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 — in honor of Ugo Bruzzo’s 60th birthday 17–22 August 2015, Guarujá, São Paulo, Brazil, mostly based on Bawane et al. (0000) and Bershtein et al. (0000).

VL - 118 UR - http://www.sciencedirect.com/science/article/pii/S0393044017300165 N1 - Interactions between Geometry and Physics. A Special Issue in Honor of Ugo Bruzzo’s 60th Birthday ER - TY - JOUR T1 - Exact results for N=2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants JF - Journal of High Energy Physics Y1 - 2016 A1 - Mikhail Bershtein A1 - Giulio Bonelli A1 - Massimiliano Ronzani A1 - Alessandro Tanzini AB -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.

VL - 2016 UR - https://doi.org/10.1007/JHEP07(2016)023 ER - TY - JOUR T1 - Symmetry enhancements via 5d instantons, qW-algebrae and (1,0) superconformal index JF - Journal of High Energy Physics Y1 - 2016 A1 - Benvenuti, Sergio A1 - Giulio Bonelli A1 - Massimiliano Ronzani A1 - Alessandro Tanzini AB -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.

VL - 2016 UR - https://doi.org/10.1007/JHEP09(2016)053 ER - TY - JOUR T1 - N=2 supersymmetric gauge theories on S^2xS^2 and Liouville Gravity JF - Journal of High Energy Physics Y1 - 2015 A1 - Aditya Bawane A1 - Giulio Bonelli A1 - Massimiliano Ronzani A1 - Alessandro Tanzini AB -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.

VL - 2015 UR - https://doi.org/10.1007/JHEP07(2015)054 ER - TY - JOUR T1 - Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics Y1 - 2014 A1 - Giulio Bonelli A1 - Antonio Sciarappa A1 - Alessandro Tanzini A1 - Petr Vasko AB - 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. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34546 U1 - 34771 U2 - Physics U4 - 2 ER - TY - JOUR T1 - The stringy instanton partition function Y1 - 2014 A1 - Giulio Bonelli A1 - Antonio Sciarappa A1 - Alessandro Tanzini A1 - Petr Vasko AB - 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. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34589 U1 - 34796 U2 - Physics U4 - 2 ER - TY - JOUR T1 - Vortex Partition Functions, Wall Crossing and Equivariant Gromov–Witten Invariants Y1 - 2014 A1 - Giulio Bonelli A1 - Antonio Sciarappa A1 - Alessandro Tanzini A1 - Petr Vasko AB - In this paper we identify the problem of equivariant vortex counting in a (2,2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov–Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the I and J-functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov–Witten theory follow just by deforming the integration contour. In particular, we apply our formalism to compute Gromov–Witten invariants of the C3/Zn orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on C2, and of An and Dn singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34652 U1 - 34859 U2 - Physics ER - TY - RPRT T1 - N=2 gauge theories on toric singularities, blow-up formulae and W-algebrae Y1 - 2013 A1 - Giulio Bonelli A1 - Kazunobu Maruyoshi A1 - Alessandro Tanzini A1 - Futoshi Yagi AB - We compute the Nekrasov partition function of gauge theories on the\r\n(resolved) toric singularities C^2/\\Gamma in terms of blow-up formulae. We\r\ndiscuss the expansion of the partition function in the \\epsilon_1,\\epsilon_2\r\n\\to 0 limit along with its modular properties and how to derive them from the\r\nM-theory perspective. On the two-dimensional conformal field theory side, our\r\nresults can be interpreted in terms of representations of the direct sum of\r\nHeisenberg plus W_N-algebrae with suitable central charges, which can be\r\ncomputed from the fan of the resolved toric variety.We provide a check of this\r\ncorrespondence by computing the central charge of the two-dimensional theory\r\nfrom the anomaly polynomial of M5-brane theory. Upon using the AGT\r\ncorrespondence our results provide a candidate for the conformal blocks and\r\nthree-point functions of a class of the two-dimensional CFTs which includes\r\nparafermionic theories. PB - SISSA UR - http://hdl.handle.net/1963/6577 N1 - 33 pages, 1 figure; v2: discussions on U(1) gauge theory and\r\n Frenkel-Kac construction have been added in section 5.1, and typos corrected;\r\n v3: published version; v4: typos corrected U1 - 6522 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - Gauge Theories on ALE Space and Super Liouville Correlation Functions JF - Lett. Math. Phys. 101 (2012) 103-124 Y1 - 2012 A1 - Giulio Bonelli A1 - Kazunobu Maruyoshi A1 - Alessandro Tanzini AB - We present a relation between N=2 quiver gauge theories on the ALE space O_{P^1}(-2) and correlators of N=1 super Liouville conformal field theory, providing checks in the case of punctured spheres and tori. We derive a blow-up formula for the full Nekrasov partition function and show that, up to a U(1) factor, the N=2^* instanton partition function is given by the product of the character of \\\\hat{SU}(2)_2 times the super Virasoro conformal block on the torus with one puncture. PB - SISSA UR - http://hdl.handle.net/1963/4305 N1 - 21 pages U1 - 4068 U2 - Physics U3 - Elementary Particle Theory U4 - -1 ER - TY - JOUR T1 - Vertices, vortices & interacting surface operators JF - JHEP 06(2012)178 Y1 - 2012 A1 - Giulio Bonelli A1 - Alessandro Tanzini A1 - Zhao Jian AB - We show that the vortex moduli space in non-abelian supersymmetric N=(2,2) gauge theories on the two dimensional plane with adjoint and anti-fundamental matter can be described as an holomorphic submanifold of the instanton moduli space in four dimensions. The vortex partition functions for these theories are computed via equivariant localization. We show that these coincide with the field theory limit of the topological vertex on the strip with boundary conditions corresponding to column diagrams. Moreover, we resum the field theory limit of the vertex partition functions in terms of generalized hypergeometric functions formulating their AGT dual description as interacting surface operators of simple type. Analogously we resum the topological open string amplitudes in terms of q-deformed generalized hypergeometric functions proving that they satisfy appropriate finite difference equations. PB - SISSA UR - http://hdl.handle.net/1963/4134 N1 - 22 pages, 4 figures U1 - 3874 U2 - Physics U3 - Elementary Particle Theory U4 - -1 ER - TY - JOUR T1 - Wild quiver gauge theories JF - JHEP 02(2012)031 Y1 - 2012 A1 - Giulio Bonelli A1 - Kazunobu Maruyoshi A1 - Alessandro Tanzini AB -We study $N=2$ supersymmetric $SU(2)$ gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional $A_1 (2,0)$ theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the Omega-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of higher order singularities.

PB - SISSA UR - http://hdl.handle.net/1963/5184 N1 - 34 pages U1 - 4999 U2 - Physics U3 - Elementary Particle Theory U4 - -1 ER - TY - JOUR T1 - Generalized matrix models and AGT correspondence at all genera JF - JHEP, Volume 2011, Issue 7, 2011, Article number055 Y1 - 2011 A1 - Giulio Bonelli A1 - Kazunobu Maruyoshi A1 - Alessandro Tanzini A1 - Futoshi Yagib AB - We study generalized matrix models corresponding to n-point Virasoro\r\nconformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT\r\ncorrespondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge\r\ntheories with generalized quiver diagrams. We obtain the generalized matrix\r\nmodels from the perturbative evaluation of the Liouville correlation functions\r\nand verify the consistency of the description with respect to degenerations of\r\nthe Riemann surface. Moreover, we derive the Seiberg-Witten curve for the N=2\r\ngauge theory as the spectral curve of the generalized matrix model, thus\r\nproviding a check of AGT correspondence at all genera. PB - SISSA UR - http://hdl.handle.net/1963/6568 N1 - This version is published in : Journal of High Energy Physics, Volume 2011, Issue 7, 2011, Article number055 U1 - 6530 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Instantons on ALE spaces and Super Liouville Conformal Field Theories Y1 - 2011 A1 - Giulio Bonelli A1 - Kazunobu Maruyoshi A1 - Alessandro Tanzini AB - We provide evidence that the conformal blocks of N=1 super Liouville\\r\\nconformal field theory are described in terms of the SU(2) Nekrasov partition\\r\\nfunction on the ALE space O_{P^1}(-2). PB - SISSA UR - http://hdl.handle.net/1963/4262 N1 - 10 pages U1 - 3987 U2 - Physics U3 - Elementary Particle Theory U4 - -1 ER - TY - JOUR T1 - The Liouville side of the vortex JF - JHEP 09(2011)096 Y1 - 2011 A1 - Giulio Bonelli A1 - Alessandro Tanzini A1 - Jian Zhao AB - We analyze conformal blocks with multiple (semi-)degenerate field insertions in Liouville/Toda conformal field theories an show that their vector space is fully reproduced by the four-dimensional limit of open topological string amplitudes on the strip with generic boundary conditions associated to a suitable quiver gauge theory. As a byproduct we identify the non-abelian vortex partition function with a specific fusion channel of degenerate conformal blocks. PB - SISSA UR - http://hdl.handle.net/1963/4304 N1 - 25pages,11figures U1 - 4019 U2 - Physics U3 - Elementary Particle Theory U4 - -1 ER - TY - RPRT T1 - Quantum Hitchin Systems via beta-deformed Matrix Models Y1 - 2011 A1 - Giulio Bonelli A1 - Kazunobu Maruyoshi A1 - Alessandro Tanzini AB -We study the quantization of Hitchin systems in terms of beta-deformations of generalized matrix models related to conformal blocks of Liouville theory on punctured Riemann surfaces. We show that in a suitable limit, corresponding to the Nekrasov-Shatashvili one, the loop equations of the matrix model reproduce the Hamiltonians of the quantum Hitchin system on the sphere and the torus with marked points. The eigenvalues of these Hamiltonians are shown to be the epsilon1-deformation of the chiral observables of the corresponding N=2 four ndimensional gauge theory. Moreover, we find the exact wave-functions in terms of the matrix model representation of the conformal blocks with degenerate field insertions.

PB - SISSA UR - http://hdl.handle.net/1963/4181 N1 - 29 pages; v2. refs. added and typos corrected U1 - 3904 U2 - Physics U3 - Elementary Particle Theory U4 - -1 ER - TY - JOUR T1 - Hitchin systems, N=2 gauge theories and W-gravity JF - Phys. Lett. B 691 (2010) 111-115 Y1 - 2010 A1 - Giulio Bonelli A1 - Alessandro Tanzini AB - We propose some arguments supporting an M-theory derivation of the duality recently discovered by Alday, Gaiotto and Tachikawa between two-dimensional conformal field theories and N=2 superconformal gauge theories in four dimensions. We find that A_{N-1} Toda field theory is the simplest two-dimensional conformal field theory quantizing the moduli of N M5-branes wrapped on a Riemann surface. This leads us to identify chiral operators of the N=2 gauge theories with W-algebra currents. As a check of this correspondence we study some relevant OPE\\\'s obtaining that Nekrasov\\\'s partition function satisfies W-geometry constraints. UR - http://hdl.handle.net/1963/3831 U1 - 496 U2 - Physics U3 - Elementary Particle Theory ER - TY - JOUR T1 - Taming open/closed string duality with a Losev trick JF - JHEP 06(2010)063 Y1 - 2010 A1 - Giulio Bonelli A1 - Andrea Prudenziati A1 - Alessandro Tanzini AB - A target space string field theory formulation for open and closed B-model is provided by giving a Batalin-Vilkovisky quantization of the holomorphic Chern-Simons theory with off-shell gravity background. The target space expression for the coefficients of the holomorphic anomaly equation for open strings are obtained. Furthermore, open/closed string duality is proved from a judicious integration over the open string fields. In particular, by restriction to the case of independence on continuous open moduli, the shift formulas of [7] are reproduced and shown therefore to encode the data of a closed string dual. UR - http://hdl.handle.net/1963/3855 U1 - 854 U2 - Physics U3 - Mathematical Physics ER - TY - JOUR T1 - Decoupling A and B model in open string theory: topological adventures in the world of tadpoles JF - JHEP 06 (2009) 046 Y1 - 2009 A1 - Giulio Bonelli A1 - Andrea Prudenziati A1 - Alessandro Tanzini A1 - Yang Jie AB - In this paper we analyze the problem of tadpole cancellation in open topological strings. We prove that the inclusion of unorientable worldsheet diagrams guarantees a consistent decoupling of A and B model for open superstring amplitudes at all genera. This is proven by direct microscopic computation in Super Conformal Field Theory. For the B-model we explicitly calculate one loop amplitudes in terms of analytic Ray-Singer torsions of appropriate vector bundles and obtain that the decoupling corresponds to the cancellation of D-brane and orientifold charges. Local tadpole cancellation on the worldsheet then guarantees the decoupling at all loops. The holomorphic anomaly equations for open topological strings at one loop are also obtained and compared with the results of the Quillen formula. UR - http://hdl.handle.net/1963/3632 U1 - 672 U2 - Physics U3 - Mathematical Physics ER - TY - JOUR T1 - Topological branes, p-algebras and generalized Nahm equations JF - Phys. Lett. B 672 (2009) 390-395 Y1 - 2009 A1 - Giulio Bonelli A1 - Alessandro Tanzini A1 - Maxim Zabzine AB - Inspired by the recent advances in multiple M2-brane theory, we consider the generalizations of Nahm equations for arbitrary p-algebras. We construct the topological p-algebra quantum mechanics associated to them and we show that this can be obtained as a truncation of the topological p-brane theory previously studied by the authors. The resulting topological p-algebra quantum mechanics is discussed in detail and the relation with the M2-M5 system is pointed out in the p=3 case, providing a geometrical argument for the emergence of the 3-algebra structure in the Bagger-Lambert-Gustavsson theory UR - http://hdl.handle.net/1963/2702 U1 - 1398 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Topological Gauge Theories on Local Spaces and Black Hole Entropy Countings JF - Adv. Theor. Math. Phys. 12 (2008) 1429-1446 Y1 - 2008 A1 - Giulio Bonelli A1 - Alessandro Tanzini AB - We study cohomological gauge theories on total spaces of holomorphic line bundles over complex manifolds and obtain their reduction to the base manifold by U(1) equivariant localization of the path integral. We exemplify this general mechanism by proving via exact path integral localization a reduction for local curves conjectured in hep-th/0411280, relevant to the calculation of black hole entropy/Gromov-Witten invariants. Agreement with the four-dimensional gauge theory is recovered by taking into account in the latter non-trivial contributions coming from one-loop fluctuations determinants at the boundary of the total space. We also study a class of abelian gauge theories on Calabi-Yau local surfaces, describing the quantum foam for the A-model, relevant to the calculation of Donaldson-Thomas invariants. UR - http://hdl.handle.net/1963/1992 U1 - 2204 U2 - Physics U3 - Elementary Particle Theory ER - TY - RPRT T1 - Computing Amplitudes in topological M-theory Y1 - 2007 A1 - Giulio Bonelli A1 - Alessandro Tanzini A1 - Maxim Zabzine AB - We define a topological quantum membrane theory on a seven dimensional manifold of $G_2$ holonomy. We describe in detail the path integral evaluation for membrane geometries given by circle bundles over Riemann surfaces. We show that when the target space is $CY_3\\\\times S^1$ quantum amplitudes of non-local observables of membranes wrapping the circle reduce to the A-model amplitudes. \\nIn particular for genus zero we show that our model computes the Gopakumar-Vafa invariants. Moreover, for membranes wrapping calibrated homology spheres in the $CY_3$, we find that the amplitudes of our model are related to Joyce invariants. JF - JHEP 03 (2007) 023 UR - http://hdl.handle.net/1963/1901 U1 - 2335 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - The holomorphic anomaly for open string moduli JF - JHEP 10 (2007) 060 Y1 - 2007 A1 - Giulio Bonelli A1 - Alessandro Tanzini AB - We complete the holomorphic anomaly equations for topological strings with their dependence on open moduli. We obtain the complete system by standard path integral arguments generalizing the analysis of BCOV (Commun. Math. Phys. 165 (1994) 311) to strings with boundaries. We study both the anti-holomorphic dependence on open moduli and on closed moduli in presence of Wilson lines. By providing the compactification a\\\' la Deligne-Mumford of the moduli space of Riemann surfaces with boundaries, we show that the open holomorphic anomaly equations are structured on the (real codimension one) boundary components of this space. UR - http://hdl.handle.net/1963/2113 U1 - 2576 U2 - Physics U3 - Elementary Particle Theory ER - TY - RPRT T1 - On topological M-theory Y1 - 2006 A1 - Giulio Bonelli A1 - Alessandro Tanzini A1 - Maxim Zabzine AB - We construct a gauge fixed action for topological membranes on G2-manifolds such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds. JF - Adv. Theor. Math. Phys. 10 (2006) 239-260 UR - http://hdl.handle.net/1963/1765 U1 - 2779 U2 - Mathematics U3 - Mathematical Physics ER -