genus-One Gromov-Witten Invariants of Calabi-Yau Hypersurfaces ArXiv: 07052397v2, 2007 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 20/57

Setup of Conjecture (B) Let X be a compact Kähler manifold Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 21/57

Setup of Conjecture (B) Let X be a compact Kähler manifold Let = p,q be the Laplacian on (p, q) forms; Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 21/57

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 + Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 21/57

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 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 21/57

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? Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 21/57

Setup of Conjecture B Bershadsky-Ceccotti-Ooguri-Vafa defined T def = p,q (det p,q ) ( 1)p+q pq Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 22/57

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? Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 22/57

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 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 22/57

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 q d y 0 (ψ) dq where Ω is the local holomorphic section of the (3, 0) forms ) 3 2 3, Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 23/57

Conjecture B was proved by Fang-L-Yoshikawa 方浩 -L-Yoshikawa( 吉川谦一 ) Asymptotic behavior of the BCOV torsion of Calabi-Yau moduli ArXiv: 0601411 JDG (80), 2008, 175-259, Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 24/57

In fact, there is only one conjecture: By counting the instanton numbers we get the function F top 1,A, and by computing the BCOV torsion we get the function F top 1,B These two functions are the same Combining Conjecture A and B, we verified the Mirror Symmetry prediction of the case g = 1 For higher genus, the B-side of the conjectures have not been set up (Yau-Yamagochi, Huang-Klemn-Quakenbush, and many others) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 25/57

Chern classes A quick review We assume that E M is a holomorphic vector bundle over a compact complex manifold M Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 26/57

Chern classes A quick review We assume that E M is a holomorphic vector bundle over a compact complex manifold M Let h be a Hermitian metric on E Let be the matrix of the connection Γ = h h 1 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 26/57

Chern classes A quick review We assume that E M is a holomorphic vector bundle over a compact complex manifold M Let h be a Hermitian metric on E Let Γ = h h 1 be the matrix of the connection Let R = Γ be the curvature matrix Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 26/57

Let f be an invariant polynomial That is, f is a polynomial on C r2 that f(a) = f(t 1 AT) such Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 27/57

Let f be an invariant polynomial That is, f is a polynomial on C r2 that f(a) = f(t 1 AT) such The Chern-Weil form with respect to the polynomial f is defined by f(r) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 27/57

If in addition, we assume that f has integer coefficients, then we have the following results: 1 f(r) is closed This is the Chern-Weil form; Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 28/57

If in addition, we assume that f has integer coefficients, then we have the following results: 1 f(r) is closed This is the Chern-Weil form; 2 [f(r)] H (M, Z), and is independent of the choice of the connection Gauss-Bonnet-Chern Theorem Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 28/57

This is joint with Michael R Douglas Zhiqin Lu and Michael R Douglas Gauss-Bonnet-Chern theorem on moduli spaces preprint, 2008, arxiv:09023839 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 29/57

This is joint with Michael R Douglas Zhiqin Lu and Michael R Douglas Gauss-Bonnet-Chern theorem on moduli spaces preprint, 2008, arxiv:09023839 Theorem Let f be an invariant polynomial with rational coefficients Let R be the curvature tensor with respect the Weil-Petersson metric Then f(r) is a rational number M Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 29/57

Some remarks The moduli space is often non-compact Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 30/57

Some remarks The moduli space is often non-compact If the moduli space were compact, then the Chern-Weil forms define Chern classes, and the theorem follows Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 30/57

Some remarks The moduli space is often non-compact If the moduli space were compact, then the Chern-Weil forms define Chern classes, and the theorem follows If the growth of the Chern-Weil forms and the connection were mild at infinity, the theorem follows from a theorem of Mumford Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 30/57

Important! We realized that Theorem Let E M be a Hodge bundle over the moduli space of a polarized Kähler manifold Let R be the curvature tensor with respect to the Hodge bundle Then f(r) is a rational number M Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 31/57

Previous results For the first Chern class of the Weil-Petersson metric, the result is by Sun-L (CMP, 06); Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 32/57

Previous results For the first Chern class of the Weil-Petersson metric, the result is by Sun-L (CMP, 06); For general moduli space, if the dimension is 1, the result is by Peters, Zucker, Mumford (independently) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 32/57

Our result is true for the Weil-Petersson metric on Calabi-Yau moduli and the Chern-Weil forms on general moduli spaces We proved the theorems of this kind to the full generality, with the introduction of new techniques Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 33/57

The idea of Mumford Let M be the compactification of M We suppose E can be extended to M Let Γ 0 be a smooth connection of the extended bundle Let ρ be a cut-off function We can compare the two integrals: ρf(r) and ρf(r 0 ) M M where R 0 = Γ 0 is the curvature of Γ 0 When ρ 1, M ρf(r) goes to M f(r), and M ρf(r 0 ) goes to M f(r 0), which is an integer by the fact that the Chern-Weil form defines a rational class Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 34/57

Let f be the polarization of f That is, f(a,, A) = f(a) and Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 35/57

Let f be the polarization of f That is, f(a,, A) = f(a) and f(, Ai,, A j, ) = f(, A j,, A i, ) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 35/57

Let f be the polarization of f That is, f(a,, A) = f(a) and f(, Ai,, A j, ) = f(, A j,, A i, ) and f(ta1 T 1,, TA j T 1, ) = f(a 1,, A j, ) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 35/57

Let f be the polarization of f That is, f(a,, A) = f(a) and f(, Ai,, A j, ) = f(, A j,, A i, ) and f(ta1 T 1,, TA j T 1, ) = f(a 1,, A j, ) f is linear with respect to all the components Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 35/57

Let f be the polarization of f That is, f(a,, A) = f(a) and f(, Ai,, A j, ) = f(, A j,, A i, ) and f(ta1 T 1,, TA j T 1, ) = f(a 1,, A j, ) f is linear with respect to all the components For example, if f = tr(a 2 ), then f = 1 2tr(AB + BA) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 35/57

Then we have the following identity: ρ(f(r) f(r 0 )) M = ρf(r,, R, R R 0, R 0,, R 0 ) i M = ρ f(r,, R, Γ Γ 0, R 0,, R 0 ) i M Apparently, if the growth of R, R 0, Γ, Γ 0 are mild, then the right hand side will go to zero Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 36/57

Unfortunately, the Hodge metrics are not good in the sense of Mumford Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 37/57

Unfortunately, the Hodge metrics are not good in the sense of Mumford What do we need to do? Control ρ; Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 37/57

Unfortunately, the Hodge metrics are not good in the sense of Mumford What do we need to do? Control ρ; Control Γ, R; Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 37/57

Unfortunately, the Hodge metrics are not good in the sense of Mumford What do we need to do? Control ρ; Control Γ, R; Control Γ 0, R 0 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 37/57

The P-boundedness Definition Let U be a neighborhood and let z 1,, z n be the holomorphic coordinate system The divisor D = {z 1 = 0} A smooth form on U\D is called P bounded (refer to Poincaré Bounded), if it is bounded under the frame d log z 1 / log 1 z 1, d log z 1/ log 1 z 1, dz j, d z j for j > 1 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 38/57

If η 1, η 2 are P bounded, then so is η 1 η 2 Furthermore, if η is an (n, n) form which is P-bounded, then η C n i=1 1 z 2 i (log 1 z i )2 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 39/57

If η 1, η 2 are P bounded, then so is η 1 η 2 Furthermore, if η is an (n, n) form which is P-bounded, then In particular, η is finite U η C n i=1 1 z 2 i (log 1 z i )2 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 39/57

If η 1, η 2 are P bounded, then so is η 1 η 2 Furthermore, if η is an (n, n) form which is P-bounded, then In particular, η is finite U η C n i=1 1 z 2 i (log 1 z i )2 It is standard to prove that one can choose a cut-off function ρ so that ρ is P-bounded with the Lebesgue of supp ( ρ) going to zero For example, we can pick ρ = g(log 1/r) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 39/57

We consider a neighborhood U at the infinity of the moduli space, where the divisor D can be written as {z 1 = 0} By the nilpotent orbit theorem of Schmid, we have the expansion of the Hodge metric ( ) log 1 α1 r 1 h = + lower order terms ) αr (log 1r1 Without loss of generality, we assume that α 1 α 2 α r Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 40/57

We let ) α1 /2 (log 1r1 Λ = ( ) log 1 αr /2 r 1 Then we have h = Λh Λ Then we have h = I + lower order terms, and the connection is hh 1 = ΛΛ 1 + Λ h (h ) 1 Λ 1 + Λh ΛΛ 1 (h ) 1 Λ 1 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 41/57

Go back to the previous expression ρ f(r,, R, Γ Γ 0, R 0,, R 0 ) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 42/57

Since Γ 0, R 0 are smooth, they are bounded Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 43/57

Since Γ 0, R 0 are smooth, they are bounded However, even in the above simplest case, the connection and hence the curvature are not P bounded: hh 1 = ΛΛ 1 + Λ h (h ) 1 Λ 1 + Λh ΛΛ 1 (h ) 1 Λ 1 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 43/57

Since Γ 0, R 0 are smooth, they are bounded However, even in the above simplest case, the connection and hence the curvature are not P bounded: hh 1 = ΛΛ 1 + Λ h (h ) 1 Λ 1 + Λh ΛΛ 1 (h ) 1 Λ 1 But is P bounded Ad(Λ 1 )( hh 1 ) = Λ 1 Λ + h (h ) 1 + h ΛΛ 1 (h ) 1 Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 43/57

Since Γ 0, R 0 are smooth, they are bounded However, even in the above simplest case, the connection and hence the curvature are not P bounded: hh 1 = ΛΛ 1 + Λ h (h ) 1 Λ 1 + Λh ΛΛ 1 (h ) 1 Λ 1 But Ad(Λ 1 )( hh 1 ) = Λ 1 Λ + h (h ) 1 + h ΛΛ 1 (h ) 1 is P bounded In fact, this is a special case of the Cattani-Kaplan-Schmid theorem Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 43/57

The problem is that, when Ad(Λ 1 )( hh 1 ) is P bounded, then in general 0 Λ = B @ log 1 r 1 «α1 /2 1 «C α log r 1 r/2a 1 Ad(Λ 1 )(Γ 0 ) and Ad(Λ 1 )(R 0 ) are not bounded Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 44/57

Let Γ 0 = (a ij ) Then Ad(Λ 1 )(Γ 0 ) = (a ij (log 1/r 1 ) αi αj ), and they are bounded if and only if Γ 0 is of lower triangular Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 45/57

Let Then Γ 0 = (a ij ) Ad(Λ 1 )(Γ 0 ) = (a ij (log 1/r 1 ) αi αj ), and they are bounded if and only if Γ 0 is of lower triangular The technical heart of the proof is that We are able to find such a connection Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 45/57

Let Ω α be the frame defined by the nilpotent theorem and let A αβ be the transition functions We proved that the connection Γ α = ψ γ A αγ A 1 αγ is the Correct connection! Or roughly speaking, they are all of lower triangular! Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 46/57

The proof is very technical and uses the full power of the SL 2 -orbit theorem of several variables Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 47/57

The proof is very technical and uses the full power of the SL 2 -orbit theorem of several variables Continue Ṣkip Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 47/57

The key technical lemma of this section is the following: Lemma Let U = U γ be an open set such that U C Let A = A αγ Then on U C, Ad((e 1 ) t )Ad(A 1 C )( AA 1 ) is Poincaré bounded Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 48/57

Sketch of the Proof Let Ω = Ω γ be the local frame of U defined by the nilpotent orbit theorem Let C be a fixed cone of U and let Ω C be the frame of the cone We assume that C C We just need to prove the assertion of the lemma on C C because as C is running over all the cones, the whole U C will be covered Let e be the matrix under the frame Ω C Let A C be the constant matrix defined as Ω = Ω C At C Then we have Ω C = Ω C At C At (A 1 C )t Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 49/57

We let B = A 1 C AA C and let h be the metric matrix of Ω C Then Ω C = Ω C Bt Thus h = Ω t CΩ C = Bh Bt (1) It follows that hh 1 = BB 1 + Ad(B)( h (h ) 1 ) (2) Since A C, A C are constant matrices, we have BB 1 = Ad(A 1 C )( AA 1 ) Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 50/57

In order to prove the lemma, we only need to prove that Ad((e 1 ) t )Ad(B) h (h ) 1 = D(Ad(((e ) 1 ) t ))( h (h ) 1 )D 1 is Poincaré bounded, where D = (e 1 ) t B(e ) t Thus we only need to prove that D and D 1 are bounded Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 51/57

To prove the claim in the last slide, we observe that if and if then by (1) h = e t kē, h = (e ) t k ē, k = Dk Dt As a consequence of the SL 2 orbit theorem of Cattani-Kaplan-Schmid, Since both k and k are positive definite, D and D 1 must be bounded Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 52/57

Physics Background In the paper of Ashok-Douglas the index of all supersymmetric vacua was given The index is given by I vac (L L max ) = const det( R WP ω WP ), M H where M is the Calabi-Yau moduli and H is the moduli space of elliptic curves Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 53/57

In the paper of Douglas-Shiffman-Zelditch, the following strengthened result of the above was given Theorem Let K be a compact subset of M with piecewise smooth boundary Then [ ] Ind χk (L) = const(l 2m ) c m (T (M) F 3 ) + O(L 1/2 ) K Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 54/57

By our results, we have Theorem The indices I vac and Ind χk are all finite Moreover, Ind χk is bounded from above uniformly with respect to K They are all bounded, up to an absolute constant, by the Hodge volume of the Calabi-Yau moduli Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 55/57

In Layman words, if string theory is true, the the number of feasible Universes is finite Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 56/57

Thank you! Zhiqin Lu, UC Irvine Gauss-Bonnet Theorem 57/57