Moduli of ADHM Sheaves and Local Donaldson-Thomas Theory

Abstract

The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve $X$. In particular it is proven that this moduli space is virtually smooth and related byrelative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipande and Thomas. This yields a new conjectural construction for the local Donaldson-Thomas theory of curves as well as a natural higher rank generalization.

61 pages AMS Latex; v2: minor corrections, reference added; v3: some proofs corrected using the GIT construction of the moduli space due to A. Schmitt; main results unchanged; final version to appear in J. Geom. Phys

References in corpus (13)

• Supersymmetric vacua and Bethe ansatz
• Curve counting via stable pairs in the derived category
• Quantum integrability and supersymmetric vacua
• Stable pairs and BPS invariants
• Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions
• The 3-fold vertex via stable pairs
• GLSM's for partial flag manifolds
• Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory
• Riemann-Roch theorems and elliptic genus for virtually smooth Schemes
• Chamber Structure and Wallcrossing in The ADHM Theory of Curves I
• A theory of the invariants obtained from the moduli stacks of stable objects on a smooth polarized surface
• $S^1$-fixed-points in hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram
• Moduli spaces of framed instanton sheaves on projective spaces

Cited by in corpus (23)

• Curve counting via stable pairs in the derived category
• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory
• Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics
• Parabolic refined invariants and Macdonald polynomials
• Chamber Structure and Wallcrossing in The ADHM Theory of Curves I
• The Stringy Instanton Partition Function
• Instantons, Quivers and Noncommutative Donaldson-Thomas Theory
• Quantum Witten localization and abelianization for qde solutions
• Rank Two ADHM Invariants and Wallcrossing
• Quantum Cohomology and Quantum Hydrodynamics from Supersymmetric Quiver Gauge Theories
• Wallcrossing and Cohomology of The Moduli Space of Hitchin Pairs
• Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds
• Chamber Structure and Wallcrossing in the ADHM Theory of Curves II
• BPS states, torus links and wild character varieties
• The ADHM Variety and Perverse Coherent Sheaves
• BPS states and the P=W conjecture
• Seiberg-Witten monopoles with multiple spinors on a surface times a circle
• Moduli spaces of framed instanton sheaves on projective spaces
• Higgs bundles twisted by a vector bundle
• Hilbert schemes of nonreduced divisors in Calabi-Yau threefolds and W-algebras
• Moduli space of pairs over projective stacks
• On Some Computations of Higher Rank Refined Donaldson-Thomas Invariants