The idea of dynamical gauge bosons of hidden local symmetries in nonlinear sigma models
is reviewed. Starting with a fresh look at the Goldstone theorem and low energy theorems …

We review the solutions of O (N) and U (N) quantum field theories in the large N limit and as
1/N expansions, in the case of vector representations. Since invariant composite fields have …

Twenty-five years ago, Michael Green, John Schwarz, and Edward Witten wrote two
volumes on string theory. Published during a period of rapid progress in this subject, these …

We provide an overview of the string landscape and the Swampland program. Our review of
the string landscape covers the worldsheet and spacetime perspectives, including vacua …

One of the most remarkable ideas in particle physics-and an idea that has not yet found its
place in our understanding of nature-is supersymmetry [1]. Recently, there has been …

We describe two constructions of hyperkähler manifolds, one based on a Legendre
transform, and one on a sympletic quotient. These constructions arose in the context of …

We investigate the effect of duality transformations on the geometries of a subclass of two-
dimensional non-linear sigma-models. We identify the torsion which appears on dualizing …

The modifications of the classical equations of motion of the gravitational field in type II string
theory are derived by studying tree-level gravitational scattering amplitudes. The effective …

This thorough and detailed exposition is the result of an intensive month-long course on
mirror symmetry sponsored by the Clay Mathematics Institute. It develops mirror symmetry …

The geometric Langlands program can be described in a natural way by compactifying on a
Riemann surface C a twisted version of N= 4 super Yang-Mills theory in four dimensions …