Higher gauge theory and integrability

In recent years, significant progress has been made in the study of integrable systems from a gauge theoretic perspective. This development originated with the introduction of four-dimensional Chern-Simons theory with defects, which provided a systematic framework for constructing two-dimensional integrable systems. In this article, we propose a novel approach to studying higher-dimensional integrable models employing techniques from higher category theory. Starting with higher Chern-Simons theory on the 4-manifold <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="double-struck">R</a:mi><a:mo>×</a:mo><a:mi>Y</a:mi></a:math>, we complexify and compactify the real line to <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi mathvariant="double-struck">C</d:mi><d:msup><d:mi>P</d:mi><d:mn>1</d:mn></d:msup></d:math> and introduce the disorder defect <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>ω</g:mi><g:mo>=</g:mo><g:msup><g:mi>z</g:mi><g:mrow><g:mo>−</g:mo><g:mn>1</g:mn></g:mrow></g:msup><g:mi mathvariant="normal">d</g:mi><g:mi>z</g:mi></g:math>. This procedure defines a holomorphic five-dimensional variant of higher Chern-Simons theory, which, when endowed with suitable boundary conditions, allows for the localization to a three-dimensional theory on <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>Y</j:mi></j:math>. The equations of motion of the resulting model are equivalent to the flatness of a 2-connection <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mo stretchy="false">(</l:mo><l:mi>L</l:mi><l:mo>,</l:mo><l:mi>H</l:mi><l:mo stretchy="false">)</l:mo></l:math>, that we then use to construct the corresponding higher holonomies. We prove that these are invariants of homotopies relative boundary, which enables the construction of conserved quantities. The latter are labeled by both the categorical characters of a Lie crossed module and the infinite number of homotopy classes of surfaces relative boundary in <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>Y</p:mi></p:math>. Moreover, we also demonstrate that the three-dimensional theory has left and right acting symmetries whose current algebra is given by an infinite-dimensional centrally extended affine Lie 2-algebra. Both of these conditions are direct higher homotopy analogs of the properties satisfied by the two-dimensional Wess-Zumino-Witten conformal field theory, which we therefore interpret as facets of integrable structures. Published by the American Physical Society 2024