F-theory, Geometric Engineering and N=1 Dualities

We consider geometric engineering of N=1 supersymmetric QFTs with matter in terms of a local model for compactification of F-theory on Calabi-Yau fourfold. By bringing 3-branes near 7-branes we engineer N=1 supersymmetric $SU(N_c)$ gauge theory with $N_f$ flavors in the fundamental. We identify the Higgs branch of this system with the instanton moduli space on the compact four dimensional space of the 7-brane worldvolume. Moreover we show that the Euclidean 3-branes wrapped around the compact part of the 7-brane worldvolume can generate superpotential for $N_f=N_c-1$ as well as lead to quantum corrections to the moduli space for $N_f=N_c$. Finally we argue that Seiberg's duality for N=1 supersymmetric QCD may be mapped to T-duality exchanging 7-branes with 3-branes.Comment: 15 page

One Loop Effects in Various Dimensions and D-Branes

We calculate some one loop corrections to the effective action of theories in $d$ dimensions that arise on the dimensional reduction of a Weyl fermion in $D$ dimensions. The terms that we are interested in are of a topological nature. Special attention is given to the effective actions of the super Yang Mills theories that arise on dimensional reduction of the N=1 theory in six dimensions or on the dimensional reduction of the N=1 theory in ten dimensions. In the latter case we suggest an interpretation of the quantum effect as a coupling of the gauge field on the brane to a relative background gauge field.Comment: 18 pages, LaTe

A Comment on BPS States in F-theory in 8 Dimensions

We study some aspects of enhanced gauge symmetries in F-theory compactified on K3. We find open string configurations connecting various 7-branes which represent stable BPS states. In this approach we recover $D_n$ and $E_n$ gauge groups previously found from an analysis of sigularities of the moduli space of elliptically fibered K3 manifolds as well as examples of non-perturbative realizations of $A_n$ groups.Comment: harvmac tex, 8 page

F-theory and linear sigma models

We present an explicit method for translating between the linear sigma model and the spectral cover description of SU(r) stable bundles over an elliptically fibered Calabi-Yau manifold. We use this to investigate the 4-dimensional duality between (0,2) heterotic and F-theory compactifications. We indirectly find that much interesting heterotic information must be contained in the `spectral bundle' and in its dual description as a gauge theory on multiple F-theory 7-branes. A by-product of these efforts is a method for analyzing semistability and the splitting type of vector bundles over an elliptic curve given as the sheaf cohomology of a monad.Comment: 40 pages, no figures; minor cosmetic reorganization of section 4; reference [6] update

BPS States and Minimal Surfaces

It was observed recently, that the low energy effective action of the four-dimensional supersymmetric theories may be obtained as a certain limit of M Theory. From this point of view, the BPS states correspond to the minimal area membranes ending on the M Theory fivebrane. We prove that for the configuration, corresponding to the SU(2) Super Yang-Mills theory, the BPS spectrum is correctly reproduced, and develop techniques for analyzing the BPS spectrum in more general cases. We show that the type of the supermultiplet is related to the topology of the membrane: disks correspond to hypermultiplets, and cylinders to vector multiplets. We explain the relation between minimal surfaces and geodesic lines, which shows that our description of BPS states is closely related to one arising in Type II string compactification on Calabi-Yau threefolds.Comment: 31 pp., LaTeX, 6 PostScript figures. The construction of bosonic and fermionic zero modes is explained in more invariant terms and generalized to membranes of arbitrary topology. Mistake in the discussion of membrane worldsheet theory is corrected. The relation between membranes and geodesic lines is explaine

Wilson Fermions on a Randomly Triangulated Manifold

A general method of constructing the Dirac operator for a randomly triangulated manifold is proposed. The fermion field and the spin connection live, respectively, on the nodes and on the links of the corresponding dual graph. The construction is carried out explicitly in 2-d, on an arbitrary orientable manifold without boundary. It can be easily converted into a computer code. The equivalence, on a sphere, of Majorana fermions and Ising spins in 2-d is rederived. The method can, in principle, be extended to higher dimensions.Comment: 18 pages, latex, 6 eps figures, fig2 corrected, Comment added in the conclusion sectio

Supersymmetric Three-cycles and (Super)symmetry Breaking

We describe physical phenomena associated with a class of transitions that occur in the study of supersymmetric three-cycles in Calabi-Yau threefolds. The transitions in question occur at real codimension one in the complex structure moduli space of the Calabi-Yau manifold. In type IIB string theory, these transitions can be used to describe the evolution of a BPS state as one moves through a locus of marginal stability: at the transition point the BPS particle becomes degenerate with a supersymmetric two particle state, and after the transition the lowest energy state carrying the same charges is a non-supersymmetric two particle state. In the IIA theory, wrapping the cycles in question with D6-branes leads to a simple realization of the Fayet model: for some values of the CY modulus gauge symmetry is spontaneously broken, while for other values supersymmetry is spontaneously broken.Comment: 10 pages, harvmac big; v2, minor change

Evidence for S-Duality in N=4 Supersymmetric Gauge Theory

Using D-brane techniques, we compute the spectrum of stable BPS states in N=4 supersymmetric gauge theory, and find it is consistent with Montonen-Olive duality.Comment: 6 pages, harvmac (reference added

Superstrings from Hamiltonian Reduction

In any string theory there is a hidden, twisted superconformal symmetry algebra, part of which is made up by the BRST current and the anti-ghost. We investigate how this algebra can be systematically constructed for strings with $N\!-\!2$ supersymmetries, via quantum Hamiltonian reduction of the Lie superalgebras $osp(N|2)$. The motivation is to understand how one could systematically construct generalized string theories from superalgebras. We also briefly discuss the BRST algebra of the topological string, which is a doubly twisted $N\!=\!4$ superconformal algebra.Comment: 32p, LaTeX, CERN-TH.7379/9

Polynomial Structure of Topological String Partition Functions

We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special K\"ahler geometry and the ring of almost-holomorphic modular forms defined on modular curves.Comment: version 2: references fixe