Publications

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.

We present a new formulation, in twistor space, of the classical second-order Yang-Mills field equations.

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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.

No abstract is provided for this article.

In these lectures, I will describe some aspects of an approach to the Jones polynomial of knots, and its generalizations, that is based on a three-dimensional quantum Yang-Mills theory in which the usual Yang-Mills Lagrangian is replaced by a Chern-Simons action. This approach gives a manifestly three-dimensional approach to the subject, but some of the key aspects of the story, involving the Feynman path integral, are somewhat beyond the reach of present rigorous understanding. In the first two lectures, I will describe aspects of the subject that can be developed rigorously at present. This basically consists of a gauge theory approach to the Jones representations of the braid group and their generalizations. In the last lecture, I will describe the more ambitious Feynman path integral approach, which is an essential part of the way that physicists actually think about problems such as this one, and which gives the most far-reaching results.

The theoretical physics of the first quarter of the twentieth century —centering around relativity theory and nonrelativistic quantum mechanics—has had a broad influence mathematically. The main achievement of theoretical physics in the following half-century was the development of quantum field theory or QFT. Yet the mathematical influence of QFT still belongs largely to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="21 Superscript s t"> <mml:semantics> <mml:msup> <mml:mn>21</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">21^{st}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> century, because its mathematical foundations are still not well-understood.

The correspondence between string theory in Anti-de Sitter space and super Yang Mills theory is an example of the Holographic principle according to which a quantum theory with gravity must be describable by a boundary theory. However, arguments given so far are incomplete because, while the bulk theory has been related to a boundary theory, the holographic bound saying that the boundary theory has only one bit of information per Planck area has not been justified. We show here that this bound is the physical interpretation of one of the unusual aspects of the correspondence between Anti-de Sitter space and the boundary conformal field theory, which is that infrared effects in the bulk theory are reflected as ultraviolet effects in the boundary theory.

We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by

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.

Perturbative fermion anomalies in spacetime dimension $d$ have a well-known relation to Chern-Simons functions in dimension $D=d+1$. This relationship is manifested in a beautiful way in "anomaly inflow" from the bulk of a system to its boundary. Along with perturbative anomalies, fermions also have global or nonperturbative anomalies, which can be incorporated by using the $η$-invariant of Atiyah, Patodi, and Singer instead of the Chern-Simons function. Here we give a nonperturbative description of anomaly inflow, involving the $η$-invariant. This formula has been expected in the past based on the Dai-Freed theorem, but has not been fully justified. It leads to a general description of perturbative and nonperturbative fermion anomalies in $d$ dimensions in terms of an $η$-invariant in $D$ dimensions. This $η$-invariant is a cobordism invariant whenever perturbative anomalies cancel.

International Press of Boston - publishers of scholarly mathematical and scientific journals and books

In this paper, we explore supersymmetric and 2d analogs of the SYK model. We begin by working out a basis of (super)conformal eigenfunctions appropriate for expanding a four-point function. We use this to clarify some details of the 1d supersymmetric SYK model. We then introduce new bosonic and supersymmetric analogs of SYK in two dimensions. These theories consist of $N$ fields interacting with random $q$-field interactions. Although models built entirely from bosons appear to be problematic, we find a supersymmetric model that flows to a large $N$ CFT with interaction strength of order one. We derive an integral formula for the four-point function at order $1/N$, and use it to compute the central charge, chaos exponent and some anomalous dimensions. We describe a problem that arises if one tries to find a 2d SYK-like CFT with a continuous global symmetry.

Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$ symmetry along the boundary becomes on-site when viewed as an extended $H$ symmetry, via a suitable group extension $1\to K\to H\to G\to1$. Namely, a non-perturbative global (gauge/gravitational) anomaly in $G$ becomes anomaly-free in $H$. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, always existent for any SPT state in any spacetime dimension $d \geq 2$ of any finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an $H$-symmetry extended boundary of bulk $d \geq 2$, or more naturally by a topological emergent $K$-gauge theory with a global symmetry $G$ on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry $G$, described by representations of $H$. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped interfaces obtained via symmetry breaking (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension. More generally, applying our approach to SPT, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.

We analyze the u-plane contribution to Donaldson invariants of a four-manifold X. For $b_2^+(X)>1$, this contribution vanishes, but for $b_2^+=1$, the Donaldson invariants must be written as the sum of a u-plane integral and an SW contribution. The u-plane integrals are quite intricate, but can be analyzed in great detail and even calculated. By analyzing the u-plane integrals, the relation of Donaldson theory to N=2 supersymmetric Yang-Mills theory can be described much more fully, the relation of Donaldson invariants to SW theory can be generalized to four-manifolds not of simple type, and interesting formulas can be obtained for the class numbers of imaginary quadratic fields. We also show how the results generalize to extensions of Donaldson theory obtained by including hypermultiplet matter fields.

No abstract is provided for this article.

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.