On the Topological Term in the String Representation of the Wilson Loop in the Dilute Instanton Gas

A topological term related to the number of self-intersections of the string world-sheet is shown to emerge in the string representation of the Wilson loop in the dilute instanton gas. The coupling constant of this term occurs to be proportional to the topological charge of the instanton gas under consideration. Recently, a string representation of the Wilson loop in the framework of the Method of Vacuum Correlators [1-3] has been proposed [4,5]. Within this approach, the expansion of the Wilson loop in powers of the derivatives with respect to the string world-sheet coordinates has been performed. In the lowest orders, this yielded the Nambu-Goto and the rigidity terms in the effective action, whose coupling constants were expressed in terms of the bilocal correlator of the gauge field strength tensors. In this way, the bare coupling constant of the rigidity term has been obtained to be negative, which is important for the stability of the string [6]. However, as it has been explained in Ref. [7], if one considers this coupling constant as a running one, this could lead to the problem of crumpling of the string world-sheet in the infrared region. A possible way of the solution of this problem, proposed by the author of Ref. [7], is based on the introduction of the so-called topological term, which is equal to the algebraic number of self-intersections of the string world-sheet, into the string effective action. Then by adjusting the coupling constant of this term, one can eventually arrange the cancellation of contributions into the string partition E-mail addresses: [email protected] and [email protected] On leave of absence from the Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117 218, Moscow, Russia. Supported by Graduiertenkolleg Elementarteilchenphysik and DFG-RFFI, grant 436 RUS 113/309/0. E-mail address: [email protected] Supported by DFG-RFFI, grant 436 RUS 113/309/0.