On stringy invariants of GUT vacua

Abstract

We investigate aspects of certain stringy invariants of singular elliptic fibrations which arise in engineering Grand Unified Theories in F-theory. In particular, we exploit the small resolutions of the total space of these fibrations provided recently in the physics literature to compute "stringy characteristic classes", and find that numerical invariants obtained by integrating such characteristic classes are predetermined by the topology of the base of the elliptic fibration. Moreover, we derive a simple (dimension independent) formula for pushing forward powers of the exceptional divisor of a blowup, which one may use to reduce any integral (in the sense of Chow cohomology) on a small resolution of a singular elliptic fibration to an integral on the base. We conclude with a remark on the cohomology of small resolutions of GUT vacua, where we conjecture that certain simple formulas for their Hodge numbers may be given solely in terms of the first Chern class and Hodge numbers of the base.

Euler Characteristics of Crepant Resolutions of Weierstrass Models

“…We work relative to a base that we leave arbitrary. In this sense, our paper is a direct generalization of the work of Fullwood and van Hoeij on stringy invariants of Weierstrass models [35].…”

“…A direct consequence of Theorem A.2 (Jacobi's identity) and Lemma 3.4 is the following pushforward formula (see [35]): Lemma 3.7. Let Z ⊂ X be the complete intersection of d nonsingular hypersurfaces Z 1 , .…”

“…The study of crepant resolutions of Weierstrass models, their fibral structure, and their flop transitions is an area of common interest to algebraic geometers, number theorists, and string theorists [33,31,32,30,39,35,69]. The theory of elliptic surfaces has its beginnings in the 1960s, and was advanced largely by the contributions of mathematicians such as Kodaira [46]; Néron [62]; Mumford and Suominen [59], Deligne [16], and Tate [70].…”

“…The Euler characteristic of an elliptic fibration plays a central role in many physical problems such as the computation of gravitational anomalies of six dimensional supergravity theories [37,63] and the cancellation of tadpoles in four dimensional theories [66,2,3,9,14,24,29]. Unfortunately, the Euler characteristics of crepant resolutions of Weierstrass models are generally not known, although they have been computed in some special cases for Calabi-Yau threefolds and fourfolds [4,5,53,35]. For instance, the Euler characteristics of G-models for Calabi-Yau threefolds were studied in [37], and there are conjectures for the Euler characteristics of G-models for Calabi-Yau fourfolds based on heterotic string theory/F-theory duality [9].…”

“…A crucial ingredient of our results is Theorem 1.8, which is a pushforward formula for any analytic function of the class of the exceptional divisor of a blowup of a nonsingular variety along a smooth complete intersection of hypersurfaces meeting transversally. Theorem 1.8 is a generalization to arbitrary analytic functions of a result of Fullwood and van Hoeij [35,Lemma 2.2 ], which relies on a theorem of Aluffi [1] simplifying the classic formula of Porteous on Chern classes of the tangent bundle of a blowup [64]. Theorem 1.8 profoundly simplifies the algebraic manipulations necessary to compute pushforwards, and therefore has a large range of applications independently of the specific applications discussed in this paper.…”