A 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.

/pdf/cohomology-of-quiver-moduli-functional-equations-and-4f6p0fdw0n.pdf

Mirror Symmetry V

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

We prove the conjectural relations between Mahler measures and L-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for L-values of elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1+X) (1+Y )(X+Y )−αXY, α∈ℝ.

/pdf/from-l-series-of-elliptic-curves-to-mahler-measures-2mg1xdsc8x.pdf

From $L$-series of elliptic curves to Mahler measures

A 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.

/pdf/cohomology-of-quiver-moduli-functional-equations-and-2q9uuvee70.pdf

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

We show that the limit of a one-parameter admissible normal function with no singularities lies in a non-classical sub-object of the limiting intermediate Jacobian. Using this, we construct a Hausdor slit analytic space, with complex Lie group bres, which ‘graphs’ such normal functions. For singular normal functions, an extension of the sub-object by a nite group leads to the N eron models. When the normal function comes from geometry, that is, a family of algebraic cycles on a semistably degenerating family of varieties, its limit may be interpreted via the Abel{Jacobi map on motivic cohomology of the singular bre, hence via regulators on K-groups of its substrata. Two examples are worked out in detail, for families of 1-cycles on CY and abelian 3-folds, where this produces interesting arithmetic constraints on such limits. We also show how to compute the nite ‘singularity group’ in the geometric setting.

/pdf/neron-models-and-limits-of-abel-jacobi-mappings-5522oft97c.pdf

Néron models and limits of Abel–Jacobi mappings

We extend our analysis in [arXiv:0801.4782] and show that the chiral algebras of (0,2) sigma models are totally trivialized by worldsheet instantons for all complete flag manifolds of compact semisimple Lie groups. Consequently, supersymmetry is spontaneously broken. Our results verify Stolz's idea that there are no harmonic spinors on the loop spaces of these flag manifolds. Moreover, they also imply that the kernels of certain twisted Dirac operators on these spaces will be empty under a "quantum" deformation of their geometries.

Mirror Symmetry V

Citations

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

From $L$-series of elliptic curves to Mahler measures

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

Néron models and limits of Abel–Jacobi mappings

Chiral Algebras of (0,2) Sigma Models: Beyond Perturbation Theory - II

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