Riemann-Hilbert problems from Donaldson-Thomas theory

Transcription

1 Riemann-Hilbert problems from Donaldson-Thomas theory Tom Bridgeland University of Sheffield Preprints: and / 25

2 Motivation 2 / 25

3 Motivation Two types of parameters in string theory: (i) Deformation parameters. (ii) Stability parameters. Exchanged by mirror symmetry. 2 / 25

4 Motivation Two types of parameters in string theory: (i) Deformation parameters. (ii) Stability parameters. Exchanged by mirror symmetry. The deformation space carries a variation of Hodge structures. Can one construct similar geometric structures on stability space? 2 / 25

5 Motivation Two types of parameters in string theory: (i) Deformation parameters. (ii) Stability parameters. Exchanged by mirror symmetry. The deformation space carries a variation of Hodge structures. Can one construct similar geometric structures on stability space? The obvious data to use is Donaldson-Thomas invariants. 2 / 25

6 Motivation Two types of parameters in string theory: (i) Deformation parameters. (ii) Stability parameters. Exchanged by mirror symmetry. The deformation space carries a variation of Hodge structures. Can one construct similar geometric structures on stability space? The obvious data to use is Donaldson-Thomas invariants. Fundamental property: Kontsevich-Soibelman wall-crossing formula. Strong analogy with Stokes factors from differential equations. 2 / 25

7 1. BPS structures.

8 The output of (unrefined) DT theory 4 / 25

9 The output of (unrefined) DT theory A BPS structure (Γ, Z, Ω) consists of (a) An abelian group Γ = Z n with a skew-symmetric form, : Γ Γ Z (b) A homomorphism of abelian groups Z : Γ C, (c) A map of sets Ω: Γ Q. 4 / 25

10 The output of (unrefined) DT theory A BPS structure (Γ, Z, Ω) consists of (a) An abelian group Γ = Z n with a skew-symmetric form, : Γ Γ Z (b) A homomorphism of abelian groups Z : Γ C, (c) A map of sets Ω: Γ Q. satisfying the conditions: (i) Symmetry: Ω( γ) = Ω(γ) for all γ Γ, (ii) Support property: fixing a norm on the finite-dimensional vector space Γ Z R, there is a C > 0 such that Ω(γ) 0 = Z(γ) > C γ. 4 / 25

11 Example: conifold BPS structure 5 / 25

12 Example: conifold BPS structure Take Γ = Z 2 with, = 0 and Z(r, d) = ir d. Set 1 if γ = ±(1, d) for some d Z, Ω(γ) = 2 if γ = (0, d) for some 0 d Z, 0 otherwise. This arises from DT theory applied to the resolved conifold. 5 / 25

13 Poisson algebraic torus 6 / 25

14 Poisson algebraic torus Consider the algebraic torus with character lattice Γ: T + = Hom Z (Γ, C ) = (C ) n C[T + ] = γ Γ C x γ = C[x ±1 1,, x ±n n ]. The form, induces an invariant Poisson structure on T + : {x α, x β } = α, β x α x β. 6 / 25

15 Poisson algebraic torus Consider the algebraic torus with character lattice Γ: T + = Hom Z (Γ, C ) = (C ) n C[T + ] = γ Γ C x γ = C[x ±1 1,, x ±n n ]. The form, induces an invariant Poisson structure on T + : {x α, x β } = α, β x α x β. More precisely we should work with an associated torsor T = { g : Γ C : g(γ 1 + γ 2 ) = ( 1) γ 1,γ 2 g(γ 1 ) g(γ 2 ) }, which we call the twisted torus. 6 / 25

16 DT Hamiltonians 7 / 25

17 DT Hamiltonians The DT invariants DT(γ) Q of a BPS structure are defined by DT(γ) = γ=nα Ω(α) n 2. For any ray l = R >0 z C we consider the generating function DT(l) = DT(γ) x γ. Z(γ) l A ray l C is called active if this expression is nonzero. 7 / 25

18 DT Hamiltonians The DT invariants DT(γ) Q of a BPS structure are defined by DT(γ) = γ=nα Ω(α) n 2. For any ray l = R >0 z C we consider the generating function DT(l) = DT(γ) x γ. Z(γ) l A ray l C is called active if this expression is nonzero. We would like to think of the time 1 Hamiltonian flow of the function DT(l) as defining a Poisson automorphism S(l) of the torus T. 7 / 25

19 Making sense of S(l) 8 / 25

20 Making sense of S(l) Formal approach Restrict to classes γ lying in a positive cone Γ + Γ, consider C[x ±1 1,, x ±1 n ] C[x 1,, x n ] C[[x 1,, x n ]], and the automorphism S(l) = exp{dt(l), } of this completion. 8 / 25

21 Making sense of S(l) Formal approach Restrict to classes γ lying in a positive cone Γ + Γ, consider C[x ±1 1,, x ±1 n ] C[x 1,, x n ] C[[x 1,, x n ]], and the automorphism S(l) = exp{dt(l), } of this completion. Analytic approach Restrict attention to BPS structures which are convergent: R > 0 such that γ Γ Ω(γ) e R Z(γ) <. Then on suitable analytic open subsets of T the sum DT(l) is absolutely convergent and its time 1 Hamiltonian flow S(l) exists. 8 / 25

22 Birational transformations Often the maps S(l) are birational automorphisms of T. 9 / 25

23 Birational transformations Often the maps S(l) are birational automorphisms of T. Note { } x nγ exp n, (x 2 β ) = x β (1 x γ ) β,γ. n 1 9 / 25

24 Birational transformations Often the maps S(l) are birational automorphisms of T. Note { } x nγ exp n, (x 2 β ) = x β (1 x γ ) β,γ. n 1 Whenever a ray l C satisfies (i) only finitely many active classes have Z(γ i ) l, (ii) these classes are mutually orthogonal γ i, γ j = 0, (iii) the corresponding BPS invariants Ω(γ i ) Z. there is a formula S(l) (x β ) = Z(γ) l (1 x γ ) Ω(γ) β,γ. 9 / 25

25 Variation of BPS structures 10 / 25

26 Variation of BPS structures A framed variation of BPS structures over a complex manifold S is a collection of BPS structures (Γ, Z s, Ω s ) indexed by s S such that 10 / 25

27 Variation of BPS structures A framed variation of BPS structures over a complex manifold S is a collection of BPS structures (Γ, Z s, Ω s ) indexed by s S such that (i) The numbers Z s (γ) C vary holomorphically. 10 / 25

28 Variation of BPS structures A framed variation of BPS structures over a complex manifold S is a collection of BPS structures (Γ, Z s, Ω s ) indexed by s S such that (i) The numbers Z s (γ) C vary holomorphically. (ii) For any convex sector C the clockwise ordered product S s ( ) = l S s (l) Aut(T) is constant whenever the boundary of remains non-active. Part (ii) is the Kontsevich-Soibelman wall-crossing formula. 10 / 25

29 Variation of BPS structures A framed variation of BPS structures over a complex manifold S is a collection of BPS structures (Γ, Z s, Ω s ) indexed by s S such that (i) The numbers Z s (γ) C vary holomorphically. (ii) For any convex sector C the clockwise ordered product S s ( ) = l S s (l) Aut(T) is constant whenever the boundary of remains non-active. Part (ii) is the Kontsevich-Soibelman wall-crossing formula. The complete set of numbers Ω s (γ) at some point s S determines them for all other points s S. 10 / 25

30 Example: the A 2 case 11 / 25

31 Example: the A 2 case Let Γ = Z 2 = Ze 1 Ze 2 with e 1, e 2 = 1. Then C[T] = C[x ±1 1, x ±1 2 ], {x 1, x 2 } = x 1 x / 25

32 Example: the A 2 case Let Γ = Z 2 = Ze 1 Ze 2 with e 1, e 2 = 1. Then C[T] = C[x ±1 1, x ±1 2 ], {x 1, x 2 } = x 1 x 2. A central charge Z : Γ C is determined by z i = Z(e i ). Take S = h 2 = {(z 1, z 2 ) : z i h}. 11 / 25

33 Example: the A 2 case Let Γ = Z 2 = Ze 1 Ze 2 with e 1, e 2 = 1. Then C[T] = C[x ±1 1, x ±1 2 ], {x 1, x 2 } = x 1 x 2. A central charge Z : Γ C is determined by z i = Z(e i ). Take Define BPS invariants as follows: S = h 2 = {(z 1, z 2 ) : z i h}. (a) Im(z 2 /z 1 ) > 0. Set Ω(±e 1 ) = Ω(±e 2 ) = 1, all others zero. (b) Im(z 2 /z 1 ) < 0. Set Ω(±e 1 ) = Ω(±(e 1 + e 2 )) = Ω(±e 2 ) = / 25

34 Wall-crossing formula: A 2 case 12 / 25

35 Wall-crossing formula: A 2 case Two types of BPS structures appear, as illustrated below Z(e 1 +e 2 ) Z(e 1 +e 2 ) Z(e 2 ) Z(e 1 ) Z(e 1 ) Z(e 2 ) 2 active rays 3 active rays 12 / 25

36 Wall-crossing formula: A 2 case Two types of BPS structures appear, as illustrated below Z(e 1 +e 2 ) Z(e 1 +e 2 ) Z(e 2 ) Z(e 1 ) Z(e 1 ) Z(e 2 ) 2 active rays 3 active rays The wall-crossing formula is the cluster pentagon identity C (0,1) C (1,0) = C (1,0) C (1,1) C (0,1). C α : x β x β (1 x α ) α,β. 12 / 25

37 2. The Riemann-Hilbert problem.

38 The Riemann-Hilbert problem Fix a BPS structure (Γ, Z, Ω) and a point ξ T. 14 / 25

39 The Riemann-Hilbert problem Fix a BPS structure (Γ, Z, Ω) and a point ξ T. Find a piecewise holomorphic function Φ: C T satisfying: 14 / 25

40 The Riemann-Hilbert problem Fix a BPS structure (Γ, Z, Ω) and a point ξ T. Find a piecewise holomorphic function Φ: C T satisfying: (i) (Jumping): When t crosses an active ray l clockwise, Φ(t) S(l)(Φ(t)). 14 / 25

41 The Riemann-Hilbert problem Fix a BPS structure (Γ, Z, Ω) and a point ξ T. Find a piecewise holomorphic function Φ: C T satisfying: (i) (Jumping): When t crosses an active ray l clockwise, Φ(t) S(l)(Φ(t)). (ii) (Limit at 0): Write Φ γ (t)) = x γ (Φ(t)). As t 0, Φ γ (t) e Z(γ)/t x γ (ξ). 14 / 25

42 The Riemann-Hilbert problem Fix a BPS structure (Γ, Z, Ω) and a point ξ T. Find a piecewise holomorphic function Φ: C T satisfying: (i) (Jumping): When t crosses an active ray l clockwise, Φ(t) S(l)(Φ(t)). (ii) (Limit at 0): Write Φ γ (t)) = x γ (Φ(t)). As t 0, Φ γ (t) e Z(γ)/t x γ (ξ). (iii) (Growth at ): For any γ Γ there exists k > 0 with t k < Φ γ (t) < t k as t. 14 / 25

43 The A 1 example 15 / 25

44 The A 1 example Consider the following BPS structure (i) The lattice Γ = Z γ is one-dimensional. Thus, = 0. (ii) The central charge Z : Γ C is determined by z = Z(γ) C, (iii) The only non-vanishing BPS invariants are Ω(±γ) = / 25

45 The A 1 example Consider the following BPS structure (i) The lattice Γ = Z γ is one-dimensional. Thus, = 0. (ii) The central charge Z : Γ C is determined by z = Z(γ) C, (iii) The only non-vanishing BPS invariants are Ω(±γ) = 1. Then T = C and all automorphisms S(l) are the identity. Φ γ (t) = ξ exp( z/t) T = C. 15 / 25

46 The A 1 example Consider the following BPS structure (i) The lattice Γ = Z γ is one-dimensional. Thus, = 0. (ii) The central charge Z : Γ C is determined by z = Z(γ) C, (iii) The only non-vanishing BPS invariants are Ω(±γ) = 1. Then T = C and all automorphisms S(l) are the identity. Φ γ (t) = ξ exp( z/t) T = C. Now double the BPS structure: take the lattice Γ Γ with canonical skew form, and extend Z and Ω by zero. Consider y(t) = Φ γ (t): C C. 15 / 25

47 Doubled A 1 case 16 / 25

48 Doubled A 1 case Consider the case ξ = 1. The map y : C C should satisfy 16 / 25

49 Doubled A 1 case Consider the case ξ = 1. The map y : C C should satisfy (i) y is holomorphic away from the rays R >0 (±z) and has jumps y(t) y(t) (1 x(t) ±1 ) ±1, x(t) = exp( z/t), as t moves clockwise across them. 16 / 25

50 Doubled A 1 case Consider the case ξ = 1. The map y : C C should satisfy (i) y is holomorphic away from the rays R >0 (±z) and has jumps y(t) y(t) (1 x(t) ±1 ) ±1, x(t) = exp( z/t), as t moves clockwise across them. (ii) y(t) 1 as t / 25

51 Doubled A 1 case Consider the case ξ = 1. The map y : C C should satisfy (i) y is holomorphic away from the rays R >0 (±z) and has jumps y(t) y(t) (1 x(t) ±1 ) ±1, x(t) = exp( z/t), as t moves clockwise across them. (ii) y(t) 1 as t 0. (iii) there exists k > 0 such that t k < y(t) < t k as t. 16 / 25

52 Solution: the Gamma function 17 / 25

53 Solution: the Gamma function The doubled A 1 problem has the unique solution ( ) 1 ±z y(t) = where (w) = ew Γ(w), 2πit 2π w w 1 2 in the half-planes ± Im(t/z) > / 25

54 Solution: the Gamma function The doubled A 1 problem has the unique solution ( ) 1 ±z y(t) = where (w) = ew Γ(w), 2πit 2π w w 1 2 in the half-planes ± Im(t/z) > 0. This is elementary: all you need is Γ(w) Γ(1 w) = π sin(πw), Γ(w + 1) = w Γ(w), log (w) g=1 B 2g 2g(2g 1) w 1 2g. 17 / 25

55 The tau function 18 / 25

56 The tau function Suppose given a framed variation of BPS structures (Γ, Z p, Ω p ) over a complex manifold S such that π : S Hom Z (Γ, C) = C n, s Z s, is a local isomorphism. Taking a basis (γ 1,, γ n ) Γ we get local co-ordinates z i = Z s (γ i ) on S. 18 / 25

57 The tau function Suppose given a framed variation of BPS structures (Γ, Z p, Ω p ) over a complex manifold S such that π : S Hom Z (Γ, C) = C n, s Z s, is a local isomorphism. Taking a basis (γ 1,, γ n ) Γ we get local co-ordinates z i = Z s (γ i ) on S. Suppose we are given analytically varying solutions Φ γ (z i, t) to the Riemann-Hilbert problems associated to (Γ, Z s, Ω s ). 18 / 25

58 The tau function Suppose given a framed variation of BPS structures (Γ, Z p, Ω p ) over a complex manifold S such that π : S Hom Z (Γ, C) = C n, s Z s, is a local isomorphism. Taking a basis (γ 1,, γ n ) Γ we get local co-ordinates z i = Z s (γ i ) on S. Suppose we are given analytically varying solutions Φ γ (z i, t) to the Riemann-Hilbert problems associated to (Γ, Z s, Ω s ). Define a function τ = τ(z i, t) by the relation t log Φ γ k (z i, t) = n j=1 ɛ jk log τ(z i, t), ɛ jk = γ j, γ k. z j 18 / 25

59 Solution in uncoupled case 19 / 25

60 Solution in uncoupled case In the A 1 case the τ-function is essentially the Barnes G-function. log τ(z, t) g 1 B 2g ( 2πit 2g(2g 2) z ) 2g / 25

61 Solution in uncoupled case In the A 1 case the τ-function is essentially the Barnes G-function. log τ(z, t) g 1 B 2g ( 2πit 2g(2g 2) z Whenever our BPS structures are uncoupled Ω(γ i ) 0 = γ 1, γ 2 = 0, ) 2g 2. we can try to solve the RH problem by superposition of A 1 solutions. This works precisely if only finitely many Ω(γ) / 25

62 Solution in uncoupled case In the A 1 case the τ-function is essentially the Barnes G-function. log τ(z, t) g 1 B 2g ( 2πit 2g(2g 2) z Whenever our BPS structures are uncoupled Ω(γ i ) 0 = γ 1, γ 2 = 0, ) 2g 2. we can try to solve the RH problem by superposition of A 1 solutions. This works precisely if only finitely many Ω(γ) 0. log τ(z, t) g 1 γ Γ Ω(γ) B 2g ( ) 2g 2 2πit 2g(2g 2) Z(γ) 19 / 25

63 Geometric case: curves on a CY 3 Can apply this to coherent sheaves on a compact Calabi-Yau threefold supported in dimension / 25

64 Geometric case: curves on a CY 3 Can apply this to coherent sheaves on a compact Calabi-Yau threefold supported in dimension 1. We have Γ = H 2 (X, Z) Z, Z(β, n) = 2π(β ω C n). Ω(β, n) = GV 0 (β), Ω(0, n) = χ(x ). 20 / 25

65 Geometric case: curves on a CY 3 Can apply this to coherent sheaves on a compact Calabi-Yau threefold supported in dimension 1. We have Γ = H 2 (X, Z) Z, Z(β, n) = 2π(β ω C n). Ω(β, n) = GV 0 (β), Ω(0, n) = χ(x ). Since χ(, ) = 0 these BPS structures are uncoupled. 20 / 25

66 Geometric case: curves on a CY 3 Can apply this to coherent sheaves on a compact Calabi-Yau threefold supported in dimension 1. We have Γ = H 2 (X, Z) Z, Z(β, n) = 2π(β ω C n). Ω(β, n) = GV 0 (β), Ω(0, n) = χ(x ). Since χ(, ) = 0 these BPS structures are uncoupled. τ(ω C, t) pos. deg g 2 χ(x ) B 2g B 2g 2 4g (2g 2) (2g 2)! (2πt)2g 2 + β H 2 (X,Z) k 1 GV 0 (β) e2πiω kβ 4k sin 2 (iπtk). Matches degenerate contributions from genus 0 GV invariants. 20 / 25

67 Resolved conifold again 21 / 25

68 Resolved conifold again Take Γ = Z 2 with, = 0 and 1 if γ = ±(1, d) for some d Z, Ω(γ) = 2 if γ = (0, d) for some 0 d Z, 0 otherwise. 21 / 25

69 Resolved conifold again Take Γ = Z 2 with, = 0 and 1 if γ = ±(1, d) for some d Z, Ω(γ) = 2 if γ = (0, d) for some 0 d Z, 0 otherwise. We get a variation of BPS structures over { (v, w) C 2 : w 0 and v + dw 0 for all d Z } C 2 by setting Z(r, d) = rv + dw. 21 / 25

70 Non-perturbative partition function The corresponding RH problems have unique solutions, which can be written explicitly in terms of Barnes double and triple sine functions. 22 / 25

71 Non-perturbative partition function The corresponding RH problems have unique solutions, which can be written explicitly in terms of Barnes double and triple sine functions. τ(v, w, t) = H(v, w, t) exp(r(v, w, t)), ( ) e vs 1 H(v, w, t) = exp e ws 1 e ts (e ts 1) ds, 2 s R(v, w, t) = ( w 2πit R+iɛ ) 2( Li3 (e 2πiv/w ) ζ(3) ) + iπ 12 v w. The function H is a non-perturbative closed-string partition function. 22 / 25

72 Finite-dimensional analogy 23 / 25

73 Finite-dimensional analogy Matrix differential equation for X : C G = GL n (C) ( d U dt X (t) = t + V ) X (t), U, V g = gl 2 t n (C). Take U = diag(u 1,, u n ) with u i u j and V skew-symmetric. 23 / 25

74 Finite-dimensional analogy Matrix differential equation for X : C G = GL n (C) ( d U dt X (t) = t + V ) X (t), U, V g = gl 2 t n (C). Take U = diag(u 1,, u n ) with u i u j and V skew-symmetric. The Stokes rays are l ij = R >0 (u i u j ). Fact: in any half-plane centered on a non-stokes ray there exists a unique solution such that X (t) exp(u/t) 1 as t / 25

75 Finite-dimensional analogy Matrix differential equation for X : C G = GL n (C) ( d U dt X (t) = t + V ) X (t), U, V g = gl 2 t n (C). Take U = diag(u 1,, u n ) with u i u j and V skew-symmetric. The Stokes rays are l ij = R >0 (u i u j ). Fact: in any half-plane centered on a non-stokes ray there exists a unique solution such that X (t) exp(u/t) 1 as t 0. The Stokes factors S lij G describe how these solutions jump. 23 / 25

76 Finite-dimensional analogy Matrix differential equation for X : C G = GL n (C) ( d U dt X (t) = t + V ) X (t), U, V g = gl 2 t n (C). Take U = diag(u 1,, u n ) with u i u j and V skew-symmetric. The Stokes rays are l ij = R >0 (u i u j ). Fact: in any half-plane centered on a non-stokes ray there exists a unique solution such that X (t) exp(u/t) 1 as t 0. The Stokes factors S lij G describe how these solutions jump. Iso-Stokes deformation: as U varies we can vary V in a unique way so that the product of Stokes factors in any fixed sector is constant. 23 / 25

77 Wall-crossing formula = iso-stokes 24 / 25

78 Wall-crossing formula = iso-stokes Differential equation for X : C G = Aut(T) ( d Z dt X (t) = t + F ) X (t), 2 t with Z Hom Z (Γ, C) and F = γ Γ f γ (x γ + x γ ), where g = Vect(T) = Hom Z (Γ, C) γ Γ C x γ. 24 / 25

79 Wall-crossing formula = iso-stokes Differential equation for X : C G = Aut(T) ( d Z dt X (t) = t + F ) X (t), 2 t with Z Hom Z (Γ, C) and F = γ Γ f γ (x γ + x γ ), where g = Vect(T) = Hom Z (Γ, C) γ Γ C x γ. Have Stokes rays are R >0 Z(γ) and Stokes factors S(l) G. 24 / 25

80 Wall-crossing formula = iso-stokes Differential equation for X : C G = Aut(T) ( d Z dt X (t) = t + F ) X (t), 2 t with Z Hom Z (Γ, C) and F = γ Γ f γ (x γ + x γ ), where g = Vect(T) = Hom Z (Γ, C) γ Γ C x γ. Have Stokes rays are R >0 Z(γ) and Stokes factors S(l) G. Wall-crossing formula is iso-stokes condition. 24 / 25

81 Wall-crossing formula = iso-stokes Differential equation for X : C G = Aut(T) ( d Z dt X (t) = t + F ) X (t), 2 t with Z Hom Z (Γ, C) and F = γ Γ f γ (x γ + x γ ), where g = Vect(T) = Hom Z (Γ, C) γ Γ C x γ. Have Stokes rays are R >0 Z(γ) and Stokes factors S(l) G. Wall-crossing formula is iso-stokes condition. Note that given ξ T there is a map eval ξ : G T. 24 / 25

82 Further directions (i) Theories of class S with G = SL 2 (C). Variation of BPS structures over space of meromorphic quadratic differentials. Monodromy of projective structures gives map to space of framed local systems. Fock-Goncharov co-ordinates give solutions to RH problem (joint with D. Allegretti). 25 / 25

83 Further directions (i) Theories of class S with G = SL 2 (C). Variation of BPS structures over space of meromorphic quadratic differentials. Monodromy of projective structures gives map to space of framed local systems. Fock-Goncharov co-ordinates give solutions to RH problem (joint with D. Allegretti). (ii) Analogy with Stokes data in finite-dimensional case. Allow ξ to vary to get RH problem with values in G = Aut(T). Uncoupled case can be solved following Gaiotto (joint with A. Barbieri). 25 / 25

84 Further directions (i) Theories of class S with G = SL 2 (C). Variation of BPS structures over space of meromorphic quadratic differentials. Monodromy of projective structures gives map to space of framed local systems. Fock-Goncharov co-ordinates give solutions to RH problem (joint with D. Allegretti). (ii) Analogy with Stokes data in finite-dimensional case. Allow ξ to vary to get RH problem with values in G = Aut(T). Uncoupled case can be solved following Gaiotto (joint with A. Barbieri). (iii) Our current formalism gives the partition function without the terms in t 2g 2 for g = 0, 1. In examples, these additional terms make τ satisfy a difference equation. How to understand this? Can we quantize the RH problem? (J. Calabrese). 25 / 25