Non-critical strings, del Pezzo singularities and Seiberg-Witten curves

Abstract

We study limits of four-dimensional type II Calabi-Yau compactifications with vanishing four-cycle singularities, which are dual to T 2 compactifications of the six-dimensional non-critical string with E 8 symmetry. We define proper subsectors of the full string theory, which can be consistently decoupled. In this way we obtain rigid effective theories that have an intrinsically stringy BPS spectrum. Geometrically the moduli spaces correspond to special geometry of certain non-compact Calabi-Yau spaces of an intriguing form. An equivalent description can be given in terms of Seiberg-Witten curves, given by the elliptic simple singularities together with a peculiar choice of meromorphic differentials. We speculate that the moduli spaces describe non-perturbative noncritical string theories.

Mahler measure variations, Eisenstein series and instanton expansions

“…This is what Rodriguez Villegas was investigating when he noticed that some of his formulas looked similar to formulas in the literature on Mirror Symmetry; see the final remarks in [11]. The modest goal of the present paper is to point out that actual instanton expansions matching his examples do occur in the physics literature [8,6,7,4]. The relation is, however, more intriguing and less straightforward than one might expect from the remarks in [11]:…”

“…That Examples 1, 2, 3 of [11] line up with Examples E 6 , E 5 , E 8 in [6,8] is immediately obvious from the formulas for the elliptic pencils and their Picard-Fuchs equations which are explicitly given in [6,8,11]. Looking at how the various authors get from the Picard-Fuchs equations to their expansions directly leads to Relation (3); for more details see Section 1.…”

“…The expansions 1 − b n n 2 q n 1−q n of the weight 3 modular forms in Examples 1, 2, 3 of [11] are related to the instanton expansions κ(1 − a n n 3 Q n 1−Q n ) in Examples E 6 , E 5 , E 8 , respectively, in [6,8], via the inverse function relations…”

“…In Section 2 we present six such examples including the above ones and the E 7 case of [6,8]. In each case the computer output suggests that the numbers a n and b n are integers.…”

“…I want to thank Don Zagier for bringing [11] to my attention (somewhen around 1998) and for keeping me well informed about his work in progress [15]. I thank Bogdan Florea for discussing with me in November 2001 some parts of his work, thereby also providing references to [6,8]. I want to thank Noriko Yui for the invitation to the Banff Workshop on Mirror Symmetry with its stimulating atmosphere to report on this work.…”