Geometry, Topology, and Physics Seminar, Fall 2011

Part of the NSF/UCSB ‘Research Training Group’ in Topology and Geometry

Organizer: Dave Morrison.
Meets 4:00 - 5:30 p.m. on selected Fridays in South Hall 6635.

Sep. 30

### Two-manifolds, three-manifolds, and supersymmetric gauge theory

Abstract: During the past two and a half years, much new progress has been made in studying supersymmetric gauge field theories by using techniques from low-dimensional topology. The starting point from the physics side is a class of mysterious physical theories in 6 dimensions with maximal supersymmetry, which should lead to supersymmetric gauge theories in 4 dimensions or 3 dimensions if appropriately "compactified." The "compactification" involves a 2-manifold or a 3-manifold, respectively, and the topology and geometry of these compactification spaces turns out to be closely related to the physics. For example, with Drukker and Okuda I found a close connection between a result in 2-manifold topology called the Dehn-Thurston theorem and certain aspects of 4-dimensional supersymmetric gauge theories. Much work has been done on the case of 2-manifolds (Riemann surfaces) and gauge theories in 4 dimensions; more recently, work has begun on 3-manifolds and theories in 3 dimensions.

We will have a series of lectures on this general topic this quarter (perhaps continuing into next quarter); this first lecture will be introductory. I will sketch the key idea: cut the compactifying manifold into simple pieces which can be described geometrically, and learn how both the pieces and the gluing data for reassembling pieces can be interpreted in both mathematics and physics.

For an introductory reference, see: Gauge Theories Labelled by Three-Manifolds by Tudor Dimofte, Davide Gaiotto and Sergei Gukov, arXiv:1108.4389 [hep-th].

Audio [ mp3, wma ]; Lecture notes.

Oct. 7

Oct. 14

### Ideal triangulations of hyperbolic manifolds

Abstract: In dimensions 2 and 3 hyperbolic manifolds can be "triangulated" with ideal simplices whose vertices are at infinity. The situation is a bit subtle since the union of these simplices omits a certain subset of the manifold of measure zero. This subset is a geodesic lamination, a generalization of the idea of finite closed (periodic) geodesic. In dimension 2 there is only one shape of ideal 2-simplex (triangle) and there are parameters that describe how these are "glued" giving a parameterization of Teichmuller space. In dimension 3 an ideal 3-simplex (tetrahedron) has shape which is a complex number and these must satisfy certain glueing equations: one per edge of the triangulation. These are related to ideas of Fock and Goncharov for studying higher Teichmuller spaces, and triangulations are used by Dimofte, Gaiotto and Gukov to construct Gauge theories.

Audio [ mp3, wma ]; Slides.

Oct. 21

### Topological Phases of Matter: modeling and classification

Abstract: Motivated by application to quantum computing, we will discuss theoretical modeling and mathematical classification of topological phases of matter. Topological quantum systems are closely related to unitary stable topological quantum field theories. While a general classification is difficult, progress has been made for (2+1)-dimension and short-ranged entangled states. In particular, non-interacting fermion topological quantum systems can be classified completely via K-theory by A. Kitaev.

Audio [ mp3, wma ]; Slides.

Oct. 28

### Gauge Theories Labelled by Three-Manifolds

Abstract: We will discuss various operations on three-dimensional supersymmetric quantum field theories which are parallel to features of the decomposition of a three-manifold into ideal hyperbolic simplices. The discussion will include extensive background information on these three-dimensional field theories, in hopes of making the discussion accessible to mathematicians as well as physicists.

The first two operations, called $S$ and $T$, were discussed by Witten for 3D conformal field theories with a $U(1)$ flavor symmetry, and together give an $SL(2,\mathbb{Z})$ action on the space of such theories. The other "operation" is known as mirror symmetry for 3D theories: like the more familiar mirror symmetry in 2D, it exchanges certain parts of the moduli spaces when passing from one theory to its mirror dual.

Audio [ mp3, wma ]; Lecture notes.

Nov. 4

Nov. 11

Nov. 18

### Ideal triangulations of hyperbolic 3-manifolds, II

Abstract: We will describe some examples of triangulations of hyperbolic 3-manifolds using ideal simplices (vertices on the light cone) and the resulting gluing equations that give a hyperbolic structure. Followed by a description of the Pachner moves which enable one to move between different triangulations.

Audio [ mp3, wma ]; Slides; other notes.

Dec. 2

Dec. 9