We show that equivariant Donaldson polynomials of compact toric surfaces can be calculated as residues of suitable combinations of Virasoro conformal blocks, by building on AGT correspondence between $\mathcal{N}=2$\ supersymmetric gauge theories and two-dimensional conformal field theory. Talk presented by A.T. at the conference Interactions between Geometry and Physics {\textemdash} in honor of Ugo Bruzzo{\textquoteright}s 60th birthday 17{\textendash}22 August 2015, Guaruj{\'a}, S{\~a}o Paulo, Brazil, mostly based on Bawane et al. (0000) and Bershtein et al. (0000).

}, keywords = {AGT, Donaldson invariants, Equivariant localization, Exact partition function, Supersymmetry, Virasoro conformal blocks}, issn = {0393-0440}, doi = {https://doi.org/10.1016/j.geomphys.2017.01.012}, url = {http://www.sciencedirect.com/science/article/pii/S0393044017300165}, author = {Mikhail Bershtein and Giulio Bonelli and Massimiliano Ronzani and Alessandro Tanzini} } @article {Bershtein2016, title = {Exact results for N=2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants}, journal = {Journal of High Energy Physics}, volume = {2016}, number = {7}, year = {2016}, month = {Jul}, pages = {23}, abstract = {We provide a contour integral formula for the exact partition function of $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for $U(2)\; \mathcal{N}=2^\star$ theory on $\mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the $\mathcal{N}=4$ supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a longstanding conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.

}, issn = {1029-8479}, doi = {10.1007/JHEP07(2016)023}, url = {https://doi.org/10.1007/JHEP07(2016)023}, author = {Mikhail Bershtein and Giulio Bonelli and Massimiliano Ronzani and Alessandro Tanzini} } @mastersthesis {2016, title = {Instanton counting on compact manifolds}, year = {2016}, school = {SISSA}, abstract = {In this thesis we analyze supersymmetric gauge theories on compact manifolds and their relation with representation theory of infinite Lie algebras associated to conformal field theories, and with the computation of geometric invariants and superconformal indices. The thesis contains the work done by the candidate during the doctorate programme at SISSA under the supervision of A. Tanzini and G. Bonelli. {\textbullet} in Chapter 2, we consider N = 2 supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S2 {\texttimes} S2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity. {\textbullet} in Chapter 3, we provide a contour integral formula for the exact partition function of N = 2 supersymmetric U(N) gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for U(2) N = 2* theory on P2 for all instanton numbers. In the zero mass case, corresponding to the N = 4 supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a long-standing conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new. {\textbullet} in Chapter 4, we explore N = (1, 0) superconformal six-dimensional theories arising from M5 branes probing a transverse Ak singularity. Upon circle compactification to five dimensions, we describe this system with a dual pq-web of five-branes and propose the spectrum of basic five-dimensional in- stanton 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 which relations among vertex correlators of qW algebrae are implied by the S-duality of the pq-web.}, keywords = {Supersymmetry}, url = {http://urania.sissa.it/xmlui/handle/1963/35219}, author = {Massimiliano Ronzani} } @article {Benvenuti2016, title = {Symmetry enhancements via 5d instantons, qW-algebrae and (1,0) superconformal index}, journal = {Journal of High Energy Physics}, volume = {2016}, number = {9}, year = {2016}, month = {Sep}, pages = {53}, abstract = {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 {Bawane2015, title = {N=2 supersymmetric gauge theories on S^2xS^2 and Liouville Gravity}, journal = {Journal of High Energy Physics}, volume = {2015}, number = {7}, year = {2015}, month = {Jul}, pages = {54}, abstract = {We consider $\mathcal{N}=2$ supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a $U(1)$ isometry. This is used to explicitly compute the supersymmetric path integral on $S^2 \times S^2$ via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.

}, issn = {1029-8479}, doi = {10.1007/JHEP07(2015)054}, url = {https://doi.org/10.1007/JHEP07(2015)054}, author = {Aditya Bawane and Giulio Bonelli and Massimiliano Ronzani and Alessandro Tanzini} }