Four-dimensional avatars of two-dimensional RCFT

Abstract

We investigate a 4D analog of 2D WZW theory. The theory turns out to have surprising finiteness properties and an infinite-dimensional current algebra symmetry. Some correlation functions are determined by this symmetry. One way to define the theory systematically proceeds by the quantization of moduli spaces of holomorphic vector bundles over algebraic surfaces. We outline how one can define vertex operators in the theory. Finally, we define four-dimensional "conformal blocks" and present an analog of the Verlinde formula.

The dynamical structure of higher dimensional Chern-Simons theory

“…Indeed, a four dimensional analog of the WZW Lagrangian exists [6]. This model is characterized by a symmetry algebra that generalizes the familiar Kac-Moody algebra in two dimensions, and which has been called WZW 4 algebra [6,7]. It is a natural question to see whether this algebra can be obtained from a Chern-Simons theory in five dimensions just as the Kac-Moody algebra is generated from the 2+1 Chern-Simons theory [4] (see also Ref.…”

“…[For a recent work dealing with the WZW 4 algebra see Ref. [7].] This paper is organized as follows.…”

Bethe/gauge correspondence on curved spaces

“…One then applies the corresponding version of the Riemann-Roch-Grothendieck formula (here the contribution of Grothendieck is crucial). In physics language this corresponds to the idea, exploited in [20][21][22], that one can choose (generically) a gauge σ(x) ∈ t, thereby reducing the gauge group locally from G to T . In this way the non-abelian localization of [11] becomes the usual localization with respect to the infinite dimensional gauge group of T -valued gauge transformations.…”

Noncommutative multi-solitons in 2+1 dimensions

“…The noncommutative U (n) sigma model may be obtained by a reduction of the Nair-Schiff sigma-model-type action [32,33] from four to three dimensions, 5) where Greek indices include the extra coordinate ρ, and λµνσ denotes the totally antisymmetric tensor in R 4 . The field Φ(t, x, y) is group-valued, Φ † = Φ −1 , with an extension Φ(t, x, y, ρ) interpolating between Φ(t, x, y, 0) = const and Φ(t, x, y, 1) = Φ(t, x, y) , (2.6) and 'tr' implies the trace over the U (n) group space.…”