Journal of the Institute of Mathematics of Jussieu

Research Article

Instantons beyond topological theory. I

E. Frenkela1, A. Loseva2 and N. Nekrasova3

a1 Department of Mathematics, University of California, Berkeley, CA 94720, USA (

a2 Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow 117259, Russia (

a3 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.

(Received October 07 2010)

(Revised March 12 2011)

(Accepted March 16 2011)


  • instanton;
  • quantum mechanics;
  • topological field theory;
  • Morse theory;
  • tempered distribution;
  • Epstein–Glaser regularization

AMS 2010 Mathematics subject classification

  • Primary 81Q35; 81T45;
  • Secondary 81T40; 81T60; 81Q12


To Pierre Schapira