In this article we use 5-brane junctions to study the 5D T N SCFTs corresponding to the 5D N = 1 uplift of the 4D N = 2 strongly coupled gauge theories, which are obtained by compactifying N M5 branes on a sphere with three full punctures. Even though these theories have no Lagrangian description, by using the 5-brane junctions proposed by Benini, Benvenuti and Tachikawa, we are able to derive their Seiberg-Witten curves and Nekrasov partition functions. We cross-check our results with the 5D superconformal index proposed by Kim, Kim and Lee. Through the AGTW correspondence, we discuss the relations between 5D superconformal indices and n-point functions of the q-deformed W N Toda theories.

In this article we explore the duality between the low energy effective theory of five-dimensional N=1 SU(N)^{M-1} and SU(M)^{N-1} linear quiver gauge theories compactified on S^1. The theories we study are the five-dimensional uplifts of four-dimensional superconformal linear quivers. We study this duality by comparing the Seiberg-Witten curves and the Nekrasov partition functions of the two dual theories. The Seiberg-Witten curves are obtained by minimizing the worldvolume of an M5-brane with nontrivial geometry. Nekrasov partition functions are computed using topological string theory. The result of our study is a map between the gauge theory parameters, i.e., Coulomb moduli, masses and UV coupling constants, of the two dual theories. Apart from the obvious physical interest, this duality also leads to compelling mathematical identities. Through the AGTW conjecture these five-dimentional gauge theories are related to q-deformed Liouville and Toda SCFTs in two-dimensions. The duality we study implies the relations between Liouville and Toda correlation functions through the map we derive.Comment: 58 pages, 17 figures; v2: minor corrections, references adde

The hypermultiplet moduli space in Type IIA string theory compactified on a rigid Calabi-Yau threefold X , corresponding to the "universal hypermultiplet", is described at tree-level by the symmetric space SU (2, 1)/(SU (2) × U (1)). To determine the quantum corrections to this metric, we posit that a discrete subgroup of the continuous tree-level isometry group SU (2, 1), namely the Picard modular group SU (2, 1; Z[i]), must remain unbroken in the exact metric -including all perturbative and non-perturbative quantum corrections. This assumption is expected to be valid when X admits complex multiplication by Z[i]. Based on this hypothesis, we construct an SU (2, 1; Z[i])-invariant, non-holomorphic Eisenstein series, and tentatively propose that this Eisenstein series provides the exact contact potential on the twistor space over the universal hypermultiplet moduli space. We analyze its non-Abelian Fourier expansion, and show that the Abelian and non-Abelian Fourier coefficients take the required form for instanton corrections due to Euclidean D2branes wrapping special Lagrangian submanifolds, and to Euclidean NS5-branes wrapping the entire Calabi-Yau threefold, respectively. While this tentative proposal fails to reproduce the correct one-loop correction, the consistency of the Fourier expansion with physics expectations provides strong support for the usefulness of the Picard modular group in constraining the quantum moduli space. E SL(2,Z) 3/2 as a function of the "axio-dilaton" C (0) + ie −φ , valued on the fundamental domain M = SL(2, Z)\SL(2, R)/SO(2) of the Poincaré upper half plane. This proposal reproduced the known tree-level and one-loop corrections [6,7], predicted the absence of higher loop corrections, later verified by an explicit two-loop computation [8], and suggested the exact form of D(-1)-instanton contributions, later corroborated by explicit matrix model computations [9,10]. From the mathematical point of view, perturbative corrections and instanton contributions correspond, respectively, to the constant terms and Fourier coefficients of the automorphic form E SL(2,Z) 3/2 . This work was extended to toroidal compactifications of Mtheory, where the R 4 -type corrections were argued to be given by Eisenstein series of the respective U-duality group [11][12][13], predicting the contributions of Euclidean Dp-brane instantons, and, when n ≥ 6, NS5-branes. Unfortunately, extracting the constant terms and Fourier coefficients of Eisenstein series is not an easy task, and it has been difficult to put the conjecture to the test. Part of our motivation is to develop the understanding of Eisenstein series beyond the relatively well understood case of G(Z) = SL(n, Z). The Hypermultiplet Moduli Space of N = 2 SupergravityCompactifications with fewer unbroken supersymmetries (N ≤ 2 in D = 4) lead to moduli spaces which are generically not symmetric spaces. An interesting example is type IIA string theory compactified on a Calabi-Yau threefold X , leading to N = 2 supergravity in four dimensions coupled to ...

We discuss various aspects of dimensional reduction of gravity with the Einstein-Hilbert action supplemented by a lowest order deformation formed as the Riemann tensor raised to powers two, three or four. In the case of R 2 we give an explicit expression, and discuss the possibility of extended coset symmetries, especially SL(n + 1, Z) for reduction on an n-torus to three dimensions. Then we start an investigation of the dimensional reduction of R 3 and R 4 by calculating some terms relevant for the coset formulation, aiming in particular towards E 8(8) /(Spin(16)/Z 2 ) in three dimensions and an investigation of the derivative structure. We emphasise some issues concerning the need for the introduction of non-scalar automorphic forms in order to realise certain expected enhanced symmetries.

Type IIA string theory compactified on a rigid Calabi-Yau threefold gives rise to a classical moduli space that carries an isometric action of U (2, 1). Various quantum corrections break this continuous isometry to a discrete subgroup. Focussing on the case where the intermediate Jacobian of the Calabi-Yau admits complex multiplication by the ring of quadratic imaginary integers O d , we argue that the remaining quantum duality group is an arithmetic Picard modular group P U (2, 1; O d ). Based on this proposal we construct an Eisenstein series invariant under this duality group and study its non-Abelian Fourier expansion. This allows the prediction of non-perturbative effects, notably the contribution of D2-and NS5-brane instantons. The present work extends our previous analysis in 0909.4299 which was restricted to the special case of the Gaussian integers O1 = Z[i].