Lectures on branes in curved backgrounds

Abstract

These lectures provide an introduction to the microscopic description of branes in curved backgrounds. After a brief reminder of the flat space theory, the basic principles and techniques of (rational) boundary conformal field theory are presented in the second lecture. The general formalism is then illustrated through a detailed discussion of branes on compact group manifolds. In the final lecture, many more recent developments are reviewed, including some results for non-compact target spaces.

Normal ordering and boundary conditions in open bosonic strings

“…So we will introduce a different kind of normal ordered product satisfying both equation of motion and boundary conditions. The mathematical problem posed by defining the normal ordering is related to that of calculating Green's functions [10,11,12,13]. The normal ordered product is defined by subtracting out the corresponding Green's functions.…”

A mathematical formalism for the Kondo effect in Wess-Zumino-Witten branes

“…We refer the reader to [32] for details on loop groups. Now Cardy [11,34,17,29,30] argues that an RCFT always has the following D-brane category: the 2-vector spaces B of branes is equal to the 2-vector space of the closed labels λ of its chiral theory. B is free on a set of "elementary branes" B (in the case of the WZW model, B = P k ).…”

“…[29], [34]) is to actually consider the cylinder A τ /2 with finite width τ /2 and glue it with the opposite cylinder A ′ τ /2 where the string boundary component is oriented the opposite way. Let C τ be the cylinder with two D-brane components obtained by gluing A τ /2 and A ′ τ /2 , with D-brane component labelled by another D-brane λ ′ .…”

“…The theory of WZW branes has several components and has been previously worked out quite satisfactorily physically (see e.g. [11,1,2,34,17,29,30,13,36,4,9]). The physical answer is that stable D-branes (at least on CFT level) in the level k WZW model corresponding to a group G are classified by a suitable twisted K-theory group 1…”

Level-rank duality of D-branes on the SU(N) group manifold

“…(For a review, see ref. [1].) One approach to this question is to study D-branes on group manifolds [2]- [21], where the background is highly symmetric, and the associated conformal field theory (the WZW model) exactly solvable.…”