T $$ \overline{T} $$ deformation in SCFTs and integrable supersymmetric theories

We calculate the $$ \mathcal{S} $$ S -multiplets for two-dimensional Euclidean $$ \mathcal{N} $$ N = (0, 2) and $$ \mathcal{N} $$ N = (2, 2) superconformal field theories under the T$$ \overline{T} $$ T ¯ deformation at leading order of perturbation theory in the deformation coupling. Then, from these $$ \mathcal{N} $$ N = (0, 2) deformed multiplets, we calculate two- and three-point correlators. We show the $$ \mathcal{N} $$ N = (0, 2) chiral ring’s elements do not flow under the T$$ \overline{T} $$ T ¯ deformation. Specializing to integrable supersymmetric seed theories, such as $$ \mathcal{N} $$ N = (2, 2) Landau-Ginzburg models, we use the thermodynamic Bethe ansatz to study the S-matrices and ground state energies. From both an S-matrix perspective and Melzer’s folding prescription, we show that the deformed ground state energy obeys the inviscid Burgers’ equation. Finally, we show that several indices independent of D-term perturbations including the Witten index, Cecotti-Fendley-Intriligator-Vafa index and elliptic genus do not flow under the T$$ \overline{T} $$ T ¯ deformation.

Multi-centered black holes, scaling solutions and pure-Higgs indices from localization

The Coulomb Branch Formula conjecturally expresses the refined Witten index for N=4 Quiver Quantum Mechanics as a sum over multi-centered collinear black hole solutions, weighted by so-called `single-centered' or `pure-Higgs' indices, and suitably modified when the quiver has oriented cycles. On the other hand, localization expresses the same index as an integral over the complexified Cartan torus and auxiliary fields, which by Stokes' theorem leads to the famous Jeffrey-Kirwan residue formula. Here, by evaluating the same integral using steepest descent methods, we show the index is in fact given by a sum over deformed multi-centered collinear solutions, which encompasses both regular and scaling collinear solutions. As a result, we confirm the Coulomb Branch Formula for Abelian quivers in the presence of oriented cycles, and identify the origin of the pure-Higgs and minimal modification terms as coming from collinear scaling solutions. For cyclic Abelian quivers, we observe that part of the scaling contributions reproduce the stacky invariants for trivial stability, a mathematically well-defined notion whose physics significance had remained obscure.

On the Supersymmetry of the Klein–Gordon Oscillator

The three-dimensional Klein–Gordon oscillator exhibits an algebraic structure known from supersymmetric quantum mechanics. The supersymmetry is unbroken with a vanishing Witten index, and it is utilized to derive the spectral properties of the Klein–Gordon oscillator, which is closely related to that of the nonrelativistic harmonic oscillator in three dimensions. Supersymmetry also enables us to derive a closed-form expression for the energy-dependent Green’s function.