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Verlinde formulae on complex surfaces I: K-theoretic invariants
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Abstract
We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by the first named author and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by the first and second named authors. We verify our conjectures in many examples (e.g. on K3 surfaces).
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We compute generating functions for elliptic genera with values in line
bundles on Hilbert schemes of points on surfaces. As an application we also
compute generating functions for elliptic genera with values in determinant
line bundles on moduli spaces of sheaves on K3 surfaces.
For a projective algebraic surface X X with an ample line bundle H H , let M H X ( c ) M_H^X(c) be the moduli space H H -semistable sheaves E \mathcal {E} of class c c in the Grothendieck group K ( X ) K(X) . We write c = ( r , c 1 , c 2 ) c=(r,c_1,c_2) or c = ( r , c 1 , χ ) c=(r,c_1,\chi ) with r r the rank, c 1 , c 2 c_1,c_2 the Chern classes, and χ \chi the holomorphic Euler characteristic. We also write M H X ( 2 , c 1 , c 2 ) = M X X ( c 1 , d ) M_H^X(2,c_1,c_2)=M_X^X(c_1,d) , with d = 4 c 2 − c 1 2 d=4c_2-c_1^2 . The K K -theoretic Donaldson invariants are the holomorphic Euler characteristics χ ( M H X ( c 1 , d ) , μ ( L ) ) \chi (M_H^X(c_1,d),\mu (L)) , where μ ( L ) \mu (L) is the determinant line bundle associated to a line bundle on X X . More generally for suitable classes c ∗ ∈ K ( X ) c^*\in K(X) there is a determinant line bundle D c , c ∗ \mathcal {D}_{c,c^*} on M H X ( c ) M^X_H(c) . We first compute some generating functions for K K -theoretic Donaldson invariants on P 2 \mathbb {P}^2 and rational ruled surfaces, using the wallcrossing formula of [Pure Appl. Math. Q. 5 (2009), pp. 1029–1111].
Then we show that Le Potier’s strange duality conjecture relating H 0 ( M H X ( c ) , D c , c ∗ ) H^0(M^X_H(c),\mathcal {D}_{c,c^*}) and H 0 ( M H X ( c ∗ ) , D c ∗ , c ) H^0(M^X_H(c^*),\mathcal {D}_{c^*,c}) holds for the cases c = ( 2 , c 1 = 0 , c 2 > 2 ) c=(2,c_1=0,c_2>2) and c ∗ = ( 0 , L , χ = 0 ) c^{*}=(0,L,\chi =0) with L = − K X L=-K_X on P 2 \mathbb {P}^2 , and L = − K X L=-K_X or − K X + F -K_X+F on P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 and P 2 ^ \widehat {\mathbb {P}^2} with F F the fibre class of the ruling, and also the case c = ( 2 , H , c 2 ) c=(2,H,c_2) and c ∗ = ( 0 , 2 H , χ = − 1 ) c^*=(0,2H,\chi =-1) on P 2 \mathbb {P}^2 .
We propose an explicit formula connecting Donaldson invariants and
Seiberg-Witten invariants of a 4-manifold of simple type via Nekrasov's
deformed partition function for the N=2 SUSY gauge theory with a single
fundamental matter. This formula is derived from Mochizuki's formula, which
makes sense and was proved when the 4-manifold is complex projective. Assuming
our formula is true for a 4-manifold of simple type, we prove Witten's
conjecture and sum rules for Seiberg-Witten invariants (superconformal simple
type condition), conjectured by Mari\~no, Moore and Peradze.
We study the moduli problem of pairs consisting of a rank 2 vector bundle and a nonzero section over a fixed smooth curve. The stability condition involves a parameter; as it varies, we show that the moduli space undergoes a sequence of flips in the sense of Mori. As applications, we prove several results about moduli spaces of rank 2 bundles, including the Harder-Narasimhan formula and the SU(2) Verlinde formula. Indeed, we prove a general result on the space of sections of powers of the ideal sheaf of a curve in projective space, which includes the Verlinde formula.
Let S be a smooth projective surfaces and S^[n] the Hilbert scheme of zero-dimensional subschemes of S of length n. We proof that the class of S^[n] in the complex cobordism ring depends only on the class of the surface itself. Moreover, we compute the cohomology and holomorphic Euler characterisitcs of certain tautological sheaves on S^[n] and prove results on the general structure of certain integrals over polynomials in Chern classes of tautological sheaves.
In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as K-theoretic versions of the Donaldson invariants. In particular, if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the physics literature). Using the results of math.AG/0606180 we give an explicit generating function for the wallcrossing of these invariants in terms of elliptic functions and modular forms.
For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture math.AG/0306198, hep-th/0306238, math.AG/0409441 and its refinement math.AG/0311058, we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with in terms of modular forms. This formula was proved earlier in alg-geom/9506018 more generally for simply connected 4-manifolds with , assuming the Kotschick-Morgan conjecture and it was also derived by physical arguments in hep-th/9709193.
For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves.
We conjecture a formula for the virtual elliptic genera of moduli spaces of rank 2 sheaves on minimal surfaces S of general type. We express our conjecture in terms of the Igusa cusp form and Borcherds type lifts of three quasi-Jacobi forms which are all related to the Weierstrass elliptic function. We also conjecture that the generating function of virtual cobordism classes of these moduli spaces depends only on and via two universal functions, one of which is determined by the cobordism classes of Hilbert schemes of points on K3. We present generalizations of these conjectures, e.g. to arbitrary surfaces with and . We use a result of J. Shen to express the virtual cobordism class in terms of descendent Donaldson invariants. In a prequel we used T. Mochizuki's formula, universality, and toric calculations to compute such Donaldson invariants in the setting of virtual -genera. Similar techniques allow us to verify our new conjectures in many cases.
Let S be a projective simply connected complex surface and be a line bundle on S. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of . The moduli space admits a -action induced by scaling the fibers of . We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on S. We define the localized Donaldson-Thomas invariants of by virtual localization in the case that twisted by the anti-canonical bundle of S admits a nonzero global section. When , in combination with Mochizuki's formulas, we are able to express the localized DT invariants in terms of the invariants of the nested Hilbert schemes defined by the authors in [GSY17a], the Seiberg-Witten invariants of S, and the integrals over the products of Hilbert schemes of points on S. When is the canonical bundle of S, the Vafa-Witten invariants defined recently by Tanaka-Thomas, can be extracted from these localized DT invariants. VW invariants are expected to have modular properties as predicted by S-duality.
Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.
Let C be a Riemann surface, L a line bundle over C, and n a natural number. Then there is a moduli space of stable n-dimensional vector bundles E over C with determinant bundle Λn
(E) ≡ L; this moduli space is smooth but in general non-compact and can be compactified by the suitable addition of semi-stable bundles to a projective, but in general singular, variety N
c,n,L
.The topology of this variety depends only on the genus g of C and the degree d of L (in fact, only on d modulo n, since tensoring E with a fixed line bundle L
1 replaces L by L ⊗ L
1n
), so we will also use the notation N
g,n,d
. We will be studying only the case n = 2, and hence will drop the n and replace d by ε = (−1)d in the notation. Thus for each g we have two moduli spaces of stable 2-dimensional bundles N
g− and N
g+, both projective varieties of complex dimension 3g − 3. We will be looking
mostly at the smooth space N
g− and will often denote it simply N
g
.
By studying the partition function of N=4 topologically twisted supersymmetric Yang-Mills on four-manifolds, we make an exact strong coupling test of the Montonen-Olive strong-weak duality conjecture. Unexpected and exciting links are found with two-dimensional rational conformal field theory.
Given a projective G-variety V (G a reductive group) and an ample bundle gq linearising the G-action, we relate the invariant cohomology (with values in gq) of V with that of the open set Vss of semistable points. We use this to extend established results, to prove the Verlinde formula for the dimension of the space of sections of these bundles on the moduli spaces of (parabolic, rank 2) vector bundles on a complex projective curve.
We study conformal field theories with a finite number of primary fields with respect to some chiral algebra. It is shown that the fusion rules are completely determined by the behavior of the characters under the modular group. We illustrate with some examples that conversely the modular properties of the characters can be derived from the fusion rules. We propose how these results can be used to find restrictions on the values of the central charge and conformal dimensions.
this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks B c (r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as follows: choose a point p 2 X, and let AX be the ring of algebraic functions on X p. To each integer c 0 is associated a representation V c of the Lie algebra sl r
We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.
We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.
Stony Brook conference in honour of the sixtieth birthday of
- S Donaldson
S. Donaldson, Gluing techniques in the cohomology of moduli spaces, in: Topological
methods in modern mathematics (Proc. of 1991 Stony Brook conference in honour
of the sixtieth birthday of J. Milnor), Publish or Perish.
- J E Andersen
- S Gukov Du
J. E. Andersen S. Gukov Du Pei, The Verlinde formula for Higgs bundles,
arXiv:1608.01761.
- A Gholampour
- A Sheshmani
- S.-T Yau
A. Gholampour, A. Sheshmani, and S.-T. Yau, Nested Hilbert schemes on surfaces:
Virtual fundamental class, to appear in J. Diff. Geom., arXiv:1701.08899.
- A Gholampour
- R P Thomas
A. Gholampour and R. P. Thomas, Degeneracy loci, virtual cycles and nested Hilbert
schemes I, arXiv:1709.06105.
- A Gholampour
- R P Thomas
A. Gholampour and R. P. Thomas, Degeneracy loci, virtual cycles and nested Hilbert
schemes II, arXiv:1902.04128.
- L Göttsche
L. Göttsche, Verlinde-type formulas for rational surfaces, arXiv:1609.07327.
- L Göttsche
- M Kool
L. Göttsche and M. Kool, Virtual refinements of the Vafa-Witten formula,
arXiv:1703.07196.
- L Göttsche
- M Kool
L. Göttsche and M. Kool, Refined SU(3) Vafa-Witten invariants and modularity,
arXiv:1808.03245.
- D Halpern-Leistner
D. Halpern-Leistner, The equivariant Verlinde formula on the moduli of Higgs bundles, arXiv:1608.01754.
- T Laarakker
T. Laarakker, Monopole contributions to refined Vafa-Witten invariants,
arXiv:1810.00385.
Vafa-Witten invariants for projective surfaces I: stable case
- Y Tanaka
- R P Thomas
Y. Tanaka and R. P. Thomas, Vafa-Witten invariants for projective surfaces I:
stable case, to appear in Jour. Alg. Geom., arXiv:1702.08487.
- R P Thomas
R. P. Thomas, Equivariant K-theory and refined Vafa-Witten invariants,
arXiv:1810.00078.