W-algebras for Argyres-Douglas theories
Thomas Creutzig
TL;DR
This work identifies vertex operator algebras whose vacuum characters realize the Schur-indices of Argyres-Douglas theories, establishing B_p for (A1,A_{2p-3}) and W_p for (A1,D_{2p}) as the respective VOAs. It uses boundary-admissible quantum Hamiltonian reduction to relate D_n theories to A_{n-3} theories, and shows singlet M(p) and triplet W(p) arise as reductions, with B_p realized as a logarithmic extension tied to these structures. The module X_p for L_k(\mathfrak{sl}_2)\otimes \mathcal{H} is constructed so its character matches the D_{2p} Schur-index, and Appell-Lerch sums illuminate the modular properties of the resulting characters. For D_{2p} theories, the W_p algebra is produced via a non-regular QHR of V_k(\mathfrak{sl}_{p+1}) at k=-(p^2-1)/p, and conformal embeddings along with cosets produce subsumed VOAs C_p and Y_p, which recover B_p, M(p), and W(p) after plus-reduction. Collectively, the results deepen the link between 4d N=2 SCFTs and logarithmic VOAs, enabling explicit character and modular data and suggesting broader implications for VOA tensor categories in this physical context.
Abstract
The Schur-index of the $(A_1, X_n)$-Argyres-Douglas theory is conjecturally a character of a vertex operator algebra. Here such vertex algebras are found for the $A_{\text{odd}}$ and $D_{\text{even}}$-type Argyres-Douglas theories. The vertex operator algebra corresponding to $A_{2p-3}$-Argyres-Douglas theory is the logarithmic $\mathcal B_p$-algebra of [1], while the one corresponding to $D_{2p}$, denoted by $\mathcal W_p$, is realized as a non-regular Quantum Hamiltonian reduction of $L_{k}(\mathfrak{sl}_{p+1})$ at level $k=-(p^2-1)/p$. For all $n$ one observes that the quantum Hamiltonian reduction of the vertex operator algebra of $D_n$ Argyres-Douglas theory is the vertex operator algebra of $A_{n-3}$ Argyres-Douglas theory. As corollary, one realizes the singlet and triplet algebras (the vertex algebras associated to the best understood logarithmic conformal field theories) as Quantum Hamiltonian reductions as well. Finally, characters of certain modules of these vertex operator algebras and the modular properties of their meromorphic continuations are given.