Duality in Complex sine-Gordon Theory

New aspects of the complex sine-Gordon theory are addressed through the reformulation of the theory in terms of the gauged Wess-Zumino-Witten action. A dual transformation between the theory for the coupling constant β > 0 and the theory for β < 0 is given which agrees with the Krammers-Wannier duality in the context of perturbed conformal field theory. The Bäcklund transform and the nonlinear superposition rule for the complex sine-Gordon theory are presented and from which, exact solutions, solitons and breathers with U(1) charge, are derived. We clarify topological and nontopological nature of neutral and charged solitons respectively, and discuss about the duality between the vector and the axial U(1) charges. 1 E-mail address; [email protected] 2 E-mail address; [email protected] The complex sine-Gordon theory, which generalizes the well-known sine-Gordon theory with an internal U(1) degree of freedom, first appeared as a model of relativistic vortices in a superfluid [1], and also independently in a treatment of O(4) nonlinear sigma model[2]. Recently, Bakas has shown that the complex sine-Gordon theory may be reformulated in terms of the gauged Wess-Zumino-Witten(WZW) action and interpreted the theory as the integrably deformed SU(2)/U(1)-coset model for the parafermions in the large N limit for the level N[3]. This led to subsequent generalizations of the sine-Gordon and the complex sine-Gordon theories to other coset cases[4][5][6] as well as generalizations of solitons and breathers with internal degrees of freedom[7]. Reformulation of the theory as the deformed coset model also provides a natural explanation for the behavior of exact factorizable Smatrix[8]. In this Letter, we report new aspects of the complex sine-Gordon theory which arises from the reformulation of the theory in terms of the gauged WZW action and a specific choice of the gauge described later. We find the exact duality between the theories for the coupling constant β > 0 and β < 0. We show that this agrees precisely with the Krammers-Wannier duality between the spin variables sj and the dual spin variables μj of ZN -parafermion theory[9]. We derive the Bäcklund transform and the nonlinear superposition rule for the complex sine-Gordon theory from the gauged WZW action in the gauge A = Ā = 0, from which exact solutions, solitons and breathers with U(1) charge, are obtained. We clarify the topological nature of soliton solutions. It is shown that charged solitons are in general nontopological solitons and become topological only when they become neutral. We also address the duality between the axial and the vector U(1) charges of the complex sine-Gordon theory. The complex sine-Gordon theory in terms of the gauged WZW action is given by I(g, A, Ā, β) = IWZW (g) + 1 2π ∫ Tr(−A∂̄gg−1 + Āg−1∂g + AgĀg−1 − AĀ) − β 2π ∫ TrgTg−1T̄ (1) where IWZW (g) is the SU(2)-WZW action for a map g : M → SU(2) on two-dimensional Minkowski space M . The connection A, Ā gauge the anomaly free diagonal subgroup U(1) of SU(2) and T = −T̄ = iσ3 = diag(i,−i) where σi; i = 1, 2, 3 are Pauli matrices. This action possesses the local U(1) vector symmetry as well as the global U(1) axial symmetry. The equation of motion of the action (1) takes a zero curvature form, δgI = − 1 2π ∫ Tr[ ∂ + g−1∂g + g−1Ag + βλT , ∂̄ + Ā+ 1 λ g−1T̄ g ]g−1δg = 0 (2) which, together with the constraint equation δAI(g, A, Ā) = 1 2π ∫ Tr( −∂̄gg−1 + gĀg−1 − Ā )δA = 0 The changing sign of coupling constants β under the Krammers-Wannier duality has been pointed out by Bakas and the duality between the two theories was suggested but without an explicit duality transform rule[3].