Modular anomaly equation, heat kernel and S-duality in $ \mathcal{N}=2 $ theories

Abstract

We investigate ǫ-deformed N = 2 superconformal gauge theories in four dimensions, focusing on the N = 2 * and N f = 4 SU(2) cases. We show how the modular anomaly equation obeyed by the deformed prepotential can be efficiently used to derive its non-perturbative expression starting from the perturbative one. We also show that the modular anomaly equation implies that S-duality is implemented by means of an exact Fourier transform even for arbitrary values of the deformation parameters, and then we argue that it is possible, perturbatively in the deformation, to choose appropriate variables such that it reduces to a Legendre transform.

Modular anomaly equations in N $$ \mathcal{N} $$ =2* theories and their large-N limit

“…and building on previous results obtained both in the U(N ) undeformed N = 2 * theory [4] and in the SU(2) deformed one [17][18][19][20][21], we expect that the f n 's (with n > 1) are almost…”

“…(2.26) is a generalization to U(N ) of the recursion relation found in [19,21] for the -deformed SU(2) theory, and a generalization in presence of the Ω-background of the relation found in [4] for the undeformed U(N ) theory. 5 As discussed in those references, the recursion relation obeyed by the prepotential coefficients can be regarded as a direct consequence of the modular anomaly equation which, in turn, encodes the same information as the holomorphic anomaly equation of the topological string amplitudes [1][2][3] and its generalizations [17][18][19][20][21]. Indeed, the holomorphic anomaly equation implies an anomalous modular behavior of the prepotential coefficients with respect to τ which can only occur through the Eisenstein series E 2 .…”

“…With the introduction of the so-called Ω-background [5][6][7] and its interpretation in terms of Neveu-Schwarz/Neveu-Schwarz or Ramond/Ramond field strengths of closed string theory [8][9][10][11], the connection between the gauge theory prepotential and the topological string amplitudes has become more clear, and recently there has been a renewed interest both in computing such amplitudes [12][13][14][15][16] and in adapting the previous works to this new scenario [17][18][19][20][21]. In particular, in [19,21] the N = 2 * theory with gauge group SU (2) has been studied in a generic Ω-background characterized by two independent parameters, 1 and 2 , and its prepotential, including all its non-perturbative instanton corrections, has been computed in a small mass expansion.…”

“…In particular, in [19,21] the N = 2 * theory with gauge group SU (2) has been studied in a generic Ω-background characterized by two independent parameters, 1 and 2 , and its prepotential, including all its non-perturbative instanton corrections, has been computed in a small mass expansion. Moreover, it has been shown that the resulting quantum prepotential obeys a modular anomaly equation with a term proportional to 1 2 , which generalizes the previously known equation of the underformed theory [4].…”

Modular properties of 6d (DELL) systems

“…These tori could become equivalent after a series of T -dualities and appropriate changes of the spacetime. We expect that the 6d modular anomaly equations can be lifted to the level of Nekrasov functions, as it was done for the 4d case in [77][78][79][80] and to the level of 2d conformal field theories in [81,82]. Note that, in the recent paper by S. Kim and J. Nahmgoong, [75], the S-duality in 6d (2, 0) theory was studied.…”