Publications

Based on a simple adiabatic argument and by considering the heterotic string counterpart of certain symmetries of Type II superstrings such as (-1)FL and orientation reversal, we construct orbifold candidates for dual pairs of heterotic and Type II string theories with N = 2 and N = 1 supersymmetry. We also analyze from a similar point of view the K3 fibrations that enter in recently proposed N = 2 candidates and use this structure together with certain orientation-reversing symmetries to construct N = 1 dual pairs. These pairs involve generalizations of Type I vacua which can be equivalent to E 8 × E 8 heterotic strings, while standard Type I vacua are related to SO(32).

It is shown that in CP non-conserving theories, the electric charge of an 't Hooft-Polyakov magnetic monopole will not ordinarily be integral, or even rational in units of the fundamental charge e. If a non-zero vacuum angle θ is the only mechanism for CP violation, the electric charge of the monopole is exactly calculable and is −eθ/2π, plus an integer. If there are additional CP violating interactions, the monopole charge must be computed as a power series in the coupling constant. These results apply in realistic theories such as SU(5).

The title of this article refers to analytic continuation of three-dimensional Chern-Simons gauge theory away from integer values of the usual coupling parameter k, to explore questions such as the volume conjecture, or analytic continuation of three-dimensional quantum gravity (to the extent that it can be described by gauge theory) from Lorentzian to Euclidean signature. Such analytic continuation can be carried out by rotating the integration cycle of the Feynman path integral. Morse theory or Picard-Lefschetz theory gives a natural framework for describing the appropriate integration cycles. An important part of the analysis involves flow equations that turn out to have a surprising four-dimensional symmetry. After developing a general framework, we describe some specific examples (involving the trefoil and figure-eight knots in S^3). We also find that the space of possible integration cycles for Chern-Simons theory can be interpreted as the 'physical Hilbert space' of a twisted version of N=4 super Yang-Mills theory in four dimensions.

Supersymmetry is a remarkable subject that has fascinated particle physicists since it was originally introduced. Although supersymmetry is no longer a new idea, we still do not know in what form, if any, it plays a role in the proper description of nature.

Some rigorous inequalities governing hadron masses in QCD are proved. One states that the electromagnetic mass shift of the pion is positive. The other states that if ${m}_{A\overline{B}}$ is the mass of the lightest meson made from a quark of type $A$ and an antiquark of type $B$, then---under conditions such that annihilation into gluons can be ignored---$2{m}_{A\overline{B}}>~{m}_{A\overline{A}}+{m}_{B\overline{B}}$. These inequalities agree with experimental data and have analogs for arbitrary vectorlike gauge theories. The first inequality has applications to the vacuum alignment problem in vectorlike technicolor theories.

We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.

No abstract is provided for this article.

We show that the Veneziano and Virasoro-Shapiro four-particle scattering amplitudes can be factored in terms of an infinite product of non-archimedean string amplitudes. This factorization is equivalent to the functional equation for the Riemann zeta function.

Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining four-manifold invariants by counting $SU(2)$ instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group. This is a ``dual'' equation in which the gauge group is the dual of the maximal torus of $SU(2)$. The new viewpoint suggests many new results about the Donaldson invariants.

The Gopakumar-Vafa (GV) formula expresses certain couplings that arise in Type IIA compactification to four dimensions on a Calabi-Yau manifold in terms of a counting of BPS states in M-theory. The couplings in question have applications to topological strings and supersymmetric black holes. In this paper, we reconsider the GV formula, taking a close look at the Schwinger-like computation that was suggested in the original GV work. The goal is to understand the background that must be used in this computation, the role played by the extended supersymmetry of this background, and how the computation gives a holomorphic result though superficially depending only on particle masses. We also examine in a similar way the Ooguri-Vafa (OV) formula, which is an extension of the GV formula to include D4-branes.

A new proof is given of the positive energy theorem of classical general relativity. Also, a new proof is given that there are no asymptotically Euclidean

Perturbative superstring theory is revisited, with the goal of giving a simpler and more direct demonstration that multi-loop amplitudes are gauge-invariant (apart from known anomalies), satisfy space-time supersymmetry when expected, and have the expected infrared behavior. The main technical tool is to make the whole analysis, including especially those arguments that involve integration by parts, on supermoduli space, rather than after descending to ordinary moduli space.

No abstract is provided for this article.

We re-examine the black hole solutions in classical theories of dilaton gravity in two dimensions. We consider an arbitrary dilaton potential such that there are black hole solutions asymptotic at infinity to the nearly $\mathrm{AdS}_2$ solutions of JT gravity, and such that the black hole energy and entropy are bounded below. We show that if there is a black hole solution with negative specific heat at some temperature $T$, then at the same temperature there is a black hole solution with lower free energy and positive specific heat. As the temperature is increased from 0 to infinity, the black hole energy and entropy increase monotonically but not necessarily continuously; there can be first order phase transitions, similar to the Hawking-Page transition. These theories can also have solutions corresponding to closed universes.

We generalize the super period matrix of a super Riemann surface to the case that Ramond punctures are present. For a super Riemann surface of genus g with 2 r Ramond punctures, we define, modulo certain choices that generalize those in the classical theory (and assuming a certain generic condition is satisfied), a g | r × g | r period matrix that is symmetric in the Z 2 -graded sense. As an application, we analyze the genus 2 vacuum amplitude in string theory compactifications to four dimensions that are supersymmetric at tree level. We find an explanation for a result that has been found in orbifold examples in explicit computations by D’Hoker and Phong: with their integration procedure, the genus 2 vacuum amplitude always vanishes “pointwise” after summing over spin structures, and hence is given entirely by a boundary contribution.

'The index of the dirac operator in loop space' published in 'Elliptic Curves and Modular Forms in Algebraic Topology'

Geometric Langlands duality relates a representation of a simple Lie group G v to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL 2 subgroup of G v makes an unexpected appearance. This can be explained using gauge theory, as this paper will show, with the help of the equations of Nahm and Bogomolny.

We present a gauge-theoretic interpretation of the “analytic” version of the geometric Langlands program, in which Hitchin Hamiltonians and Hec