Refined Donaldson-Thomas theory and Nekrasov s formula

Transcription

1 Refined Donaldson-Thomas theory and Nekrasov s formula Balázs Szendrői, University of Oxford Maths of String and Gauge Theory, City University and King s College London 3-5 May 2012

2 Geometric engineering Geometric engineering starts with a product X R 4, where X is a (local) Calabi Yau threefold.

3 Geometric engineering Geometric engineering starts with a product X R 4, where X is a (local) Calabi Yau threefold. For appropriate X, integrating out the X-directions results in a gauge theory on C 2 = R 4, with gauge group G = G(X). The partition function Z C 2 is (a version of) Nekrasov s partition function

4 Geometric engineering Geometric engineering starts with a product X R 4, where X is a (local) Calabi Yau threefold. For appropriate X, integrating out the X-directions results in a gauge theory on C 2 = R 4, with gauge group G = G(X). The partition function Z C 2 is (a version of) Nekrasov s partition function Integrating out the R 4 -directions results in a U(1) gauge theory on the threefold X. The partition function Z X is (a version of) the topological string partition function of X

5 Geometric engineering Geometric engineering starts with a product X R 4, where X is a (local) Calabi Yau threefold. For appropriate X, integrating out the X-directions results in a gauge theory on C 2 = R 4, with gauge group G = G(X). The partition function Z C 2 is (a version of) Nekrasov s partition function Integrating out the R 4 -directions results in a U(1) gauge theory on the threefold X. The partition function Z X is (a version of) the topological string partition function of X The aim of the talk is to discuss the precise relationship Z C 2 Z X in the simplest example, and study the question whether there is anything more to this than an equality of generating series.

6 Gauge-theoretic moduli spaces on C 2 Assume that X is chosen so that G = U(n). Then we have the moduli space M n,k = { (E, ) framed finite action U(n)-instanton of charge k on R 4}.

7 Gauge-theoretic moduli spaces on C 2 Assume that X is chosen so that G = U(n). Then we have the moduli space M n,k = { (E, ) framed finite action U(n)-instanton of charge k on R 4}. A finite action instanton extends to S 4 R 4, and framing is an isomorphism of the fibre at infinity with a fixed vector space.

8 Gauge-theoretic moduli spaces on C 2 Assume that X is chosen so that G = U(n). Then we have the moduli space M n,k = { (E, ) framed finite action U(n)-instanton of charge k on R 4}. A finite action instanton extends to S 4 R 4, and framing is an isomorphism of the fibre at infinity with a fixed vector space. By the Kobayashi Hitchin correspondence (Donaldson), this space can also be obtained as the moduli space M n,k = { (E, φ) framed rank-n vector bundle on P 2, c 2 (E) = k }.

9 Gauge-theoretic moduli spaces on C 2 Assume that X is chosen so that G = U(n). Then we have the moduli space M n,k = { (E, ) framed finite action U(n)-instanton of charge k on R 4}. A finite action instanton extends to S 4 R 4, and framing is an isomorphism of the fibre at infinity with a fixed vector space. By the Kobayashi Hitchin correspondence (Donaldson), this space can also be obtained as the moduli space M n,k = { (E, φ) framed rank-n vector bundle on P 2, c 2 (E) = k }. Here C 2 P 2 with complement l = P 1, and the framing is an isomorphism φ: E l = O n P 1

10 More gauge-theoretic moduli spaces on C 2 This space suffers from divergencies. A better behaved space, taking account of singular instantons, is obtained by considering M n,k = { (E, φ) framed rank-n torsion-free sheaf on P 2, c 2 (E) = k }.

11 More gauge-theoretic moduli spaces on C 2 This space suffers from divergencies. A better behaved space, taking account of singular instantons, is obtained by considering M n,k = { (E, φ) framed rank-n torsion-free sheaf on P 2, c 2 (E) = k }. (The corresponding instantons really live on a non-commutative R 4.) This is still noncompact, but nonsingular and complete in a natural metric.

12 More gauge-theoretic moduli spaces on C 2 This space suffers from divergencies. A better behaved space, taking account of singular instantons, is obtained by considering M n,k = { (E, φ) framed rank-n torsion-free sheaf on P 2, c 2 (E) = k }. (The corresponding instantons really live on a non-commutative R 4.) This is still noncompact, but nonsingular and complete in a natural metric. Now the U(n) Nekrasov partition function is Z U(n) (Λ) = Λ k 1. C 2 k 0 M n,k

13 More gauge-theoretic moduli spaces on C 2 This space suffers from divergencies. A better behaved space, taking account of singular instantons, is obtained by considering M n,k = { (E, φ) framed rank-n torsion-free sheaf on P 2, c 2 (E) = k }. (The corresponding instantons really live on a non-commutative R 4.) This is still noncompact, but nonsingular and complete in a natural metric. Now the U(n) Nekrasov partition function is This is an ill-defined expression. Z U(n) (Λ) = Λ k 1. C 2 k 0 M n,k

14 Symmetries of the gauge-theoretic moduli spaces Note that all M n,k carry an action of the torus T = (C ) 2 (C ) n 1.

15 Symmetries of the gauge-theoretic moduli spaces Note that all M n,k carry an action of the torus T = (C ) 2 (C ) n 1. The first component acts on E via its action on C 2 P 2.

16 Symmetries of the gauge-theoretic moduli spaces Note that all M n,k carry an action of the torus T = (C ) 2 (C ) n 1. The first component acts on E via its action on C 2 P 2. The second component (the maximal torus of the complex gauge group G C ) acts on the framing φ.

17 Symmetries of the gauge-theoretic moduli spaces Note that all M n,k carry an action of the torus T = (C ) 2 (C ) n 1. The first component acts on E via its action on C 2 P 2. The second component (the maximal torus of the complex gauge group G C ) acts on the framing φ. Crucial fact: the fixed point set M T n,k is a finite set for all n, k.

18 Symmetries of the gauge-theoretic moduli spaces Note that all M n,k carry an action of the torus T = (C ) 2 (C ) n 1. The first component acts on E via its action on C 2 P 2. The second component (the maximal torus of the complex gauge group G C ) acts on the framing φ. Crucial fact: the fixed point set M T n,k is a finite set for all n, k. Thus T -equivariant integrals make sense on the moduli space M n,k.

19 The equivariant index and the K-theoretic partition function We will be particularly interested in a K-theoretic interpretation of the partition function (M-theory).

20 The equivariant index and the K-theoretic partition function We will be particularly interested in a K-theoretic interpretation of the partition function (M-theory). The integrand 1 is interpreted as the (K-theory class of) the trivial line bundle O Mn,k.

21 The equivariant index and the K-theoretic partition function We will be particularly interested in a K-theoretic interpretation of the partition function (M-theory). The integrand 1 is interpreted as the (K-theory class of) the trivial line bundle O Mn,k. Integration gets replaced by cohomology (pushforward to the point).

22 The equivariant index and the K-theoretic partition function We will be particularly interested in a K-theoretic interpretation of the partition function (M-theory). The integrand 1 is interpreted as the (K-theory class of) the trivial line bundle O Mn,k. Integration gets replaced by cohomology (pushforward to the point). Thus Nekrasov defines Z U(n) C 2 (Λ, q 1, q 2, a 1,..., a n 1 ) = k 0 Λ k char T H (M n,k, O Mn,k )

23 The equivariant index and the K-theoretic partition function We will be particularly interested in a K-theoretic interpretation of the partition function (M-theory). The integrand 1 is interpreted as the (K-theory class of) the trivial line bundle O Mn,k. Integration gets replaced by cohomology (pushforward to the point). Thus Nekrasov defines Z U(n) C 2 (Λ, q 1, q 2, a 1,..., a n 1 ) = k 0 Λ k char T H (M n,k, O Mn,k ) Here, for a representation V of T, char T V Z[q i, a j ] denotes its T -character.

24 Nekrasov s partition function for U(1) Assume now that in fact G = U(1). There is only one line bundle on P 2 with trivial determinant, so

25 Nekrasov s partition function for U(1) Assume now that in fact G = U(1). There is only one line bundle on P 2 with trivial determinant, so M 1,k = Hilb k (C 2 ) the Hilbert scheme of k points on the plane.

26 Nekrasov s partition function for U(1) Assume now that in fact G = U(1). There is only one line bundle on P 2 with trivial determinant, so M 1,k = Hilb k (C 2 ) the Hilbert scheme of k points on the plane. It can be shown that there is no higher cohomology H i ( Hilb k (C 2 ), O ) = 0 for i > 0

27 Nekrasov s partition function for U(1) Assume now that in fact G = U(1). There is only one line bundle on P 2 with trivial determinant, so M 1,k = Hilb k (C 2 ) the Hilbert scheme of k points on the plane. It can be shown that there is no higher cohomology and that H i ( Hilb k (C 2 ), O ) = 0 for i > 0 H 0 ( Hilb k (C 2 ), O ) = H 0 ( S k (C 2 ), O ) where S k (C 2 ) is the k-th symmetric power of C 2.

28 Nekrasov s partition function for U(1): the computation Now we can finish the computation: Z U(1) (Λ, q C 2 1, q 2 ) = Λ k char T H ( Hilb k (C 2 ), O ) k 0 = k 0 Λ k char T H 0 (S k (C 2 ), O)

29 Nekrasov s partition function for U(1): the computation Now we can finish the computation: Z U(1) (Λ, q C 2 1, q 2 ) = Λ k char T H ( Hilb k (C 2 ), O ) k 0 = k 0 Λ k char T H 0 (S k (C 2 ), O)

30 Nekrasov s partition function for U(1): the computation Now we can finish the computation: Z U(1) (Λ, q C 2 1, q 2 ) = Λ k char T H ( Hilb k (C 2 ), O ) k 0 = k 0 Λ k char T H 0 (S k (C 2 ), O) = k 0 Λ k char T S k C[x, y]

31 Nekrasov s partition function for U(1): the computation Now we can finish the computation: Z U(1) (Λ, q C 2 1, q 2 ) = Λ k char T H ( Hilb k (C 2 ), O ) k 0 = k 0 Λ k char T H 0 (S k (C 2 ), O) = k 0 Λ k char T S k C[x, y] = char T C S C[x, y]

32 Nekrasov s partition function for U(1): the computation Now we can finish the computation: Z U(1) (Λ, q C 2 1, q 2 ) = Λ k char T H ( Hilb k (C 2 ), O ) k 0 = k 0 Λ k char T H 0 (S k (C 2 ), O) = k 0 Λ k char T S k C[x, y] = char T C S C[x, y] = (1 q1q i j 2 Λ) 1. i,j 0

33 Relation to the conifold We have Z U(1) (Λ, q C 2 1, q 2 ) = (1 q1q i j 2 Λ) 1 i,j 0

34 Relation to the conifold We have Z U(1) (Λ, q C 2 1, q 2 ) = (1 q1q i j 2 Λ) 1 i,j 0 and thus, setting q 1 = q 2 = q and T = q 1 Λ

35 Relation to the conifold We have Z U(1) (Λ, q C 2 1, q 2 ) = (1 q1q i j 2 Λ) 1 i,j 0 and thus, setting q 1 = q 2 = q and T = q 1 Λ, Z U(1) C 2 (Λ, q = q 1 = q 2 ) 1 = k 1(1 q k T ) k =Z X(q, T ) the reduced topological string partition function of the resolved conifold X.

36 Relation to the conifold We have Z U(1) (Λ, q C 2 1, q 2 ) = (1 q1q i j 2 Λ) 1 i,j 0 and thus, setting q 1 = q 2 = q and T = q 1 Λ, Z U(1) C 2 (Λ, q = q 1 = q 2 ) 1 = k 1(1 q k T ) k = Z X(q, T ) the reduced topological string partition function of the resolved conifold X.

37 Relation to the conifold We have Z U(1) (Λ, q C 2 1, q 2 ) = (1 q1q i j 2 Λ) 1 i,j 0 and thus, setting q 1 = q 2 = q and T = q 1 Λ, Z U(1) C 2 (Λ, q = q 1 = q 2 ) 1 = k 1(1 q k T ) k = Z X(q, T ) the reduced topological string partition function of the resolved conifold X. Indeed, log Z U(1) C 2 ( Λ, q1 = q 2 = e i ) = g 0 2g 2 F X g (T ) where F X g (T ) is the genus-g Gromov-Witten potential of X.

38 Gauge-theoretic interpretation on X The reduced topological string partition function Z X(q, T ) = k 1 (1 q k T ) k = n,l q n T l P n,l corresponds under the MNOP correspondence (gauge/string duality), to a version of U(1) gauge theory on the resolved conifold X (Pandharipande-Thomas stable pairs theory).

39 Gauge-theoretic interpretation on X The reduced topological string partition function Z X(q, T ) = k 1 (1 q k T ) k = n,l q n T l P n,l corresponds under the MNOP correspondence (gauge/string duality), to a version of U(1) gauge theory on the resolved conifold X (Pandharipande-Thomas stable pairs theory).

40 Gauge-theoretic interpretation on X The reduced topological string partition function Z X(q, T ) = k 1 (1 q k T ) k = n,l q n T l P n,l corresponds under the MNOP correspondence (gauge/string duality), to a version of U(1) gauge theory on the resolved conifold X (Pandharipande-Thomas stable pairs theory). The coefficients P n,l are the (virtual) numbers of U(1)-sheaves on X of a certain kind.

41 Gauge-theoretic interpretation on X The reduced topological string partition function Z X(q, T ) = k 1 (1 q k T ) k = n,l q n T l P n,l corresponds under the MNOP correspondence (gauge/string duality), to a version of U(1) gauge theory on the resolved conifold X (Pandharipande-Thomas stable pairs theory). The coefficients P n,l are the (virtual) numbers of U(1)-sheaves on X of a certain kind. The corresponding moduli spaces N n,l are singular gauge-theoretic moduli spaces associated to X.

42 More about gauge-theoretic moduli spaces on X Recall that Z X (q, T ) was a specialization of the full U(1) partition function of C 2 at q 1 = q 2 = q. It is natural to ask what is the geometric interpretation of the full partition function on the conifold X.

43 More about gauge-theoretic moduli spaces on X Recall that Z X (q, T ) was a specialization of the full U(1) partition function of C 2 at q 1 = q 2 = q. It is natural to ask what is the geometric interpretation of the full partition function on the conifold X. The coefficients P n,l are a kind of Euler characteristic of the moduli spaces N n,l.

44 More about gauge-theoretic moduli spaces on X Recall that Z X (q, T ) was a specialization of the full U(1) partition function of C 2 at q 1 = q 2 = q. It is natural to ask what is the geometric interpretation of the full partition function on the conifold X. The coefficients P n,l are a kind of Euler characteristic of the moduli spaces N n,l. Theorem (Sz., Nagao-Nakajima) The spaces N n,l are global critical loci N n,l = Zeros (df n,l ) of smooth functions f n,l on smooth manifolds N n,l.

45 More about gauge-theoretic moduli spaces on X Recall that Z X (q, T ) was a specialization of the full U(1) partition function of C 2 at q 1 = q 2 = q. It is natural to ask what is the geometric interpretation of the full partition function on the conifold X. The coefficients P n,l are a kind of Euler characteristic of the moduli spaces N n,l. Theorem (Sz., Nagao-Nakajima) The spaces N n,l are global critical loci N n,l = Zeros (df n,l ) of smooth functions f n,l on smooth manifolds N n,l. This uses the quiver description of the conifold and the Klebanov-Witten superpotential.

46 Refining the numerical gauge theoretic invariants of X Using the critical locus interpretation, one gets a topological coefficient system φ n,l on the singular moduli spaces N n,l, and a corresponding cohomology theory with mixed Hodge structure H (N n,l, φ n,l ).

47 Refining the numerical gauge theoretic invariants of X Using the critical locus interpretation, one gets a topological coefficient system φ n,l on the singular moduli spaces N n,l, and a corresponding cohomology theory with mixed Hodge structure H (N n,l, φ n,l ). This gives a weight polynomial W Nn,l (t) Z[t ±1 2].

48 Refining the numerical gauge theoretic invariants of X Using the critical locus interpretation, one gets a topological coefficient system φ n,l on the singular moduli spaces N n,l, and a corresponding cohomology theory with mixed Hodge structure H (N n,l, φ n,l ). This gives a weight polynomial W Nn,l (t) Z[t ±1 2]. This has the property that ) W Nn,l (t2 1 = 1 = P n,l, the numerical coefficient in the reduced partition function Z X (q, T ).

49 Refining the numerical gauge theoretic invariants of X Using the critical locus interpretation, one gets a topological coefficient system φ n,l on the singular moduli spaces N n,l, and a corresponding cohomology theory with mixed Hodge structure H (N n,l, φ n,l ). This gives a weight polynomial W Nn,l (t) Z[t ±1 2]. This has the property that ) W Nn,l (t2 1 = 1 = P n,l, the numerical coefficient in the reduced partition function Z X (q, T ). This weight polynomial refinement of the Euler characteristic is equivalent to the motivic refinement introduced by Kontsevich-Soibelman and studied by Behrend-Bryan-Sz., Dimofte-Gukov and others.

50 Interpretation of the full partition function on X Let Z X (q, t, T ) = n,l q n T l W Nn,l (t) be the refined partition function of the conifold.

51 Interpretation of the full partition function on X Let Z X (q, t, T ) = n,l q n T l W Nn,l (t) be the refined partition function of the conifold. Theorem (Morrison-Mozgovoy-Nagao-Sz.) Under the change of variables q 1 = qt2, 1 q 2 = qt 1 2, Λ = qt, we have Z X (q, t, T ) = (1 q1q i j 2Λ)= ZU(1) (Λ, q C 2 1, q 2 ) 1 i,j 0

52 Interpretation of the full partition function on X Let Z X (q, t, T ) = n,l q n T l W Nn,l (t) be the refined partition function of the conifold. Theorem (Morrison-Mozgovoy-Nagao-Sz.) Under the change of variables q 1 = qt2, 1 q 2 = qt 1 2, Λ = qt, we have Z X (q, t, T ) = (1 q1q i j 2Λ) = ZU(1) (Λ, q C 2 1, q 2 ) 1 i,j 0

53 Interpretation of the full partition function on X Let Z X (q, t, T ) = n,l q n T l W Nn,l (t) be the refined partition function of the conifold. Theorem (Morrison-Mozgovoy-Nagao-Sz.) Under the change of variables q 1 = qt2, 1 q 2 = qt 1 2, Λ = qt, we have Z X (q, t, T ) = (1 q1q i j 2Λ) = ZU(1) (Λ, q C 2 1, q 2 ) 1 i,j 0 Thus we obtain a cohomological interpretation on X of the full Nekrasov partition function in this case.

54 Relationship between the underlying vector spaces Recall from the computation that Z U(1) (Λ, q C 2 1, q 2 ) = char T C S C[x, y], just the Hilbert series of a triply-graded vector space, the symmetric space of the space C[x, y] of functions on C 2. Then, the corresponding exterior space! Z U(1) (Λ, q C 2 1, q 2 ) 1 = char T C Λ C[x, y],

55 Relationship between the underlying vector spaces Recall from the computation that Z U(1) (Λ, q C 2 1, q 2 ) = char T C S C[x, y], just the Hilbert series of a triply-graded vector space, the symmetric space of the space C[x, y] of functions on C 2.

56 Relationship between the underlying vector spaces Recall from the computation that Z U(1) (Λ, q C 2 1, q 2 ) = char T C S C[x, y], just the Hilbert series of a triply-graded vector space, the symmetric space of the space C[x, y] of functions on C 2. Then, the corresponding exterior space! Z U(1) (Λ, q C 2 1, q 2 ) 1 = char T C Λ C[x, y],

57 Relationship between the underlying vector spaces Recall from the computation that Z U(1) (Λ, q C 2 1, q 2 ) = char T C S C[x, y], just the Hilbert series of a triply-graded vector space, the symmetric space of the space C[x, y] of functions on C 2. Then, Z U(1) (Λ, q C 2 1, q 2 ) 1 = char T C Λ C[x, y], the corresponding exterior space! Assuming a purity conjecture, this suggests that there should be a natural isomorphism n,l H (N n,l, φ n,l ) = Λ C[x, y].

58 Relationship between the underlying vector spaces Recall from the computation that Z U(1) (Λ, q C 2 1, q 2 ) = char T C S C[x, y], just the Hilbert series of a triply-graded vector space, the symmetric space of the space C[x, y] of functions on C 2. Then, Z U(1) (Λ, q C 2 1, q 2 ) 1 = char T C Λ C[x, y], the corresponding exterior space! Assuming a purity conjecture, this suggests that there should be a natural isomorphism n,l H (N n,l, φ n,l ) = Λ C[x, y]. I don t know how to construct this, but SL(2)-action on the left hand side... should come from a Lefschetz SL(2)-action on the cohomology!

59 Relationship between the underlying vector spaces Recall from the computation that Z U(1) (Λ, q C 2 1, q 2 ) = char T C S C[x, y], just the Hilbert series of a triply-graded vector space, the symmetric space of the space C[x, y] of functions on C 2. Then, Z U(1) (Λ, q C 2 1, q 2 ) 1 = char T C Λ C[x, y], the corresponding exterior space! Assuming a purity conjecture, this suggests that there should be a natural isomorphism n,l H (N n,l, φ n,l ) = Λ C[x, y]. I don t know how to construct this, but SL(2)-action on the left hand side... should come from a Lefschetz SL(2)-action on the cohomology!

60 Towards higher rank invariants of the conifold Keep X to be the resolved conifold, but consider geometric engineering on Y X instead, where Y C 2 /Z n is the minimal resolution of the Type A n 1 surface singularity (Nekrasov).

61 Towards higher rank invariants of the conifold Keep X to be the resolved conifold, but consider geometric engineering on Y X instead, where Y C 2 /Z n is the minimal resolution of the Type A n 1 surface singularity (Nekrasov). On Y, this should engineer U(1) gauge theory. The corresponding partition function is still easy to write down, the answer is

62 Towards higher rank invariants of the conifold Keep X to be the resolved conifold, but consider geometric engineering on Y X instead, where Y C 2 /Z n is the minimal resolution of the Type A n 1 surface singularity (Nekrasov). On Y, this should engineer U(1) gauge theory. The corresponding partition function is still easy to write down, the answer is Z U(1) Y (Λ, q 1, q 2 ) = char T C S C[x, y] Z n = (1 q1q i j 2 Λ) 1 i,j 0 i j(n)

63 Towards higher rank invariants of the conifold Keep X to be the resolved conifold, but consider geometric engineering on Y X instead, where Y C 2 /Z n is the minimal resolution of the Type A n 1 surface singularity (Nekrasov). On Y, this should engineer U(1) gauge theory. The corresponding partition function is still easy to write down, the answer is Z U(1) Y (Λ, q 1, q 2 ) = char T C S C[x, y] Z n = i,j 0 i j(n) (1 q i 1q j 2 Λ) 1 On the conifold X, this should correspond to some version of U(n) Donaldson- Thomas theory. I don t know an interpretation at present!