Four-Dimensional Chern-Simons and Gauged Sigma Models

Abstract

In this paper we introduce a new method for generating gauged sigma models from four-dimensional Chern-Simons theory and give a unified action for a class of these models. We begin with a review of recent work by several authors on the classical generation of integrable sigma models from four dimensional Chern-Simons theory. This approach involves introducing classes of two-dimensional defects into the bulk on which the gauge field must satisfy certain boundary conditions. One finds integrable sigma models from four-dimensional Chern-Simons theory by substituting the solutions to its equations of motion back into the action. The integrability of these sigma models is guaranteed because the gauge field is gauge equivalent to the Lax connection of the sigma model. By considering a theory with two four-dimensional Chern-Simons fields coupled together on two-dimensional surfaces in the bulk we are able to introduce new classes of 'gauged' defects. By solving the bulk equations of motion we find a unified action for a set of genus zero integrable gauged sigma models. The integrability of these models is guaranteed as the new coupling does not break the gauge equivalence of the gauge fields to their Lax connections. Finally, we consider a couple of examples in which we derive the gauged Wess-Zumino-Witten and nilpotent gauged Wess-Zumino-Witten models. This latter model is of note given one can find the conformal Toda models from it.

Non-Abelian Toda field theories from a 4D Chern-Simons theory

“…We have presented a 4d CS theory with two order defects which lead to NATFTs. In contrast to the case of the Faddeev-Reshetikhin model [21], the kinetic term of the 2d fields This would be possible by considering a doubled 4d CS theory [31] (with disorder defects).…”

3-dimensional mixed BF theory and Hitchin's integrable system

“…The 4-dimensional gauge theoretic origin of 2-dimensional integrable field theories, as proposed by Costello and Yamazaki in [CY], has been extensively studied over the past couple of years, see for instance [DLMV3,Sc1,FSY1,FSY2,CS,T,THC,BSV,BS,LV,CSV,FSY3,ABW,FSY4,D,St,Sc2,L2,FSY5].…”

Integrable degenerate $\mathcal E$-models from 4d Chern-Simons theory

“…It has been extensively applied to reproduce many existing 2d integrable field theories and also construct a wide variety of new ones [2,[8][9][10][11][12][13][14][15][16][17][18][19][20][21]. See also [1,[22][23][24][25][26][27][28][29][30] for further developments in relation to 4d Chern-Simons theory.…”