Rank $Q$ E-string on a torus with flux

Abstract

We discuss compactifications of rank QQ E-string theory on a torus with fluxes for abelian subgroups of the E_8E8 global symmetry of the 6d6d SCFT. We argue that the theories corresponding to such tori are built from a simple model we denote as E[USp(2Q)]E[USp(2Q)]. This model has a variety of non trivial properties. In particular the global symmetry is USp(2Q)\times USp(2Q)\times U(1)^2USp(2Q)×USp(2Q)×U(1)2 with one of the two USp(2Q)USp(2Q) symmetries emerging in the IR as an enhancement of an SU(2)^QSU(2)Q symmetry of the UV Lagrangian. The E[USp(2Q)]E[USp(2Q)] model after dimensional reduction to 3d3d and a subsequent Coulomb branch flow is closely related to the familiar 3d3dT[SU(Q)]T[SU(Q)] theory, the model residing on an S-duality domain wall of 4d4d\mathcal{N}=4𝒩=4SU(Q)SU(Q) SYM. Gluing the E[USp(2Q)]E[USp(2Q)] models by gauging the USp(2Q)USp(2Q) symmetries with proper admixtures of chiral superfields gives rise to systematic constructions of many examples of 4d4d theories with emergent IR symmetries. We support our claims by various checks involving computations of anomalies and supersymmetric partition functions. Many of the needed identities satisfied by the supersymmetric indices follow directly from recent mathematical results obtained by E. Rains.

Sequences of 6d SCFTs on generic Riemann surfaces

“…In particular it suggests that if one is to understand the generalization of our statements for non minimal conformal matter one could look for SU (k) N −1 × U (1) N −1 punctures such that gluing is performed by first gauging the SU (k) N −1 symmetry and then the U (1) N −1 enhances to SU (N ) and subsequently gauged. 15 In this section we have argued that combining k minimal punctures leads to a new maximal puncture of symmetry SU (k) × U (1) such that we find a trinion with three maximal punctures with minimal flux. We discussed a concrete procedure of S-gluing together k free trinions.…”

“…A similarly looking, though different, enhancement of SU (2) N symmetry to U Sp(2N ) was recently considered in[15].…”

“…For example (2, 0) theories on surfaces with irregular punctures in some cases give Argyres-Douglas theories, for which Lagrangians have been worked out in [8][9][10]. Other examples involve (2, 0) SCFTs probing ADE singularities and higher rank E-string on surfaces of genus zero with at most two punctures or tori [11][12][13][14][15], some more esoteric 6d SCFTs on tori [16,17], as well as (2, 0) on spheres with special collections of punctures [7]. Such dictionaries between four and six dimensions lead to many insights into duality properties of four dimensional theories as well as to some systematic understanding of emergent IR symmetries.…”

“…This is a recurring theme in many constructions of dictionaries between six and four dimensions. There are numerous examples by now [2,4,6,24] with some of the more recent ones being the sphere with two puncture compactifications of rank Q > 1 E-string [15]. There one of the two U Sp(2Q) puncture symmetries is an enhanced symmetry in the IR from the SU (2) Q UV Lagrangian symmetry.…”

Anomaly inflow, accidental symmetry, and spontaneous symmetry breaking

“…The reduction of 6d (2,0) theories to 4d N = 2 QFTs was first analyzed in [10,11], and reduction to 4d N = 1 QFTs has been studied in [12][13][14][15][16]. The compactification of 6d (1,0) theories has been addressed in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”