We show that manifolds of fixed points, which are generated by exactly marginal operators, are common in N=1 supersymmetric gauge theory. We present a unified and simple prescription for identifying these operators, using tools similar to those employed in two-dimensional N=2 supersymmetry. In particular we rely on the work of Shifman and Vainshtein relating the β-function of the gauge coupling to the anomalous dimensions of the matter fields. Finite N=1 models, which have marginal operators at zero coupling, are easily identified using our approach. The method can also be employed to find manifolds of fixed points which do not include the free theory; these are seen in certain models with product gauge groups and in many non-renormalizable effective theories. For a number of our models, S-duality may have interesting implications. Using the fact that relevant perturbations often cause one manifold of fixed points to flow to another, we propose a specific mechanism through which the N=1 duality discovered by Seiberg could be associated with the duality of finite N=2 models. (Submitted to Nuclear Physics B)

We show how to make a topological string theory starting from an N = 4 superconformal theory. The critical dimension for this theory is ĉ = 2 (c = 6). It is shown that superstrings (in both the RNS and GS formulations) and critical N = 2 strings are special cases of this topological theory. Applications for this new topological theory include: 1) Proving the vanishing to all orders of all scattering amplitudes for the self-dual N = 2 string with flat background, with the exception of the three-point function and the closed-string partition function; 2) Showing that the topological partition function of the N = 2 string on the K3 background may be interpreted as computing the superpotential in harmonic superspace generated upon compactification of type II superstrings from 10 to 6 dimensions; and 3) Providing a new prescription for calculating superstring amplitudes which appears to be free of total-derivative ambiguities. July

Abstract. We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the “mirror symmetry ” phenomenon recently observed by string theorists.

The condition of having an N = 1 spacetime supersymmetry for heterotic string leads to 4 distinct possibilities for compactifications namely compactifications down to 6,4,3 and 2 dimensions. Compactifications to 6 and 4 dimensions have been studied extensively before (corresponding to K3 and a Calabi-Yau threefold respectively). Here we complete the study of the other two cases corresponding to compactification down to 3 on a 7 dimensional manifold of G2 holonomy and compactification down to 2 on an 8 dimensional manifold of Spin(7) holonomy. We study the extended chiral algebra and find the space of exactly marginal deformations. It turns out that the role the U(1) current plays in the N = 2 superconformal theories, is played by tri-critical Ising model in the case of G2 and Ising model in the case of Spin(7) manifolds. Certain generalizations of mirror symmetry are found for these two cases. We also discuss the topological twisting in each case.

These lectures review some of the basic properties of N = 2 superconformal field theories and the corresponding topological field theories. One of my basic aims is to show how the techniques of topological field theory can be used to compute effective Landau-Ginzburg potentials for perturbed N = 2 superconformal field theories. In particular, I will briefly discuss the application of these ideas to N = 2 supersymmetric quantum integrable models. Dedicated to the memmory of Brian Warr USC-93/001 hep-th/9301088

In this review we study BPS D-branes on Calabi–Yau threefolds. Such D-branes naturally divide into two sets called A-branes and B-branes which are most easily understood from topological field theory. The main aim of this paper is to provide a self-contained guide to the derived category approach to B-branes and the idea of Π-stability. We argue that this mathematical machinery is hard to avoid for a proper understanding of B-branes. A-branes and B-branes are related in a very complicated and interesting way which ties in with the “homological mirror symmetry ” conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3-fold, flops and orbifolds are discussed at some length. In the latter

Superconformal sigma models with Calabi--Yau target spaces described as complete intersection subvarieties in toric varieties can be obtained as the low-energy limit of certain abelian gauge theories in two dimensions. We formulate mirror symmetry for this class of Calabi--Yau spaces as a duality in the abelian gauge theory, giving the explicit mapping relating the two Lagrangians. The duality relates inequivalent theories which lead to isomorphic theories in the low-energy limit. This formulation suggests that mirror symmetry could be derived using abelian duality. The application of duality in this context is complicated by the presence of nontrivial dynamics and the absence of a global symmetry. We propose a way to overcome these obstacles, leading to a more symmetric Lagrangian. The argument, however, fails to produce a derivation of the conjecture. Introduction Two dimensional conformal field theories with N=2 supersymmetry have been extensively studied as candidate vacua for pe...

Looking for string vacua with fixed moduli, we study compactifications of type IIA string theory on Calabi-Yau fourfolds in the presence of generic Ramond-Ramond fields. We explicitly derive the (super)potential induced by Ramond-Ramond fluxes performing a Kaluza-Klein reduction of the ten-dimensional effective action. This can be conveniently achieved in a formulation of the massive type IIA supergravity where all Ramond-Ramond fields appear in a democratic way. The result agrees with the general formula for the superpotential written in terms of calibrations. We further notice that for generic Ramond-Ramond fluxes all geometric moduli are stabilized and one finds non-supersymmetric vacua at positive values of the scalar potential. March

Abstract. The geometric aspects of mirror symmetry are reviewed, with an eye towards future developments. Given a mirror pair (X, Y) of Calabi–Yau threefolds, the best-understood mirror statements relate certain small corners of the moduli spaces of X and of Y. We will indicate how one might go beyond such statements, and relate the moduli spaces more globally. In fact, in the boldest version of mirror symmetry (the Strominger–Yau–Zaslow conjecture), the Calabi–Yau threefolds X and Y should be directly related to each other through a very geometric construction.

Abstract. The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi–Yau manifolds have mirror partners. The geometric description—that one Calabi–Yau manifold should serve as a compactified, complexified moduli space for special Lagrangian tori on the other Calabi–Yau manifold—is rather surprising. We formulate some precise mathematical conjectures concerning how these moduli spaces are to be compactified and complexified, as well as a definition of geometric mirror pairs (in arbitrary dimension) which is independent of those conjectures. We investigate how this new geometric description ought to be related to the mathematical statements which have previously been extracted from mirror symmetry. In particular, we discuss how the moduli spaces of the ‘mirror ’ Calabi–Yau manifolds should be related to one another, and how appropriate subspaces of