We consider topological solitons in the CP^n sigma models in two space dimensions. In particular, we study "kinks", which are independent of one coordinate up to a rotation of the target space, and "chains", which are periodic in one coordinate up to a rotation of the target space. Kinks and chains both exhibit constituents, similar to monopoles and calorons in SU(n) Yang-Mills-Higgs and Yang-Mills theories. We examine the constituent structure using Lie algebras.
We describe the kink solitary waves of a massive non-linear sigma model with an ${\mathbb S}^2$ sphere as the target manifold. Our solutions form a moduli space of non-relativistic solitary waves in the long wavelength limit of ferromagnetic linear spin chains.
The stability of the kinks of the non-linear ${\mathbb S}^2$-sigma model discovered in Phys. Rev. Lett. 101(2008)131602 is discussed from several points of view. After a direct estimation of the spectra of the second-order fluctuation operators around topological kinks, first-order field equations are proposed to distinguish between BPS and non-BPS kinks. The one-loop mass shifts caused by quantum fluctuations around the topological kinks are computed using the Cahill-Comtet-Glauber formula proposed in Phys. Lett. 64B(1976)283. The (lack of) stability of the non-topological kinks is unveiled b…
In this work, we investigate the solutions of vortices in the O(3)-sigma model with the gauge field governed by the Chern-Simons term and subject to a hyperbolic self-dual potential. We show that this model admits both topological and nontopological solitons solutions. By means of numerical analysis, we realize that the topological solutions of the model can be transformed into compacton-like solutions. On the other hand, after modifying the model by the introduction of a dielectric constant, an interesting feature appears; namely, the nontopological solutions can be transformed into kink-like…
Massive maximally-supersymmetric sigma models are shown to exhibit multiple static kink-domain wall solutions that preserve 1/2 of the supersymmetry. The kink moduli space admits a natural Kahler metric. We examine in some detail the case when the target of the sigma model is given by the co-tangent bundle of CP^n equipped with the Calabi metric, and we show that there exist BPS solutions corresponding to n kinks at arbitrary separation. We also describe how 1/4-BPS charged and intersecting domain walls are described in the low-energy dynamics on the kink moduli space. We comment on the simila…
Massive D=4 N=2 supersymmetric sigma models typically admit domain wall (Q-kink) solutions and string (Q-lump) solutions, both preserving 1/2 supersymmetry. We exhibit a new static 1/4 supersymmetric `kink-lump' solution in which a string ends on a wall, and show that it has an effective realization as a BIon of the D=4 super DBI-action. It is also shown to have a time-dependent Q-kink-lump generalization which reduces to the Q-lump in a limit corresponding to infinite BI magnetic field. All these 1/4 supersymmetric sigma-model solitons are shown to be realized in M-theory as calibrated, or `Q…
We identify the kinks of a deformed O(3) linear Sigma model as the solutions of a set of first-order systems of equations; the above model is a generalization of the MSTB model with a three-component scalar field. Taking into account certain kink energy sum rules we show that the variety of kinks has the structure of a moduli space that can be compactified in a fairly natural way. The generic kinks, however, are unstable and Morse Theory provides the framework for the analysis of kink stability.
It is shown how a integrable mechanical system provides all the localized static solutions of a deformation of the linear O(N)-sigma model in two space-time dimensions. The proof is based on the Hamilton-Jacobi separability of the mechanical analogue system that follows when time-independent field configurations are being considered. In particular, we describe the properties of the different kinds of kinks in such a way that a hierarchical structure of solitary wave manifolds emerges for distinct N.
In this paper the whole kink varieties arising in several massive non-linear Sigma models whose target space is the torus ${\mathbb S}^1\times{\mathbb S}^1$ are analytically calculated. This possibility underlies the construction of first-order differential equations by adapting the Bogomolny procedure to non-Euclidean spaces. Among the families of solutions non-topological kinks connecting the same vacuum are found. This class of solutions are usually considered to be not globally stable. However, in this context the topological constraints obtained by the non-simply connectedness of the targ…
One-loop mass shifts to the classical masses of stable kinks arising in a massive non-linear ${\mathbb S}^2$-sigma model are computed. Ultraviolet divergences are controlled using the heat kernel/zeta function regularization method. A comparison between the results achieved from exact and high-temperature asymptotic heat traces is analyzed in depth.
The solitary waves of massive (1+1)-dimensional nonlinear S^N-sigma models are unveiled. It is shown that the solitary waves in these systems are in one-to-one correspondence with the separatrix trajectories in the repulsive N-dimensional Neumann mechanical problem. There are topological (heteroclinic trajectories) and non-topological (homoclinic trajectories) kinks. The stability of some embedded sine-Gordon kinks is discussed by means of the direct estimation of the spectra of the second-order fluctuation operators around them, whereas the instability of other topological and non-topological…
We use the generalized zeta function regularization method to compute the one-loop quantum correction to the masses of the TK1 and TK2 kinks in a deformation of the O(N) linear sigma model on the real line.
The complete one-loop planar dilatation operator of N=4 supersymmetric Yang-Mills is isomorphic to the hamiltonian of an integrable PSU(2,2|4) quantum spin chain. We construct the non-linear sigma models describing the continuum limit of the SU(1|3) and SU(2|3) sectors of the complete N=4 chain. We explicitely identify the spin chain sigma model with the one for a superstring moving in AdS_5xS^5 with large angular momentum along the five-sphere.
The ferromagnetic integrable SU(3) spin chain provides the one loop anomalous dimension of single trace operators involving the three complex scalars of N=4 supersymmetric Yang-Mills. We construct the non-linear sigma model describing the continuum limit of the SU(3) spin chain. We find that this sigma model corresponds to a string moving with large angular momentum in the five-sphere in AdS_5xS^5. The energy and spectrum of fluctuations for rotating circular strings with angular momenta along three orthogonal directions of the five-sphere is reproduced as a particular case from the spin chain…
In this paper, we have studied the kink and antikink solutions in several neutral scalar models in 1+1 dimension. We follow the standard approach to write down the leading order and the second order force between long distance separated kink and antikink. The leading order force is proportional to exponential decay with respect to the distance between the two nearest kinks or antikinks. The second order force have a similar behavior with the larger decay factor, namely $3\over 2$. We make use of these properties to construct the kink lattice. The dynamics of the kink lattice with leading order…
From the SU(2) spin chain sigma model at the one-loop and two-loop orders we recover the classical circular string solution with two S^5 spins (J_1, J_2) in the AdS_5 x S^5 string theory. In the SL(2) sector of the one-loop spin chain sigma model we explicitly construct a solution which corresponds to the folded string solution with one AdS_5 spin S and one S^5 spin J. In the one-loop general sigma model we demonstrate that there exists a solution which reproduces the energy of the circular constant-radii string solution with three spins (S_1, S_2, J).
We derive the coherent state representation of the integrable spin chain Hamiltonian with non-compact supersymmetry group G=SU(1,1|1). By passing to the continuous limit, we find a spin chain sigma model describing a string moving on the supercoset G/H, H being the stabilizer group. The action is written in a manifestly G-invariant form in terms of the Cartan forms and the string coordinates in the supercoset. The spin chain sigma model is shown to agree with that following from the Green-Schwarz action describing two-charged string spinning on AdS_5 x S^5.
We derive the coherent state representation of the integrable spin chain Hamiltonian with supersymmetry group $SU(1,1|2)$. By the use of a projected Hamiltonian onto bosonic states, we give explicitly the action of the Hamiltonian on $SU(2)\times SL(2)$ coherent states. Passing to the continuous limit, we find that the corresponding bosonic sigma model is the sum of the known SU(2) and SL(2) ones, and thus it gives a string spinning fast on $S^1_{ϕ_1}\times S^1_{ϕ_1}\times S^1_{ϕ_2}$ in $\rm{AdS}_5 \times S^5$. The full sigma model on the supercoset $SU(1,1|2)/SU(1|1)^2$ is given.
We propose that the spin-chain with the PSU(2|2)xU(1)^3 symmetry is equivalent to the non-linear sigma-model on PSU(2|2)xU(1)^3/{HxU(1)} with a certain subgroup. To this end we show that the spin-variable of the former theory is identified as the Killing scalar of the latter and their correlation functions can have the same integrability. It is crucial to think that the respective theory gets the PSU(2|2)xU(1)^3 symmetry by a symmetry reduction the exceptional supergroup D(2,1;gamma), rather than by an extension of PSU(2|2).
We derive the asymptotic Bethe ansatz (AFS equations) for the string on S^3 x R sector of AdS_5 x S^5 from the integrable nonhomogeneous dynamical spin chain for the string sigma model proposed in GKSV. It is clear from the derivation that AFS equations can be viewed only as an effective model describing a certain regime of a more fundamental inhomogeneous spin chain.
We consider lambda and anisotropic deformations of the SU(2) principal chiral model and show how they can be quantized in the Hamiltonian formalism on a lattice as a suitable spin chain. The spin chain is related to the higher spin XXZ Heisenberg chain and can be solved by using the Bethe Ansatz. This yields the spectrum and S-matrix of the excitations. In particular, we find the S-matrix in the gapped anti-ferromagnetic regime. In this regime, a continuum limit does not exist and this suggests that the field theories in this regime, precisely ones with a cyclic RG like the Yang-Baxter deforma…
We argue that the world-sheet dynamics of magnetic k-strings in the Higgs phase of the mass-deformed N=4 theory, is controlled by a bosonic O(3) sigma model with anisotropy and a topological theta term. The theory interpolates between a massless O(2) symmetric regime, a massive O(3) symmetric phase and another massive phase with a spontaneously broken Z_2 symmetry. The first two phases are separated by a Kosterlitz-Thouless transition. When θ= πthe O(3) symmetric phase flows to an interacting fixed point; sigma model kinks and their dyonic partners become degenerate, mirroring the behaviour of…
Classical strings propagating in $AdS_{3}\times S^{3}\times T^{4}$ supported with Neveu-Schwarz-Neveu-Schwarz flux are described by a Wess-Zumino-Witten model. In this note, we study the emergence of their semiclassical $SU(2)$ spectrally flowed sectors as the Landau-Lifshitz limit of the underlying quantum spin chain. We consider the propagator in the coherent state picture, and find that the time interval is discretized proportionally to the lattice spacing. In the Landau-Lifshitz limit, where both time and space become continuous, we derive a path integral representation of the propagator f…
We derive the coherent state representation of the integrable spin chain Hamiltonian with symmetry group SL(2,R). By passing to the continuum limit, we find a spin chain sigma model describing a string moving on the hyperboloid SL(2,R)/U(1). The same sigma model is found by considering strings rotating with large angular momentum in AdS_5xS^5. The spinning strings are identified with semiclassical coherent states built out of SL(2,R) spin chain states.
We define and study a lattice model which we argue is in the universality class of the $OSp(2S+2|2S)$ supercoset sigma model for a large range of values of the coupling constant $g_σ^2$. In this first paper, we analyze in details the symmetries of this lattice model, in particular the decomposition of the space of the quantum spin chain $V^{\otimes L}$ as a bimodule over $OSp(2S+2|2S)$ and its commutant, the Brauer algebra $B_L(2)$. It turns out that $V^{\otimes L}$ is a nonsemisimple module for both $OSp(2S+2|2S)$ and $B_L(2)$. The results are used in the companion paper to elucidate the stru…
We propose a new multiple integral representation for the correlation function <sigma_1^z sigma_{m+1}^z> of the XXZ spin-1/2 Heisenberg chain in the disordered regime. We show that for Delta=1/2 the integrals can be separated and computed exactly. As an example we give the explicit results up to the lattice distance m=8. It turns out that the answer is given as integer numbers divided by 2^[(m+1)^2].
This work concerns the quantum Lorentzian and Euclidean black hole non-linear sigma models. For the Euclidean black hole sigma model an equilibrium density matrix is proposed, which reproduces the modular invariant partition function from the 2001 paper of Maldacena, Ooguri and Son. For the Lorentzian black hole sigma model, using its formulation as a gauged ${\rm SL}(2,\mathbb{R})$ WZW model, we describe the linear and Hermitian structure of its space of states and also propose an expression for the equilibrium density matrix. Our analysis is guided by the results of the study of a certain cr…
The sigma models on projective superspaces CP^{N+M-1|N} with topological angle theta=pi mod 2pi flow to non-unitary, logarithmic conformal field theories in the low-energy limit. In this paper, we determine the exact spectrum of these theories for all open boundary conditions preserving the full global symmetry of the model, generalizing recent work on the particular case M=0 [C. Candu et al, JHEP02(2010)015]. In the sigma model setting, these boundary conditions are associated with complex line bundles, and are labelled by an integer, related with the exact value of theta. Our approach relies…
One dimensional SU($n$) chains with the same irreducible representation $\mathcal{R}$ at each site are considered. We determine which $\mathcal{R}$ admit low-energy mappings to a $\text{SU}(n)/[\text{U}(1)]^{n-1}$ flag manifold sigma model, and calculate the topological angles for such theories. Generically, these models will have fields with both linear and quadratic dispersion relations; for each $\mathcal{R}$, we determine how many fields of each dispersion type there are. Finally, for purely linearly-dispersing theories, we list the irreducible representations that also possess a $\mathbb{…
We introduce a spin chain based on finite-dimensional spin-1/2 SU(2) representations but with a non-hermitian `Hamiltonian' and show, using mostly analytical techniques, that it is described at low energies by the SL(2,R)/U(1) Euclidian black hole Conformal Field Theory. This identification goes beyond the appearance of a non-compact spectrum: we are also able to determine the density of states, and show that it agrees with the formulas in [J. Math. Phys. 42, 2961 (2001)] and [JHEP 04, 014 (2002)], hence providing a direct `physical measurement' of the associated reflection amplitude.
N=2 SQED with several flavors admits multiple, static BPS domain wall solutions. We determine the explicit two-kink metric and examine the dynamics of colliding domain walls. The multi-kink metric has a toric Kahler structure and we reduce the Kahler potential to quadrature. In the second part of this paper, we consider semi-local vortices compactified on circle. We argue that, in the presence of a suitable Wilson line, the vortices separate into domain wall constituents. These play the role of fractional instantons in two-dimensional gauge theories and sigma-models.
This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar $CP^{n-1}$ and Grassmannian models. They naturally arise in the description of continuum limits of spin chains, and their phase structure is sensitive to the values of the topological angles, which are determined by the representations of spins in the chain. Gapless phases can in certain cases be explained by the presence of discrete 't Hooft anomalies in the continuum theory. We also discuss integrable flag manifold sigma models, which provide a generalization o…
We review the non linear sigma model approach (NLSM) to spin chains and spin ladders, presenting new results. The generalization of the Haldane's map to ladders in the Hamiltonian approach, give rise to different values of the $θ$ parameter depending on the spin S, the number of legs $n_{\ell}$ and the choice of blocks needed to built up the NLSM fields. For rectangular blocks we obtain $θ= 0 $ or $2 πS$ depending on wether $n_{\ell}$, is even or odd, while for diagonal blocks we obtain $θ= 2 πS n_{\ell}$. Both results agree modulo $2 π$, and yield the same prediction, namely that even ( resp.…
The well known Haldane map from spin chains into the $O(3)$ non linear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, exactly in the same way it explains the difference between integer and half-integer spin chains. Namely, for even ladders the topological term in the sigma model action is absent, while for odd ladders the $θ$ parameter, which multiplies the topological term, is equal to $2 πS$, where $S$ is the spin of the ladder. Hence even ladders should have a dynamically generated spin …
Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯特克洛夫数学研究所）；Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University（莫斯科国立大学理论与数学物理研究所）；HSE University（高等经济大学）；Moscow Center of Fundamental and Applied Mathematics, Lomonosov Moscow State University（莫斯科国立大学莫斯科基础与应用数学中心）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）
It was previously established that the critical staggered XXZ spin chain provides a lattice regularization of the black hole CFT. We reconsider the continuum limit of this spin chain with the exact method of non-linear integral equations (NLIEs), paying particular attention to the effects of a singular integration kernel. With the help of the NLIEs, we rederive the continuous black hole spectrum, but also numerically match the density of states of the spin chain with that of the CFT, which is a new result. Finally, we briefly discuss the integrable structure of the black hole CFT and the ident…
Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains
