arXiv : Quasi-Jacobi Forms, Elliptic Genera and Strings in Four Dimensions

Lee, Seung-Joo; Lockhart, Guglielmo; Lerche, Wolfgang; Weigand, Timo

doi:10.1007/JHEP01(2021)162

Article
Report number	arXiv:2005.10837
Title	Quasi-Jacobi Forms, Elliptic Genera and Strings in Four Dimensions
Author(s)	Lee, Seung-Joo (IBS, Daejeon, CTPU) ; Lerche, Wolfgang (CERN) ; Lockhart, Guglielmo (CERN) ; Weigand, Timo (U. Mainz, PRISMA ; Mainz U., Inst. Phys.)
Publication	2021-01-26
Imprint	2020-05-21
Number of pages	95
Note	90 pages, 4 figures; v2: minor comments added to match published version
In:	JHEP 2101 (2021) 162
DOI	10.1007/JHEP01(2021)162
Subject category	hep-th ; Particle Physics - Theory
Abstract	We investigate the interplay between the enumerative geometry of Calabi-Yau fourfolds with fluxes and the modularity of elliptic genera in four-dimensional string theories. We argue that certain contributions to the elliptic genus are given by derivatives of modular or quasi-modular forms, which encode BPS invariants of Calabi-Yau or non-Calabi-Yau threefolds that are embedded in the given fourfold. As a result, the elliptic genus is only a quasi-Jacobi form, rather than a modular or quasi-modular one in the usual sense. This manifests itself as a holomorphic anomaly of the spectral flow symmetry, and in an elliptic holomorphic anomaly equation that maps between different flux sectors. We support our general considerations by a detailed study of examples, including non-critical strings in four dimensions. For the critical heterotic string, we explain how anomaly cancellation is restored due to the properties of the derivative sector. Essentially, while the modular sector of the elliptic genus takes care of anomaly cancellation involving the universal B-field, the quasi-Jacobi one accounts for additional B-fields that can be present. Thus once again, diverse mathematical ingredients, namely here the algebraic geometry of fourfolds, relative Gromow-Witten theory pertaining to flux backgrounds, and the modular properties of (quasi-)Jacobi forms, conspire in an intriguing manner precisely as required by stringy consistency.
Copyright/License	publication: © 2021-2024 The Authors (License: CC-BY-4.0), sponsored by SCOAP³ preprint: (License: arXiv nonexclusive-distrib 1.0)

$A sketch of the interplay between elliptic holomorphic anomalies and (almost holomorphic) partition functions, $\hat Z_{w,m}(q,\xi)$, pertaining to the various fluxes as classified in (\ref{H22vertdecomp}). Here $\hat Z_{-1,m}(q,\xi)$ is (the non-holomorphic cousin of) the elliptic genus in four dimensions. Moreover $\hat Z_{-2,m}(q,\xi)$ encodes relative invariants of certain embedded threefolds $\mathbb Y_3^i$, which can, at least formally, be associated to elliptic genera in six dimensions.$ $The derivative contribution to the elliptic genus of a four-dimensional string is captured, in suitable situations, by the relative BPS invariants of certain threefolds $\mathbb Y_3^i$ inside the given elliptic fourfold $Y_4$. Here we present a sketch of how these threefolds are embedded for the example of a heterotic string. The heterotic string arises in this F-theory geometry from a D3-brane that wraps the rational fiber $C^0$ in the base threefold, $B_3\subset Y_4$. The extra two-form fields $B_i$ that are needed for reinstating Green-Schwarz anomaly cancellation arise by expanding the ten-dimensional four-form, ${\mathbf C}_4$, with respect to the divisor classes $p^\ast(C^i)$ for $i=1,\dots h^{1,1}(B_2)$.$ $This figure shows the component of the matter curve $\Sigma_r$ (drawn in red) along a base curve $C_i$. It refers to a geometry where the fourfold $Y_4$ is fibered over $C_i$ with generic fiber $\mathbb Y_3^i$. The issue is to determine the multiplicity $N^i(\Cr)$ of $C_i$ in the decomposition (\ref{Sigmar-relation-2}), and from the picture we see that it coincides with the number of points on $\mathbb Y_3^i$ over which the elliptic fiber $\mathbb E_\tau$ degenerates. The remaining components of $\Sigma_r$ in the decomposition (\ref{Sigmar-relation-2}) lie within special fibers of the fibration $\rho_i$ (\ref{pi3fold}) and are not depicted.$ Show more plots

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