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Instantons beyond topological theory. I

Published online by Cambridge University Press:  12 May 2011

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


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.

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Copyright © Cambridge University Press 2011

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