Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities

The chiral superstring measure constructed in the earlier papers of this series for general gravitino slices is examined in detail for slices supported at two points x_\alpha. In this case, the invariance of the measure under infinitesimal changes of gravitino slices established previously is strengthened to its most powerful form: the measure is shown, point by point on moduli space, to be locally and globally independent from the points x_\alpha, as well as from the superghost insertion points p_a, q_\alpha introduced earlier as computational devices. In particular, the measure is completely unambiguous. The limit x_\alpha = q_\alpha is then well defined. It is of special interest, since it elucidates some subtle issues in the construction of the picture-changing operator Y(z) central to the BRST formalism. The formula for the chiral superstring measure in this limit is derived explicitly.Comment: 20 pages, no figure

Topics in Two-Loop Superstring Perturbation Theory

In this contribution to the Proceedings of the Conference on Analysis, Complex Geometry, and Mathematical Physics, an expository overview of superstring perturbation theory to two loop order is presented to an audience of mathematicians and physicists. Recent results on perturbative supersymmetry breaking effects in Heterotic string theory compactified on Z_2 \times Z_2 Calabi-Yau orbifolds, and the calculation of the two-loop vacuum energy in these theories are discussed in detail, and the appearance of a new modular identity with respect to Sp(4,Z)/Z_4 is reviewed.Comment: 27 pages, 5 figures, Dedicated to D.H. Phong on the occasion of his 60-th birthday; version 2 includes minor changes and added references in section 1

Exact M-Theory Solutions, Integrable Systems, and Superalgebras

In this paper, an overview is presented of the recent construction of fully back-reacted half-BPS solutions in 11-dimensional supergravity which correspond to near-horizon geometries of M2 branes ending on, or intersecting with, M5 and M5$'$ branes along a self-dual string. These solutions have space-time manifold ${\rm AdS}_3 \times S^3 \times S^3$ warped over a Riemann surface $\Sigma$, and are invariant under the exceptional Lie superalgebra $D(2,1;\gamma) \oplus D(2,1;\gamma)$, where $\gamma$ is a real continuous parameter and $|\gamma|$ is governed by the ratio of the number of M5 and M5$'$ branes. The construction proceeds by mapping the reduced BPS equations onto an integrable field theory on $\Sigma$ which is of the Liouville sine-Gordon type. Families of regular solutions are distinguished by the sign of $\gamma$, and include a two-parameter Janus solution for $\gamma >0$, and self-dual strings on M5 as well as asymptotically ${\rm AdS}_4/{\mathbb Z}_2$ solutions for $\gamma <0$.Comment: This paper is dedicated to Luc Vinet on the occasion of his 60-th birthda

Integrable systems from supergravity BPS equations

Integrable systems of the sine-Gordon/Liouville type, which arise from reducing the BPS equations for solutions invariant under 16 supersymmetries in Type IIB supergravity and M-theory, are shown to be special cases of an infinite family of integrable systems, parametrized by an arbitrary real function $f$ of a real variable. It is shown that, for each function $f$, this generalized integrable system may be mapped onto a system of linear equations, which in turn may be integrated in terms of the two linearly independent solutions of an ordinary linear second order differential equation which depends only on the function $f$.Comment: 7 page