77 research outputs found
Virtual refinements of the Vafa-Witten formula
We conjecture a formula for the generating function of virtual
-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with
holomorphic 2-form. Specializing the conjecture to minimal surfaces of general
type and to virtual Euler characteristics, we recover (part of) a formula of C.
Vafa and E. Witten.
These virtual -genera can be written in terms of descendent Donaldson
invariants. Using T. Mochizuki's formula, the latter can be expressed in terms
of Seiberg-Witten invariants and certain explicit integrals over Hilbert
schemes of points. These integrals are governed by seven universal functions,
which are determined by their values on and . Using localization we calculate these functions up to some
order, which allows us to check our conjecture in many cases.
In an appendix by H. Nakajima and the first named author, the virtual Euler
characteristic specialization of our conjecture is extended to include
-classes, thereby interpolating between Vafa-Witten's formula and Witten's
conjecture for Donaldson invariants.Comment: 44 pages. Published version. Appendix C by first named author and H.
Nakajim
Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex
We compute the Donaldson-Thomas invariants of a local elliptic surface with
section. We introduce a new computational technique which is a mixture of
motivic and toric methods. This allows us to write the partition function for
the invariants in terms of the topological vertex. Utilizing identities for the
topological vertex proved in arXiv:1603.05271, we derive product formulas for
the partition functions. The connected version of the partition function is
written in terms of Jacobi forms. In the special case where the elliptic
surface is a K3 surface, we get a derivation of the Katz-Klemm-Vafa formula for
primitive curve classes which is independent of the computation of
Kawai-Yoshioka.Comment: 43 pages, 3 figures. Formal methods replaced by much simpler
stratification according to location of embedded points. Published versio
Higher rank sheaves on threefolds and functional equations
We consider the moduli space of stable torsion free sheaves of any rank on a
smooth projective threefold. The singularity set of a torsion free sheaf is the
locus where the sheaf is not locally free. On a threefold it has dimension
. We consider the open subset of moduli space consisting of sheaves
with empty or 0-dimensional singularity set.
For fixed Chern classes and summing over , we show that the
generating function of topological Euler characteristics of these open subsets
equals a power of the MacMahon function times a Laurent polynomial. This
Laurent polynomial is invariant under (upon
replacing ). For some choices of these open
subsets equal the entire moduli space.
The proof involves wall-crossing from Quot schemes of a higher rank reflexive
sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret
this sublocus in terms of the singularities of the reflexive sheaf.Comment: 29 pages. Published versio
A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula
We conjecture a formula for the virtual elliptic genera of moduli spaces of
rank 2 sheaves on minimal surfaces 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 . 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.Comment: 24 pages. In order to keep the paper self-contained, we recall the
necessary material from the prequel arXiv:1703.07196, which results in some
overlap. Published version. Typo fixe
Stable reflexive sheaves and localization
We study moduli spaces of rank 2 stable reflexive sheaves on
. Fixing Chern classes , , and summing over , we
consider the generating function of Euler
characteristics of such moduli spaces. The action of the torus on
lifts to and we classify all sheaves in
. This leads to an explicit expression for
. Since is bounded below and above,
is a polynomial. We find a simple formula for
its leading term when .
Next, we study moduli spaces of rank 2 stable torsion free sheaves on
and consider the generating function of Euler characteristics of
such moduli spaces. We give an expression for this generating function in terms
of and Euler characteristics of Quot schemes of
certain -equivariant reflexive sheaves, which are studied elsewhere. Many
techniques of this paper apply to any toric 3-fold. In general,
depends on the choice of polarization which
leads to wall-crossing phenomena. We briefly illustrate this in the case of
.Comment: 27 pages. Published version. Typo's correcte
Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
Recently, Cao-Maulik-Toda defined stable pair invariants of a compact
Calabi-Yau 4-fold . Their invariants are conjecturally related to the
Gopakumar-Vafa type invariants of defined using Gromov-Witten theory by
Klemm-Pandharipande. In this paper, we consider curve counting invariants of
using Hilbert schemes of curves and conjecture a DT/PT correspondence which
relates these to stable pair invariants of .
After providing evidence in the compact case, we define analogous invariants
for toric Calabi-Yau 4-folds using a localization formula. We formulate a
vertex formalism for both theories and conjecture a relation between the (fully
equivariant) DT/PT vertex, which we check in several cases. This relation
implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary
insertions.Comment: 28 pages. Published versio
Trace Identities for the Topological Vertex
The topological vertex is a universal series which can be regarded as an
object in combinatorics, representation theory, geometry, or physics. It
encodes the combinatorics of 3D partitions, the action of vertex operators on
Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the
open string partition function of .
We prove several identities in which a sum over terms involving the
topological vertex is expressed as a closed formula, often a product of simple
terms, closely related to Fourier expansions of Jacobi forms. We use purely
combinatorial and representation theoretic methods to prove our formulas, but
we discuss applications to the Donaldson-Thomas invariants of elliptically
fibered Calabi-Yau threefolds at the end of the paper.Comment: 21 pages. Published versio
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