# Geometric Unification from Six-Dimensional Physics (15w5154)

Arriving in Banff, Alberta Sunday, May 24 and departing Friday May 29, 2015

## Organizers

Michael Hopkins (Harvard University)

David Nadler (University of California, Berkeley)

Andrew Neitzke (University of Texas at Austin)

Thomas Nevins (University of Illinois at Urbana-Champaign)

## Objectives

There is a long history of ideas from physics which have served as powerful organizing principles for mathematical investigation. A storied example is the classical self-dual Yang-Mills equations in four dimensions. These instanton equations not only revolutionized the theory of smooth four-manifolds, but also in dimensionally reduced form -- as monopole equations in three dimensions, Hitchin equations in two dimensions, Nahm equations in one dimension -- have had profound consequences in many directions in algebraic geometry, differential geometry, low-dimensional topology, integrable systems, nonlinear PDE, etc. Mirror symmetry, which has its origin in two-dimensional quantum field theory, has had an equally important impact on algebraic geometry and symplectic topology in the last two decades.

In the last few years a new idea has appeared in physics, which appears poised to play a similarly prominent role in mathematics. The star of the story is a six-dimensional quantum field theory, known to physicists as "the 6-dimensional (2,0) superconformal field theory", which we dub here "Theory X". In short, Theory X appears to be connected to many of the central ideas in twenty-first-century geometry, topology and geometric representation theory.

Many groups are currently working on ideas that can be traced back to this theory in an "in vitro" form, separated from the connecting fabric present in the underlying physics. One of the goals of the workshop is to introduce mathematicians to the rich "in vivo" ideas of Theory X as a new organizing principle connecting diverse structures in gauge theory, integrable systems, geometric representation theory, holomorphic symplectic geometry, knot invariants. Recent work by Witten, Moore-Tachikawa, Freed-Teleman and others has begun to flesh out the axiomatic structures implied by the existence of Theory X and the new symmetries they predict for familiar mathematical objects, and we expect this workshop to lead to more activity along these lines and a much broader appreciation of the potential impacts. Another major theme is expected to be the use of the Theory X perspective to generalize the geometric Langlands program, which can be seen (starting from the work of Kapustin-Witten) as the study of Theory X compactified on the product of a torus and a Riemann surface.

In order to achieve these goals, the format of the workshop will diverge from that of a typical research conference, with an emphasis on longer lecture series by leading physicists with a proven track record of communicating well to mathematicians, as well as mathematicians who have been working directly on Theory X.

The organizers -- together with David Ben-Zvi, Dan Freed, Edward Frenkel, Gregory Moore, and Constantin Teleman -- are principal investigators on an NSF Focused Research Grant which centers around Theory X and its mathematical applications. This grant provides funds which can be used as supplementary support for the workshop.

In the last few years a new idea has appeared in physics, which appears poised to play a similarly prominent role in mathematics. The star of the story is a six-dimensional quantum field theory, known to physicists as "the 6-dimensional (2,0) superconformal field theory", which we dub here "Theory X". In short, Theory X appears to be connected to many of the central ideas in twenty-first-century geometry, topology and geometric representation theory.

Many groups are currently working on ideas that can be traced back to this theory in an "in vitro" form, separated from the connecting fabric present in the underlying physics. One of the goals of the workshop is to introduce mathematicians to the rich "in vivo" ideas of Theory X as a new organizing principle connecting diverse structures in gauge theory, integrable systems, geometric representation theory, holomorphic symplectic geometry, knot invariants. Recent work by Witten, Moore-Tachikawa, Freed-Teleman and others has begun to flesh out the axiomatic structures implied by the existence of Theory X and the new symmetries they predict for familiar mathematical objects, and we expect this workshop to lead to more activity along these lines and a much broader appreciation of the potential impacts. Another major theme is expected to be the use of the Theory X perspective to generalize the geometric Langlands program, which can be seen (starting from the work of Kapustin-Witten) as the study of Theory X compactified on the product of a torus and a Riemann surface.

In order to achieve these goals, the format of the workshop will diverge from that of a typical research conference, with an emphasis on longer lecture series by leading physicists with a proven track record of communicating well to mathematicians, as well as mathematicians who have been working directly on Theory X.

The organizers -- together with David Ben-Zvi, Dan Freed, Edward Frenkel, Gregory Moore, and Constantin Teleman -- are principal investigators on an NSF Focused Research Grant which centers around Theory X and its mathematical applications. This grant provides funds which can be used as supplementary support for the workshop.