Elliptic quantum curves of class $$ {\mathcal{S}}_k $$

Abstract

Quantum curves arise from Seiberg-Witten curves associated to 4d $$ \mathcal{N} $$ N = 2 gauge theories by promoting coordinates to non-commutative operators. In this way the algebraic equation of the curve is interpreted as an operator equation where a Hamiltonian acts on a wave-function with zero eigenvalue. We find that this structure generalises when one considers torus-compactified 6d $$ \mathcal{N} $$ N = (1, 0) SCFTs. The corresponding quantum curves are elliptic in nature and hence the associated eigenvectors/eigenvalues can be expressed in terms of Jacobi forms. In this paper we focus on the class of 6d SCFTs arising from M5 branes transverse to a ℂ2/ℤk singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d $$ \mathcal{N} $$ N = 2 theories are known as class $$ {\mathcal{S}}_k $$ S k . We explicitly show that the eigenvectors associated to the quantum curve are expectation values of codimension 2 surface operators, while the corresponding eigenvalues are codimension 4 Wilson surface expectation values.

5d/6d Wilson loops from blowups

“…where we used the identities (A.8) and (A.10) to simplify the solution. We compared this solution against the result from the ADHM calculation in [62] in the e −τ expansion and verified that two results, although they look completely different, perfectly match up to e −5τ order. We also checked that the solution at 2-string also matches the 2-string elliptic genus from the ADHM calculation up to e −5τ order.…”

“…We first compute the expectation value of this 6d/2d coupled system and then relate it to the Wilson loop operator. The codimension-4 defect in the 6d SU(2) theory with 4 fundamentals has been recently studied in [62]. The defect is constructed by coupling a 2d fundamental fermion to the 6d bulk SU(2) gauge field.…”

“…The analogous observables in 6d SCFTs are the BPS Wilson surface operators realized by probe strings carrying tensor charges as well as gauge charges [57,58]. The partition functions of the Wilson surfaces have been evaluated by generalizing the ADHM approaches in [59][60][61][62]. Such loop (and surface) operators 1 have been used for non-trivial checks of dualities among a large class of 5d/6d gauge theories [55,56,61].…”

Bootstrapping BPS spectra of 5d/6d field theories

“…Throughout the paper, we refer to them as the BPS partition functions, though it may be an abuse of notation. Protected by the supersymmetry, the BPS spectra encoded in the partition function can be used to confirm non-trivial dualities and symmetry enhancements at various points on the moduli space [20][21][22][23][24][25][26][27][28][29][30][31]. Also, the Seiberg-Witten prepotential in the Coulomb phase can be reproduced from the BPS partition function by taking the limit 1 , 2 → 0 of two Ω-deformation parameters [32,33].…”

E-string Quantum Curve

“…A particularly interesting class of quantum curves is the class of "elliptic quantum curves" which arise in compactifications of 6d superconformal field theories (SCFTs) with 8 supercharges on elliptic curves [9]. Compactifications of 6d SCFTs give rise to a wide plethora of four and five-dimensional quantum field theories where the 6d origin sheds light on many dualities between the lower dimensional theories [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] and thus it is expected that elliptic quantum curves mirror these dualities.…”