Publication Date

2012-04-21

Availability

Open access

Embargo Period

2012-04-21

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2012-03-30

First Committee Member

Orlando Alvarez

Second Committee Member

Nikolai Saveliev

Third Committee Member

Alexander Dvorsky

Fourth Committee Member

Rafael Nepomechie

Abstract

String theoretic considerations imply the existence of a Dirac-like operator, known as the Dirac-Ramond operator, on the free loop space of a closed string manifold. We study the index bundle of the Dirac-Ramond operator associated with a family π : Z → X of closed spin manifolds. We work instead with a formal version of the operator, the usual Dirac operator twisted by a certain formal q-series of vector bundles. Its index bundle is an element of K(X)[[q]]. In the case where the total space Z is a string manifold, we show that the Chern character of this index bundle has certain modular properties. We then use the modularity to derive some explicit formulas for the Chern character of this index bundle. We also show that these formulas identify the index bundle with an L(E₈) bundle in a special case.

Keywords

Algebraic Topology; Index Theory; Modular Forms

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