[Paper Review] The Index Bundle for a Family of Dirac-Ramond Operators

Christopher L Harris|arXiv (Cornell University)|Feb 9, 2012

Homotopy and Cohomology in Algebraic Topology22 references3 citations

TL;DR

This paper studies the index bundle of the Dirac-Ramond operator for a family of compact spin manifolds, treating it as a formal sum of twisted Dirac operators with coefficients in power series in $ q $. Using modular properties of the Chern character and Jacobi-like forms, the author derives explicit formulas for the index bundle in $ K(X)[[q]] $, and identifies it with an $ L(E_8) $ bundle in the case of 8-dimensional string manifolds with trivial first Pontryagin class.

ABSTRACT

We study the index bundle of the Dirac-Ramond operator associated with a family $π: Z o X$ of compact spin manifolds. We view this operator as the formal twisted Dirac operator $\dd \otimes \bigotimes_{n=1}^{\infty}S_{q^n}TM_{\C}$ so that its index bundle is an element of $K(X)[[q]]$. When $p_1 (Z) = 0$, we derive some explicit formulas for the Chern character of this index bundle using its modular properties. We also use the modularity to identify our index bundle with an $L(E_8)$ bundle in a special case.

Motivation & Objective

• To define and study the index bundle of the Dirac-Ramond operator in the context of families of compact spin manifolds.
• To express the index bundle as a formal sum of twisted Dirac operator indices in $ K(X)[[q]] $.
• To use modular invariance and Jacobi-like forms to derive explicit Chern character formulas for the index bundle.
• To identify the index bundle with an $ L(E_8) $-bundle in the case of 8-dimensional string manifolds under $ p_1(Z) = 0 $.

Proposed method

• Treat the Dirac-Ramond operator as a formal twisted Dirac operator $ \not{\partial} \otimes \bigotimes_{n=1}^\infty S_{q^n} TM_\mathbb{C} $, leading to an index in $ K(X)[[q]] $.
• Use the cohomological family index theorem to relate the index to characteristic classes via the $ \widehat{A} $-genus and Chern character.
• Apply modular properties of Eisenstein series $ E_2, E_4, E_6 $ and Jacobi-like forms to derive explicit expressions for the Chern character of the index bundle.
• Employ Newton identities to express power sums of Chern roots in terms of Pontryagin classes, enabling computation of $ \operatorname{ch}(W_n) $.
• Use the generating series $ \sum_{n=0}^\infty q^n W_n = \bigotimes_{j=1}^\infty S_{q^j} W_\mathbb{C} $ to define the formal bundle coefficients.
• Verify consistency with known anomaly cancellation formulas by comparing index components under $ p_1(Z) = 0 $.

Experimental results

Research questions

• RQ1 How can the index bundle of the Dirac-Ramond operator on a family of spin manifolds be systematically described in terms of $ K $-theory with $ q $-adic coefficients?
• RQ2 What modular properties govern the Chern character of the index bundle, and how can they be used to derive explicit formulas?
• RQ3 In what cases does the index bundle of the Dirac-Ramond operator coincide with the $ L(E_8) $-bundle associated with the basic representation of the $ E_8 $ loop group?
• RQ4 How do the index components $ \operatorname{ch}_0 $, $ \operatorname{ch}_2 $, etc., transform under nontrivial $ p_1 $-classes, and what relations emerge?

Key findings

• When $ p_1(Z) = 0 $, the Chern character of the index bundle of the Dirac-Ramond operator is given by a modular expression involving $ E_4(q) $, $ E_6(q) $, and Pontryagin classes.
• The index bundle satisfies $ \operatorname{ch}_0(\operatorname{Ind} \not{\partial}^{V_1}) = 248 \operatorname{ch}_0(\operatorname{Ind} \not{\partial}) $ and $ \operatorname{ch}_2(\operatorname{Ind} \not{\partial}^{V_1}) = -496 \operatorname{ch}_2(\operatorname{Ind} \not{\partial}) $, matching known anomaly cancellation formulas.
• For 8-dimensional string manifolds with $ \dim X < 16 $, the index bundle in $ (K(X) \otimes \mathbb{Q})[[q]] $ is equivalent to the $ L(E_8) $-bundle associated with the basic representation of the $ E_8 $ loop group.
• Explicit formulas for $ \operatorname{ch}_0 $ and $ \operatorname{ch}_2 $ of the index bundle are derived as integrals over the fiber: $ \int_Y \frac{-31p_2(V)}{180} $ and $ \int_Y \frac{-13p_1(V)p_2(V) + 62p_3(V)}{7560} $, respectively.
• The generating series $ \bigotimes_{j=1}^\infty S_{q^j} W_\mathbb{C} $ allows the index bundle to be expressed as a formal sum of twisted Dirac operator indices, enabling modular analysis.
• The Chern character expansion reveals that $ \widehat{a}(V,\tau) e^{G_2(\tau)p_1(V)} $ encodes the index via $ \exp\left( \sum_{n=1}^\infty \frac{2}{(2n)!} G_{2n}(\tau) \operatorname{Tr}(y^{2n}) \right) $, which is rewritten using Eisenstein series and Newton identities.