The freckled instantons (Q2755531)

The problem studied in this paper can be formulated as follows. Let us consider for simplicity the case of the pure \(\text{SU}(N)\) (twisted) supersymmetric gauge theory with \(\Phi\) be the complex adjoint scalar in the vector multiplet. Let \({\mathcal O}=\text{Tr}_V\Phi\) being the local operator in the gauge theory for an irreducible representation \(V\) of \(\text{SU}(N)\). The theory has low-energy effective description in terms of the \({\mathcal N}=2\) theory with \(r\equiv N-1\) abelian vector multiplets. The problem is to find a representative for the operators \({\mathcal O}_V\) and their descendents \(\int_{C_i} {\mathcal O}_V^{(i)}\) in terms of \(u_k\), where the parameters \(u_k\) are identified with the traces of \(\Phi\) in the representations \(V_k=\Lambda^{k+1}\mathbb{C}^N\), and other data of the low-energy theory. The authors expect to get a relation like: NEWLINE\[NEWLINE {\mathcal O}_V\rightarrow P_V(u_1,\ldots,u_r;\Lambda), NEWLINE\]NEWLINE where \(P_V(u_1,\ldots,u_r;\Lambda)\) is the polynomial, and the correspondence determined by the above relation can be considered as the four dimensional generalization of the well-known quantum cohomology rings of two dimensional supersymmetric sigma models. The authors propose to solve the above mentioned problem by means of the following scenario: if we have a theory I which has instantons of all sizes, and a theory II which has both instantons of all sizes and some other type of topological defects (which the authors call freckles) whose characteristic size is bounded from above by some parameter \(\rho\), then we integrate out all the fluctuations of the wavelengths smaller than \(\rho\) obtaining the identical instantonic field configurations in both theories (in this case the authors say that the theories belong to the same universality class). Now if the theory II is simpler than the theory I the correlation functions of the theory II can be used to compute the correlation functions of the theory I. In this paper the authors carry out this program for the supersymmetric \(\mathbb{C}\mathbb{P}^{N-1}\) model in two dimensions and then generalize to the four dimensional gauge theory. The theories I and II in these cases are respectively: nonlinear and gauged linear sigma model in two dimensions, non-abelian gauge theory and (conjecturally) the gauge theory on the noncommutative space in four dimensions.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00052].