Abstract
We test in (An−1, Am−1) Argyres-Douglas theories with gcd(n, m) = 1 the proposal of Song’s in [1] that the Macdonald index gives a refined character of the dual chiral algebra. In particular, we extend the analysis to higher rank theories and Macdonald indices with surface operator, via the TQFT picture and Gaiotto-Rastelli-Razamat’s Higgsing method. We establish the prescription for refined characters in higher rank minimal models from the dual (An−1, Am−1) theories in the large m limit, and then provide evidence for Song’s proposal to hold (at least) in some simple modules (including the vacuum module) at finite m. We also discuss some observed mismatch in our approach for surface operators with large vortex number.
Highlights
Theories with no Lagrangian description, such as Argyres-Douglas (AD) theories discovered in [5]
The dual chiral algebras of (A1, A2k) theories are known to be Virasoro minimal models [13], and we count the number of Virasoro generators in each basis of the Verma module with weight T to compute the refined character.The relation between the Macdonald index and the refined character of the vacuum module has been further studied in [14, 15] from the viewpoint of the vertex operator algebra (VOA), and the refinement described above was found to be identified with the Kazhdan filtration in the context of VOA
We tested the proposal of Song’s in [1] that the Macdonald index can be interpreted as a refined character of the corresponding chiral algebra with the refinement parameter T = t/q counting the number of “basic” generators in each state, for simple examples in (A1, A1) and (An−1, Am−1) theories with gcd(n, m) = 1
Summary
The wavefunction for the (full) regular puncture in the Hall-Littlewood (HL) limit is given by. The wavefunctions of the irregular singularities In,−n+1 and In,0, as they can be used to construct certain (non-conformal) Lagrangian theories, can be worked out through gauging the flavor symmetry attached to the full regular puncture used in an equivalent construction [9], f In,−n+1 λ (q, t) fλIn,0 (a; q, t) zα)Ivec(z)fλ(z), i zα)Ivec(z)Ihyp(z, a; q, t)fλ(z). One can compute the wavefunction for I2,0 by evaluating the integral fλI2,0 (a; t). It is conjectured in [9] that the general wavefunction for I2,2n is given by the following scaling rule,. One can check the expressions of wavefunction for irregular punctures by comparing the HL index with the 3d Coulomb branch index of the 3d mirror theory developed in [27].
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