Homotopical Analysis of 4d Chern-Simons Theory and Integrable Field Theories
Marco Benini, Alexander Schenkel, Benoît Vicedo
Abstract
Abstract This paper provides a detailed study of 4-dimensional Chern-Simons theory on $$\mathbb {R}^2\times \mathbb {C}P^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> for an arbitrary meromorphic 1-form $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> on $$\mathbb {C}P^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> . Using techniques from homotopy theory, the behaviour under finite gauge transformations of a suitably regularised version of the action proposed by Costello and Yamazaki is investigated. Its gauge invariance is related to boundary conditions on the surface defects located at the poles of $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> that are determined by isotropic Lie subalgebras of a certain defect Lie algebra. The groupoid of fields satisfying such a boundary condition is proved to be equivalent to a groupoid that implements the boundary condition through a homotopy pullback, leading to the appearance of edge modes. The latter perspective is used to clarify how integrable field theories arise from 4-dimensional Chern-Simons theory.