D-Branes and Topological Field Theories

In the presence of a D-brane a string theory develops a new subsector. We show that for curved D-branes the corresponding sector is a (partially twisted) topological field theory. We use this result to compute the degeneracy of 2-branes wrapped around $K3$ cycles as well as 3-branes wrapped around CY threefold vanishing 3-cycles. In both cases we find the degeneracy is in accord with expectation. The counting of BPS states of a gas of 0-branes in the presence of a 4-brane in $K3$ is considered and it is noted that the effective 0-brane charge is shifted by 1, due to a quantum correction. This is in accord with string duality and the fact that left-moving ground state energy of heterotic string starts at $-1$. We also show that all the three different topological twistings of four dimensional $N=4$ Yang-Mills theory do arise from curved three-branes embedded in different spaces (Calabi-Yau manifolds and manifolds with exceptional holonomy groups).Comment: 22 page

tt* Geometry and Closed String Tachyon Potential

We propose a closed string tachyon action including kinetic and potential terms for non-supersymmetric orbifolds. The action is given in terms of solutions to $tt^*$ equations which captures the geometry of vacua of the corresponding N=2 worldsheet theory. In certain cases the solutions are well studied. In case of tachyons of ${\bf C}/Z_n$, solutions to affine toda equations determine the action. We study the particular case of ${\bf C}/Z_3\to {\bf C}$ in detail and find that the Tachyon action is determined in terms of a solution to Painleve III equation.Comment: 10 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

Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes

We develop techniques to compute higher loop string amplitudes for twisted $N=2$ theories with $\hat c=3$ (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of the $N=2$ theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira--Spencer theory, which may be viewed as the closed string analog of the Chern--Simon theory. Using the mirror map this leads to computation of the `number' of holomorphic curves of higher genus curves in Calabi--Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the corresponding $N=2$ theory. Relations with $c=1$ strings are also pointed out.Comment: 178 pages, 20 figure

On ADE Quiver Models and F-Theory Compactification

Based on mirror symmetry, we discuss geometric engineering of N=1 ADE quiver models from F-theory compactifications on elliptic K3 surfaces fibered over certain four-dimensional base spaces. The latter are constructed as intersecting 4-cycles according to ADE Dynkin diagrams, thereby mimicking the construction of Calabi-Yau threefolds used in geometric engineering in type II superstring theory. Matter is incorporated by considering D7-branes wrapping these 4-cycles. Using a geometric procedure referred to as folding, we discuss how the corresponding physics can be converted into a scenario with D5-branes wrapping 2-cycles of ALE spaces.Comment: 21 pages, Latex, minor change