4d Theories, Fivebranes, and M-Theory

AbstractIn this section, we study line operators in the 4d $$\mathcal {N}=2^*$$ N = 2 ∗ theories. A 4d $$\mathcal {N}=2$$ N = 2 theory of class $$\mathcal {S}$$ S arises from a compactification of 6d $$\mathcal {N}=(2, 0)$$ N = ( 2 , 0 ) theory on a once-punctured torus C. The spectrum of line operators in the theory depends on additional discrete data, a maximal isotropic lattice $$\textsf{L}\subset H^1(C,Z(G))$$ L ⊂ H 1 ( C , Z ( G ) ) where line operators must be invariant under the discrete group $$\textsf{L}$$ L . Therefore, we will show that a non-commutative algebra of line operators of a 4d $$\mathcal {N}=2^*$$ N = 2 ∗ theory on $$S^1\times \mathbb {R}\times _q \mathbb {R}^2$$ S 1 × R × q R 2 with the $$\Omega $$ Ω -background is the $$\textsf{L}$$ L -invariant subalgebra of spherical DAHA.