Han-Tarantello-2014-Cal.Var.PDE
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Xiaosen Han · Gabriella Tarantello
Received: 25 September 2012 / Accepted: 2 March 2013 / Published online: 31 March 2013 © Springer-Verlag Berlin Heidelberg 2013
Calc. Var. (2014) 49:1149–1176 DOI 10.1007/s00526-013-0615-7
Calculus of Variations
Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model
123
1150
X. Han, G. Tarantello
nonlinearity and Dirac source terms. Within this framework we mention for example the (2 + 1)-dimensional Abelian Chern–Simons model of Hong et al. [23] and Jackiw–Weinberg [27], for which Taubes’ approach has led to the existence of topological multivortices (as described in [42,52]), non-topological multivortics (as constructed in [8,9,11,13,43]) and doubly periodic vortices (as given in [7,14,15,31,38,47,49,53]). In the non-Abelian context, rigorous existence results are established in [4,10,44,45], while a series of sharp existence results have been obtained in [12,30,32,33,48] for non-Abelian models proposed in connection with the quark confinement phenomenon [24,25,34,35]. For more results about self-dual vortices, we refer the readers to the monographs [46,55]. Here, we are going to analyze a relativistic (self-dual) non-Abelian Chern–Simons model proposed by Dunne in [16,17]. For this model, Yang [54] first established the existence of topological solutions in a very general situation. Subsequently, for the gauge group SU (3), Nolasco and Tarantello [37] proved a multiplicity result about the existence of doubly periodic vortices. The purpose of this paper is to establish analogous multiplicity results for theories that involve more general gauge groups. More precisely, we focus on gauge groups with a semi-simple Lie algebra of rank 2. From the technical point of view, we need to handle a 2 × 2 nonlinear elliptic system on the flat 2–torus, with coupling matrix given by the Cartan matrix associated to the gauge group. Clearly, this more general situation poses new analytical difficulties compared to the (already nontrivial) case analyzed in [37], where the authors handle a (specific) symmetric 2 × 2 system. Actually, we manage to resolve such difficulties for a larger class of 2 × 2 systems, where our vortex problem is included as a particular case. 2 Derivation of a general 2 × 2 nonlinear elliptic system and statement of the main results The non-Abelian Chern–Simons model introduced by Dunne in [16,17], is formulated over the Minkowski space R1+2 with metric tensor: diag(1, −1, −1), that will be used in the usual way to raise and lower indices. Using the summation convention over repeated lower and upper indices ( ranging over 0, 1, 2), we consider the Lagrangian density: κ 2 L = − Tr μνα ∂μ Aν Aα + Aμ Aν Aα 2 3 + Tr [ Dμ φ ]† [ D μ φ ] − V municated by P. Rabinowitw. X. Han Institute of Contemporary Mathematics, School of Mathematics, Henan University, Kaifeng 475004, Henan, People’s Republic of China G. Tarantello (B) Dipartimento di Matematica, Unversità di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy e-mail: tarantel@axp.mat.uniroma2.it
where Dμ = ∂μ + [ Aμ , ·] is the gauge-covariant derivative applied to the Higgs field φ in the adjoint representation of the gauge group G . The associated semi-simple Lie algebra is denoted by G , with [·, ·] the corresponding Lie bracket. Moreover, ( Aμ )μ=0,1,2 denotes the G -valued gauge fields and Tr refers to the trace in the matrix representation of G . As usual, we denote by κ > 0 the Chern–Simons coupling parameter, μνα the Levi–Civita totally skew-symmetric tensor with ε 012 = 1 and we let V be the Higgs potential. The Euler-Lagrange equations corresponding to (2.1) are given by Dμ D μ φ = κ Fμν with the strength tensor: Fμν = ∂μ Aν − ∂ν Aμ + [ Aμ , Aν ], (2.4) ∂V , ∂φ † = μνα J α , (2.2) (2.3)
Abstract In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions in a 2 × 2 nonlinear elliptic system arising in the study of self-dual nonAbelian Chern–Simons vortices. We show that the system admits at least two solutions when the Chern–Simons coupling parameter κ > 0 is sufficiently small; while no solution exists for κ > 0 sufficiently large. As in Nolasco and Tarantello (Commun Math Phys 213:599–639, 2000), we use a variational formulation of the problem. Thus, we obtain a first solution via a constrained minimization method and show that it is asymptotically gauge-equivalent to the (broken) principal embedding vacuum of the system, as κ → 0. Then we obtain a second solution by a min-max procedure of “mountain pass” type. Mathematics Subject Classification 1 Introduction As well known vortices play an important role in many areas of physics, including superconductivity [1,20,28], optics [5], cosmology [22,29,51], the quantum Hall effect [41], and quark confinement [24,25,34–36]. After the pioneering work of Bogomol’nyi [6] and Prasad– Sommerfield [40], rigorous mathematical results about the existence of vortices have been pursued in various self-dual gauge field theories on the basis of an analytical approach that Taubes introduced in [50] to treat the Abelian–Higgs model. Indeed following [50], one is able to reduce the vortex problem to second-order elliptic equations with exponential 35J47 · 35J50 · 35Q51 · 37K40
Received: 25 September 2012 / Accepted: 2 March 2013 / Published online: 31 March 2013 © Springer-Verlag Berlin Heidelberg 2013
Calc. Var. (2014) 49:1149–1176 DOI 10.1007/s00526-013-0615-7
Calculus of Variations
Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model
123
1150
X. Han, G. Tarantello
nonlinearity and Dirac source terms. Within this framework we mention for example the (2 + 1)-dimensional Abelian Chern–Simons model of Hong et al. [23] and Jackiw–Weinberg [27], for which Taubes’ approach has led to the existence of topological multivortices (as described in [42,52]), non-topological multivortics (as constructed in [8,9,11,13,43]) and doubly periodic vortices (as given in [7,14,15,31,38,47,49,53]). In the non-Abelian context, rigorous existence results are established in [4,10,44,45], while a series of sharp existence results have been obtained in [12,30,32,33,48] for non-Abelian models proposed in connection with the quark confinement phenomenon [24,25,34,35]. For more results about self-dual vortices, we refer the readers to the monographs [46,55]. Here, we are going to analyze a relativistic (self-dual) non-Abelian Chern–Simons model proposed by Dunne in [16,17]. For this model, Yang [54] first established the existence of topological solutions in a very general situation. Subsequently, for the gauge group SU (3), Nolasco and Tarantello [37] proved a multiplicity result about the existence of doubly periodic vortices. The purpose of this paper is to establish analogous multiplicity results for theories that involve more general gauge groups. More precisely, we focus on gauge groups with a semi-simple Lie algebra of rank 2. From the technical point of view, we need to handle a 2 × 2 nonlinear elliptic system on the flat 2–torus, with coupling matrix given by the Cartan matrix associated to the gauge group. Clearly, this more general situation poses new analytical difficulties compared to the (already nontrivial) case analyzed in [37], where the authors handle a (specific) symmetric 2 × 2 system. Actually, we manage to resolve such difficulties for a larger class of 2 × 2 systems, where our vortex problem is included as a particular case. 2 Derivation of a general 2 × 2 nonlinear elliptic system and statement of the main results The non-Abelian Chern–Simons model introduced by Dunne in [16,17], is formulated over the Minkowski space R1+2 with metric tensor: diag(1, −1, −1), that will be used in the usual way to raise and lower indices. Using the summation convention over repeated lower and upper indices ( ranging over 0, 1, 2), we consider the Lagrangian density: κ 2 L = − Tr μνα ∂μ Aν Aα + Aμ Aν Aα 2 3 + Tr [ Dμ φ ]† [ D μ φ ] − V municated by P. Rabinowitw. X. Han Institute of Contemporary Mathematics, School of Mathematics, Henan University, Kaifeng 475004, Henan, People’s Republic of China G. Tarantello (B) Dipartimento di Matematica, Unversità di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy e-mail: tarantel@axp.mat.uniroma2.it
where Dμ = ∂μ + [ Aμ , ·] is the gauge-covariant derivative applied to the Higgs field φ in the adjoint representation of the gauge group G . The associated semi-simple Lie algebra is denoted by G , with [·, ·] the corresponding Lie bracket. Moreover, ( Aμ )μ=0,1,2 denotes the G -valued gauge fields and Tr refers to the trace in the matrix representation of G . As usual, we denote by κ > 0 the Chern–Simons coupling parameter, μνα the Levi–Civita totally skew-symmetric tensor with ε 012 = 1 and we let V be the Higgs potential. The Euler-Lagrange equations corresponding to (2.1) are given by Dμ D μ φ = κ Fμν with the strength tensor: Fμν = ∂μ Aν − ∂ν Aμ + [ Aμ , Aν ], (2.4) ∂V , ∂φ † = μνα J α , (2.2) (2.3)
Abstract In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions in a 2 × 2 nonlinear elliptic system arising in the study of self-dual nonAbelian Chern–Simons vortices. We show that the system admits at least two solutions when the Chern–Simons coupling parameter κ > 0 is sufficiently small; while no solution exists for κ > 0 sufficiently large. As in Nolasco and Tarantello (Commun Math Phys 213:599–639, 2000), we use a variational formulation of the problem. Thus, we obtain a first solution via a constrained minimization method and show that it is asymptotically gauge-equivalent to the (broken) principal embedding vacuum of the system, as κ → 0. Then we obtain a second solution by a min-max procedure of “mountain pass” type. Mathematics Subject Classification 1 Introduction As well known vortices play an important role in many areas of physics, including superconductivity [1,20,28], optics [5], cosmology [22,29,51], the quantum Hall effect [41], and quark confinement [24,25,34–36]. After the pioneering work of Bogomol’nyi [6] and Prasad– Sommerfield [40], rigorous mathematical results about the existence of vortices have been pursued in various self-dual gauge field theories on the basis of an analytical approach that Taubes introduced in [50] to treat the Abelian–Higgs model. Indeed following [50], one is able to reduce the vortex problem to second-order elliptic equations with exponential 35J47 · 35J50 · 35Q51 · 37K40