We explore $\mathcal{N}=(1,0)$ superconformal six-dimensional theories arising from M5 branes probing a transverse $A_k$ singularity. Upon circle compactification to 5 dimensions, we describe this system with a dual pq-web of five-branes and propose the spectrum of basic five-dimensional instanton operators driving global symmetry enhancement. For a single M5 brane, we find that the exact partition function of the 5d quiver gauge theory matches the 6d (1, 0) index, which we compute by letter counting. We finally show that S-duality of the pq-web implies new relations among vertex correlators of $q\mathcal{W}$ algebrae.

}, issn = {1029-8479}, doi = {10.1007/JHEP09(2016)053}, url = {https://doi.org/10.1007/JHEP09(2016)053}, author = {Benvenuti, Sergio and Giulio Bonelli and Massimiliano Ronzani and Alessandro Tanzini} } @article {2014, title = {Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics}, number = {Journal of high energy physics;volume 2014; issue 7; article number 141;}, year = {2014}, publisher = {Springer}, abstract = {We show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on C^2 x S^2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S^2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W_N algebrae, thus providing a gauge theoretical proof of AGT correspondence.}, doi = {10.1007/JHEP07(2014)141}, url = {http://urania.sissa.it/xmlui/handle/1963/34546}, author = {Giulio Bonelli and Antonio Sciarappa and Alessandro Tanzini and Petr Vasko} } @article {2014, title = {The stringy instanton partition function}, number = {Journal of high energy physics;volume 2014; issue 1; article number 038;}, year = {2014}, publisher = {Springer}, abstract = {We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A_1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C^2/Z_2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants of P^1 x C^2.}, doi = {10.1007/JHEP01(2014)038}, url = {http://urania.sissa.it/xmlui/handle/1963/34589}, author = {Giulio Bonelli and Antonio Sciarappa and Alessandro Tanzini and Petr Vasko} }