Exceptional String: Instanton Expansions and Seiberg–witten Curve

Abstract

We investigate instanton expansions of partition functions of several toric Estring models using local mirror symmetry and elliptic modular forms. We also develop a method to determine the Seiberg-Witten curve of E-string with the help of elliptic functions.

Fusing E-strings to heterotic strings:E+E→H

“…Using these modified theta functions one can check easily using (3.12) that the elliptic genus of n M-strings, which is no longer holomorphic, satisfies the following holomorphic anomaly equation: 17) where ǫ + = ǫ 1 +ǫ 2 . Said differently, we are trading here the modular anomaly of (3.14) with the holomorphic anomaly of (3.17).…”

“…The free energy of a single E-string is known to arbitrary genus (see the discussion in Section 6.3); in the case of several strings, topological string techniques have been employed to compute the free energy to high genus (for instance, in [17] the free energy of up to five E-strings is computed up to g = 5). Recently, a similar approach was successfully developed [22] in the refined case (where ǫ 1 , ǫ 2 are taken to be arbitrary), generalizing techniques that were employed in the unrefined limit.…”

“…In this context string dualities [14] relate this system to topological strings on a CY 3-fold in the vicinity of the 1 2 K3 surface. The topological string partition function for this theory has been studied in [14][15][16][17][18][19][20][21][22] and, even though major progress has been made, the result is still incomplete. Our main aim is to build on these partial results to obtain explicit formulas for the elliptic genus of two E-strings, Z…”

Seiberg-Witten Curve for theE-String Theory

“…(1.1) as the six-dimensional Seiberg-Witten (SW) curve: the six-dimensional curve determines the prepotential of E-string theory and gives a complete description of its low-energy dynamics. Previously there were attempts at deriving the six-dimensional curve [4,5,6,7], however, only partial results with a few non-vanishing mass parameters {m i } were obtained. In this paper we will determine all the coefficient functions a j , b j and the curve (1.1) for arbitrary values of {m i }.…”

Seiberg-Witten prepotential for E-string theory and random partitions

“…Among others, compactification down to four dimensions is of particular interest [4][5][6][7][8][9][10][11]. In this case, the low energy effective theory admits a description in terms of Seiberg-Witten theory [12,13].…”