4-dimensional Chern-Simons theory and integrable field theories

These lecture notes concern the semi-holomorphic 4d Chern-Simons theory and its applications to classical integrable field theories in 2d and in particular integrable sigma-models. After introducing the main properties of the Chern-Simons theory in 3d, we will define its 4d analogue and explain how it is naturally related to the Lax formalism of integrable 2d theories. Moreover, we will explain how varying the boundary conditions imposed on this 4d theory allows to recover various occurences of integrable sigma-models through this construction, in particular illustrating this on two simple examples: the Principal Chiral Model and its Yang-Baxter deformation. These notes were written for the lectures delivered at the school “Integrability, Dualities and Deformations”, that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

Symmetric space λ-model exchange algebra from 4d holomorphic Chern-Simons theory

We derive, within the Hamiltonian formalism, the classical exchange algebra of a lambda deformed string sigma model in a symmetric space directly from a 4d holomorphic Chern-Simons theory. The explicit forms of the extended Lax connection and R-matrix entering the Maillet bracket of the lambda model are explained from a symmetry principle. This approach, based on a gauge theory, may provide a mechanism for taming the non-ultralocality that afflicts most of the integrable string theories propagating in coset spaces.

Batalin-Vilkovisky formality for Chern-Simons theory

We prove that the differential graded Lie algebra of functionals associated to the Chern-Simons theory of a semisimple Lie algebra is homotopy abelian. For a general field theory, we show that the variational complex in the Batalin-Vilkovisky formalism is a differential graded Lie algebra.

Exact solutions in higher-dimensional Lovelock and AdS 5 Chern-Simons gravity

Lovelock gravity in D-dimensional space-times is considered adopting Cartan's structure equations. In this context, we find out exact solutions in cosmological and spherically symmetric backgrounds. In the latter case, we also derive horizons and the corresponding Bekenstein-Hawking entropies. Moreover, we focus on the topological Chern-Simons theory, providing exact solutions in 5 dimensions. Specifically, it is possible to show that Anti-de Sitter invariant Chern-Simons gravity can be framed within Lovelock-Zumino gravity in 5 dimensions, for particular choices of Lovelock parameters.

Topological entanglement and hyperbolic volume

The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the Rényi entropy of index m, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with S3 complements of a two-component link which is a connected sum of a knot $$ \mathcal{K} $$ K and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the m-moment of the reduced density matrix as a three-manifold invariant Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ), which is the partition function of $$ {M}_{{\mathcal{K}}_m} $$ M K m . Here $$ {M}_{{\mathcal{K}}_m} $$ M K m is a closed 3-manifold associated with the knot $$ \mathcal{K} $$ K m, where $$ \mathcal{K} $$ K m is a connected sum of m-copies of $$ \mathcal{K} $$ K (i.e., $$ \mathcal{K} $$ K #$$ \mathcal{K} $$ K . . . #$$ \mathcal{K} $$ K ) which mimics the well-known replica method. We analayse the partition functions Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling k. For SU(2) group, we show that Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) can grow at most polynomially in k. On the contrary, we conjecture that Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) is the hyperbolic volume of the knot complement S3\$$ \mathcal{K} $$ K m. We further propose that the Rényi entropies associated with SO(3) group converge to a finite value in the large k limit. We present some examples to validate our conjecture and proposal.

DNA Mutations via Chern–Simons Currents

We test the validity of a possible schematization of DNA structure and dynamics based on the Chern–Simons theory, that is a topological field theory mostly considered in the context of effective gravity theories. By means of the expectation value of the Wilson Loop, derived from this analogue gravity approach, we find the point-like curvature of genomic strings in KRAS human gene and COVID-19 sequences, correlating this curvature with the genetic mutations. The point-like curvature profile, obtained by means of the Chern–Simons currents, can be used to infer the position of the given mutations within the genetic string. Generally, mutations take place in the highest Chern–Simons current gradient locations and subsequent mutated sequences appear to have a smoother curvature than the initial ones, in agreement with a free energy minimization argument.