Two dimensional quantum field theory and string theory are mathematically very interesting but still not precisely defined. Two dimensional field theories are algebraic and analytic structures associated to geometric surfaces. String theories are averages or integrals of two dimensional field theories over all surface geometries. In this project we study various two dimensional field theories , their integrals and their common underlying structure. For example we will study a supersymmetric refinement of the Atiyah-Segal-Witten notion of 2D field theories with a nontrivial internal stucture related to modular forms. The point here is to construct examples and relate these ideas to the topological modular form cohomology theory. We will explore a new infinity categorical version of 2D field theory. There are many natural examples to investigate and categorifications of two dimensional theories relate to three dimensional topological quantum field theories. We intend to clarify the "string topology" two dimensional field theories associated to any "target manifold" . These theories feature a master equation solution, dX + X*X = 0, required to form the analogue of the string theory integral. The algebraic formalism resonates with the J holomorphic curves of symplectic topology. More generally, we will research the conceptual meaning of solutions to master equations, deformations of structures with duality, and the formalism related to multilinear functions or operations called correlators.
The different parts of science and mathematics are woven together in a rich tapestry and this phenomenon is well illustrated by the above. The studies of this project will impact a wide range of younger researchers in the universities as well as PhD students and undergraduates. Other impacts might eventually include improved technology. Practical applications by scientists are sometimes accidental like penicillin and synthetic rubber. Sometimes however, like the discovery of transistors and microelectronics, they depend on a deep understanding of subjects like quantum theory. A very speculative application of a deeper understanding of two dimensional quantum field theory, the subject of this project, could be to the physical realization of quantum computers. One knows quantum computers can theoretically solve problems not known to be solved theoretically by non quantum computing systems. One such problem is the theoretical possibility of cracking the factorization part of the security systems used by financial systems and government agencies. The technical difficulties to realizing quantum computers can be recast according to Michael Freedman, using the three dimensional topological theories alluded to above. Furthermore the relationship of these three dimensional theories with two dimensional theories suggests a direction to look to solve the technical difficulties: the experimental physics that takes place in two dimensions, the very active area of condensed matter research. There is an opportunity just now to organize these different perspectives and energies of the participants of the project into a coherent campaign to illuminate the area. The circle of ideas from two dimensional field theory, string theory, and deformations of structures with duality may very well become an important organizing center for twenty first century mathematics in the way that topology influenced the twentieth century.
Exotic physical processes in two dimensions are conjecturally related to quantum computing.The modeling of such exotic processes in two dimensions in terms of mathematics is illuminated by the study of a rich set of systems called two dimensional field theories. Many of these mathematical systems are imagined to be described by probes of higher dimensional spaces by curves and surfaces in these models of space-time. Algebraic structures related to topology, the study of position and deformation, emerge in a wide array of possibilities. These emerging structures promise to keep generations of upcoming mathematicians busy. The current project was conceived to join several possible sectors of such a development under the umbrella title " How Poincare Duality in the Algebraic Topology of a Closed Manifold relates to 2D Field Theory". By the last Meeting-Workshop the braver souls among algebraically minded, the analytically minded and the topologically minded were reporting on their attempts to fit the others' achievements into their own settings.This was especially true of a large number of the younger participants. Notable examples of achievements and cross currents were provided by 1) field theories related to algebras invented by von Neumann 2) the cobordism conjecture and algebraic deformation theory, 3) rigorous quantum field theory 4) unexpected results in a new version of string topology. In summary, an eclectic array of math phenomona were studied in the goal to find the important patterns illuminating enough to satisfy the mathematician's desire to understand "what is really going on" and simple enough to facilitate applications. First steps were definitely achieved. On a more concrete side, a group of undergraduate and high school students took their first steps in research, working in understanding certain statistical aspects of the above theories.
