Classification of 6d N = 1 0 $$ \mathcal{N}=\left(1,0\right) $$ gauge theories

Abstract

We delineate a procedure to classify 6d N = (1, 0) gauge theories composed, in part, of a semi-simple gauge group and hypermultiplets. We classify these theories by requiring that satisfy some consistency conditions. The primary consistency condition is that the gauge anomaly can be cancelled by adding tensor multiplets which couple to the gauge fields by acting as sources of instanton strings. Based on the number of tensor multiplets required to cancel the anomaly, we conjecture that the UV completion of these consistent gauge theories (if it exists) should be either a 6d N = (1, 0) SCFT or a 6d N = (1, 0) little string theory.

Geometric engineering, mirror symmetry and 6 d 1 0 → 4 d N = 2 $$ 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} $$

“…In the above expression, mass and Coulomb branch deformations are encoded in the terms in parenthesis: the leading order pole gives the mass deformation, and the subleading gives the contribution to Coulomb branch. The pole structure of this puncture is (1,4,5,7,8,11). The same…”

“…The crucial data is to identify the local pole structures to the various differentials i (z). This data has been worked out in [73] for E 6 class S theory, and the result is: the pole structure of i near the full puncture is (1,4,5,7,8,11) and the order of pole near the minimal puncture is (1, 1, 2, 2, 2, 3) [73].…”

“…The order of poles near the three punctures are: y 1 = 0 and y 2 = 0 : (1,5,7,9,11,13,17), 2, 2, 2, 3, 3, 4). (6.10) The N = 2 geometry for E 7 class S theory can be written in the following form:…”

“…The full puncture is labeled using the regular Nilpotent orbit of E 7 lie algebra, and its order of pole to the differential i (z) is (1,5,7,9,11,13,17), and the simple puncture is labeled by the minimal Nilpotent orbit of E 7 , and it has pole structure (1, 2, 2, 2, 3, 3, 4). 4 So the 4d theory we find is a class S[E 7 ] on a sphere with two full punctures and one simple puncture.…”

“…Similarly, the S duality of four dimensional N = 2 class S theories could be derived from compactifying 6d (2, 0) theory on a punctured Riemann surface [2]. Recently, a classification of 6d (1,0) theories has been proposed which is a surprisingly rich set [3,4] (see also [5]). It is natural to ask what kind of 4d theory we can get and what kind of interesting 4d dynamics we can learn from their compactification.…”