Elliptic blowup equations for 6d SCFTs. Part III. E-strings, M-strings and chains

Abstract

We establish the elliptic blowup equations for E-strings and M-strings and solve elliptic genera and refined BPS invariants from them. Such elliptic blowup equations can be derived from a path integral interpretation. We provide toric hypersurface construction for the Calabi-Yau geometries of M-strings and those of E-strings with up to three mass parameters turned on, as well as an approach to derive the perturbative prepotential directly from the local description of the Calabi-Yau threefolds. We also demonstrate how to systematically obtain blowup equations for all rank one 5d SCFTs from E-string by blow-down operations. Finally, we present blowup equations for E-M and M string chains.

Bootstrapping ADE M-strings

“…This prevents us from using many powerful techniques such as mirror symmetry. The other is that how to mass deform the geometry away from the JHEP02(2021)057 trivial fibration is not known, except for the A 1 case [38]. For some relevant discussions, see [38,39].…”

“…The other is that how to mass deform the geometry away from the JHEP02(2021)057 trivial fibration is not known, except for the A 1 case [38]. For some relevant discussions, see [38,39]. Last but not least, even if one finds ways to overcome the two issues, it is perhaps equally hard to show a useful vanishing condition based on the geometry.…”

“…Third, our result could provide hints to other methods of computing the elliptic genera of (2,0) SCFTs. In particular, it would be nice to see if one could write down 6d blow-up equations that the elliptic genera are supposed to satisfy, along the line of [36][37][38][39].…”

“…B-model technique is explored in [34,35], but restricted to the first few genus expansions. Recently, [36][37][38][39] found a powerful way to compute the elliptic genera using 6d blow-up equations, which in particular covers all the theories in the non-higgsable clusters [40].…”

Quantum periods and spectra in dimer models and Calabi-Yau geometries

“…The nonperturbative parts of the two types of exact quantization conditions in [15,16] are related by certain constrains on the BPS invariants known as the blowup equations [22,23]. The blowup equations originally come from studies of Seiberg-Witten gauge theories [24] (see also [25,26]), but have now become a very effective tool for computing topological string amplitudes on various Calabi-Yau manifolds [27][28][29][30][31]. The exact quantization conditions have also been applied to related condensed matter systems, e.g., in [32][33][34][35].…”