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The geometric Langlands correspondence was described some years ago in terms of $S$-duality of $\N=4$ super Yang-Mills theory. Some additional matters relevant to this story are described here. The main goal is to explain directly why an $A$-brane of a certain simple kind can be an eigenbrane for the action of 't Hooft operators. To set the stage, we review some facts about Higgs bundles and the Hitchin fibration. We consider only the simplest examples, in which many technical questions can be avoided.
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We discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue of the timelike tube theorem and the Reeh-Schlieder property hold with respect to states satisfying a weak form of microlocal analyticity condition. The former means the von Neumann algebra of observables of a spacelike tube equals the von Neumann algebra of observables of a significantly bigger region, that is obtained by deforming the boundary of the tube in a timelike manner. This generalizes theorems by Borchers and Araki to curved spacetimes.
Our discussions of string scattering amplitudes in the first volume of this book were limited to tree diagrams. These are the lowest-order approximations to string scattering amplitudes. In principle, quantum corrections to the tree level or classical results should be obtained by a perturbation expansion derived from string quantum field theory. Our present state of knowledge does not make this possible. Historically, loop diagrams were constructed by using unitarity to construct loop diagrams from tree diagrams. This unitarization of the tree diagrams led, in time, to the topological expansion, as sketched in chapter 1.
The effect of modular transformations which change the spin structure of the string world-sheet is discussed. Invariance of string perturbation theory under such transformations forces us to sum over different spin structures with well defined coefficients. The sum over spin structures amounts to performing the GSO projections. We explain why the GSO projections is compatible with unitarit; that is, why wrong G-parity states are not produced in pairs. On the contrary, we show that unitarity requires either the GSO projection, which gives a supersymmetric spectrum, or another projection which removes the massless spin- 3 2 particle from the spectrum. This is in agreement with the familiar idea that a consistent theory with a massless spin- 3 2 particle must be supersymmetric. These considerations lead us to find four new string theories in ten dimensions. Although they all have tachyons, they are otherwise consistent, modular invariant and unitary. One of these theories is particularly interesting since it has anomaly free chiral fermions in ten dimensions.
These notes are based on lectures at the PSSCMP/PiTP summer school that was held at Princeton University and the Institute for Advanced Study in July, 2015. They are devoted largely to topological phases of matter that can be understood in terms of free fermions and band theory. They also contain an introduction to the fractional quantum Hall effect from the point of view of effective field theory.
We derive a condition under which (0,2) linear sigma models possess a “left-moving” conformal stress tensor in Q + cohomology (i.e. which leaves invariant the “right-moving” ground states) even away from their critical points. At the classical level this enforces quasihomogeneity of the superpotential terms. The persistence of this structure at the quantum level on the worldsheet is obstructed by an anomaly unless the charges and superpotential degrees satisfy a condition which is equivalent to the condition for the cancellation of the anomaly in a particular “right-moving” U(1) R-symmetry.
The timelike tube theorem asserts that in quantum field theory without gravity, the algebra of observables in an open set U is the same as the corresponding algebra of observables in its ``timelike envelope'' E(U), which is an open set that is in general larger. The theorem was originally proved in the 1960's by Borchers and Araki for quantum fields in Minkowski space. Here we sketch the proof of a version of the theorem for quantum fields in a general real analytic spacetime. Details have appeared elsewhere.
We derive two-dimensional topological gravity, which previously has been discussed in a BRST framework, by gauge fixing of an underlying gauge invariant system. The underlying lagrangian is zero.
Naive imitation of the usual formulas for compact gauge group in quantizing three dimensional Chern-Simons gauge theory with non-compact gauge group leads
We compute static properties of baryons in an SU(2) × SU(2) chiral theory (the Skyrme model) whose solitons can be interpreted as the baryons of QCD. Our results are generally within about 30% of experimental values. We also derive some relations that hold generally in soliton models of baryons, and therefore, serve as tests of the 1 N expansion.
We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras $${\mathcal A_V}$$ on the Minkowski half-plane M +
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The recent discovery of an explicit conformal field theory description of Type II p-branes makes it possible to investigate the existence of bound states of such objects. In particular, it is possible with reasonable precision to verify the prediction that the Type IIB superstring in ten dimensions has a family of soliton and bound state strings permuted by SL(2,Z). The space-time coordinates enter tantalizingly in the formalism as non-commuting matrices.
The SYK model is a quantum mechanical model that has been proposed to be holographically dual to a $1+1$-dimensional model of a quantum black hole. An emergent "gravitational" mode of this model is governed by an unusual action that that has been called the Schwarzian action. It governs a reparametrization of a circle. We show that the path integral of the Schwarzian theory is one-loop exact. The argument uses a method of fermionic localization, even though the model itself is purely bosonic.
To understand in detail duality between heterotic string and F theory compactifications, it is important to understand the construction of holomorphic G bu
We develop an approach to Khovanov homology of knots via gauge theory (previous physics-based approches involved other descriptions of the relevant spaces of BPS states). The starting point is a system of D3-branes ending on an NS5-brane with a nonzero theta-angle. On the one hand, this system can be related to a Chern-Simons gauge theory on the boundary of the D3-brane worldvolume; on the other hand, it can be studied by standard techniques of $S$-duality and $T$-duality. Combining the two approaches leads to a new and manifestly invariant description of the Jones polynomial of knots, and its generalizations, and to a manifestly invariant description of Khovanov homology, in terms of certain elliptic partial differential equations in four and five dimensions.
