Revisiting the quantum geometry of torus-fibered Calabi-Yau threefolds

Abstract

About ten years ago, Katz, Klemm and Huang conjectured that topological string amplitudes on compact, elliptically fibered Calabi-Yau threefolds at fixed base degree could be expressed in terms of meromorphic Jacobi forms for SL(2,\mathbb{Z}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="double-struck">ℤ</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , giving access to Gromov-Witten invariants at arbitrary genus. This was later generalized to torus-fibered CY threefolds with N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> -sections, where topological string amplitudes are conjecturally governed by meromorphic Jacobi forms under the congruence subgroup \Gamma_1(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> . In this work, we show that these modularity properties follow from (and are equivalent to) the wave-function property of the topological string partition function Z_{\rm top} under a relative conifold monodromy, implementing a particular Fourier-Mukai transformation on the derived category of coherent sheaves. In particular, we introduce a variant of Z_{\rm top} which is both holomorphic and modular covariant. Under the same relative conifold monodromy, the generating series of genus 0 Gopakumar-Vafa invariants at fixed base degree is mapped to the generating series of rank 0 Donaldson-Thomas indices counting D4-D2-D0-brane bound states wrapped on the torus fiber. We show that the quasimodularity of the generating series of GV invariants matches the expected mock-modular behavior of the generating series of D4-D2-D0 indices, despite having different multi-cover contributions. We analyze and tabulate a large number of CY threefolds fibered over del Pezzo surfaces, with an N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> -section for N≤ 5 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> , including several new examples beyond the realm of toric geometry.