The moduli space of flat SU(2)-bundles over a nonorientable surface

Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve

We consider the moduli space of rank two, odd degree, semi-stable Real vector bundles over a real curve, calculating the singular cohomology ring in odd and zero characteristic for most examples.

The real locus of an involution map on the moduli space of flat connections on a Riemann surface

It is known that every nonorientable surface $Σ$ has an orientable double cover $\tildeΣ$. The covering map induces an involution on the moduli space $\tilde{\M}$ of gauge equivalence classes of flat $G$-connections on $\tildeΣ$. We identify the relation between the moduli space $\M$ and the fixed point set of the moduli space $\tilde{\M}$. In particular, $\M$ is isomorphic to the fixed point set of $\tilde{\M}$ if and only if the order of the center of $G$ is odd. One important application is that we give a way to construct a minimal Lagrangian submanifold of the moduli space $\tilde{\M}$.

Anti-Perfect Morse Stratification

For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface is locally equ…

Prequantization of the Moduli Space of Flat ${\rm PU}(p)$-Bundles with Prescribed Boundary Holonomies

Using the framework of quasi-Hamiltonian actions, we compute the obstruction to prequantization for the moduli space of flat ${\rm PU}(p)$-bundles over a compact orientable surface with prescribed holonomies around boundary components, where $p>2$ is prime.

Moduli spaces of vector bundles over a real curve: Z/2-Betti numbers

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces, proving formulas recently obtained by Liu and Schaffhauser.

Variation of moduli of parabolic Higgs bundles

We study moduli spaces of parabolic Higgs bundles on a curve and their dependence on the choice of weights. We describe the chamber structure on the space of weights and show that, when a wall is crossed, the moduli space undergoes an elementary transformation in the sense of Mukai.

A Finite Rank Bundle over $J$-Holomorphic map Moduli Spaces

We study a finite rank bundle $\mathbf{F}$ over a neighborhood of $J$-Holomorphic map Moduli Spaces, prove the exponential decay of the derivative of the gluing maps for $\mathbf{F}$ with respect to the gluing parameter.

The moduli space of flat G-bundles on a compact hyper-Kahler manifold

We prove that if G is a compact connected Lie group and X is a compact connected hyper-Kahler manifold, then the L^2 metric on (the smooth locus of) the moduli space of flat G-bundles on X is a hyper-Kahler metric.

Motivic classes of moduli of Higgs bundles and moduli of bundles with connections

Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero. We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approxima…

抛物向量丛模空间的生成元

Generators for the moduli space of parabolic bundle

本文为抛物向量丛模空间构造deRham上同调生成元的显式代表元，并据此计算其上同调的相交配对。

Witten's formulas for intersection pairings on moduli spaces of flat $G$-bundles

In a 1992 paper, Witten gave a formula for the intersection pairings of the moduli space of flat $G$-bundles over an oriented surface, possibly with markings. In this paper, we give a general proof of Witten's formula, for arbitrary compact, simple groups, and any markings for which the moduli space has at most orbifold singularities.

The Picard group of the universal moduli stack of principal bundles on pointed smooth curves

For any smooth connected linear algebraic group G over an algebraically closed field k, we describe the Picard group of the universal moduli stack of principal G-bundles over pointed smooth k-projective curves.

Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces

We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves of the latter are classified by restrictions of the bundles to certain divisors. This can be regarded as fixing a "complex analogue of the holonomy" of a connection along a "complex analogue of the boundary" in analogy with the real case.

Bundle gerbes and moduli spaces

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analytically constructed index bundle gerbe. We apply these constructions to certain moduli spaces associated to compact Riemann surfaces, constructing on these moduli spaces, natural bundle gerbes with connection and curving, whose 3-curvature represent Dixmier-Douady classes that are…