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A note on modular forms and generalized anomaly cancellation formulas 2

Siyao Liu, Yong Wang

TL;DR

The paper addresses the extension of modular anomaly cancellation formulas to $(a,b)$-type cancellations on even-dimensional manifolds and the construction of modular secondary forms via transgression on odd-dimensional manifolds. It employs characteristic forms, Jacobi theta functions, and modular form theory to build generating functions $Q_1(\\tau)$, $Q_2(\\tau)$ (and their twists with line bundles or additional bundles) and proves their modularity on subgroups $\\Gamma_{0}(2)$, $\\Gamma^{0}(2)$, and $\\Gamma_{\\theta}$, yielding concrete 4d and 8d dimensional corollaries and integrality/divisibility results. The approach centers on expressing twisted $\\widehat{A}$-genera in terms of theta-function data, deriving transformation laws, and translating these into explicit cancellation formulas that generalize previous Han-Liu-Zhang results. The transgression framework further provides modularity properties for odd-dimensional manifolds, enriching the landscape of elliptic-genus-like invariants under bundle twists with potential implications for anomaly cancellation in geometric settings.

Abstract

In [7], Liu and Wang generalized the Han-Liu-Zhang cancellation formulas to the (a, b) type cancellation formulas. In this note, we prove some another (a, b) type cancellation formulas for even-dimensional Riemannian manifolds. And by transgression, we obtain some characteristic forms with modularity properties on odd-dimensional manifolds.

A note on modular forms and generalized anomaly cancellation formulas 2

TL;DR

The paper addresses the extension of modular anomaly cancellation formulas to -type cancellations on even-dimensional manifolds and the construction of modular secondary forms via transgression on odd-dimensional manifolds. It employs characteristic forms, Jacobi theta functions, and modular form theory to build generating functions , (and their twists with line bundles or additional bundles) and proves their modularity on subgroups , , and , yielding concrete 4d and 8d dimensional corollaries and integrality/divisibility results. The approach centers on expressing twisted -genera in terms of theta-function data, deriving transformation laws, and translating these into explicit cancellation formulas that generalize previous Han-Liu-Zhang results. The transgression framework further provides modularity properties for odd-dimensional manifolds, enriching the landscape of elliptic-genus-like invariants under bundle twists with potential implications for anomaly cancellation in geometric settings.

Abstract

In [7], Liu and Wang generalized the Han-Liu-Zhang cancellation formulas to the (a, b) type cancellation formulas. In this note, we prove some another (a, b) type cancellation formulas for even-dimensional Riemannian manifolds. And by transgression, we obtain some characteristic forms with modularity properties on odd-dimensional manifolds.
Paper Structure (6 sections, 11 theorems, 65 equations)

This paper contains 6 sections, 11 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Characteristic forms and modular forms
  3. The generalized cancellation formulas for even-dimensional Riemannian manifolds
  4. The generalized cancellation formulas involving a complex line bundle for even-dimensional Riemannian manifolds
  5. Transgressed forms and modularities on odd-dimensional Riemannian manifolds
  6. Acknowledgements

Key Result

Lemma 2.2

(L) $\delta_{1}(\tau)$ (resp. $\varepsilon_{1}(\tau)$) is a modular form of weight 2 (resp. 4) over $\Gamma_{0}(2),$$\delta_{2}(\tau)$ (resp. $\varepsilon_{2}(\tau)$) is a modular form of weight 2 (resp. 4) over $\Gamma^{0}(2),$ while $\delta_{3}(\tau)$ (resp. $\varepsilon_{3}(\tau)$) is a modular f

Theorems & Definitions (18)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Corollary 3.5
  • ...and 8 more