Line defect Schur indices, Verlinde algebras and U(1)rU(1)_r fixed points

Given an $\mathcal{N}=2$ superconformal field theory, we reconsider the Schur index $\mathcal{I}_L(q)$ in the presence of a half line defect $L$. Recently Cordova-Gaiotto-Shao found that $\mathcal{I}_L(q)$ admits an expansion in terms of characters of the chiral algebra $\mathcal{A}$ introduced by Beem et al., with simple coefficients $v_{L,\beta}(q)$. We report a puzzling new feature of this expansion: the $q \to 1$ limit of the coefficients $v_{L_,\beta}(q)$ is linearly related to the vacuum expectation values $\langle L \rangle$ in $U(1)_r$-invariant vacua of the theory compactified on $S^1$. This relation can be expressed algebraically as a commutative diagram involving three algebras: the algebra generated by line defects, the algebra of functions on $U(1)_r$-invariant vacua, and a Verlinde-like algebra associated to $\mathcal{A}$. Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type $(A_1,A_2)$, $(A_1,A_4)$, $(A_1, A_6)$, $(A_1, D_3)$ and $(A_1, D_5)$. In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.Comment: 64 pages, 21 figures. v2 published version, references update

Mass and Mean Velocity Dispersion Relations for Supermassive Black Holes in Galactic Bulges

Growing evidence indicate supermassive black holes (SMBHs) in the mass range of $M_{\rm BH}$$\sim 10^6-10^{10}M_{\odot}$ lurking in central bulges of many galaxies. Extensive observations reveal fairly tight power laws of $M_{\rm BH}$ versus the mean stellar velocity dispersion $\sigma$ of the host bulge. The dynamic evolution of a bulge and the formation of a central SMBH should be physically linked by various observational clues. In this contribution, we reproduce the empirical $M_{\rm BH}-\sigma$ power laws based on a self-similar general polytropic quasi-static bulge evolution and a sensible criterion of forming a SMBH surrounding the central density singularity of a general singular polytropic sphere (SPS) \cite{loujiang2008}. Other properties of host bulges and central SMBHs are also examined. Based on our model, we discuss the intrinsic scatter of the $M_{\rm BH}-\sigma$ relation and a scenario for the evolution of SMBHs in different host bulges.Comment: 8 pages, 2 figures, accepted for publication in the Proceedings of Science for VII Microquasar Workshop: Microquasars and Beyon