Elliptic genera and $SL(2,Z)$ modular forms for fibre bundles

This paper utilizes family index theory to generalize well-known $SL(2,Z)$ modular forms to the family setting, thereby deriving new anomaly cancellation formulas for determinant line bundles, index gerbes, eta invariants, and residue Chern forms.

The Gist

Imagine you are an architect trying to build a massive, multi-story skyscraper. In the world of mathematics and physics, this "skyscraper" is a complex shape called a manifold (a space that can curve and twist in many dimensions).

Usually, when we study these shapes, we look at them as single, static objects. But in this paper, the author, Yong Wang, is looking at something more dynamic: a family of these shapes. Think of it not as one building, but as a whole city of buildings, where each building is slightly different, and they are all connected by a network of roads (the "base space").

Here is a simple breakdown of what the paper does, using everyday analogies:

1. The Problem: "Glitches" in the System (Anomalies)

In physics and math, when you try to calculate things about these shapes (like how particles move or how the shape curves), you sometimes hit a "glitch." In physics, this is called an anomaly. It's like trying to bake a cake where the ingredients don't quite add up, or a bridge that looks perfect on paper but wobbles when you put weight on it.

For a long time, mathematicians found a "magic trick" to fix these glitches for single shapes. They discovered that if you mix certain mathematical ingredients together in a specific way, the errors cancel each other out perfectly. This is called a "Miracle Cancellation."

2. The New Challenge: The "Family" Problem

The problem with the old "magic tricks" is that they only worked for single, static shapes. They broke down when you tried to apply them to a whole family of shapes (like our city of buildings). The "glitches" became more complicated because the buildings were changing as you moved from one to another.

Yong Wang's paper asks: "Can we find a new magic trick that works for the whole family, not just one building?"

3. The Tool: Modular Forms (The "Universal Remote")

To solve this, the author uses a powerful mathematical tool called Modular Forms.

• The Analogy: Imagine a "Universal Remote Control" for the universe. No matter how you twist, turn, or zoom in on the shape, this remote keeps the signal stable. It's a mathematical function that stays consistent even when the world around it changes.
• In this paper, the author takes these "Universal Remotes" (specifically ones related to a group called $SL(2, Z)$) and adapts them to work for families of shapes.

4. The Discovery: New Cancellation Formulas

By using these adapted "Remotes," Wang discovers new formulas.

• What they do: These formulas show exactly how to mix different mathematical ingredients (like the "determinant line bundle" and "index gerbes"—which are fancy names for the "blueprints" and "structural supports" of the shapes) so that the glitches cancel out.
• The Result: Just like the original "Miracle Cancellation" saved the day for single shapes, these new formulas save the day for families of shapes. They prove that even in a complex, changing system, the universe still balances out.

5. The "Higher Degree" Twist

The paper also looks at "higher degree" cases.

• The Analogy: Imagine you are not just looking at the floor plan of a building (2D), but you are also analyzing the sound waves bouncing off the walls, the air pressure in the rooms, and the vibration of the foundation all at once.
• Wang extends his "magic trick" to handle these more complex, multi-layered analyses, providing formulas for these "residue Chern forms" (which are like the detailed fingerprints of the shape's geometry).

Why Does This Matter?

You might ask, "Who cares about balancing equations for imaginary shapes?"

• Physics Connection: These shapes often represent the fabric of spacetime in theories like String Theory. If the math doesn't balance (if there's an anomaly), the theory of the universe breaks down.
• The Impact: By proving that these "Miracle Cancellations" work for families of shapes, Wang is giving physicists and mathematicians a stronger, more flexible toolkit. It helps them understand how the universe might behave when things are changing or moving, rather than just sitting still.

Summary

In short, Yong Wang took a famous mathematical "glitch-fixing" formula, upgraded it to handle moving, changing families of shapes, and proved that the universe still balances perfectly even in these complex scenarios. He did this by using a special type of mathematical "remote control" (modular forms) to ensure that no matter how the shapes twist and turn, the errors always cancel out.

Technical Summary

Here is a detailed technical summary of the paper "Elliptic Genera and SL(2, Z) Modular Forms for Fibre Bundles" by Yong Wang.

1. Problem Statement

The paper addresses the generalization of "miraculous cancellation" formulas and anomaly cancellation phenomena from single manifolds to families of manifolds (fibre bundles).

• Context: In theoretical physics and differential geometry, anomaly cancellation formulas relate characteristic classes (like the $\hat{A}$-genus and $L$-genus) to modular forms. These formulas, originally discovered for single manifolds (e.g., by Alvarez-Gaumé, Witten, and Liu), explain the vanishing of gravitational anomalies in string theory.
• The Gap: While these formulas are well-established for individual manifolds, their extension to families (where the base space $B$ parametrizes a family of fibres $Z$) requires handling determinant line bundles, index gerbes, and eta invariants in a fibered context.
• Objective: The author aims to generalize known $SL(2, \mathbb{Z})$ modular forms to the family case to derive new anomaly cancellation formulas for:
  1. Determinant line bundles (associated with even-dimensional fibres).
  2. Index gerbes (higher analogues for odd-dimensional fibres).
  3. Eta invariants (spectral asymmetry).
  4. Residue Chern forms (higher-degree cases).

2. Methodology

The paper employs a combination of Family Index Theory, Modular Form Theory, and Characteristic Classes.

• Family Index Theory: The author utilizes the Atiyah-Singer family index theorem.
  • For even-dimensional fibres, the curvature of the Bismut-Freed connection on the determinant line bundle corresponds to the 2-form component of the fiber integration of the $\hat{A}$-form.
  • For odd-dimensional fibres, Lott's index gerbe is used, where the curvature is a closed 3-form derived from the fiber integration of the $\hat{A}$-form.
  • For eta invariants, the Bismut-Freed theorem for reduced eta invariants is applied.
• Modular Forms Construction:
  • The author constructs specific modular forms over $SL(2, \mathbb{Z})$ by defining generating functions involving Jacobi theta functions ($\theta, \theta_1, \theta_2, \theta_3$) and infinite tensor products of symmetric and exterior powers of the complexified vertical tangent bundle ($T^{\mathbb{C}}Z$).
  • Key objects include $Q(TZ, \tau)$, $\tilde{Q}(TZ, \tau)$, and $\hat{Q}(TZ, \tau)$, which are shown to be modular forms of specific weights ($2k$).
• Expansion and Coefficient Comparison:
  • The constructed modular forms are expanded in powers of $q = e^{2\pi i \tau}$.
  • By comparing the coefficients of the expansion (which represent characteristic forms of the bundles) with the known polynomial structure of modular forms (specifically in terms of Eisenstein series $E_4$ and $E_6$), the author derives linear relations between these characteristic forms.
• Generalization to $spinc$ and Complex Structures: The methodology is extended to $spinc$ manifolds and families with almost complex structures, introducing two-variable elliptic genera.

3. Key Contributions and Results

A. Anomaly Cancellation for Determinant Line Bundles (Even-dimensional Fibres)

The paper generalizes results from [22] to the family case. For a fibre bundle with a closed spin fibre $Z$ of dimension $4k-2$:

• Modular Form: Defined $Q(TZ, \tau)$ as a modular form of weight $2k$.
• Results: Derived explicit cancellation formulas relating the curvatures of determinant line bundles twisted by various bundles (spinor bundles, tangent bundles, and their exterior/symmetric powers).
  • Examples:
    • For $\dim(Z) = 6$ ($k=2$): $R_{LD} \otimes \dots = 120(R_{LD} \otimes \dots)$.
    • For $\dim(Z) = 10, 14, 18$: Similar formulas with coefficients derived from Eisenstein series (e.g., $-252, 240, -132$).
• Significance: These formulas represent the cancellation of gravitational anomalies in the family setting, generalizing the "miracle cancellation" of Alvarez-Gaumé and Witten.

B. Anomaly Cancellation for Index Gerbes (Odd-dimensional Fibres)

For odd-dimensional spin fibres ($4k-3$):

• Modular Form: Defined $\tilde{Q}(TZ, \tau)$ as a modular form of weight $2k$.
• Results: Established cancellation formulas for the curvature of the index gerbe ($R_{GD}$).
  • The formulas relate the curvature of the gerbe twisted by spinor bundles to the curvature of the gerbe twisted by other characteristic bundles.
  • Specific cases for dimensions 5, 9, 13, and 17 are provided, mirroring the even-dimensional results but adapted for the 3-form curvature of the gerbe.

C. Eta Invariants

For families of odd-dimensional manifolds:

• Results: Derived formulas for the reduced eta invariants ($\bar{\eta}$).
• The formulas express linear combinations of eta invariants of twisted Dirac operators as multiples of the eta invariant of the untwisted operator (plus a constant).
• These results provide new divisibility properties and relations for spectral asymmetry in families.

D. $spinc$ and Complex Manifolds

• Extended the theory to $spinc$ manifolds and families with almost complex structures.
• Defined two-variable elliptic genera $Ell(TZ, W, \tau, z)$ and $\tilde{Ell}(TZ, W, \tau, z)$.
• Proved that under specific conditions (vanishing first Chern classes and matching Pontryagin classes), these genera are weak Jacobi forms.
• Derived cancellation formulas for the curvatures of determinant line bundles and index gerbes in the $spinc$ case, involving coefficients like $240, 2160, -504$, etc.

E. Higher Degree Cases (Residue Chern Forms)

• Utilized theorems by Mickelsson and Paycha regarding Wodzicki residues and superconnections.
• Defined higher-degree modular forms $Q_1$ and $Q_2$.
• Result: Established anomaly cancellation formulas for residue Chern forms ($str(A^{2j})$ and $sres(|A|^{2j-1})$). This extends the cancellation phenomenon to higher-degree differential forms, not just the topological invariants.

4. Significance

• Unification of Geometry and Physics: The paper bridges the gap between the modular invariance properties of characteristic forms (mathematics) and the cancellation of anomalies in quantum field theory (physics) within the context of families of manifolds.
• New Divisibility Results: The derived formulas imply new divisibility results for the indices of twisted Dirac operators and $spinc$ Dirac operators in families.
• Generalization of Known Theorems: It successfully lifts the "miracle cancellation" formulas from single manifolds to the more complex setting of fibre bundles, providing a robust framework for studying global anomalies.
• Foundation for Future Work: The introduction of residue Chern form cancellation and the extension to $spinc$ structures opens new avenues for research in non-commutative geometry (Kaster-Kalau-Walze theorem extension) and higher spectral flow.

Conclusion

Yong Wang's paper provides a comprehensive framework for understanding anomaly cancellation in families of manifolds. By constructing specific $SL(2, \mathbb{Z})$ modular forms and applying family index theory, the author derives a series of powerful formulas that relate the curvatures of determinant line bundles, index gerbes, and eta invariants. These results generalize previous work on single manifolds and offer new insights into the topological and geometric structures of fibre bundles.