Communications in Number Theory and Physics 

About: Communications in Number Theory and Physics is an academic journal published by International Press of Boston, Inc.. The journal publishes majorly in the area(s): Calabi–Yau manifold & Modular form. It has an ISSN identifier of 1931-4523. Over the lifetime, 253 publications have been published receiving 9343 citations. The journal is also known as: CNTP.

Electric-Magnetic Duality And The Geometric Langlands Program

Abstract: The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.

Invariants of algebraic curves and topological expansion

Abstract: For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a tau function attached to an algebraic curve. These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e. the (p,q) minimal models of conformal field theory. As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV tau-function.

Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

Abstract: We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Donaldson-Thomas invariants.

Exact results for perturbative Chern-Simons theory with complex gauge group.

Abstract: We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of “arithmetic TQFT” and conjecture (with supporting numerical evidence) that SL(2,C) Chern-Simons theory is an example of such a theory.

Anomalies and the euler characteristic of elliptic calabi-yau threefolds

Abstract: We investigate the delicate interplay between the types of singular fibers in elliptic fibrations of Calabi-Yau threefolds (used to formulate F-theory) and the "matter" representation of the associated Lie algebra. The main tool is the analysis and the appropriate interpretation of the anomaly formula for six-dimensional supersymmetric theories. We find that this anomaly formula is geometrically captured by a relation among codimension two cycles on the base of the elliptic fibration, and that this relation holds for elliptic fibrations of any dimension. We introduce a "Tate cycle" which efficiently describes thisrelation- ship, and which is remarkably easy to calculate explicitly from the Weierstrass equation of the fibration. We check the anomaly cancellation formula in a num- ber of situations and show how this formula constrains the geometry (and in particular the Euler characteristic) of the Calabi-Yau threefold.