Euler Characteristic Of Moduli Space Research Articles

Chen–Ruan cohomology and orbifold Euler characteristic of moduli spaces of parabolic bundles

We consider the moduli space of stable parabolic Higgs bundles of rank r and fixed determinant, and having full flag quasi-parabolic structures over an arbitrary parabolic divisor on a smooth connected complex projective curve X of genus g, with g≥2. The group Γ of r-torsion points of the Jacobian of X acts on this moduli space. We describe the connected components of the various fixed point loci of this moduli under non-trivial elements from Γ. When the Higgs field is zero, or in other words when we restrict ourselves to the moduli of stable parabolic bundles, we also compute the orbifold Euler characteristic of the corresponding global quotient orbifold. We also describe the Chen–Ruan cohomology groups of this orbifold under certain conditions on the rank and degree, and describe the Chen–Ruan product structure in special cases.

Generalisations of the Harer–Zagier recursion for 1-point functions

Harer and Zagier proved a recursion to enumerate gluings of a 2d-gon that result in an orientable genus g surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer–Zagier recursion, but our methodology also applies to the enumeration of dessins d’enfant, to Bousquet-Mélou–Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs single Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion.

Moduli Spaces of Point Configurations and Plane Curve Counts

Abstract We identify certain Gromov–Witten invariants counting rational curves with given incidence and tangency conditions with the Euler characteristics of moduli spaces of point configurations in projective spaces. On the Gromov–Witten side, S. Fomin and G. Mikhalkin established a recurrence relation via tropicalization, which is realized on the moduli space side using Donaldson–Thomas invariants of subspace quivers.

Stable Pairs on Nodal $K3$ Fibrations

We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.

Permutation combinatorics of worldsheet moduli space

Light-cone string diagrams have been used to reproduce the orbifold Euler characteristic of moduli spaces of punctured Riemann surfaces at low genus and with few punctures. Nakamura studied the meromorphic differential introduced by Giddings and Wolpert to characterise light-cone diagrams and introduced a class of graphs related to this differential. These Nakamura graphs were used to parametrise the cells in a light-cone cell decomposition of moduli space. We develop links between Nakamura graphs and realisations of the worldsheet as branched covers. This leads to a development of the combinatorics of Nakamura graphs in terms of permutation tuples. For certain classes of cells, including those of top dimension, there is a simple relation to Belyi maps, which allows us to use results from Hermitian and complex matrix models to give analytic formulae for the counting of cells at arbitrarily high genus. For the most general cells, we develop a new equivalence relation on Hurwitz classes which organises the cells and allows efficient enumeration of Nakamura graphs using the group theory software GAP.

Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces

Given a smooth toric variety \(X\), the action of the torus \(T\) lifts to the moduli space \({\mathcal {M}}\) of stable sheaves on \(X\). Using the pioneering work of Klyacho, a fairly explicit combinatorial description of the fixed point locus \({\mathcal {M}}^T\) can be given (as shown by earlier work of the author). In this paper, we apply this description to the case of torsion free sheaves on a smooth toric surface \(S\). A general expression for the generating function of the Euler characteristics of such moduli spaces is obtained. The generating function is expressed in terms of Euler characteristics of certain moduli spaces of stable configurations of linear subspaces appearing in classical GIT. The expression holds for any choice of \(S\), polarization, rank, and first Chern class. Specializing to various examples allows us to compute some new as well as known generating functions.

The equivariant Euler characteristic of moduli spaces of curves

We derive a formula for the Sn-equivariant Euler characteristic of the moduli space Mg,n of genus g curves with n marked points.

Stable pairs on local $K3$ surfaces

We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local $K3$ surfaces to counting invariants of semistable sheaves on them. Our formula generalizes Kawai- Yoshioka’s formula for stable pairs with irreducible curve classes to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with our main result, leads to an Euler characteristic version of Katz- Klemm-Vafa conjecture for stable pairs.

On Euler characteristics for large Kronecker quivers

We study Euler characteristics of moduli spaces of stable representations of m-Kronecker quivers for m≫0. In particular, we study an asymptotic log formula of Euler characteristics and a normalized asymptotic log formula of Euler characteristic, motivated by so-called Douglas conjecture.

Refined GW/Kronecker correspondence

Gromov–Witten invariants of weighted projective planes and Euler characteristics of moduli spaces of representations of bipartite quivers are related via the tropical vertex, a group of formal automorphisms of a torus. On the Gromov–Witten side, this uses the work of Gross, Pandharipande and Siebert. The quiver moduli side features quiver wall-crossing formulas, functional equations for Euler characteristics, and localization techniques. We derive several explicit formulas for Gromov–Witten invariants.

Fixed point loci of moduli spaces of sheaves on toric varieties

Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves of any dimension on an arbitrary nonsingular toric variety X. Using geometric invariant theory (GIT), this allows us to construct explicit moduli spaces of pure equivariant sheaves on X corepresenting natural moduli functors (similar to work of Payne in the case of equivariant vector bundles). The action of the algebraic torus on X lifts to the moduli space of all Gieseker stable sheaves on X and we express its fixed point locus explicitly in terms of moduli spaces of pure equivariant sheaves on X. One of the problems arising is to find an equivariant line bundle on the side of the GIT problem, which precisely recovers Gieseker stability. In the case of torsion free equivariant sheaves, we can always construct such equivariant line bundles. As a by-product, we get a combinatorial description of the fixed point locus of the moduli space of μ-stable reflexive sheaves on X. As an application, we show in a sequel Kool (2009) [25] how these methods can be used to compute generating functions of Euler characteristics of moduli spaces of μ-stable torsion free sheaves on nonsingular complete toric surfaces.

Cohomology of quiver moduli, functional equations, and integrality of Donaldson–Thomas type invariants

AbstractA system of functional equations relating the Euler characteristics of moduli spaces of stable representations of quivers and the Euler characteristics of (Hilbert-scheme-type) framed versions of quiver moduli is derived. This is applied to wall-crossing formulas for the Donaldson–Thomas type invariants of M. Kontsevich and Y. Soibelman, in particular confirming their integrality.

Torus fixed points of moduli spaces of stable bundles of rank three

By a result of Klyachko the Euler characteristic of moduli spaces of stable bundles of rank two on the projective plane is determined. Using similar methods we extend this result to bundles of rank three. The fixed point components correspond to moduli spaces of the subspace quiver. Moreover, the stability condition is given by a certain system of linear inequalities so that the generating function of the Euler characteristic can be determined explicitly.

Euler characteristics of moduli spaces of curves

Let M n be the moduli space of n-pointed Riemann surfaces of genus g. Denote by M n g the Deligne-Mumford compactification of M n. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of M n g for any g and n such that n > 2 2g.

On the $S_n$-equivariant Euler characteristic of moduli spaces of hyperelliptic curves

The generating function for $S_n$-equivariant Euler characteristics of moduli spaces of pointed hyperelliptic curves for any genus g>1 is calculated. This answer generalizes the known ones for genera 2 and 3 and answers obtained by J. Bergstrom for any genus and n<8 points.

A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves

We determine an expression 9 (1X) for the virtual Euler characteristics of the moduli spaces of s-pointed real (-y = 1/2) and complex (-y = 1) algebraic curves. In particular, for the space of real curves of genus g with a fixed point free involution, we find that the Euler characteristic is (-2)s-1(1-29-1)(g?s-2)!Bg/g! where Bg is the gth Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is (-1) S (g+s-2)!Bg+ /(g+1) (g -1) ! The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to counit cells. The approach involves a parameter y that permits specialization of the formula to the real and complex cases. This suggests that 4 (-y) itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.

A possible IIB superstring matrix model with Euler characteristic and a double scaling limit

We show that a recently proposed Yang-Mills matrix model with an auxiliary field, which is a candidate for a non-perturbative description of type IIB superstrings, captures the Euler characteristic of moduli space of Riemann surfaces. This happens at the saddle point for the Yang-Mills field. It turns out that the large-n limit in this matrix model corresponds to a double scaling limit in the Penner model.

TOPOLOGICAL GAUGE THEORIES FROM SUPERSYMMETRIC QUANTUM MECHANICS ON SPACES OF CONNECTIONS

We rederive the recently introduced N=2 topological gauge theories, representing the Euler characteristic of moduli spaces ℳ of connections, from supersymmetric quantum mechanics on the infinite-dimensional spaces [Formula: see text] of gauge orbits. To that end we discuss variants of ordinary supersymmetric quantum mechanics which have meaningful extensions to infinite-dimensional target spaces, and introduce supersymmetric quantum mechanics actions modeling the Riemannian geometry of submersions and embeddings, relevant to the projections [Formula: see text] and inclusions [Formula: see text] respectively. We explain the relation between Donaldson theory and the gauge theory of flat connections in three dimensions and illustrate the general construction by other two- and four-dimensional examples.

A CRITICAL MATRIX MODEL FOR NON-ORIENTED STRING

The Penner-like model of real symmetric matrices with logarithmic potential is solved. We suggest that the double-scaling limit free energy of this model can be interpreted as a generating function for virtual Euler characteristics of moduli space of non-orientable surfaces. The asymptotic expansion in the double-scaling limit is found to coincide with the R=1/4 Gross–Klebanov c=1 model. We also found some amusing correspondences between this model and exactly solvable six-vertex model on square lattice.

A CRITICAL MATRIX MODEL AT c=1

By taking the critical limit of Penner’s matrix model we obtain a continuum theory whose free energy at genus-g is the Euler characteristic of moduli space of Riemann surfaces of genus-g. The exponents, and the appearance of logarithmic corrections suggest that we are dealing with a theory at c=1.