M. Ronen Plesser

New N= 2 superconformal field theories in four dimensions

New examples of N = 2 supersymmetric conformal field theories are found as fixed points of SU(2) N = 2 supersymmetric QCD. Relations among the scaling dimensions of their relevant chiral operators, global symmetries, and Higgs branches are understood in terms of the general structure of relevant deformations of non-trivial N = 2 conformal field theories. The spectrum of scaling dimensions found are all those compatible with relevant deformations of a y2 = x3 singular curve.

Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties

Abstract We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety V or a Calabi-Yau hypersurface M ⊂ V . In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth V , our results reproduce and clarify an algebraic solution of the V model due to Batyrev. In addition, we find an algebraic relation determining the solution for M in terms of that for V . Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the M model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured “monomial-divisor mirror map” of Aspinwall, Greene and Morrison.

The moduli space of vacua of N = 2 SUSY QCD and duality in N = 1 SUSY QCD

Abstract We analyze in detail the moduli space of vacua of N = 2 SUSY QCD with n c colors and n f flavors. The Coulomb branch has submanifolds with non-Abelian gauge symmetry. The massless quarks and gluons at these vacua are smoothly connected to the underlying elementary quarks and gluons. Upon breaking N = 2 by an N = 1 preserving mass term for the adjoint field the theory flows to N = 1 SUSY QCD. Some of the massless quarks and gluons on the moduli space of the N = 2 theory become the magnetic quarks and gluons of the N = 1 theory. In this way we derive the duality in N = 1 SUSY QCD by identifying its crucial building blocks — the magnetic degrees of freedom — using only semiclassical physics and the non-renormalization theorem.

The Coulomb phase of N=2 supersymmetric QCD

We present an explicit nonperturbative solution of N=2 supersymmetric SU({ital N}) gauge theory with {ital N}{sub {ital f}}{le}2{ital N} flavors generalizing results of Seiberg and Witten for {ital N}=2.

Enhanced gauge symmetry in type II string theory

Abstract We show how enhanced gauge symmetry in type II string theory compactified on a Calabi-Yau threefold arises from singularities in the geometry of the target space. When the target space of the type IIA string acquires a genus g curve C of AN−1 singularities, we find that an SU(N) gauge theory with g adjoint hypermultiplets appears at the singularity. The new massless states correspond to solitons wrapped about the collapsing cycles, and their dynamics is described by a twisted supersymmetric gauge theory on C × R 4 . We reproduce this result from an analysis of the S-dual D-manifold. We check that the predictions made by this model about the nature of the Higgs branch, the monodromy of period integrals, and the asymptotics of the one-loop topological amplitude are in agreement with geometrical computations. In one of our examples we find that the singularity occurs at strong coupling in the heterotic dual proposed by Kachru and Vafa.

Mirror manifolds in higher dimension

We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We isolate certain simplifying features present only in dimension three, and supply alternative methods that do not rely on these special characteristics and hence can be generalized to other dimensions. Although the moduli spaces for Calabi-Yaud-folds are not “special Kähler manifolds” whend>3, they still have a restricted geometry, and we indicate the new geometrical structures which arise. We formulate and apply procedures which allow for the construction of mirror maps and the calculation of order-by-order instanton corrections to Yukawa couplings. Mathematically, these corrections are expected to correspond to calculating Chern classes of various parameter spaces (Hilbert schemes) for rational curves on Calabi-Yau manifolds. Our mirror-aided calculations agree with those Chern class calculations in the limited number of cases for which the latter can be carried out with current mathematical tools. Finally, we make explicit some striking relations between instanton corrections for various Yukawa couplings, derived from the associativity of the operator product algebra.

Exact S-matrix for two-dimensional string theory

Abstract We formulate simple graphical rules which allow explicit calculation of nonperturbative c = 1 S -matrices. This allows us to investigate the constraint of nonperturbative unitarity, which indeed rules out some theories. Nevertheless, we show that there is an infinite parameter family of nonperturbatively unitary c = 1 S -matrices. We investigate the dependence of the S -matrix on one of these nonperturbative parameters. In particular, we study the analytic structure, background dependence and high-energy behavior of some nonperturbative c = 1 S -matrices. The scattering amplitudes display interesting resonant behavior both at high energies and in the complex energy plane.

TOWARDS MIRROR SYMMETRY AS DUALITY FOR TWO DIMENSIONAL ABELIAN GAUGE THEORIES

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.

N = 2 moduli spaces and N = 1 dualities for SO(nc) and USp(2nc) super-QCD

Abstract We determine the exact global structure of the moduli space of N = 2 supersymmetric SO ( n ) and USp (2 n ) gauge theories with matter hypermultiplets in the fundamental representations, using the non-renormalization theorem for the Higgs branches and the exact solutions for the Coulomb branches. By adding an ( N = 2)-breaking mass term for the adjoint chiral field and varying the mass, the N = 2 theories can be made to flow to either an “electric” N = 1 supersymmetric QCD or its N = 1 dual “magnetic” version. We thus obtain a derivation of the N = 1 dualities of Seiberg.

Topological Landau-Ginzburg formulation and integrable structure of two-dimensional string theory

We construct a topological Landau-Ginzburg formulation of the two-dimensional string at the self-dual radius. The model is an analytic continuation of the