Nested Donaldson-Thomas invariants and the Elliptic genera in String theor

Transcription

1 Nested Donaldson-Thomas invariants and the Elliptic genera in String theory String-Math 2012, Hausdorff Center for Mathematics July 18, 2012

2 Elliptic Genus in String theory: 1 Gaiotto, Strominger, Yin (hep-th/ v) 2 Gaiottoe, Yin (hep-th/ v1) 3 Frederik Denef, Gregory W. Moore (hep-th/ ) 4 Hirosi Ooguri, Andrew Strominger, Cumrun Vafa (hep-th/ v2) Study of D4-D2-D0 systems over a threefold.

3 Elliptic Genus in String theory: 1 Gaiotto, Strominger, Yin (hep-th/ v) 2 Gaiottoe, Yin (hep-th/ v1) 3 Frederik Denef, Gregory W. Moore (hep-th/ ) 4 Hirosi Ooguri, Andrew Strominger, Cumrun Vafa (hep-th/ v2) Study of D4-D2-D0 systems over a threefold. X, D4-D2-D0 (D X, K D ) a divisor in X (p and C) D X, Nested subschemes An integral linear combination of holomorphic curves C D X

4 Definition Definitions and motivation from String theory The modified elliptic genus of an M5 Brane is given by the generating series of the invariants of D4-D2-D0 bound states in an M5-Brane background: Z X,D (τ, τ, y A ) = X Z δ (τ)θ Λ+δ (τ, τ, y A ). δ Λ /Λ Λ H 2 (D, Z) is the image of H 2 (X, Z) H 2 (D, Z), Θ Λ+δ = Complicated!, but well known modular forms (Jacobi Theta functions!): Θ Λ+δ (τ, τ, y A ) = X q Λ+δ+ J 2 Z δ (τ) are given by holomorphic modular vectors. ( ) J q exp[ πiτq 2 (J q)2 + πi(τ τ) + 2πiy q]. J J Z X,D (τ, τ, y A ) is a modular form! It is difficult to compute Z X,D (τ, τ, y A ) due to singularities of the moduli space of D4-D2-D0 systems.

5 Main Goal: Definitions and motivation from String theory Use Algebraic Geometry to compute the corresponding generating series.

6 A D4-D2-D0 system of (p and C) D X can be characterized with a torsion 2 dimensional sheaf F such that Ch(F ) = (0, [D], β = [C], n[pt]). 1 Need the condition that [D] is irreducible. 2 We consider the moduli space of stable sheaves over X with Ch(F ) = (0, D, β, n) denote it M := M (0,D,β,n) X. 3 Assume that F K X = F, then Ext 3 (F, F ) = Hom(F, F ) = C, perfect obstruction theory of dim=0. 4 We compute invariants Z over M using Deformation-Obstruction theories and virtual classes: 1. We call them Nested DT invariants (NDT X ). [M] vir 5 When X :=CY3, these are also given by weighted Euler characteristic and Behrend s functions. 6 Toda studied similar torsion sheaves with (0, n[d], β, n) when n 0, using wallcrossing and Behrend s function. For us n = 1.

7 Example Definitions and motivation from String theory Let j : D X be a divisor. Fix a linear system of divisors with representative D (i.e D ) over X. For an element S D (a surface) let C S with [C] = β be a curve lying scheme theoretically on S. The sheaf F := j I C/S satisfies the property that Ch(F ) = (0, D, β, n).

8 Example Definitions and motivation from String theory Let j : D X be a divisor. Fix a linear system of divisors with representative D (i.e D ) over X. For an element S D (a surface) let C S with [C] = β be a curve lying scheme theoretically on S. The sheaf F := j I C/S satisfies the property that Ch(F ) = (0, D, β, n). Relationship to Joyce-Song pairs theory: 0 O X ( S) s I C/X j I C/S 0 The tuple (s, I C/X ) is a stable Joyce-Song pair. However if S is not very ample then (s, I C/X ) and j I C/S do not have the same deformation theory! As objects in derived category I := [O X ( S) s I C/X ] = j I C/S. Therefore if I and (s, I C/X ) deform in the same way then a theory for j I C/S can be computed via Joyce-Song stable pairs theory.

9 Lemma Definitions and motivation from String theory Let the curve C satisfy the property that H i (I C/X (S)) = 0 for all i > 0. Then every deformation I over B of I 0 is quasi-isomorphic to a complex O X ( S) O B φ ĨC/X where Ĩ C/X is a B-flat deformation of (Ĩ C/X ) 0 with section φ. O X ( S) s I C/X I C/X In that case the NDT invariants can be related to MNOP invariants. Other wise a new theory is required for 2 dimensional torsion sheaves.

10 K3 Fibrations over a curve K3 fibrations with nodal fibers X Other geometries such as Quintic threefold

11 K3 Fibrations over a curve K3 fibrations with nodal fibers X Other geometries such as Quintic threefold Fix a smooth K3 fibration X Z and an ample polarization L over X. We are following the work of Pandharipande-Maulik (Gromov-Witten theory and Noether-lefschetz theory): GW(X ) = GW(X /Z) NL(X Z)

12 K3 Fibrations over a curve K3 fibrations with nodal fibers X Other geometries such as Quintic threefold Fix a smooth K3 fibration X Z and an ample polarization L over X. We are following the work of Pandharipande-Maulik (Gromov-Witten theory and Noether-lefschetz theory): GW(X ) = GW(X /Z) NL(X Z) Our Goal: DT(X ) = DT(X /Z) NL(X Z)

13 Let M and M/Z denote the absolute and relative moduli spaces. Let i : S X be a smooth surface. Consider a stable torsion free sheaf G over S and let F = i G. The points in M parameterize F and points in M/Z parameterize G.

14 Let M and M/Z denote the absolute and relative moduli spaces. Let i : S X be a smooth surface. Consider a stable torsion free sheaf G over S and let F = i G. The points in M parameterize F and points in M/Z parameterize G. 1 Need deformation-obstruction theories for M and M/Z.

15 Let M and M/Z denote the absolute and relative moduli spaces. Let i : S X be a smooth surface. Consider a stable torsion free sheaf G over S and let F = i G. The points in M parameterize F and points in M/Z parameterize G. 1 Need deformation-obstruction theories for M and M/Z. 2 Need to relate them!

16 Let M and M/Z denote the absolute and relative moduli spaces. Let i : S X be a smooth surface. Consider a stable torsion free sheaf G over S and let F = i G. The points in M parameterize F and points in M/Z parameterize G. 1 Need deformation-obstruction theories for M and M/Z. 2 Need to relate them! Theorem There exists a map in the derived category Rπ M (RH om X M (F, F) ω πm [2]) L M out of which one can obtain a perfect relative deformation obstruction theory of amplitude [ 1, 0] for M, i.e E := τ 1 Rπ M (RH om X M (F, F) [ 1] L M is a perfect deformation-obstruction theory of amplitude [ 1, 0] for M.

17 Theorem Definitions and motivation from String theory Let ĩ : X Z M X M be the natural inclusion. Denote by π M the composition map π M : π M ĩ (note that π M is relative dimension 2). There exists a map in the derived category RH om X Z M(G, G) ω π M [1] L Rπ M M/Z out of which one can obtain a perfect deformation obstruction theory of amplitude [ 1, 0] for M/Z, i.e F := τ 1 Rπ M (RH om X Z M(G, G) [ 1] L M/Z is a perfect deformation-obstruction theory of amplitude [ 1, 0] for M/Z.

18 Theorem Definitions and motivation from String theory Let ĩ : X Z M X M be the natural inclusion. Denote by π M the composition map π M : π M ĩ (note that π M is relative dimension 2). There exists a map in the derived category RH om X Z M(G, G) ω π M [1] L Rπ M M/Z out of which one can obtain a perfect deformation obstruction theory of amplitude [ 1, 0] for M/Z, i.e F := τ 1 Rπ M (RH om X Z M(G, G) [ 1] L M/Z is a perfect deformation-obstruction theory of amplitude [ 1, 0] for M/Z. In order to relate the two deformation-obstruction theories we study the exact triangle in the derived category of M

19 Lemma Definitions and motivation from String theory Let ĩ : X Z M X M be the inclusion as above and G be the universal sheaf over X Z M. By definition F = ĩ G. Then there exists an exact triangle Rπ M ĩ RH om X Z M(G, G) Rπ M (RH om X M (F, F)) Rπ M ĩ RH om X Z M(G, G O X Z M(X Z M)) [ 1] (1) In the level of obstruction bundles we have: Ext 2 S(G, G) Ext 2 X (F, F ) Ext 1 S(G, G O S (S))

20 Theorem Definitions and motivation from String theory Denote by E the perfect obstruction theory over M and by F the perfect absolute obstruction theory over M/Z. Let [M, E ] vir and [M, F ] vir denote the induced virtual fundamental classes of M. Then the following identity holds true over A (M): [M, E ] vir = [M, F ] vir c top(ext 1 π M (G, G(S))).

21 Theorem Definitions and motivation from String theory Denote by E the perfect obstruction theory over M and by F the perfect absolute obstruction theory over M/Z. Let [M, E ] vir and [M, F ] vir denote the induced virtual fundamental classes of M. Then the following identity holds true over A (M): [M, E ] vir = [M, F ] vir c top(ext 1 π M (G, G(S))). Vanishing Issue: h 1 (F ) = Ext 2 X z ( G m, G m ω Xz ) = C

22 Theorem Definitions and motivation from String theory Denote by E the perfect obstruction theory over M and by F the perfect absolute obstruction theory over M/Z. Let [M, E ] vir and [M, F ] vir denote the induced virtual fundamental classes of M. Then the following identity holds true over A (M): [M, E ] vir = [M, F ] vir c top(ext 1 π M (G, G(S))). Vanishing Issue: h 1 (F ) = Ext 2 X z ( G m, G m ω Xz ) = C. Need to compute the reduced deformation obstruction theory over the surfaces! (F ) h 1 (F ) [ 1] tr R 2 π M O X Z M[ 1] = K [ 1] The fiber of K over a point z Z is given by H 0 (X z, ω X Xz N Xz /X ) = H 0 (X z, K Xz ) where N Xz /X and K Xz denote the normal bundle and the canonical bundle (respectively) of the fiber of X Z over a point z Z

23 Given the dual map in the derived category «Ψ : K[1] h 1 (F ) [ 1] F The reduced obstruction theory is obtained by coning off the map Ψ: Definition Let D := Cone(Ψ)[ 1]. It can be shown that D is an absolute obstruction theory over M (with rather a different obstruction bundle). Theorem The following identity holds true over A (M): Z Z 1 = c top(ext 1 π (G, G(S))) c1(k ) [M,E ] vir [M,D ] vir M Z Z = c top(t M(K3) ) c 1(ɛ K ) [M(K3)] red B(m,h,γ) (2)

24 Given the dual map in the derived category «Ψ : K[1] h 1 (F ) [ 1] F The reduced obstruction theory is obtained by coning off the map Ψ: Definition Let D := Cone(Ψ)[ 1]. It can be shown that D is an absolute obstruction theory over M (with rather a different obstruction bundle). Theorem The following identity holds true over A (M): Z Z 1 = c top(ext 1 π (G, G(S))) c1(k ) [M,E ] vir [M,D ] vir M Z Z = c top(t M(K3) ) c 1(ɛ K ) [M(K3)] red B(m,h,γ) = (Reduced NDT invariants) (Noether-Lefschetz numbers) (3)

25 Given Ch(G) = (0, D, β, n), note that if G M is supported on the smooth surface i : S X then G = M I for some M Pic(S) and ideal sheaf of n points on S, denoted by I. Let i β = γ and β 2 = 2h 2. Then NDT(X, P) = X X χ(hilb h+1 P2L(0) (S)) NL π h,γ h Z γ H 2(X ) 2L γ = P 2L(m) m m=0 Z + δ 0, m P L (0) χ(hilb2 P L(0) (S)) c 1(K ) (4) where P L = PL F (m) = L2 2 m2 + (L β)m + β2 2 n + 2 is the Hilbert polynomial of F with respect to polarization L. C

26 Given Ch(G) = (0, D, β, n), note that if G M is supported on the smooth surface i : S X then G = M I for some M Pic(S) and ideal sheaf of n points on S, denoted by I. Let i β = γ and β 2 = 2h 2. Then NDT(X, P) = X X χ(hilb h+1 P2L(0) (S)) NL π h,γ h Z γ H 2(X ) 2L γ = P 2L(m) m m=0 Z + δ 0, m P L (0) χ(hilb2 P L(0) (S)) c 1(K ) (5) where P L = PL F (m) = L2 2 m2 + (L β)m + β2 2 n + 2 is the Hilbert polynomial of F with respect to polarization L. C

27 Given Ch(G) = (0, D, β, n), note that if G M is supported on the smooth surface i : S X then G = M I for some M Pic(S) and ideal sheaf of n points on S, denoted by I. Let i β = γ and β 2 = 2h 2. Then NDT(X, P) = X X χ(hilb h+1 P2L(0) (S)) NL π h,γ h Z γ H 2(X ) 2L γ = P 2L(m) m m=0 Z + δ 0, m P L (0) χ(hilb2 P L(0) (S)) c 1(K ) (6) C NDT (X ) = (Goettsche s Invariants) (Noether-Lefschetz numbers).

28 Given Ch(G) = (0, D, β, n), note that if G M is supported on the smooth surface i : S X then G = M I for some M Pic(S) and ideal sheaf of n points on S, denoted by I. Let i β = γ and β 2 = 2h 2. Then NDT(X, P) = X X χ(hilb h+1 P2L(0) (S)) NL π h,γ h Z γ H 2(X ) 2L γ = P 2L(m) m m=0 Z + δ 0, m P L (0) χ(hilb2 P L(0) (S)) c 1(K ) (7) C NDT (X ) = (Goettsche s Invariants) (Noether-Lefschetz numbers). 1 Goettsche invariants Modular

29 Given Ch(G) = (0, D, β, n), note that if G M is supported on the smooth surface i : S X then G = M I for some M Pic(S) and ideal sheaf of n points on S, denoted by I. Let i β = γ and β 2 = 2h 2. Then NDT(X, P) = X X χ(hilb h+1 P2L(0) (S)) NL π h,γ h Z γ H 2(X ) 2L γ = P 2L(m) m m=0 Z + δ 0, m P L (0) χ(hilb2 P L(0) (S)) c 1(K ) (8) C NDT (X ) = (Goettsche s Invariants) (Noether-Lefschetz numbers). 1 Goettsche invariants Modular 2 NL numbers Fourier coefficients of Modular forms [Borcherds]

30 Given Ch(G) = (0, D, β, n), note that if G M is supported on the smooth surface i : S X then G = M I for some M Pic(S) and ideal sheaf of n points on S, denoted by I. Let i β = γ and β 2 = 2h 2. Then NDT(X, P) = X X χ(hilb h+1 P2L(0) (S)) NL π h,γ h Z γ H 2(X ) 2L γ = P 2L(m) m m=0 Z + δ 0, m P L (0) χ(hilb2 P L(0) (S)) c 1(K ) (9) C NDT (X ) = (Goettsche s Invariants) (Noether-Lefschetz numbers). 1 Goettsche invariants Modular 2 NL numbers Fourier coefficients of Modular forms [Borcherds] Modularity of NDT invariants!

31 We obtain the generating series for the NDT invariants Z NDT (X, P)(q) = 2 q P(0) Θ(q) Y (1 q h ) 24 Where Θ is a vector valued modular form. Example In case where X C is obtained by a pencil of quartics over P 1, Θ is given by a vector valued modular form of degree 21 and level 8: h 1 Θ(q) = q + 320q q 3 2 +

32 Nodal Fibrations

33 Nodal Fibrations X x x x x x t 1 t2 Z

34 Nodal Fibrations X x x x x x t 1 t2 Z Main idea: To compute the NDT invariants of X via relating X to smooth fibrations.

35 Nodal Fibrations X x x x x x t 1 t2 Z Main idea: Conifold Transition!

36 2:1 Let ɛ 0 : Z 0 Z be the double cover of Z ramified at t i. Then ɛ X has conifold singularities at t i. We resolve the singularities and obtain X ɛ 0 X. 2:1 Let ɛ t : Z t Z be the double cover of Z at s i (t) such that s i (t) t i as t 0. Then ɛ t X is smooth as t 0 and ɛ t X = ɛ 0 X at t = 0. Relate ɛ t X to X via degenerations BL {s1 (0), s r (0)}(ɛ t X ): ɛ t X Y 1 P 1 D 1 A 1 t 0 t = 0 BL {e1, e r } 0( X A 1 ): X Y 2 P 2 D 2 A 1 t 0 t = 0

37 BL {s1 (0), s r (0)}(ɛ t X ): ɛ t X Y 1 P 1 D 1 A 1 t 0 t = 0 BL {e1, e r } 0( X A 1 ): X Y 2 P 2 D 2 A 1 t 0 t = 0 1 P 1: A double cover of P 3 branched over a smooth quartic surface ( = P 1 P 1 ) 2 D 1 = P 1 P 1 = D2 3 P 2 = P(OP 1( 1) O P 1( 1) O P 1) 4 Y 1 = Y2 = BL t1, t r (ɛ 0 X )

38 It turns out that P 1 and P 2 do not contribute to our invariants!

39 It turns out that P 1 and P 2 do not contribute to our invariants! Obtain a conifold transition identity Z NDT (M ɛ t X ) = Z NDT (M Y1 /D 1 ) = Z NDT (M Y2 /D 2 ) = Z NDT (M X )

40 It turns out that P 1 and P 2 do not contribute to our invariants! Obtain a conifold transition identity Z NDT (M ɛ t X ) = Z NDT (M Y1 /D 1 ) = Z NDT (M Y2 /D 2 ) = Z NDT (M X ) Theory over Nodal fibrations Theory over smooth fibrations

41 It turns out that P 1 and P 2 do not contribute to our invariants! Obtain a conifold transition identity Z NDT (M ɛ t X ) = Z NDT (M Y1 /D 1 ) = Z NDT (M Y2 /D 2 ) = Z NDT (M X ) Theory over Nodal fibrations Theory over smooth fibrations Application: When the fibration X Z is CY3

42 It turns out that P 1 and P 2 do not contribute to our invariants! Obtain a conifold transition identity Z NDT (M ɛ t X ) = Z NDT (M Y1 /D 1 ) = Z NDT (M Y2 /D 2 ) = Z NDT (M X ) Theory over Nodal fibrations Theory over smooth fibrations Application: When the fibration X Z is CY3 Example 1 χ(hilb [n] (Nodal Surface), ν): χ(hilb [n] (X ɛ(pi )), ν β M Hilb [n] (X )) = 1 1 ɛ(pi ) X r 2 χ(hilb[n] ( X / c)) deg(k C ) «χ(c {p 1,, p r }) χ(hilb [n] (ɛ X q)). (10)

43 Example Definitions and motivation from String theory 1 χ(hilb [n] (Nodal Surface), ν): χ(hilb [n] (X ɛ(pi )), ν β M Hilb [n] (X )) = 1 1 ɛ(pi ) X r 2 χ(hilb[n] ( X / c)) deg(k C ) «χ(c {p 1,, p r }) χ(hilb [n] (ɛ X q)). (11) 2 χ(m/z, ν) for γ 0, for a nodal fiber: χ(m β X ɛ(pi ), ν M β X 1 1 r X 2 h Z β ) = M X ɛ(pi ) X γ H 2( X ) 2L γ = P 2L(m) m m=0 χ(hilb h+1 P 2L(0) ( X / c)) NL π C h,γ «χ(c {p 1,, p r }) χ(hilb [n] (ɛ X q)). (12)

44 Work in Progress: NDT invariants of Quintic Threeofold. Following work of Pandharipande-Maulik (Topological view of GW theory)

45 Figure: Degeneration of Quintic

46 Thank you