751 research outputs found
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
Topological Field Theory and Quantum Holonomy Representations of Motion Groups
Canonical quantization of abelian BF-type topological field theory coupled to
extended sources on generic d-dimensional manifolds and with curved line
bundles is studied. Sheaf cohomology is used to construct the appropriate
topological extension of the action and the topological flux quantization
conditions, in terms of the Cech cohomology of the underlying spatial manifold,
as required for topological invariance of the quantum field theory. The
wavefunctions are found in the Hamiltonian formalism and are shown to carry
multi-dimensional representations of various topological groups of the space.
Expressions for generalized linking numbers in any dimension are thereby
derived. In particular, new global aspects of motion group presentations are
obtained in any dimension. Applications to quantum exchange statistics of
objects in various dimensionalities are also discussed.Comment: 45 pages LaTe
Crystals, instantons and quantum toric geometry
We describe the statistical mechanics of a melting crystal in three
dimensions and its relation to a diverse range of models arising in
combinatorics, algebraic geometry, integrable systems, low-dimensional gauge
theories, topological string theory and quantum gravity. Its partition function
can be computed by enumerating the contributions from noncommutative instantons
to a six-dimensional cohomological gauge theory, which yields a dynamical
realization of the crystal as a discretization of spacetime at the Planck
scale. We describe analogous relations between a melting crystal model in two
dimensions and N=4 supersymmetric Yang-Mills theory in four dimensions. We
elaborate on some mathematical details of the construction of the quantum
geometry which combines methods from toric geometry, isospectral deformation
theory and noncommutative geometry in braided monoidal categories. In
particular, we relate the construction of noncommutative instantons to deformed
ADHM data, torsion-free modules and a noncommutative twistor correspondence.Comment: 33 pages, 5 figures; Contribution to the proceedings of "Geometry and
Physics in Cracow", Jagiellonian University, Cracow, Poland, September 21-25,
2010. To be published in Acta Physica Polonica Proceedings Supplemen
Higher Quantum Geometry and Non-Geometric String Theory
We present a concise overview of the physical and mathematical structures
underpinning the appearence of nonassociative deformations of geometry in
non-geometric string theory. Starting from a quick recap of the appearence of
noncommutative product and commutator deformations of geometry in open string
theory with -fields, we argue on physical principles that closed strings
should instead probe triproduct and tribracket deformations in backgrounds of
locally non-geometric fluxes. After describing the toy model of electric
charges moving in fields of smooth distributions of magnetic charge as a
physical introduction to the notions of nonassociative geometry, we review the
description of non-geometric fluxes in generalized geometry and double field
theory, and the worldsheet calculations suggesting the appearence of
nonassociative deformations, together with their caveats. We discuss how
algebroids and their associated AKSZ sigma-models give a description of
non-geometric backgrounds in terms of higher geometry, and consider the
quantization of the membrane sigma-model which geometrizes closed strings with
-flux. From this we derive an explicit nonassociative star product for the
quantum geometry of the closed string phase space, and apply it to derive the
triproducts that appear in conformal field theory correlation functions, to
describe a consistent treatment of nonassociative quantum mechanics, to
demonstrate quantitatively the coarse-graining of spacetime due to -flux,
and to describe the quantization of Nambu brackets. We also briefly review how
these constructions lead to a nonassociative theory of gravity, their uplifts
to non-geometric M-theory, and the role played by -algebras in these
developments.Comment: 44 pages; v2: some discussions expanded and references added. Based
on Lectures at the School "Quantum Spacetime and Physics Models", Corfu
Summer Institute on Elementary Particle Physics and Gravity, September 2-28,
2017, Corfu, Greece; Final version to be published in Proceedings of Scienc
Strings, Gauge Fields and Membranes
We present an overview of the intimate relationship between string and
D-brane dynamics, and the dynamics of gauge and gravitational fields in three
spacetime dimensions. The successes, prospects and open problems in describing
both perturbative and nonperturbative aspects of string theory in terms of
three-dimensional quantum field theory are highlighted.Comment: 53 pages, 8 figures; To be published in the Ian Kogan Memorial
Collection ``From Fields to Strings: Circumnavigating Theoretical Physics'',
World Scientific, 200
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