The geometric approach to quantisation is described in which one starts with a symplectic manifold (M, ω) and then constructs a prequantum Hermitian line bundle B over M such that its first Chern Class is integral and the curvature is ω. This leads to a prequantum wave function that should be viewed as the section of the line bundle B.
Then to quantise one imposes a polarisation such that operators form irreducible representations of the algebra of observables. The subsequent role of determining a measure compatible with the polarisation and how to relate different polarisation choices is discussed.
Finally, time permitting, we describe the relation to various contemporary topics in string and M-theory.
Schedule:16/05/2018 11:00-12:00 G.O. Jones 610 24/05/2018 11:00-12:00 G.O. Jones 410A25/05/2018 10:30-11:30 G.O. Jones 610
These lectures will be focused on aspects of combinatorics relevant to gauge-string duality (holography). The physical theories we will discuss include two dimensional Yang Mills theory, N=4 super Yang Mills theory with U(N) gauge group, Matrix and tensor models. The key mathematical concepts include : Schur Weyl-duality, permutation equivalence classes and associated discrete Fourier transforms as an approach to counting problems and, branched covers and Hurwitz spaces. Schur-Weyl duality is a powerful relation between representations of U(N) and representations of symmetric groups. Representation theory of symmetric groups offers a method to define nice bases for functions on equivalence classes of permutations. These bases are useful in counting gauge invariant functions of matrices or tensors, as well as computing their correlators in physical theories. In AdS/CFT these bases have proved useful in identifying local operators in gauge-theory dual to giant gravitons in AdS. In the simplest cases of gauge-string duality, the known mathematics of branched covers and Hurwitz spaces provide the mechanism for the holographic correspondence between gauge invariants and stringy geometry.
Lecture notes: HolographicCombinatorics1 [PDF 11,019KB], HolographicCombinatorics2 [PDF 7,631KB], HolographicCombinatorics3 [PDF 10,005KB]
Schedule:24/11/2017 11:00-13:00 G.O. Jones 610 01/12/2017 11:00-13:00 G.O. Jones 610 08/12/2017 11:00-13:00 G.O. Jones 610