Grassmannian Sigma Models Research Articles

Grassmannian Sigma Models

Grassmannian Sigma Models

Topological electrostatics

We present a theory of optimal topological textures in nonlinear sigma-models with degrees of freedom living in the Grassmannian manifold. These textures describe skyrmion lattices of N-component fermions in a quantising magnetic field, relevant to the physics of graphene, bilayer and other multicomponent quantum Hall systems near integer filling factors ν > 1. We derive analytically the optimality condition, minimizing topological charge density fluctuations, for a general Grassmannian sigma model on a sphere and a torus, together with counting arguments which show that for any filling factor and number of components there is a critical value of topological charge above which there are no optimal textures. Below a solution of the optimality condition on a torus is unique, while in the case of a sphere one has, in general, a continuum of solutions corresponding to new non-Goldstone zero modes, whose degeneracy is not lifted (via a order from disorder mechanism) by any fermion interactions depending only on the distance on a sphere. We supplement our general theoretical considerations with the exact analytical results for the case of , appropriate for recent experiments in graphene.

Grassmannian and flag sigma models on interval: phase structure and L-dependence

We discuss the two-dimensional Grassmannian SU(N)/S(U(N − 2) × U(2)) and the flag SU(N )/S(U(N − 2) × U(1) × U(1)) sigma models on a finite interval and construct analytical solutions of gap equations in the large-N limit. We show that the flag model admits a homogeneous solution for “mixed” Dirichlet-Neumann (DN) boundary conditions only for sufficiently large length L and undergoes a phase transition from the phase of partly broken gauge symmetry U(1) to the symmetric phase U(1) × U(1) for large L. On the other hand, the Grassmannian model has a detached phase with one massive and one massless non-zero condensates that completely break U(2) gauge symmetry. This phase lives on a region of L bounded from above and has to use the Robin boundary conditions. We also examine the L-dependence of the total energy and detect the linear growth inherent to confining string in all phases.

Grassmannian heterotic sigma model

We study the non-minimal supersymmetric heterotically deformed $\mathcal{N}=(0,2)$ sigma model with the Grassmannian target space $\mathcal{G}_{M,N}$. To develop the appropriate superfield formalism, we begin with a simplified model with flat target space, find its beta function up to two loops, and prove a non-renormalization theorem. Then we generalize the results to the full model with the Grassmannian target space. Using the geometric formulation, we calculate the beta functions and discuss the 't Hooft and Veneziano limits.

Quantum distillation of Hilbert spaces, semi-classics and anomaly matching

A symmetry-twisted boundary condition of the path integral provides a suitable framework for the semi-classical analysis of nonperturbative quantum field theories (QFTs), and we reinterpret it from the viewpoint of the Hilbert space. An appropriate twist with the unbroken symmetry can potentially produce huge cancellations among excited states in the state-sum, without affecting the ground states; we call this effect “quantum distillation”. Quantum distillation can provide the underlying mechanism for adiabatic continuity, by preventing a phase transition under S1 compactification. We revisit this point via the ’t Hooft anomaly matching condition when it constrains the vacuum structure of the theory on ℝd and upon compactification. We show that there is a precise relation between the persistence of the anomaly upon compactification, the Hilbert space quantum distillation, and the semi-classical analysis of the corresponding symmetry-twisted path integrals. We motivate quantum distillation in quantum mechanical examples, and then study its non-trivial action in QFT, with the example of the 2D Grassmannian sigma model Gr(N, M). We also discuss the connection of quantum distillation with large-N volume independence and flavor-momentum transmutation.

Holomorphic solutions of the susy Grassmannian σ-model and gauge invariance

We study the gauge invariance of the supersymmetric Grassmannian sigma model . It is richer then its purely bosonic submodel and we show how to use it in order to reduce some constant curvature holomorphic solutions of the model into simpler expressions.

Grassmannian sigma model on a finite interval

We discuss the two-dimensional Grassmannian sigma model $\mathbb{G}_{N, M}$ on a finite interval $L$. The different boundary conditions which allow to obtain analytical solutions by the saddle-point method in the large $N$ limit are considered. The nontrivial phase structure of the model on the interval similar to $\mathbb{C}P(N)$ model is found.

Non-BPS exact solutions and their relation to bions in ℂ P N − 1 models

We investigate non-BPS exact solutions in ${\mathbb C}P^{N-1}$ sigma models on ${\mathbb R}^1 \times S^{1}$ with twisted boundary conditions, by using the Din-Zakrzewski projection method. We focus on the relation of the non-BPS solutions to the ansatz of multi-instanton (bion) configurations and discuss their significance in the context of the resurgence theory. We find that the transition between seemingly distinct configurations of multi-instantons occur as moduli changes in the non-BPS solutions, and the simplest non-BPS exact solution corresponds to multi-bion configurations with fully-compressed double fractional instantons in the middle. It indicates that the non-BPS solutions make small but nonzero contribution to the resurgent trans-series as special cases of the multi-bion configurations. We observe a generic pattern of transitions between distinct multi-bion configurations (flipping partners), leading to the three essential properties of the non-BPS exact solution: (i) opposite sign for terms corresponding to the left and right infinities, (ii) symmetric location of fractional instantons, and (iii) the transition between distinct bion configurations. By studying the balance of forces, we show that the relative phases between the instanton constituents play decisive roles in stability and instability of the muli-instanton configurations. We discuss local and global instabilities of the solutions such as negative modes and the flow to the other saddle points, by considering the deformations of the non-BPS exact solutions within our multi-instanton ansatz. We also briefly discuss some classes of the non-BPS exact solutions in Grassmann sigma models.

Semilocal fractional instantons

We find semi-local fractional instantons of codimension four in Abelian and non-Abelian gauge theories coupled with scalar fields and the corresponding ${\mathbb C}P^{N-1}$ and Grassmann sigma models at strong gauge coupling. They are 1/4 BPS states in supersymmetric theories with eight supercharges, carry fractional (half) instanton charges characterized by the fourth homotopy group $\pi_4 (G/H)$, and have divergent energy in infinite spaces. We construct exact solutions for the sigma models and numerical solutions for the gauge theories. Small instanton singularity in sigma models is resolved at finite gauge coupling (for the Abelian gauge theory). Instantons in Abelian and non-Abelian gauge theories have negative and positive instantons charges, respectively, which are related by the Seiberg-like duality that changes the sign of the instanton charge.

Resurgence and dynamics of O(N) and Grassmannian sigma models

We study the non-perturbative dynamics of the two dimensional O(N ) and Grassmannian sigma models by using compactification with twisted boundary conditions on $$ \mathbb{R}\times {S}^1 $$ , semi-classical techniques and resurgence. While the O(N) model has no instantons for N > 3, it has (non-instanton) saddles on $$ {\mathbb{R}}^2 $$ , which we call 2d-saddles. On $$ \mathbb{R}\times {S}^1 $$ , the resurgent relation between perturbation theory and non-perturbative physics is encoded in new saddles, which are associated with the affine root system of the o(N ) algebra. These events may be viewed as fractionalizations of the 2d-saddles. The first beta function coefficient, given by the dual Coxeter number, can then be intepreted as the sum of the multiplicities (dual Kac labels) of these fractionalized objects. Surprisingly, the new saddles in O(N ) models in compactified space are in one-to-one correspondence with monopole-instanton saddles in SO(N ) gauge theory on $$ {\mathbb{R}}^3\times {S}^1 $$ . The Grassmannian sigma models Gr(N, M ) have 2d instantons, which fractionalize into N kink-instantons. The small circle dynamics of both sigma models can be described as a dilute gas of the one-events and two-events, bions. One-events are the leading source of a variety of non-perturbative effects, and produce the strong scale of the 2d theory in the compactified theory. We show that in both types of sigma models the neutral bion emulates the role of IR-renormalons. We also study the topological theta angle dependence in both the O(3) model and Gr(N, M ), and describe the multi-branched structure of the observables in terms of the theta-angle dependence of the saddle amplitudes, providing a microscopic argument for Haldane’s conjecture.

Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes

We classify bions in the Grassmann |$Gr_{N_{\rm F},N_{\rm C}}$| sigma model (including the |${\mathbb C}P^{N_{\rm F}-1}$| model) on |${\mathbb R}^{1}\times S^{1}$| with twisted boundary conditions. We formulate these models as |$U(N_{\rm C})$| gauge theories with |$N_{\rm F}$| flavors in the fundamental representations. These theories can be promoted to supersymmetric gauge theories and, further, can be embedded into D-brane configurations in type-II superstring theories. We focus on specific configurations composed of multiple fractional instantons, termed neutral bions and charged bions, which are identified as perturbative infrared renormalons by Ünsal and his collaborators [G. V. Dunne and M. Ünsal, J. High Energy Phys. 1211, 170 (2012); G. V. Dunne and M. Ünsal, Phys. Rev. D 87, 025015 (2013)]. We show that D-brane configurations, as well as the moduli matrix, offer a very useful tool to classify all possible bion configurations in these models. In contrast to the |${\mathbb C}P^{N_{\rm F}-1}$| model, there exist Bogomol’nyi–Prasad–Sommerfield (BPS) fractional instantons with topological charges greater than unity (of order |$N_{\rm C}$|⁠) that cannot be reduced to a composite of an instanton and fractional instantons. As a consequence, we find that the Grassmann sigma model admits neutral bions made of BPS and anti-BPS fractional instantons, each of which has a topological charge greater (less) than one (minus one), that are not decomposable into an instanton–anti-instanton pair and the rest. The |${\mathbb C}P^{N_{\rm F}-1}$| model is found to have no charged bions. In contrast, we find that the Grassmann sigma model admits charged bions, for which we construct exact non-BPS solutions of the field equations.

Geometry of surfaces associated to Grassmannian sigma models

We investigate the geometric characteristics of constant Gaussian curvature surfaces obtained from solutions of the G(m, n) sigma model. Most of these solutions are related to the Veronese sequence. We show that we can distinguish surfaces with the same Gaussian curvature using additional quantities like the topological charge and the mean curvature. The cases of G(1,n) = CPn-1 and G(2,n) are used to illustrate these characteristics.

Abelian and non-Abelian Hopfions in all odd dimensions

We extend the definition of the topological charge pertaining to the CP1 (i.e. O(3)) Skyrme-Fadde'ev Hopfion on ℝ3 to candidates for topological charges of ℂPn sigma models on ℝ2n+1 for all n. For this, the Abelian composite connections of the ℂPn sigma models are employed. In higher dimensions (n ≥ 1) it turns out that such charges, described by the nonAbelian composite connections of suitable Grassmannian sigma models, can also be constructed. A concrete discussion of the non-Abelian case for n = 2 is presented.

Constant curvature solutions of Grassmannian sigma models: (2) Non-holomorphic solutions

We generalize here our general procedure for constructing constant curvature maps of 2-spheres into Grassmannian manifolds G(m,n) this time concentrating our attention on maps which are non-holomorphic. We present some expressions describing these solutions in the general case and discuss how to use these results to construct solutions of constant curvature. We also discuss possible values of this constant curvature.

Constant curvature solutions of Grassmannian sigma models: (1) Holomorphic solutions

We present a general procedure for constructing constant curvature holomorphic maps of 2-spheres into Grassmannian manifolds G(m,n). Our procedure allows us to make a couple of conjectures as to the possible values of this curvature. We prove our conjectures for G(2,4), G(2,5), present explicit formulae for the relevant maps and show that they agree with those found by other methods. We also make some comments about the maps into G(2,n) for n≥6.

Canonical surfaces associated with projectors in Grassmannian sigma models

We discuss the construction of higher-dimensional surfaces based on the harmonic maps of S2 into CPN−1 and other Grassmannians. We show that there are two ways of implementing this procedure—both based on the use of the relevant projectors. We study various properties of such projectors and show that the Gaussian curvature of these surfaces, in general, is not constant. We look in detail at the surfaces corresponding to the Veronese sequence of such maps and show that for all of them this curvature is constant but its value depends on which mapping is used in the construction of the surface.

Quantum SUSY algebra of Q-lumps in the massive Grassmannian sigma model

We compute the SUSY algebra of the massive Grassmannian sigma model in (2+1) dimensions. We first rederive the action of the model by using Scherk–Schwarz dimensional reduction from theory in (3+1) dimensions. Then, we perform canonical quantization by using the Dirac method. We find that a particular choice of the operator ordering yields the quantum SUSY algebra of Q-lumps with central extension.

The Grassmannian sigma model in SU(2) Yang–Mills theory

Spin–charge separation in pure SU(2) Yang–Mills theory was recently found to involve the dynamics of an O(3) nonlinear sigma model and, seemingly, a Grassmannian nonlinear sigma model. In this paper we explicitly construct the Grassmannian sigma model of the form appearing in the spin–charge separated SU(2) theory through a quaternionic decomposition of the manifold, thus verifying its relevance in this context. The coupling between this model and the O(3) nonlinear sigma model is further commented upon.

Sigma models with N=8 supersymmetries in 2+1 and 1+1 dimensions

Abstract We introduce an N = 8 supersymmetric extension of the Bogomolny-type model for Yang–Mills–Higgs fields in 2 + 1 dimensions related with twistor string theory. It is shown that this model is equivalent to an N = 8 supersymmetric U ( n ) chiral model in 2 + 1 dimensions with a Wess–Zumino–Witten-type term. Further reduction to 1 + 1 dimensions yields N = ( 8 , 8 ) supersymmetric extensions of the standard U ( n ) chiral model and Grassmannian sigma models.

Surfaces Associated with Sigma Models

We present a unified method of construction of surfaces associated with Grassmannian sigma models, expressed in terms of an orthogonal projector. This description leads to compact formulae for structural equations of two‐dimensional surfaces immersed in the su(N) algebra. In the special case of the sigma model, we obtain constant negative Gaussian curvature surfaces. As a consequence, this leads us to an explicit relation between the sigma model and the sine‐Gordon equation.