524 research outputs found
Geometric Langlands Twists of N = 4 Gauge Theory from Derived Algebraic Geometry
We develop techniques for describing the derived moduli spaces of solutions
to the equations of motion in twists of supersymmetric gauge theories as
derived algebraic stacks. We introduce a holomorphic twist of N=4
supersymmetric gauge theory and compute the derived moduli space. We then
compute the moduli spaces for the Kapustin-Witten topological twists as its
further twists. The resulting spaces for the A- and B-twist are closely related
to the de Rham stack of the moduli space of algebraic bundles and the de Rham
moduli space of flat bundles, respectively. In particular, we find the
unexpected result that the moduli spaces following a topological twist need not
be entirely topological, but can continue to capture subtle algebraic
structures of interest for the geometric Langlands program.Comment: 55 pages; minor correction
Instantons, Topological Strings and Enumerative Geometry
We review and elaborate on certain aspects of the connections between
instanton counting in maximally supersymmetric gauge theories and the
computation of enumerative invariants of smooth varieties. We study in detail
three instances of gauge theories in six, four and two dimensions which
naturally arise in the context of topological string theory on certain
non-compact threefolds. We describe how the instanton counting in these gauge
theories are related to the computation of the entropy of supersymmetric black
holes, and how these results are related to wall-crossing properties of
enumerative invariants such as Donaldson-Thomas and Gromov-Witten invariants.
Some features of moduli spaces of torsion-free sheaves and the computation of
their Euler characteristics are also elucidated.Comment: 61 pages; v2: Typos corrected, reference added; v3: References added
and updated; Invited article for the special issue "Nonlinear and
Noncommutative Mathematics: New Developments and Applications in Quantum
Physics" of Advances in Mathematical Physic
Curve counting, instantons and McKay correspondences
We survey some features of equivariant instanton partition functions of
topological gauge theories on four and six dimensional toric Kahler varieties,
and their geometric and algebraic counterparts in the enumerative problem of
counting holomorphic curves. We discuss the relations of instanton counting to
representations of affine Lie algebras in the four-dimensional case, and to
Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For
resolutions of toric singularities, an algebraic structure induced by a quiver
determines the instanton moduli space through the McKay correspondence and its
generalizations. The correspondence elucidates the realization of gauge theory
partition functions as quasi-modular forms, and reformulates the computation of
noncommutative Donaldson-Thomas invariants in terms of the enumeration of
generalized instantons. New results include a general presentation of the
partition functions on ALE spaces as affine characters, a rigorous treatment of
equivariant partition functions on Hirzebruch surfaces, and a putative
connection between the special McKay correspondence and instanton counting on
Hirzebruch-Jung spaces.Comment: 79 pages, 3 figures; v2: typos corrected, reference added, new
summary section included; Final version to appear in Journal of Geometry and
Physic
- …