In these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and five-dimensional manifolds with boundary, with a rather subtle boundary condition that encodes the knots and links. The construction is formally analogous to Floer and Donaldson theory in three and four dimensions. It was discovered using quantum field theory arguments but can be described and understood purely in terms of classical gauge theory. (Based on a lecture at the conference Low-Dimensional Manifolds and High-Dimensional Categories, University of California at Berkeley, June 2011).
Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold $C$. But understanding these statements is extremely difficult without picking a complex structure on $C$ and using Hitchin's equations. We sketch the essential statements both for the ``unramified'' case that $C$ is a compact oriented two-manifold without boundary, and the ``ramified'' case that one allows punctures. We also give a few indications of why a more precise description requires a starting point in four-dimensional gauge theory.
We prove that for genus g � 5, the moduli space of super Rie- mann surfaces is not projected (and in particular is not split): it cannot be holomorphically projected to its underlying reduced manifold. Physically, this means that certain approaches to superstring perturbation theory that are very powerful in low orders have no close analog in higher orders. Mathemat- ically, it means that the moduli space of super Riemann surfaces cannot be constructed in an elementary way starting with the moduli space of ordinary Riemann surfaces. It has a life of its own.
In toroidal compactification of string theory, enlarged symmetry groups arise at special radii of the compactified dimensions. Here an orbifold-like construction is considered in which the degrees of freedom describing the compact spatial dimensions are twisted by symmetries which exist only for special values of the radius. Constraints on the possible twists arising from the requirements of worldsheet supersymmetry and modular invariance are described, and some simple examples are given. One motivation for this work is that in twisting by a symmetry which exists only at a special value of the radius, one of the dilaton-like fields can acquire a mass, since the radius of the compact space is fixed.
We prove that for genus greater than or equal to 5, the moduli space of super Riemann surfaces is not projected (and in particular is not split): it cannot be holomorphically projected to its underlying reduced manifold. Physically, this means that certain approaches to superstring perturbation theory that are very powerful in low orders have no close analog in higher orders. Mathematically, it means that the moduli space of super Riemann surfaces cannot be constructed in an elementary way starting with the moduli space of ordinary Riemann surfaces. It has a life of its own.
Making use of known facts about tensor models, it is possible to construct a quantum system without quenched disorder that has the same large $n$ limit for its correlation functions and thermodynamics as the SYK model. This might be useful in further probes of this approach to holographic duality.
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 ex- pected 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.
Chromodynamics with n flavors of massless quarks is invariant under chiral U(n) × U(n). We show that in the limit of large number of colors, under reasonable assumptions, this symmetry group must spontaneously break down to diagonal U(n).
We study four dimensional N = 2 supersymmetric gauge theories with matter multiplets. For all such models for which the gauge group is SU(2), we derive the exact metric on the moduli space of quantum vacua and the exact spectrum of the stable massive states. A number of new physical phenomena occur, such as chiral symmetry breaking that is driven by the condensation of magnetic monopoles that carry global quantum numbers. For those cases in which conformal invariance is broken only by mass terms, the formalism automatically gives results that are invariant under electric-magnetic duality. In one instance, this duality is mixed in an interesting way with SO(8) triality.
The conjecture that N=2 minimal models in two dimensions are critical points of a superrenormalizable Landau-Ginzburg model can be tested by computing the path integral of the Landau-Ginzburg model with certain twisted boundary conditions. This leads to simple expressions for certain characters of the N=2 models which can be verified at least at low levels. An N=2 superconformal algebra can in fact be found directly in the noncritical Landau-Ginzburg system, giving further support for the conjecture.
The standard boundary state of a topological insulator in 3+1 dimensions has gapless charged fermions. We present model systems that reproduce this standard gapless boundary state in one phase, but also have gapped phases with topological order. Our models are weakly coupled and all the dynamics is explicit. We rederive some known boundary states of topological insulators and construct new ones. Consistency with the standard spin/charge relation of condensed matter physics places a nontrivial constraint on models.
The theta dependent of pure gauge theories in four dimensions can be studied using a duality of large N gauge theories with string theory on a certain spacetime. Via this duality, one can argue that for every theta, there are infinitely many vacua that are stable in the large N limit. The true vacuum, found by minimizing the energy in this family, is a smooth function of theta except at theta equal to pi, where it jumps. This jump is associated with spontaneous breaking of CP symmetry. Domain walls separating adjacent vacua are described in terms of wrapped sixbranes.
In N = 1 super Yang-Mills theory in three spacetime dimensions, with a simple gauge group G and a Chern-Simons interaction of level k, the supersymmetric index Tr (-1) F can be computed by making a relation to a pure Chern-Simons theory or microscopically by an explicit Born-Oppenheimer calculation on a two-torus.The result shows that supersymmetry is unbroken if |k| ≥ h/2 (with h the dual Coxeter number of G) and suggests that dynamical supersymmetry breaking occurs for |k| < h/2.The theories with large |k| are massive gauge theories whose universality class is not fully described by the standard criteria.
On the world-volume of an M-theory five-brane propagates a two-form with self-dual field strength. As this field is non-Lagrangian, there is no obvious framework for determining its partition function. An analogous problem exists in Type IIB superstring theory for the self-dual five-form. The resolution of these problems and definition of the partition function is explained. A more complete analysis of perturbative anomaly cancellation for M-theory five-branes is also presented, uncovering some surprising details.
The Feynman path integral of ordinary quantum mechanics is complexified and it is shown that possible integration cycles for this complexified integral are associated with branes in a two-dimensional A-model. This provides a fairly direct explanation of the relationship of the A-model to quantum mechanics; such a relationship has been explored from several points of view in the last few years. These phenomena have an analog for Chern-Simons gauge theory in three dimensions: integration cycles in the path integral of this theory can be derived from N=4 super Yang-Mills theory in four dimensions. Hence, under certain conditions, a Chern-Simons path integral in three dimensions is equivalent to an N=4 path integral in four dimensions.
