The behavior of the chiral symmetry order parameter ($$) in the (1 + 1)-dimensional Schwinger model at finite chemical potential is studied. We find an inhomogeneous chiral condensate.

W e study the one-loop contribution t o the induced parity-odd part ~, , ,~~a '~' (~) o f the gauge field self-energy in ( 2 + 1 )-dimensional gauge theories at a finite temperature T . local Chern-Simons term. It is pointed out that a'O'(p) is not an analytic function at zero frequency and momentum when T f O . Therefore, at T f O , there exist no induced localChern-Simons terms other than those due to regularization.PACS number(s): 11.15.Bt

By quantizing an open string ending on a D-brane in a nontrivial supergravity background, we argue that there is a new kind of uncertainty relation on a D-brane worldvolume. Furthermore, we fix the form of the uncertainty relations and their dependence on the string coupling constant by requiring them to be consistent with various string theory and M theory dualities. In this way we find a web of uncertainties of spacetime for all kinds of brane probes, including fundamental strings, D-branes of all dimensions as well as M theory membranes and fivebranes.

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We show that no topological mass terms are induced radiatively to the two-loop order in either Abelian or non-Abelian gauge theories of 2 + 1 dimensions with massive ferrnions if the Pauli-Villars regularization is used consistently for ferrnion loops. We study the case where gauge bosons have no topological mass at the tree level.Deser, Jackiw, and Templeton' expounded the significance of the topological mass terms (Chern-Simons invariants) for gauge fields in 2 + 1 dimension^.^ They pointed out, among other things, that the topological mass terms could be generated in general through fermion loop corrections even if the bare Lagrangians do not contain such terms. The conclusion was, however, dependent on how to regularize divergent integrals. It was shown1 that for the massive fermions no topological mass is induced for gauge bosons in the one-loop order if the Pauli-Villars regularization is used with the regulator mass having the same sign as the physical fermion mass.3 Whether a gauge boson mass is generated in higher-order loops or not4 has an important implication in the nature of confining forces in ( 2 + 1)dimensional gauge theories. It may also have a direct relevance to condensed matter physics of thin films and possibly to the quantized Hall e f f e~t .~ In this Brief Report we present our results of a two-loop calculation for the topological mass terms in the Abelian and non-Abelian gauge theories of 2 + 1 dimensions, where fermions have nonvanishing masses. Starting with a vanishing gauge boson mass at the tree level, we have found that n o gauge boson masses are induced radiatively to the two-loop order when the Pauli-Villars regularization is used consistently for all fermion loops. There are a few general arguments to suggest that the coefficients of the Chern-Simons invariants should not be subject to radiative corrections. We will briefly look into these arguments in the conclusion.We consider the Abelian and non-Abelian gauge theories with massive fermions, but without topological mass term at the tree level. They are described by the Lagrangians in the standard notations. The diagrams to be calculated are shown in Fig. 1. In the Abelian case, only diagrams (a) and (b) need to be calculated. The proper self-energy parts of the gauge bosons II,, ( k ) take the form n,,(k)-(g,,k2--k,k,)II'"(k2) + i~, ,~ ( k2) + k, , k,II(3)( k2) .If 1 1 ( 2 ) (~) ;to, the gauge boson acquires mass. The quantity n ( 2 ) ( 0 ) appears in effective actions as the coefficients of S d 3 x ( 1/4e2) E,,,FP"A~ in the Abelian case and in the non-Abelian case.