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These meetings are designed to encourage interaction, both social and intellectual, between the mathematics divisions (pure, applied and statistics) and theoretical physics. They consist of a short and accessible talk by a member of one of the divisions, or by an invited guest, followed (or interrupted) by discussion. In the academic year 2010/2011 we plan to have a combination of stand-alone talks volunteered by participants, starting with a seminar by Alon Feraggi on vector-spinor duality, and talks which continue the previous seminars on stability conditions in physics and geometry. The later will be based on the book by P. Aspinwall et al, Dirichlet Branes and Mirror Symmetry, AMS, Clay Mathematics Monographs Vol 4. This is an edited volume written jointly by physicists and mathematicians, which seems particularly suitable for `bridging the gap' between the two communities. (T.M. has a copy available for short term borrowing.) In the second semester we plan to have two talks (one by Jon Woolf, one by Thomas Mohaupt) on material from Chapters 3 and 4 of this book, and a talk by Radu Tatar on his recent work.
For confirmation please contact Jonathan Woolf on (0151) 794 4052 or Thomas Mohaupt on (0)151 795 5177.
Alon Faraggi - "Vector - Spinor Duality"
Abstract: Over the last few years a new duality has been uncovered in the space of heterotic string compactifications under the exchange of spinor and vector representations of an SO(10) group. The duality can be seen to arise from the breaking of the enhanced E6 symmetry in vacua with (2,2) world-sheet supersymmetry. The duality was initially observed by using the free fermionic formalism. In this talk I will discuss a representation of the duality in terms of discrete torsion in an orbifold compactification. I will speculate that the duality, which is akin to mirror symmetry, may arise from properties of K3 surfaces plus a non-trivial action on the vector bundle.Thomas Mohaupt - "A geometrical approach to particles, strings and fields"
Abstract: Quantum theories, whether based on particles, strings or fields, can be understood as rules which assign vector spaces and linear maps to geometrical data (manifolds and cobordisms). A modern approach to quantum theories, which is used in the monograph `Dirichlet Branes and Mirror Symmetry' edited by Paul Aspinwall et al, formalizes this idea by regarding quantum theories as functors between geometric and linear categories. In the seminar I'll try to relate this idea (as far as I'll understand it by then) to the conventional Hamiltonian and Lagrangian formulations of quantum theories. I'll use simple examples, which will also provide some background to the models which occured in Alon Faraggi's seminar. The talk will be self-contained and (hopefully) pedagogical. Some of the material is taken from Chapters 2 and 3 of the aforementioned book, and I plan to fill in the details for the sections I have picked.Jon Woolf - "Open and Closed Topological Field Theories"
Abstract: A 2d Topological Field Theory (TFT) assigns a complex number, the partition function, to each closed oriented surface, in such a way that this number can be computed by `cutting the surface into pieces'. The simplest way one could do this would be to cut along smooth curves so that the surface is divided into a number of components each of which is an oriented surface with boundary. The gluing rules for such a subdivision are governed by a `closed' TFT. Mathematically, this is an assignment of a vector space to each boundary and of a linear map between these to each surface component, satisfying certain rules. There is a well-known one-to-one correspondence between 2d closed TFTs and commutative Frobenius algebras. A more complicated way of cutting a closed surface into pieces might result in surfaces with boundary and corners. This results in more complicated gluing rules, which are governed by an `open and closed' TFT. These have a similar, but more complicated, algebraic description. Roughly this can be broken into two pieces, a commutative Frobenius algebra describing the corresponding closed theory, and a Frobenius (or Calabi-Yau) `category of boundary conditions' describing the open part of the theory. I will try to explain the above, using a mainly pictorial approach.Jon Woolf - "Open and Closed Topological Field Theories (Part 2)"
Abstract: Jon will continue his seminar on topological field theores.Your contribution is welcome!