A few remarks on chiral theories with sophisticated topology

Golo, V. L.; Perelomov, A. M.

doi:10.1007/bf00398500

Cited by 46 publications

(31 citation statements)

References 13 publications

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“…Nonlinear sigma models such as the CP N −1 model in 1+1 dimensions [1][2][3][4] are known to share a number of phenomena common with 3+1 dimensional QCD, e.g. asymptotic freedom, dynamical mass generation, confinement, and instantons [5][6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Self-consistent large-N analytical solutions of inhomogeneous condensates in quantum ℂPN − 1 model

Nitta

Yoshii

2017

J. High Energ. Phys.

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We give, for the first time, self-consistent large-N analytical solutions of inhomogeneous condensates in the quantum CP N −1 model in the large-N limit. We find a map from a set of gap equations of the CP N −1 model to those of the Gross-Neveu (GN) model (or the gap equation and the Bogoliubov-de Gennes equation), which enables us to find the self-consistent solutions. We find that the Higgs field of the CP N −1 model is given as a zero mode of solutions of the GN model, and consequently only topologically nontrivial solutions of the GN model yield nontrivial solutions of the CP N −1 model. A stable single soliton is constructed from an anti-kink of the GN model and has a broken (Higgs) phase inside its core, in which CP N −1 modes are localized, with a symmetric (confining) phase outside. We further find a stable periodic soliton lattice constructed from a real kink crystal in the GN model, while the Ablowitz-Kaup-Newell-Segur hierarchy yields multiple solitons at arbitrary separations.

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Section: Introductionmentioning

confidence: 99%

Self-consistent large-N analytical solutions of inhomogeneous condensates in quantum ℂPN − 1 model

Nitta

Yoshii

2017

J. High Energ. Phys.

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“…They can exist only for finite volumes. The explicit expression for the first sphaleron on a finite circle S 1 is given by the sigma field configuration induced by the two-dimensional instanton on the circle centered at the center of the instanton [21] and with radius a equal to the size of the instanton r [22]. In stereographic projection of polar coordinates, it reads z sph ͑x͒ ͑u 1 ve 2i arctan͑x͞2a͒ ͒͞ p 2 for any pair u and v of orthogonal unit vectors of C N 11 , i.e., u y u v y v 1 and u y v 0.…”

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confidence: 99%

Vacuum Structure ofCPNSigma Models atθ=π

Asorey

Falceto

1998

Phys. Rev. Lett.

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We show that parity symmetry is not spontaneously broken in the CP N sigma model for any value of N when the coefficient of the u term becomes u p ͑mod 2p͒. The result follows from a nonperturbative analysis of the nodal structure of the vacuum functional c 0 ͑z͒. The dynamical role of sphalerons turns out to be very important for the argument. The result introduces severe constraints on the possible critical behavior of the models at u p ͑mod 2p͒ [S0031-9007(97) The effect of a CP-violating u-term interaction on the infrared behavior of quantum field theories generated substantial changes in the low energy spectrum [1]. For this reason, phenomenological requirements impose severe bounds on the actual value of the u parameter. The analysis of the response of the systems to such a topological perturbation is, however, a very rich source of information on the vacuum structure of the quantum systems at u 0 [2]. This structure is very hard to analyze by direct methods because it dwells entirely on the strong coupling regime and involves deep nonperturbative behaviors. The most relevant changes appear at u p ͑mod 2p͒. In this case, the classical Lagrangian is also CP invariant as in the absence of the u term, u 0 ͑mod 2p͒, but the behavior of the system is completely different in these two regimes.Quantum chromodynamics (QCD), the archetype of such systems, is still inaccessible for analytic studies of its prominent physical effects: confinement and chiral symmetry breaking. CP N sigma models share many similar features, such as dynamical mass generation, confinement, and asymptotic freedom, but they are simpler to analyze. In particular, for some of those models there is exact analytic information on their quantum spectrum. It is, therefore, interesting to analyze the u-vacuum effects in those systems in order to gain some insight into the similar effects in QCD 311 .The simplest model, CP 1 , is integrable for u 0 and u p, the only two values of u for which the system is classically CP invariant. However, the behavior of the system is very different in those cases. At u 0 the system is confining and exhibits a mass gap; at u p the model is massless and its critical exponents are those of the SU(2) Wess-Zumino-Witten conformal invariant model at level 1 [3,4]. In both cases, the CP symmetry is not spontaneously broken. In the second case, this can be rigorously shown in a discrete regularization of the model, where it turns out to be equivalent to a spin 1͞2 chain by Haldane transformation [5]; for those chains the Lieb-Schulz-Mattis theorem [6] establishes that there are only two possibilities: Either parity is spontaneously broken and there is a mass gap or the theory is gapless and parity is not spontaneously broken. Since it is known that the mass gap is zero for u p [3-5,7], then parity is not spontaneously broken.For higher values of N the integrability of the CP N model is lost and the only nonperturbative information comes from either large N [8] and strong coupling expansions [9] or Monte Carlo numerical si...

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“…From (4.4) and (3.14) one sees at once that the vector ψ u transforms in the same way as ψ when either x 1 or x 2 is translated by 1/k, 6) where the matrices S µ have been introduced in (3.15) .…”

Section: Nahm Transform Of Gauge Fields On 2-dimensional Torus Tmentioning

confidence: 99%

“…The model has also been used to analyze the sphaleron induced fermion-number violation at high temperature [4]. By setting n = Ψ † σ Ψ with a normalized Ψ ∈ C 2 , the action (1.1) can be rewritten in the equivalent form [6]. In contrast to the O(N ) models they possess instanton solutions for all N, and a θ term can be added to the action so that their topological properties can be explored [7].…”

Section: Introductionmentioning

confidence: 99%

Nahm Transform and Moduli Spaces of PN-Models on the Torus

2002

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There is a Nahm transform for two-dimensional gauge fields which establishes a oneto-one correspondence between the orbit space of U (N ) gauge fields with topological charge k defined on a torus and that of U (k) gauge fields with charge N on the dual torus. The main result of this paper is to show that a similar duality transform cannot exist for CP analyzing the role of overlapping instantons in the infrared sector of CP N sigma models.

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A few remarks on chiral theories with sophisticated topology

Cited by 46 publications

References 13 publications

Self-consistent large-N analytical solutions of inhomogeneous condensates in quantum ℂPN − 1 model

Self-consistent large-N analytical solutions of inhomogeneous condensates in quantum ℂPN − 1 model

Vacuum Structure ofCPNSigma Models atθ=π

Nahm Transform and Moduli Spaces of PN-Models on the Torus

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