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

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

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

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

On the Lagrangian Realization of Non-Critical W{\cal W}-Strings

A large class of non-critical string theories with extended worldsheet gauge symmetry are described by two coupled, gauged Wess-Zumino-Witten Models. We give a detailed analysis of the gauge invariant action and in particular the gauge fixing procedure and the resulting BRST symmetries. The results are applied to the example of ${\cal W}_3$ strings.Comment: 19 pages, LaTeX (REVTEX macro's

Gauge theory, topological strings, and S-duality

We offer a derivation of the duality between the topological U(1) gauge theory on a Calabi-Yau 3-fold and the topological A-model on the same manifold. This duality was conjectured recently by Iqbal, Nekrasov, Okounkov, and Vafa. We deduce it from the S-duality of the IIB superstring. We also argue that the mirror version of this duality relates the topological B-model on a Calabi-Yau 3-fold and a topological sector of the Type IIA Little String Theory on the same manifold.Comment: 9 pages, latex. v2: a footnote has been added. The footnote corrects an inaccuracy in the original argument; the results are unchanged. v3: exposition improve

Euler number of Instanton Moduli space and Seiberg-Witten invariants

We show that a partition function of topological twisted N=4 Yang-Mills theory is given by Seiberg-Witten invariants on a Riemannian four manifolds under the condition that the sum of Euler number and signature of the four manifolds vanish. The partition function is the sum of Euler number of instanton moduli space when it is possible to apply the vanishing theorem. And we get a relation of Euler number labeled by the instanton number $k$ with Seiberg-Witten invariants, too. All calculation in this paper is done without assuming duality.Comment: LaTeX, 34 page

Duality of Non-Supersymmetric Large N Gauge Theories

Starting from Seiberg's electric-magnetic duality for supersymmetric QCD, we construct dual pairs of non-supersymmetric gauge theories. This is accomplished by first taking the large N limit of supersymmetric QCD and its dual partner and then performing a special ``orbifold projection'' recently introduced by Kachru and Silverstein. We argue that in the large N limit the two projected theories remain dual. The non-supersymmetric gauge theories which can be studied in this fashion have non-supersymmetric field content, chiral fermions and exactly massless scalar matter.Comment: 27 pages, 4 figure

Anti-de Sitter/CFT Correspondence in Three-Dimensional Supergravity

Anti-de Sitter supergravity models are considered in three dimensions. Precise asymptotic conditions involving a chiral projection are given on the Rarita-Schwinger fields. Together with the known boundary conditions on the bosonic fields, these ensure that the asymptotic symmetry algebra is the superconformal algebra. The classical central charge is computed and found to be equal to the one of pure gravity. It is also indicated that the asymptotic degrees of freedom are described by 2D "induced supergravity" and that the boundary conditions "transmute" the non-vanishing components of the WZW supercurrent into the supercharges.Comment: Additional remarks in the extended case, added references, and small misprints corrected. To appear in Phys. Rev. D. Latex, 19 pages, no figure