Exact Kähler potential for Calabi-Yau fourfolds

Abstract

We study quantum Kähler moduli space of Calabi-Yau fourfolds. Our analysis is based on the recent work by Jockers et al. which gives a novel method to compute the Kähler potential on the quantum Kähler moduli space of Calabi-Yau manifold. In contrast to Calabi-Yau threefold, the quantum nature of higher dimensional Calabi-Yau manifold is yet to be fully elucidated. In this paper we focus on the Calabi-Yau fourfold. In particular, we conjecture the explicit form of the quantum-corrected Kähler potential. We also compute the genus zero Gromov-Witten invariants and test our conjecture by comparing the results with predictions from mirror symmetry. Local toric Calabi-Yau varieties are also discussed. * yhonma@hri.res.in † masahidemanabe@gmail.com Topological string theory gives us a framework to study "topological quantum quantities" as Gromov-Witten invariants corresponding to the worldsheet instanton numbers.Strongly motivated from the string compactification, Calabi-Yau threefolds has been well-studied. The quantum-corrected Kähler potential on the Kähler moduli space of Calabi-Yau threefold can be determined by a single holomorphic function called prepotential which encodes the information about the genus zero Gromov-Witten invariants.This simplification is due to the special Kähler structure of the moduli space of Calabi-Yau threefold [1]. Mirror symmetry is an efficient tool to compute the prepotential as first demonstrated in [2] to the quintic Calabi-Yau threefold. Although moduli spaces of Calabi-Yau manifolds whose complex dimension is greater than three do not have such a special Kähler structure, the higher dimensional mirror symmetry is still useful to compute Gromov-Witten invariants [3, 4, 5] (see also [6]). As pioneered in [7], various Calabi-Yau manifolds can be realized at the IR fixed points of the two dimensional N = (2, 2) gauged linear sigma model (GLSM). Mirror symmetry for Calabi-Yau manifolds described by abelian GLSMs is well understood [8,9] and also proved physically in [10]. However, for general cases with non-abelian gauge groups, comprehensive study of mirror symmetry is yet to come.Recently, it was proposed in [11] that Gromov-Witten invariants of Calabi-Yau manifolds can be computed without using mirror symmetry. More precisely, they proposed the relation e −K = Z GLSM between the exact Kähler potential K on the quantum Kähler moduli space of a Calabi-Yau manifold and the exact partition function Z GLSM of an N = (2, 2) GLSM on S 2 which was computed in [12,13]. This means that we can extract the genus zero Gromov-Witten invariants only from the GLSM calculation. They also checked the consistency of their proposal for some known examples and two physical proofs are given in [14]. Their method is applicable to non-abelian GLSMs and the authors of [11] also made predictions for Gromov-Witten invariants of a determinantal Calabi-Yau threefold. Note that, as mentioned in [11,14], their method has a close relationship to the toric mirror symmetry.The aim of this paper is to study the quant...

The stringy instanton partition function

“…6 Similar arguments appeared also in [32]. 7 The normalization here has been chosen having in mind the M2,1 case; see the next paragraph.…”

Onα′corrections inN=1F-theory compactifications

“…Therefore, we immediately see that the volume correction in (26) should arise from a string amplitude that involves the sum over two topologies: The disk (g = c = 0, b = 1) and the projective plane (g = b = 0, c = 1). They correspond to the tree-level of orientable open strings and non-orientable closed strings, respectively.…”

“…We stress that the sign difference in the two contributions arises due to their origin in distinct R 4 couplings in 10d [8,9]. The ζ(3)-part can also be derived using mirror symmetry or localization techniques as done in [25,26]. However, it vanishes in the M-theory limit g IIA → ∞, and hence is of no relevance for the present purposes.…”

Universal properties of type IIB and F-theory flux compactifications at large complex structure

“…Here κ ijkl denote the classical intersection numbers of the divisors of the mirror dual four-fold, as discussed in [34][35][36][37]. We will consider flux compactifications of F-theory in which the four-form flux G 4 on the corresponding cycle does not vanish.…”