UC Davis

We study the category of line operators in specific topological, or holomorphic-topological twists of supersymmetric quantum field theories in dimension $3$ and $4$. More specifically, we focus on topological A and B twist of 3d $\CN=4$ gauge theories and holomorphic-topological twists of 4d $\CN=2$ theories. We use geometric and representation-theoretic tools to define and study these categories, and prove physical conjectures about these cartegories and their relations.

We study the category of line operators in the topological twist of a 3d $\CN=4$ abelian gauge theory $\CT_\rho$. We complete the analysis of the boundary vertex operator algebras of Costello-Gaiotto, which results in boundary VOAs $V_{A,\rho}$ and $V_{B,\rho}$. We obtain explicit free field realizations of these boundary VOAs and use the free field realizations to prove the isomorphism $V_{A,\rho}\cong V_{B,\rho^\vee}$, which we interpret as the mirror symmetry statement in terms of the boundary VOAs. We then use the theory of logarithmic intertwining operators to define braided tensor categories $\CL_{A,\rho}$ and $\CL_{B,\rho}$ of modules of $V_{A,\rho}$ and $V_{B,\rho}$ (as derived categories). We propose that these are the categories of line operators for the A and B twist of the 3d $\CN=4$ theory $\CT_\rho$. Using the isomorphism of VOAs, we prove equivalence of braided tensor categories $\CL_{A,\rho}\simeq \CL_{B,\rho^\vee}$, which we interpret as the mirror symmetry statement in terms of the category of line operators. Finally, we show that $V_{B,\rho}$ admit a sheafification over the Higgs branch $\CM_{H,\rho}$, whose construction is related to the tangent Lie algebra.

We study the category of line operators in the holomorphic-topological twist of a 4d $\CN=2$ gauge theory $T_{HT}[G,V]$. This category is given a geometric description by Cautis-Williams, following the work of Kapustin, as $\Coh (G(\CO)\setminus \CR_{G,V})$. Using the idea of formal geometry, we compute the derived endomorphism of the unit object $\mathbbm{1}$ and show that it is quasi-isomorphic to the Poisson vertex algebra of Oh-Yagi, and that its graded super-trace reproduces the Schur index. Using the same method, we compute the derived homomorphism between line bundles supported on the miniscule orbits of $\Gr_G$, in the case when $G=PSL(2)$. We compare the graded super-trace of the results with the defect Schur indices of Cordova-Gaiotto-Shao.