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

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

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

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 $B$-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 $R$-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 $R$-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 $L_\infty$-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