Abstract
This paper provides a detailed study of 4-dimensional Chern-Simons theory on mathbb {R}^2times mathbb {C}P^1 for an arbitrary meromorphic 1-form omega on mathbb {C}P^1. 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 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.
Highlights
The goal of the present paper is twofold
Its gauge invariance is related to boundary conditions on the surface defects located at the poles of ω that are determined by isotropic Lie subalgebras of a certain defect Lie algebra
The boundary conditions we consider are determined by a choice of Lie subalgebra k ⊂ gz of the Lie algebra gz of the product of jet groups G z = x∈z J nx −1G, where nx ≥ 1 is the order of the pole x ∈ z of ω, that is isotropic with respect to a non-degenerate symmetric invariant bilinear form ·, · ω defined in terms of ω
Summary
If ι∗g = ι∗h this vanishes by the skew-symmetry of the bilinear pairing ·, · ω : 1( , gz) × 1( , gz) → 2( ) on 1-forms It follows that ω ∧ (gh−1)∗χG = ω ∧ g∗χG − ω ∧ h∗χG (2.8). (Note that g is smooth because g is lazy.) By Proposition 2.8, we deduce ω ∧ g∗χG = ω ∧ g∗χG It remains to compute the integral on the right hand side. = −2π i k0x g∗χG = −2π i k0x g∗χGx = −2π i g∗χGz. The first equality follows from noting that g∗χG vanishes outside of In the second step, we used the fact that g∗χG is closed, exact x∈z on the.
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