String Geometry Seminar
(abstracts)

Thursdays, 3-4 P.M.
LCB 323

 University of Utah
September 18, 2003 Yuan-Pin Lee
Symplectic loop spaces and genus zero Gromov--Witten theory
To make this talk suitable for a string geometry seminar, I will start by explaining, very briefly, how Feynman integrals in topological sigma models are reduced to integrals over moduli spaces of holomorphic maps.
With this backdrop, the traditional formulation of quantum cohomology, i.e. genus zero Gromov--Witten theory, in terms of moduli spaces of stable maps will be reviewed.
The main focus of this talk is to explain how one can reformulate quantum cohomology in terms of some sort of enriched classical field theory, a.k.a. Givental's theory. This new formulation calls for Lagrangian cones in infinite dimensional symplectic loop spaces, with properties of "semi-infinite variation of Hodge structures". I will then show that two formulations are equivalent.
This will pave the way for the next talk.
October 9, 2003 Eric Sharpe
Some mathematical aspects of D-branes
In this talk we shall outline some recent work concerning boundary states and open string spectra in the B model topological field theory. General off-shell boundary states in the B model are believed to correspond to objects in the derived category of coherent sheaves, and open string modes are believed to correspond to RHom's. In this talk we shall describe how this dictionary can be checked in detail for on-shell states, and some of the many physical effects that conspire to give the stated results. Computing massless spectra in closed strings is easy, but we shall see that in open strings the story is much more interesting: nontrivial boundary conditions in open strings have the effect of physically realizing spectral sequences in BRST cohomology, and multiple anomalies in the open string B model work together to make the spectrum match Ext's. The end-result is that an extremely complicated physical computation can be translated into a comparatively easy mathematical computation, giving us powerful tools for physical computations. Time permitting, we will also describe how the same methods can be used to extend the known dictionary between on-shell boundary states and coherent sheaves.
October 23, 2003 Jian Dai
Orbifolding matrix theory and deconstructing type II string (joint with Yong-Shi Wu)
In this talk I will try to convince the audience of the following statements.
(1) A class of quiver mechanics models results from orbifolding Matrix Theory {\sl a la} BFSS with some surviving supersymmetries.
(2) At its large $N$ limit, quiver mechanics provides a (de)construction of low dimensional super Yang-Mills theory.
(3) Matrix String, hence IIA string, is a consequence of $Z_N$ orbifold.
(4) IIB string is a consequence of $Z_N^2$ orbifold.
(5) A target-worldsheet(volume) duality is shown from different angles, including a deduction in terms of discrete calculi.
(6) At intermediate step, a world volume geometry can be identified, whose significance is still to be clarified.
November 13, 2003 Aaron Bertram
Toward mirror symmetry of nonabelian quotients
It is natural to consider relationship of Gromov-Witten invariants for abelian and nonabelian quotients. We formulate some conjectures on it and explain some cases proven. This is a joint work with Ciocan-Fontanine and Kim.
November 20, 2003 William Linch III
M-theory, Fluxes, and 3D, N=1 Supergravity
We calculate the most general N=1 three-dimensional, renormalizable gauge invariant action coupled to matter in superspace and derive its component form. One example of such an action can be obtained by compactifying M-theory on a Spin(7) holonomy manifold taking non-vanishing fluxes into account. We show that the resulting three-dimensional action is indeed in agreement with our more general construction. The resulting scalar potential freezes all the moduli of the internal manifold arising from the metric except for the radial modulus. This potential can be written in terms of the superpotential previously conjectured in the literature.
January 22, 2004 Javier Fernandez
Some geometric aspects of the moduli of Calabi-Yau manifolds
In this talk I will review some classical aspects of the geometry of the moduli of Calabi-Yau manifolds. This is the first of a series of talks aimed at reading (parts of) the book "Mirror Symmetry" by C. Vafa et al.
February 12, 2004 Daniele Arcara
An introduction to Toric Varieties
This talk is a brief introduction to toric varieties. After giving the definition of a toric variety, we shall look at some examples, and then study some basic properties. We shall also define maps between toric varieties, and in particular look at blow-ups.
March 31, 2004 Bumsig Kim
Quantum Cohomology via D-module (after Guest)
Using D-module approach we will discuss how to recover small quantum product structure for flag manifolds and Fano toric manifolds. This is a survey talk on Guest's work.