Almost commuting elements in compact Lie groups

Abstract

10 The tori S(k) and S w C (g, k) and their Weyl groups 10.

6d N = 1 0 $$ \mathcal{N}=\left(1,0\right) $$ theories on T2 and class S theories. Part I

“…G = E n . The analysis is completely similar to the cases above, using the data in [11]. For G = E 6 , we have the following commuting triples:…”

“…Using the above tables for the values of v i and the groups G i , one can check (and it was indeed proved in [11]) that we always have d i,i+1 = 1 for adjacent commuting triples in the tables. This is interpreted as the fact that a fractional M5-brane has only the center-of-mass degrees of freedom.…”

“…The Chern-Simons invariant is 1/2 mod 1 [11], and the unbroken subgroup is so (2N − 7). Using this we have the following one-instanton configuration on T 3 × R:…”

Classical Yang-Mills vacua onT3:Explicit constructions

“…In all cases, the number of quantum vacuum state in pure N = 1 supersymmetric YangMills theory coincides with the so called dual Coxeter number h ∨ (or just the adjoint Casimir eigenvalue c V ) of the group. 3 The classification of flat connections with nontrivial twist for nonunitary groups was constructed in [9] (for symplectic and orthogonal groups it was pedagogically explained in the last section of recent [10]). Again, the number of quantum vacuum states always coincides with the dual Coxeter number independently of the boundary conditions chosen.…”

“…But in the twisted case the situation is different. To find (24), we substitute there U(x, y, z) in the form (5), (9). Then U −1 ∂ z U = 2πiT (x, y).…”