Black Holes and Calabi-Yau Threefolds

We compute the microscopic entropy of certain 4 and 5 dimensional extermal black holes which arise for compactification of M-theory and type IIA on Calabi-Yau 3-folds. The results agree with macroscopic predictions, including some subleading terms. The macroscopic entropy in the 5 dimensional case predicts a surprising growth in the cohomology of moduli space of holomorphic curves in Calabi-Yau threefolds which we verify in the case of elliptic threefolds.Comment: 11 pages, reference adde

Brane/anti-Brane Systems and U(N|M) Supergroup

We show that in the context of topological string theories N branes and M anti-branes give rise to Chern-Simons gauge theory with the gauge supergroup $U(N|M)$. We also identify a deformation of the theory which corresponds to brane/anti-brane annihilation. Furthermore we show that when $N=M$ all open string states are BRST trivial in the deformed theory.Comment: 10 pages, additional reference include

Extending Mirror Conjecture to Calabi-Yau with Bundles

We define the notion of mirror of a Calabi-Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi-Yau manifolds in terms of supersymmetric cycles on the mirror.Comment: 7 pages, minor corrections and additional reference

Geometric Physics

Talk given at ICM '98, Berlin, reviewing some of the recent developments in understanding of string theory for a mathematical audience (to appear in Documenta Mathematica).Comment: 25 pages, 10 figure

Fractional Quantum Hall Effect and M-Theory

We propose a unifying model for FQHE which on the one hand connects it to recent developments in string theory and on the other hand leads to new predictions for the principal series of experimentally observed FQH systems with filling fraction $\nu ={n\over 2n\pm1}$ as well as those with $\nu ={m\over m+2}$. Our model relates these series to minimal unitary models of the Virasoro and superVirasoro algebra and is based on $SL(2, {\bf C})$ Chern-Simons theory in Euclidean space or $SL(2,{\bf R})\times SL(2,{\bf R})$ Chern-Simons theory in Minkowski space. This theory, which has also been proposed as a soluble model for 2+1 dimensional quantum gravity, and its N=1 supersymmetric cousin, provide effective descriptions of FQHE. The principal series corresponds to quantized levels for the two $SL(2,{\bf R})$'s such that the diagonal $SL(2,{\bf R})$ has level $1$. The model predicts, contrary to standard lore, that for principal series of FQH systems the quasiholes possess non-abelian statistics. For the multi-layer case we propose that complex ADE Chern-Simons theories provide effective descriptions, where the rank of the ADE is mapped to the number of layers. Six dimensional $(2,0)$ ADE theories on the Riemann surface $\Sigma$ provides a realization of FQH systems in M-theory. Moreover we propose that the q-deformed version of Chern-Simons theories are related to the anisotropic limit of FQH systems which splits the zeroes of the Laughlin wave function. Extensions of the model to 3+1 dimensions, which realize topological insulators with non-abelian topologically twisted Yang-Mills theory is pointed out.Comment: 30 pages. Some connections with the standard approach to FQHE are pointed ou

Minimal Cycles, Black Holes and QFT's

We review some aspects of minimal cycles in string compactifications and their role in constructing new critical theories in six and lower dimensions as well as in accounting for black hole entropy. (Based on a talk presented at the Salam Memorial Meeting, the Abdus Salam International Center for Theoretical Physics, Fall 1997)Comment: 5 page

Geometric Origin of Montonen-Olive Duality

We show how $N=4, D=4$ duality of Montonen and Olive can be derived for all gauge groups using geometric engineering in the context of type II strings, where it reduces to T-duality. The derivation for the non-simply laced cases involves the use of some well known facts about orbifold conformal theories.Comment: 8 page

Non-Unitary Holography

We propose gauge theory/gravity duality involving conformal theories based on U(N+k|k) gauge groups. We show that to all orders in 1/N these non-unitary theories based on supergroups are indistinguishable from the corresponding unitary theories where the gauge group is replaced by U(N). This leads to non-unitary gravity duals which to all orders in 1/N are indistinguishable from their unitary cousins. They are distinguished by operators whose correlation functions differ by O(exp(-aN)). The celebrated type IIB on AdS^5 x S^5 and M-theory on AdS^4 x S^7 fall in this class and thus seem to also admit non-unitary non-perturbative completions. It is tempting to conjecture that this setup may provide a non-unitary model for black hole evaporation.Comment: 13 pages, note added and additional references and minor edit

On N=1 Yang-Mills in Four Dimensions

Extending previous work on geometric engineering of N=1 Yang-Mills in four dimensions for simply laced ($A_n,D_n,E_{6,7,8}$) gauge groups, we construct local models for all other gauge groups ($B_n,C_n,F_4,G_2$) in terms of F-theory. We compute the radius dependent superpotential upon further compactification on a circle to $d=3$ in the dual M-theory and use it to show that the number of vacua in four dimensions for each group is given by its dual coxeter number, in accordance with expectations based on gaugino condensates.Comment: 7 page

Puzzles at Large N

We consider the AdS/CFT correspondence in the context of 2d CFT and find that essentially all ``single particle'' primary fields with higher spin, predicted from the CFT side are missing from the Kaluza-Klein excitations of the AdS supergravity. The high mass extension of these missing states gives rise to the macroscopic entropy of extremal 5d black holes.Comment: 7 pages, 4d puzzle resolved by a comment of O. Aharony. The paper deals with the puzzle in the 2d cas