| dc.description.abstract | We discuss some aspects of defects and boundaries in quantum field theories (QFTs) and their applications in revealing non-perturbative aspects of QFTs in combination with other techniques, including integrability.
Firstly, we study the Kondo line defects that arise from local impurities chirally coupled to a two-dimensional conformal field theory. They have interesting defect Renormalization Group flows and integrability properties. We give a construction from four-dimensional Chern Simons theory whose two-dimensional compactification leads to a 2d CFT with Kondo line insertion. This construction will provide new perspectives into the surprising integrable properties of Kondo line defects.
Secondly, we study the ODE/IM correspondence, which states a surprising link between conformal field theories and the spectral problems of ordinary differential equations. A direct derivation of the correspondence is still unknown. We study a more refined description by directly relating the expectation values of a Kondo defect line and the generalized monodromy data of an ODE. Thanks to the 4d Chern Simons construction, we conjecture an explicit recipe for constructing the ODE corresponding to a Kondo defect. New examples we discuss include the isotropic/anisotropic Kondo defects in the multichannel $\prod_i SU(2)_{k_i}$ WZW models. We then extend the ODE/IM correspondence we find to the excited states, which provides a full solution to the spectral problems for the affine Gaudin model and the Kondo defects. In particular, by generalizing and applying techniques of exact WKB analysis, we derive the non-perturbative infra-red behaviours and wall-crossing properties of a large class of Kondo line defects.
Finally, we study the conformal boundary conditions of a four-dimensional Abelian gauge field. One starts by coupling a three-dimensional CFT with a $U(1)$ symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFT) parametrised by the gauge coupling $\tau$ in the upper-half plane and by the choice of the 3d CFT in the decoupling limit $\tau \to \infty$. The $SL(2,\mathbb{Z})$ electromagnetic transformations act on the BCFTs and relate different 3d CFTs in the various decoupling limits. We study the general properties of this BCFT and show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. We propose a new computational scheme that can be used to approximate observables in strongly coupled 3d CFTs. As an example, we consider the 3d CFT to be one Dirac fermion and compute scaling dimensions of various boundary operators and the hemisphere free-energy up to two loops. Using an $S$-duality improved ansatz, we extrapolate the perturbative results and find good approximations to the observables of the $O(2)$ model. | en |
