Transformations of Spherical Blocks

Journal of High Energy Physics

We consider the Ω-deformed N = 2 SU(2) gauge theory in four dimensions with N f = 4 massive fundamental hypermultiplets. The low energy effective action depends on the deformation parameters ε 1 , ε 2 , the scalar field expectation value a, and the hypermultiplet masses m = (m 1 , m 2 , m 3 , m 4). Motivated by recent findings in the N = 2 * theory, we explore the theories that are characterized by special fixed ratios ε 2 /ε 1 and m/ε 1 and propose a simple condition on the structure of the multi-instanton contributions to the prepotential determining the effective action. This condition determines a finite set Π N of special points such that the prepotential has N poles at fixed positions independent on the instanton number. In analogy with what happens in the N = 2 * gauge theory, the full prepotential of the Π N theories may be given in closed form as an explicit function of a and the modular parameter q appearing in special combinations of Eisenstein series and Jacobi theta functions with well defined modular properties. The resulting finite pole partition functions are related by AGT correspondence to special 4-point spherical conformal blocks of the Virasoro algebra. We examine in full details special cases where the closed expression of the block is known and confirms our Ansatz. We systematically study the special features of Zamolodchikov's recursion for the Π N conformal blocks. As a result, we provide a novel effective recursion relation that can be exactly solved and allows to prove the conjectured closed expressions analytically in the case of the Π 1 and Π 2 conformal blocks.

2017

In this thesis, we study gauge theories with N = 2 supersymmetry in four dimensions. The low energy effective action of these theories on their Coulomb branch is described by a holomorphic function called the prepotential. In the first half, we study linear conformal quiver theories with gauge group SU(2). These theories have an SU(2) gauge group at each node of the quiver, and matter arranged in the fundamental and the bi-fundamental representations, such that at each node the β-function vanishes. To compute the prepotential for these theories, we follow three different approaches. These are (i) the classic Seiberg-Witten approach, in which we consider an M-theory construction of the Seiberg-Witten curve and the associated differential, (ii) equivariant localization as developed by Nekrasov, and (iii) the 2d/4d correspondence of the four dimensional gauge theory with the two dimensional Liouville conformal field theory, as put forward by Alday, Gaiotto, and Tachikawa. Matching the ...

Journal of High Energy Physics, 2000

The results obtained by Seiberg and Witten for the low-energy Wilsonian effective actions of N = 2 supersymmetric theories with gauge group SU(2) are in agreement with instanton computations carried out for winding numbers one and two. This suggests that the instanton saddle point saturates the non-perturbative contribution to the functional integral. A natural framework in which corrections to this approximation are absent is given by the topological field theory built out of the N = 2 Super Yang-Mills theory. After extending the standard construction of the Topological Yang-Mills theory to encompass the case of a non-vanishing vacuum expectation value for the scalar field, a BRST transformation is defined (as a supersymmetry plus a gauge variation), which on the instanton moduli space is the exterior derivative. The topological field theory approach makes the so-called "constrained instanton" configurations and the instanton measure arise in a natural way. As a consequence, instanton-dominated Green's functions in N = 2 Super Yang-Mills can be equivalently computed either using the constrained instanton method or making reference to the topological twisted version of the theory. We explicitly compute the instanton measure and the contribution to u = Trφ 2 for winding numbers one and two. We then show that each non-perturbative contribution to the N = 2 low-energy effective action can be written as the integral of a total derivative of a function of the instanton moduli. Only instanton configurations of zero conformal size contribute to this result. Finally, the 8k-dimensional instanton moduli space is built using the hyperkähler quotient procedure, which clarifies the geometrical meaning of our approach.

Nuclear Physics B, 1998

Extending recent work of Kachru and Silverstein, we consider "orbifolds" of 4-dimensional N = 4 SU (n) super-Yang-Mills theories with respect to discrete subgroups of the SU (4) R-symmetry which act nontrivially on the gauge group. We show that for every discrete subgroup of SU (4) there is a canonical choice of imbedding of the discrete group in the gauge group which leads to theories with a vanishing one-loop beta-function. We conjecture that these give rise to (generically non-supersymmetric) conformal theories. The gauge group is ⊗ i SU (Nn i ) where n i denote the dimension of the irreducible representations of the corresponding discrete group; there is also bifundamental matter, dictated by associated quiver diagrams. The interactions can also be read off from the quiver diagram. For SU (3) and SU (2) subgroups this leads to superconformal theories with N = 1 and N = 2 respectively.

We apply exact WKB methods to the study of the partition function of pure N=2 epsilon_i-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in epsilon_2/epsilon_1 (i.e. at large central charge) and in an expansion in epsilon_1. We find corrections of the form ~ exp[- SW periods / epsilon_1] to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of epsilon_1 when passing from the Nekrasov form of the partition function to the topological string theory inspired form. We conjecture that such corrections should be computable from a worldsheet perspective on the partition function. Our results follow upon the determination of the Stokes graphs associated to the Mathieu equation with complex parameters and the application of exact WKB techniques to compute the Mathieu characteristic exponent.

2013

We study the non-perturbative properties of N=2 super conformal field theories in four dimensions using localization techniques. In particular we consider SU(2) gauge theories, deformed by a generic epsilon-background, with four fundamental flavors or with one adjoint hypermultiplet. In both cases we explicitly compute the first few instanton corrections to the partition function and the prepotential using Nekrasov's approach. These results allow to reconstruct exact expressions involving quasi-modular functions of the bare gauge coupling constant and to show that the prepotential terms satisfy a modular anomaly equation that takes the form of a recursion relation with an explicitly epsilon-dependent term. We then investigate the implications of this recursion relation on the modular properties of the effective theory and find that with a suitable redefinition of the prepotential and of the effective coupling it is possible, at least up to the third order in the deformation para...

International Journal of Modern Physics A, 2005

We derive the master function governing the component action of the four-dimensional non-anticommutative (NAC) and fully N=2 supersymmetric gauge field theory with a non-simple gauge group U(2) = SU(2) × U(1). We use a Lorentz-singlet NACdeformation parameter and an N=2 supersymmetric star (Moyal) product, which do not break any of the fundamental symmetries of the undeformed N=2 gauge theory. The scalar potential in the NAC-deformed theory is calculated. We also propose the non-abelian BPS-type equations in the case of the NAC-deformed N=2 gauge theory with the SU(2) gauge group, and comment on the SU(3) case too. The NACdeformed field theories can be thought of as the effective (non-perturbative) N=2 gauge field theories in a certain (scalar only) N=2 supergravity background.

Nuclear Physics B, 2013

We present the gravity dual to a class of three-dimensional N = 2 supersymmetric gauge theories on a U (1) × U (1)-invariant squashed three-sphere, with a non-trivial background gauge field. This is described by a supersymmetric solution of four-dimensional N = 2 gauged supergravity with a non-trivial instanton for the graviphoton field. The particular gauge theory in turn determines the lift to a solution of eleven-dimensional supergravity. We compute the partition function for a class of Chern-Simons quiver gauge theories on both sides of the duality, in the large N limit, finding precise agreement for the functional dependence on the squashing parameter. This constitutes an exact check of the gauge/gravity correspondence in a non-conformally invariant setting. theories may be computed by putting the theories on S 1 × S 3 [5, 6, 7] and S 1 × S 2 , respectively.

Journal of High Energy Physics

We calculate the instanton partition function of the four-dimensional N = 2 ⋆ SU(N) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from S-duality, we show that the effective twisted superpotential, which governs the infrared dynamics of the two-dimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. We also show that our results, derived for monodromy defects in the four-dimensional theory, match the effective twisted superpotential describing the infrared properties of certain two-dimensional sigma models coupled either to pure N = 2 or to N = 2 ⋆ gauge theories.

Journal of High Energy Physics

Recursion relations for the sphere 4-point and torus 1-point W 3 conformal blocks, generalizing Alexei Zamolodchikov's famous relation for the Virasoro conformal blocks are proposed. One of these relations is valid for any 4-point conformal block with two arbitrary and two special primaries with charge parameters proportional to the highest weight of the fundamental irrep of SU(3). The other relation is designed for the torus conformal block with a special (in above mentioned sense) primary field insertion. AGT relation maps the sphere conformal block and the torus block to the instanton partition functions of the N = 2 SU(3) SYM theory with 6 fundamental or an adjoint hypermultiplets respectively. AGT duality played a central role in establishing these recurrence relations, whose gauge theory counterparts are novel relations for the SU(3) partition functions with N f = 6 fundamental or an adjoint hypermultiplets. By decoupling some (or all) hypermultiplets, recurrence relations for the asymptotically free theories with 0 ≤ N f < 6 are found.