# Schwinger terms from externalfield problems

Abstract (Summary)

The current algebra for second quantized chiral fermions in an external eld
contains Schwinger terms. These are studied in two di erent ways. Both are
non-perturbative and valid for arbitrary odd dimension of the physical space,
although explicit expressions are only given for lower dimensions.
The thesis is an introductory text to the four appended research papers.
In the rst two papers, Schwinger terms are studied by realizing gauge transformations
as linear operators acting on sections of the bundle of Fock spaces
parametrized byvector potentials. Bosons and fermions are mixed in a Z2-graded
fashion. Charged particles are considered in the rst paper and neutral particles
in the second. In the the third and the fourth paper, Schwinger terms are identi
ed with cocycles obtained from the family index theorem for a manifold with
boundary. A generating form for the covariant anomaly and Schwinger term is
obtained in the third paper. The rst three papers consider Yang-Mills while
the fourth (in cooperation with Jouko Mickelsson) also includes gravitation.
Key words: Schwinger terms, external
anomaly, Z2-grading, index theory.
eld problems, higher dimensions, chiral
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iv
Preface
This thesis will be about Schwinger terms. It is terms that appear in equal time
commutators of currents in quantum eld theory. As a mathematical physicist
I nd it hard to write a thesis about this subject. Both the physical and mathematical
aspects should preferably be covered. Ihavedecided to focus on some
of the mathematical tools that the Schwinger term and the closely related chiral
anomaly have in common. This is part of what I have learned during the years
1994{1999 as a graduate student attheRoyal Institute of Technology.
The following conventions and assumptions will be made throughout the
thesis: All manifolds are assumed to be second countable and Hausdor . They
are assumed to be paracompact whenever a partition of unity argument is needed.
In nite-dimensional manifolds are also considered unless stated otherwise. The
physical space (-time) M is real while all other manifolds and (mathematical)
elds are assumed to be complex if nothing is said about them. All manifolds,
bre bundles and sections are assumed to be smooth unless explicitly stated
otherwise. The restriction operator to local neighbourhoods will be suppressed
when convenient.
The content of the thesis will now be described brie y. Chapter 1 contains
a short introduction to anomalies. Basic ideas behind index theorems and determinant
bundles are reviewed in 2. Mathematical ideas which are not very
well-known are used there, and the text can therefore be considered as quite
`heavy'. The reader who is satis ed with a short discussion about the (family)
index theorem should therefore not read this chapter but rather consult section
2inPaper IV or some of the various physics articles that reviews the matter, for
instance [1{5]. The cohomological meaning of transgression, and related homomorphisms,
is covered by chapter 3. This chapter is independent of the previous
one and is not absolutely necessary for the rest of the thesis. Then, in chapter
4, the mathematical structure of a gauge theory is developed. This part is
independent of the previous chapters. It is further explained how the family
index theorem can be applied. Using these results, the chiral anomaly and the
Schwinger term are calculated in chapter 5. Finally, inchapter 6, the Schwinger
term is de ned and discussed. It is done by viewing it as an obstruction in the
lift of the action of the gauge group from the space of gauge connections to the
Fock bundle. This chapter is independent of the previous ones.
The thesis contains four appended research papers, henceforth referred to as
Papers I{IV. Complementary material to Papers I and II can be found in chapter
6. Chapter 2{5 serves as background material for Papers III and IV.
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List of Papers
I Christian Ekstrand,
Z2-Graded Cocycles in Higher Dimensions,
Lett. Math. Phys. 43, 359 (1998)
II Christian Ekstrand,
Neutral Particles and Schwinger Terms,
Submitted for publication (hep-th/9903148)
III Christian Ekstrand,
A Simple Algebraic Derivation of the Covariant
Anomaly and Schwinger Term,
Submitted for publication (hep-th/9903147)
IV Christian Ekstrand and Jouko Mickelsson,
Gravitational Anomalies, Gerbes and Hamiltonian Quantization,
Submitted for publication (hep-th/9904189)
Bibliographical Information:

Advisor:

School:Kungliga Tekniska högskolan

School Location:Sweden

Source Type:Doctoral Dissertation

Keywords:

ISBN:91-7170-395-0

Date of Publication:01/01/1999