The main goal of the proposed research is to study the physical realization of various constructions in geometric representation theory, notably the physical framework for knot homologies and the geometric Langlands program. The physical realization is based on topological gauge theory and topological string theory which describe the supersymmetric sector of the physical gauge theory/string theory. Recently the idea of "categorification" led to a number of remarkable developments in various branches of mathematics which may well have a physical interpretation. These include a variety of problems in representation theory, as well as "categorification" of polynomial knot invariants in low-dimensional topology. Related structures recently emerged in physics. In four-dimensional gauge theory the study has led to new interesting connections with the geometric Langlands program. The realization of Chern-Simons invariants of knots and links in terms of BPS states in the dual string theory led to formulation of triply-graded knot homologies. These developments point to many new connections between various problems in physics and mathematics that are described in this proposal. Broader impact: This proposal, on the one hand, should advance the understanding of gauge theory dynamics and string theory and, on the other hand, lead to important lessons and new results in related areas of mathematics, including enumerative geometry, homological algebra, low-dimensional topology, and representation theory. The PI is actively involved in organizing seminars, workshops and conferences on various topics related to the proposed research, and in mentoring students at all levels.
The main theme of S.Gukov's recent research is the relation between the geometry of quantum group invariants and their categorification, on the one hand, and the physics of supersymmetric gauge theory and string theory, on the other. Specifically, while carrying out this exciting research program, recent work of S.Gukov can be divided in developing the following general areas: Gauge Theory and Representation Theory: Connection between gauge theory and representation theory has a long history. It goes back to the first days of gauge theory; indeed, even the formulation of gauge theory involves a choice of a compact Lie group G that becomes the group of internal symmetries, a.k.a. the gauge group. For example, the quantum field theory that describes electromagnetic interactions has the Abelian gauge group G=U(1), while QCD -- the theory of strong nuclear interactions -- has the gauge group G=SU(3). Quantum field theories with other gauge groups also appear in Nature, as theoretical frameworks describing fundamental forces and many phenomena in condensed matter physics. Moreover, gauge theory often involves other fields representing particles charged under the gauge group. For instance, in the aforementioned examples of quantum electrodynamics (QED) and QCD typical examples of such particles are the electron and quarks, respectively, which altogether make up most of the matter on Earth. The fields describing these particles transform in a certain representation of the gauge group G, and this is one of the origins of the connection between gauge theory and representation theory. Thus, physical questions such as collision of particles require studying tensor products of representations, etc. Much deeper connections and insights can be obtained in interesting examples of gauge theories which, on the one hand, have interesting physical properties and, on the other hand, a large degree of control so that one can understand quantum dynamics of the theory. Typical examples of such ``soluble'' theories are supersymmetric or topological gauge theories. Topological theories (which are formally independent on the choice of space-time metric) played a key role in the recent interaction between physics and various branches of mathematics, especially low-dimensional topology where many deep insights were motivated by developments in quantum field theory. One of the prominent examples of topological field theory, which is also a good example of the ``modern'' connection between gauge theory and representation theory, is the Chern-Simons gauge theory in three dimensions. The physical observables in this theory are the so-called Wilson lines, which can be regarded as lines traversed by particles coupled to the gauge field. Thus, just as different species of particles, the set of Wilson line observables is naturally labeled by representations of the gauge group G. As is well-known from the celebrated work of E.Witten, the partition function of the Chern-Simons theory and correlation functions of Wilson line operators compute quantum group invariants of knots and three-manifolds. Similarly, other examples of topological gauge theories (which are close cousins of supersymmetric gauge theories) compute interesting invariants of 4-manifold, such as Donaldson and Seiberg-Witten invariants. Understanding these topological invariants in gauge theory involves studying moduli spaces of solutions to certain differential equations. Depending on the particular example of gauge theory, these can be various equations on the gauge connection, such as flatness equations or instanton equations. This is another source of interaction between gauge theory, geometry, and topology, not only in the context of 3-manifolds and 4-manifolds which appear directly as space-time manifolds, but also in the case of moduli spaces whose geometry and topology encodes interesting information about topology of the space-time as well as representation theory that enters the formulation of the physical problem. Gauge Theory and Categorification: In general, gauge theory proves to be a natural framework for ``geometrization'' or ``categorification'', which is one of the origins of the modern connection between gauge theory and geometric representation theory. It is closely related to recent developments in string theory, more specifically, the homological algebra of mirror symmetry. Following the original proposal of M.Kontsevich, it has been understood by now that a mathematically accurate description of D-branes (which properly accounts for open strings) in topological string theory is in terms of objects in derived categories. This description naturally appears in gauge theory on manifolds with boundaries and corners, so that all the insights and techniques developed in string theory enter the connection between gauge theory and the geometric representation theory. There are several concrete realizations of the connection between gauge theory and geometric representation theory and a number of strong indications that many new ones are likely to emerge. The line of research described below contributes to understanding the connections among these topics, namely, (geometric) representation theory, categorification, homological algebra, enumerative geometry, low-dimensional topology, gauge theory, and non-perturbative effects in SUSY gauge theories. One particular line of development is the relation between the geometric Langlands program and a certain version of topological Yang-Mills theory.