The Callias Index Formula Revisited
These lecture notes aim at providing a purely analytical and accessible proof of the Callias index formula. In various branches of mathematics (particularly, linear and nonlinear partial differential operators, singular integral operators, etc.) and theor
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Fritz Gesztesy Marcus Waurick
The Callias Index Formula Revisited
Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Austin Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and New York Catharina Stroppel, Bonn Anna Wienhard, Heidelberg
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More information about this series at http://www.springer.com/series/304
Fritz Gesztesy • Marcus Waurick
The Callias Index Formula Revisited
123
Marcus Waurick Institut fRur Analysis TU Dresden Sachsen Dresden, Germany
Fritz Gesztesy Dept of Mathematics University of Missouri Missouri Columbia, USA
ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-29976-1 DOI 10.1007/978-3-319-29977-8
ISSN 1617-9692 (electronic) ISBN 978-3-319-29977-8 (eBook)
Library of Congress Control Number: 2016942549 Mathematics Subject Classification (2010): 47A53, 47F05, 47B25 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
Preface
We revisit the Callias index formula in connection with supersymmetric Dirac-type operators H of the form
0 L HD L 0
in odd space dimensions n, originally derived in 1978, and prove that ind.L/ D
i 8 Z
.n1/=2
n 1 1 X lim "i :::i Œ.n 1/=2Š !1 2 i ;:::;i D1 1 n 1
Sn1
(1)
n
trCd .U.x/.@i1 U/.x/ : : : .@in1 U/.x//xin dn1 .x/;
where U.x/ WD j˚.x/j1 ˚.x/ D sgn.˚.x//;
x 2 Rn :
nO
Here the closed operator L in L2 .Rn /2 d is of the form L D Q C ˚; where Q :D Q ˝ Id D
X n
j;n @j Id ;
jD1
v
vi
Preface
with j;n , j 2 f1; : : : ; ng, elements of the Euclidean Dirac algebra, such that n D 2On or n D 2On C 1. Here ˚ is identified with I ˝ ˚, satisfying ˚ 2 Cb2 Rn I
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