Spectroscopy of gauge theories based on exceptional Lie groups

Abstract

Abstract:We generate by computer a basis of invariants for the fundamental representations of the exceptional Lie groups E 6 and E 7 , up to degree 18. We discuss the relevance of this calculation for the study of supersymmetric gauge theories, and revisit the self-dual exceptional models. We study the chiral ring of G 2 to degree 13, as well as a few classical groups. The homological dimension of a ring is a natural estimator of its complexity and provides a guideline for identifying theories that have a good chance to be amenable to a solution.

Highest weight generating functions for Hilbert series

“…The HWG series 2.34 can be presented in terms of Young tableaux for U (N ), with the total number of boxes being given by the exponents of the fugacities t in a partition described by the fugacities m i . Similar results also can be obtained by decomposition of tensor products [16], however, the HWG approach has the potential advantage of generating the complete infinite series of GIOs, thereby resolving uncertainties about the multiplicities of distinct invariants and/or their appearance at higher orders. It can be noted that equivalent series can also be obtained by using the SU (N ) group in place of U (N ), since the rightmost Dynkin label of a U (N ) representation can be calculated from the Dynkin labels of the corresponding SU (N ) representation, together with the total number of Young boxes given by the fugacity t. This provides an alternative calculation schema, which can be more efficient.…”

Low-energy dynamics of 3d N $$ \mathcal{N} $$ = 2 G2 supersymmetric gauge theory

“…In 4d, such theories do not have any s-confinement phases but have some quantum-deformed moduli spaces (see for instance [20][21][22][23][24][25]). Naively we expect that this is also the case in 3d.…”

“Chiral” and “non-chiral” 3d Seiberg duality

“…The Higgs branch, which is identical to the 4d one, is described by the following gaugeinvariant composites [40][41][42] M := QQ, B := Q 3 , F := Q 4 ,…”