Moduli Space Of Representations Research Articles

Differential graded algebras for trivalent plane graphs and their representations

To any trivalent plane graph embedded in the sphere Casals and Murphy associate a differential graded algebra (dg-algebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is the construction of a generalization of the Casals–Murphy dg-algebra to non-commutative coefficients, for which we prove various functoriality properties not previously verified in the commutative setting. Our second result is to prove that rank r representations of this dg-algebra, over a field \mathbb{F} , correspond to colorings of the faces of the graph by elements of the Grassmannian \operatorname{Gr}(r,2r;\mathbb{F}) so that bordering faces are transverse, up to the natural action of \operatorname{PGL}_{2r}(\mathbb{F}) . Underlying the combinatorics, the dg-algebra is a computation of the fully non-commutative Legendrian contact dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not prove as such in this paper. The graph coloring problem verifies that, for Legendrian 2-weaves, rank r representations of the Legendrian contact dg-algebra correspond to constructible sheaves of microlocal rank r . This is the first explicit such computation of the bijection between the moduli spaces of representations and sheaves for an infinite family of Legendrian surfaces.

Derived invariance of Crawley-Boevey’s H0-Poisson structure

Crawley-Boevey introduced in [Poisson structure on moduli spaces of representations, J. Algebra 325 (2011) 205–215.], the notion of [Formula: see text]-Poisson structure for associative algebras, which is the weakest condition that induces a Poisson structure on the moduli spaces of their representations. In this paper, by using a result of Armenta and Keller in [Derived invariance of the Tamarkin-Tsygan calculus of an algebra, C. R. Math. Acad. Sci. Paris 357(3) (2019) 236–240.], we show that an [Formula: see text]-Poisson structure is preserved under derived Morita equivalence.

Tree Normal Forms for Quiver Representations

We explore methods for constructing normal forms of indecomposable quiver representations. The first part of the paper develops homological tools for recursively constructing families of indecomposable representations from indecomposables of smaller dimension vector. This is then specialized to the situation of tree modules, where the existence of a special basis simplifies computations and gives nicer normal forms. Motivated by a conjecture of Kac, we use this to construct cells of indecomposable representations as deformations of tree modules. The second part of the paper develops geometric tools for constructing cells of indecomposable representations from torus actions on moduli spaces of representations. As an application, we combine these methods to construct families of indecomposables grouped into cells. These actually give a normal form for all indecomposables of certain roots.

Topological strings, strips and quivers

We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized q-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.

Torsion and symplectic volume in Seifert manifolds

For any oriented Seifert manifold X and compact connected Lie group G with finite center, we relate the Reidemeister density of the moduli space of representations of the fundamental group of X into G to the Liouville measure of some moduli spaces of representations of surface groups into G.

Poisson structures on moduli spaces of representations

We show that a Poisson structure can be induced on the affine moduli space of (semisimple) representations of an associative algebra from a suitable Lie algebra structure on the zeroth Hochschild homology of the algebra. In particular this applies to necklace Lie algebra for path algebras of doubled quivers and preprojective algebras. We call such structures H 0 -Poisson structures, and show that they are well behaved for Azumaya algebras and under Morita equivalence.

Moduli spaces for D-branes at the tip of a cone

For physicists: We show that the quiver gauge theory derived from a Calabi-Yau cone via an exceptional collection of line bundles on the base has the original cone as a component of its classical moduli space. For mathematicians: We use data from the derived category of sheaves on a Fano surface to construct a quiver, and show that its moduli space of representations has a component which is isomorphic to the anticanonical cone over the surface.

Group cohomology and the symplectic structure on the moduli space of representations

The space of equivalence classes of irreducibleb representations of the fundamental group of a compact oriented surface of genus at least 2 in a Lie group has a natural symplectic form. In lAB], Atiyah and Bott described this symplectic structure using methods from gauge theory. Goldman [G] constructed the skew-symmetric pairing algebraically using methods from group cohomology. Using Poincar6 duality, Goldman showed that the pairing is nondegenerate and identified it with the symplectic structure given by gauge theory.