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Instantons beyond topological theory. I

Part of: Quantum field theory; related classical field theories General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  12 May 2011

E. Frenkel ,
A. Losev  and
N. Nekrasov
E. Frenkel
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA ([email protected])
A. Losev
Affiliation:
Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow 117259, Russia ([email protected])
N. Nekrasov
Affiliation:
Institut des Hautes Études Scientifiques, 35, Route de Chartres, Bures-sur-Yvette, F-91440, France ([email protected])
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Abstract

Many quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyse the corresponding models as full quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kähler manifolds the space of states exhibits holomorphic factorization. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.

In Part I we describe the class of models under study and present our results in the case of one-dimensional (quantum mechanical) models, which is quite representative and at the same time simple enough to analyse explicitly. Part II will be devoted to supersymmetric two-dimensional sigma models and four-dimensional Yang–Mills theory. In Part III we will discuss non-supersymmetric models.

Keywords

instantonquantum mechanicstopological field theoryMorse theorytempered distributionEpstein–Glaser regularization

MSC classification

Secondary: 81Q35: Quantum mechanics on special spaces 81T40: Two-dimensional field theories, conformal field theories, etc.
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Special Issue 3: A Special Issue in Honour of Louis Boutet de Monvel and Pierre Schapira , July 2011 , pp. 463 - 565
Copyright
Copyright © Cambridge University Press 2011

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