Bootstrapping ADE M-strings

Abstract

We study elliptic genera of ADE-type M-strings in 6d (2,0) SCFTs from their modularity and explore the relation to topological string partition functions. We find a novel kinematical constraint that elliptic genera should follow, which determines elliptic genera at low base degrees and helps us to conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of related geometries. Using this, we can bootstrap the elliptic genera to arbitrary base degree, including D/E-type theories for which explicit formulas are only partially known. We utilize our results to obtain the 6d Cardy formulas and the superconformal indices for (2,0) theories.

Twisted 6d (2, 0) SCFTs on a circle

“…Geometrically, this should correspond to the topological string partition function on genus one fibered CY threefolds upon which M-theory compactifies [10]. We develop novel modular ansatz for 6d twisted (2, 0) theories and determine the elliptic genera at low base degrees using conjectural vanishing conditions for the corresponding genus one fibered CY threefolds, extending the previous work [35]. We emphasize that although the twisted elliptic genera are not related to the 6d elliptic genera in an apparent way, the choice of subgroups indeed coincides nicely with one's expectation from twisting the theory, which is not at all obvious from the perspective of 5d instanton dynamics.…”

“…Instead, SL(2, Z) modular property of the elliptic genera can be used to determine them. Such procedure is called the 'modular bootstrap', and it has been studied in various 6d theories [29][30][31][32][33][34][35][36], including (2, 0) D, E-type theories [30,35]. In this paper, we will focus on the twisted circle compactification of (2, 0) theories whose elliptic genera are unknown.…”

“…where Ω is the Cartan matrix for the Lie algebra g and D is the matrix D ij = 2 δ i,j α i ,α i . For g simply-laced, D is the identity matrix and we are back to the case considered in [35], while for g non-simply-laced which is the focus of this paper, it is non-trivial such that the combined matrix Ω s = ΩD becomes…”

“…If one can find sufficiently many extra constraints either from gauge theory or geometry, one is able to fix all the unknowns and hence the elliptic genus itself. This approach is often dubbed modular bootstrap [29][30][31][32][33][34][35][36], and was successfully applied to compute the elliptic genera of various (1, 0) and (2, 0) SCFTs in 6d. This method is also applicable to compact elliptically fibered Calabi-Yau (CY) threefolds [37,38], leading to interesting results on Heterotic/Type II string duality [39,40], black hole physics [41] and swampland conjectures [40,[42][43][44], to name a few.…”

Weyl invariant Jacobi forms along Higgsing trees

“…In [22,23,25,58], these boundary conditions are imposed in the form of so-called vanishing conditions: the constraint that Gopakumar-Vafa invariants of a given curve class must vanish at sufficiently high genus. In [22], it was argued that imposing generic vanishing conditions (i.e.…”