T-duality is a symmetry of the heterotic string to all orders in string perturbation theory. This results in an effective four dimensional supergravity theory with desirable features for phenomenology. T-duality, as well as, generically, an anomalous U(1), is broken by quantum anomalies of the effective field theory. The structure of the full anomaly is presented, and the mechanisms for anomaly...

Abstract. After Zagier proved that the traces of singular moduli j(z) are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group PSL2(Z) have been investigated such as their exact formulas, limiting distribution, duality, and congruences. The purpose of this paper is to generalize t...

The theory of exact C∗-algebras was introduced by Kirchberg and has been influential in recent development of C∗-algebras. A fundamental result on exact C∗-algebras is a local characterization of exactness. The notion of weakly exact von Neumann algebras was also introduced by Kirchberg. In this paper, we give a local characterization of weak exactness. As a corollary, we prove that a discrete ...

In this paper we discuss duality theory of optimization problems with a linear objective function and subject to linear constraints with cone inclusions, referred to as conic linear problems. We formulate the Lagrangian dual of a conic linear problem and survey some results based on the conjugate duality approach where the questions of “no duality gap” and existence of optimal solutions are rel...