The M5-brane on K3 and del Pezzo's and multi-loop string amplitudes

Abstract

We study the BPS spectrum of Little String Theory for bound states of M5branes wrapped on six manifold of product topology M 4 ×Σ 2 and the apparence of multi-loop θ-functions in a supersymmetric index calculation. We find a total reconstruction of the gloop heterotic contribution in the case of a double K3 M-theory compactification. Moreover, we consider total wrapping of M5-branes on del Pezzo surfaces B k and, by studying the relevant amplitude, we notice the arising of θ-functions relative to BPS strings on T k−1 , i.e. membranes on T k . This happens because of beautiful relations between four dimensional SYM theories and CFTs in two dimensions and seems to be linked to a duality recently observed by A.Iqbal, A.Neitzke and C.Vafa in [1]. 1

Fiber-base duality and global symmetry enhancement

“…Since we have shown that the fiber-base duality, which translates to S-duality in the brane setup picture, is part of the enhanced E N f +1 symmetry, it would be also interesting to consider whether there is any relation between this observation and [71] in which the duality between del-Pezzo surfaces and U-dualities was discussed. Our result may also be related to the theta function arising in the BPS spectrum of M5-branes wrapped on del-Pezzo surfaces [72].…”

$ \mathcal{N}=2 $ gauge theories on toric singularities, blow-up formulae and W-algebrae

“…This holomorphically factorizes in chiral times anti-chiral blocks. The M5-brane partition function is extracted as one of such chiral blocks, in the very same way as chiral partition functions are obtained in two-dimensions[22,23,24,25,26]. The explicit form of this partition function is given in terms of a theta function Θ(Z) with suitable characteristics and arguments.…”

W(E10) symmetry, M-theory and Painlevé equations

“…The metric ansatz (30) is close to that for the D7-brane solutions [26], but the crucial difference is the appearance of ρ 2 in G (10) φφ . Owing to this explicit coordinate dependence ('the Weyl canonical coordinate'), τ cannot be holomorphic, and the solution does not preserve supersymmetry.…”

“…In this correspondence, the Weyl group part of the U-duality group is identified with the Cremona isometry W (E k ) for del Pezzo B k [9]. (This duality was discussed from the point of view of Little String Theory supersymmetric indices [10].) The aim of this paper is to examine the correspondence by closely looking the way how the Weyl group is realized in each case.…”