Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states

Abstract

We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi-Yau threefolds. Fieldtheoretically, these 5d SCFTs descend from 6d N = (1, 0) SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi-Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, these graphs manifestly capture strongly coupled data of the 5d SCFTs, such as the superconformal flavor symmetry, BPS states, and mass deformations.The capabilities of this approach are demonstrated by deriving all rank one and rank two 5d SCFTs. The full potential, however, becomes apparent when applied to theories with higher rank. Starting with the higher rank conformal matter theories in 6d, we are led to the discovery of previously unknown flavor symmetry enhancements and new 5d SCFTs.

Sequences of 6d SCFTs on generic Riemann surfaces

“…First, one should understand the compactification of the 6d SCFTs on a circle to five dimensions. A lot of progress have been made on this front in recent years [16,22,23,29,30,44,[47][48][49][50][51][52][53][54][55][56][57][58]. In case the five dimensional theory has a gauge theory effective description it teaches us about the punctures we can have and how to glue four dimensional theories together.…”

Complete prepotential for 5d $$ \mathcal{N} $$ = 1 superconformal field theories

“…1 and the masses of the second SU (2) (2) are labeled as m (2) 0 and m (2) i (i = 1, · · · , 5). We then take :=ã 1 − m AS , we find the complete prepotential for SU (2) 0 −SU (2)− [5] shows an E 8 global symmetry [9] 11 as follows:…”

“…For instance, 5d Sp(N ) gauge theories with N f hypermultiplets in the fundamental representations (flavors) and 5d SU (N + 1) κ gauge theories of Chern-Simons (CS) level κ = N + 3 − N f /2 with N f flavors are a typical example of such UV-duality [5]. In particular, 5d N = 1 gauge theories of rank-2 gauge groups are completely classified with their dual partners and field contents [6][7][8][9][10][11]. Such dual theories enjoy intriguing enhanced global symmetry, whose symmetry structure arises through non-trivial interplays between instanton particles and hypermultiplets, and is often checked from the index functions like superconformal index or partition function with shifted Coulomb branch parameters.…”

“…As many of such 5d N = 1 theories can be engineered via Type IIB 5-brane webs [12,13] or M-theory on Calabi-Yau (CY) threefold [14,15], brane configurations also provide a direct description of the prepotential. For instance, one can study CY geometry of 5d gauge theories to obtain their triple intersections which yield the prepotential of the 5d theories or one can also scan possible gauge theory descriptions from the geometry which lead to a classification of the UV-dual theories [6,9,11,[16][17][18][19]. Though Type IIB 5branes can be understood as dual description of the geometry, not all CY geometries can be realized as a 5-brane web.…”

Symmetry enhancement in 4d Spin(n) gauge theories and compactification from 6d

“…Recently, there has been some progress in the study of 6d SCFT lifts of 5d gauge theories using geometric methods[46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62]. It is our hope that these tools may also be useful for the cases discussed here.…”