On the elliptic genus of three E-strings and heterotic strings

Abstract

A precise formula for the elliptic genus of three E-strings is presented. The related refined free energy coincides with the result calculated from topological string on local half K3 Calabi-Yau threefold up to genus twelve. The elliptic genus of three heterotic strings computed from M9 domain walls matches with the result from orbifold formula to high orders. This confirms the n = 3 case of the recent conjecture that n pairs of E-strings can recombine into n heterotic strings.

Universal features of BPS strings in six-dimensional SCFTs

“…1would have central charge 1 or 3/2 depending on whether the SO(1) factor is realized as the trivial theory or as the Ising model. Neither choice is consistent with the prediction from equation (6.45) that the central charge be 8/3, and there does not seem to be an obvious way to recover the correct central charge, even by allowing for enhancements of the flavor symmetry (other than allowing for the possibility that SO(2) 8 × SO 1 , equation (6.45) predicts the central charge of the flavor symmetry chiral algebra, c F , to be respectively 69 7 and 111 8 . In particular, the factor of the flavor symmetry that rotates the spinorial matter is expected to have central charge respectively 2 and 33 8 .…”

Blowup equations for refined topological strings

“…The formulae for N , N and N are extremely complicated. The explicit form can be found in [105]. For more three E-strings, the elliptic genus can be computed with the methods in [106] or [58], at least in principle.…”

Topological string on elliptic CY 3-folds and the ring of Jacobi forms

“…The rational elliptic surface is the 1 2 K3. The solution has been discussed first in [53] and in the context of the (refined) holomorphic anomaly equation in [41,28,10]. A quiver description has been found in [49].…”

“…We argue that the special form of the denominator in our ansatz naturally is crucial to satisfy the Castelnuovo bounds. The scaling of the z argument in the denominator can interpreted as coming from multiple string windings of the base similar as in the elliptic genus of E-strings [28,10,49] or more general strings of 6d SCFTs [26] with gauge symmetries. On the other hand, the paper [49] constructs a 2d quiver gauge theory for E-strings and [26] for theD 4 string that can in principle compute the elliptic genus of any finite number of E-strings with the techniques of [5] 7 .…”