Fractons in Twisted Multiflavor Schwinger Model.pdf

a r X i v : h e p - t h / 9 4 0 7 0 0 7 v 1 1 J u l 1 9 9 4 Theoretical Physics Institute University of Minnesota TPI-MINN-94/24-T UMN-TH-1262-94 July 1994 Fractons in Twisted Multiflavor Schwinger Model M.A. Shifman and A.V. Smilga1 Theoretical Physics Institute, Univ. of Minnesota, Minneapolis, MN 55455 Abstract We consider two-dimensional QED with several fermion flavors on a finite spatial circle. We discuss a modified version of the model with flavor-dependent boundary conditions ψp(L) = e 2πip/Nψp(0), p = 1, . . . , N where N is the number of flavors. In this case the Euclidean path integral acquires the contribution from the gauge field configurations with fractional topological charge being an integer multiple of 1/N . The configuration with ν = 1/N is responsible for the formation of the fermion condensate 〈ψ?pψp〉0. The condensate dies out as a power of L?1 when the length L of the spatial box is sent to infinity. Implications of this result for non-abelian gauge field theories are discussed. 1On leave of absence from ITEP, B. Cheremushkinskaya 25, Moscow, 117259, Russia 1 1 Motivation. Since the pioneering work [1], it is known that the Euclidean path integrals in the gauge field theories get contributions from sectors with nonzero topological charge. In the nonabelian 4-dimensional gauge theories the topological charge coincides with the so called Pontryagin class and is given by the integral ν4 = g2 32π2 ∫ d4xGaμνG? a μν . (1.1) It is generally assumed that the field densities Gμν contributing to the path integral are not singular and fall off fast enough when x2 = τ 2 + x2 is sent to infinity. In this case, the topological charge (1.1) must be an integer which describes the mapping S3 → gauge group. However, since the beginning of the eighties, different indications have been crop- ping up that the restriction ν4 = integer is too rigid, and in some cases configurations with the fractional topological charge may be relevant. The most explicit indica