Gauged Sigma Models Research Articles

The geometry of gauged (super)conformal mechanics

Motivated by recently explored examples, we undertake a systematic study of conformal invariance in one-dimensional sigma models where an isometry group has been gauged. Perhaps surprisingly, we uncover classes of sigma models which are only scale invariant in their ungauged form and become fully conformally invariant only after gauging. In these cases the target space of the gauged sigma model satisfies a deformation of the well-known conformal geometry constraints. We consider bosonic models as well as their mathcal{N} = 1, 2, 4 supersymmetric extensions. We solve the quantum ordering ambiguities in implementing (super-) conformal symmetry on the physical Hilbert space. Examples of our general results are furnished by the D(2, 1; 0)-invariant Coulomb branch quiver models relevant for black hole physics.

Gauged sigma models and exceptional dressing cosets

Abstract The Poisson–Lie (PL) T-duality is a generalized T-duality based on the Lie algebra of the Drinfel’d double. In particular, when we consider the PL T-duality of a coset space, the dual space is found to be a generalized coset space, which is called the dressing coset. We investigate an extension of the dressing cosets to the U-duality setup. We propose the gauged actions for various branes in M-theory and type IIB theory, where the generalized metric is constructed by using the exceptional Drinfel’d algebra (EDA) and the gauge algebra is a certain isotropic subalgebra of the EDA. By eliminating the gauge fields, the gauged action reduces to the standard brane action on a certain reduced background, which we call the exceptional dressing coset. We also propose an alternative definition of the exceptional dressing cosets based on Sfetsos’s approach and reproduce a known example of non-Abelian T-duality in the U-duality-covariant formulation.

Self-dual CP(2) vortex-like solitons in the presence of magnetic impurities

We investigate the existence of vortex configurations in two gauged-$CP(2)$ models extended via the inclusion of magnetic impurities. In particular, we consider both the Maxwell-$CP(2)$ and the Chern-Simons-$CP(2)$ enlarged scenarios, separately. We choose a $CP(2)$-field configuration with a null topological charge not only in the simplest (free) case, but also when coupled to an Abelian gauge field. The implementation of the Bogomol'nyi-Prasad-Sommerfield (BPS) formalism shows that the effective models for such a configuration possess a self-dual structure which looks like those inherent to the gauged sigma models. Therefore, when the $CP(2)$ field is coupled to the Maxwell term, the corresponding total energy possesses both a well-defined Bogomol'nyi bound and a quantized magnetic flux. Further, when the $CP(2)$ scenario is gauged with the Chern-Simons action, the total electric charge is verified to be proportional to the quantized magnetic flux. In addition, the analysis verifies that the magnetic impurity contributes to the BPS potentials and appears in both of the models' BPS equations. Next, we introduce a Gaussian-type impurity and solve the self-dual equations via a finite-difference scheme. The resulting solutions present a nonmonotonic behavior that flips both the magnetic and electric fields. Finally, we discuss the topologically trivial solutions in the limit for which the impurity becomes a Dirac $\ensuremath{\delta}$ function.

A note on the third way consistent deformation of Yang-Mills theory

Three-dimensional Yang-Mills theory allows for a deformation quadratic in the field strengths which can not be integrated to a local action without auxiliary fields. Yet, its covariant divergence consistently vanishes after iterating the equation, realizing a spin-1 analogue of ‘minimal massive gravity’, which has been dubbed ‘third way consistent’. In this note, we show that after dualization of the three-dimensional gauge fields, the model possesses a natural action as a Chern-Simons coupled gauged sigma model. In this dual formulation, coupling to matter and to gravity becomes straightforward. As a direct application, we derive the coupling of the model to N=1 supergravity.

Poisson–Lie T-plurality for dressing cosets

Abstract The Poisson–Lie T-plurality is an equivalence of string theories on various cosets $\mathcal {D}/\tilde{G},\ \mathcal {D}/\tilde{G}^{\prime },\ldots$, where $\mathcal {D}$ is a Drinfel’d double and $\tilde{G}$, $\tilde{G}^{\prime },\ldots$ are maximal isotropic subgroups. This can be extended to the equivalence for dressing cosets, i.e., $F\backslash \mathcal {D}/\tilde{G},\ F\backslash \mathcal {D}/\tilde{G}^{\prime },\ldots$, where F is an isotropic subgroup of $\mathcal {D}$. We explore this extended Poisson–Lie T-plurality, clarifying the relation between several previous approaches. We propose a gauged sigma model for a general gauge group F and obtain the formula for the metric and the B-field on the dressing coset. Using this formula and an ansatz for the dilaton, we show that the Poisson–Lie (PL) T-plurality for dressing cosets (with spectator fields) is a symmetry of double field theory. The formula for the Ramond–Ramond field strength is also proposed such that the equations of motion for the Neveu–Schwarz–Neveu–Schwarz fields are transformed covariantly. In addition, we provide specific examples of the PL T-plurality for dressing cosets.

Classification of non-topological solutions of an elliptic equation arising from self-dual gauged Sigma model

<p style='text-indent:20px;'>Our purpose in this paper is to classify the non-topological solutions of equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta u +\frac{4e^u}{1+e^u} = 4\pi\sum\limits_{i = 1}^k n_i\delta_{p_i}-4\pi\sum^l\limits_{j = 1}m_j\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2,\;\;\;\;\;\;(E) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \{\delta_{p_i}\}_{i = 1}^k $\end{document}</tex-math></inline-formula> (resp. <inline-formula><tex-math id="M2">\begin{document}$ \{\delta_{q_j}\}_{j = 1}^l $\end{document}</tex-math></inline-formula>) are Dirac masses concentrated at the points <inline-formula><tex-math id="M3">\begin{document}$ \{p_i\}_{i = 1}^k $\end{document}</tex-math></inline-formula>, (resp. <inline-formula><tex-math id="M4">\begin{document}$ \{q_j\}_{j = 1}^l $\end{document}</tex-math></inline-formula>), <inline-formula><tex-math id="M5">\begin{document}$ n_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ m_j $\end{document}</tex-math></inline-formula> are positive integers. Denote <inline-formula><tex-math id="M7">\begin{document}$ N = \sum^k_{i = 1}n_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ M = \sum^l_{j = 1}m_j $\end{document}</tex-math></inline-formula> satisfying that <inline-formula><tex-math id="M9">\begin{document}$ N-M>1 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>Problem <inline-formula><tex-math id="M10">\begin{document}$ (E) $\end{document}</tex-math></inline-formula> arises from gauged sigma models and we first construct an extremal non-topological solution <inline-formula><tex-math id="M11">\begin{document}$ u $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M12">\begin{document}$ (E) $\end{document}</tex-math></inline-formula> with asymptotic behavior</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ u(x) = -2\ln |x|-2\ln\ln|x|+O(1)\quad{\rm as}\quad |x|\to+\infty $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and with total magnetic flux <inline-formula><tex-math id="M13">\begin{document}$ 4\pi (N-M-1) $\end{document}</tex-math></inline-formula>. And then we do the classification for non-topological solutions of <inline-formula><tex-math id="M14">\begin{document}$ (E) $\end{document}</tex-math></inline-formula> with finite magnetic flux. This solves a challenging long standing problem. We believe that our approach is novel and applies to other types of equations.</p>

Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.

The Geometry of the Space of BPS Vortex\u2013Antivortex Pairs

The gauged sigma model with target {mathbb {P}}^1, defined on a Riemann surface Sigma , supports static solutions in which k_{+} vortices coexist in stable equilibrium with k_{-} antivortices. Their moduli space is a noncompact complex manifold {textsf {M}}_{(k_{+},k_{-})}(Sigma ) of dimension k_{+}+k_{-} which inherits a natural Kähler metric g_{L^2} governing the model’s low energy dynamics. This paper presents the first detailed study of g_{L^2}, focussing on the geometry close to the boundary divisor D=partial , {textsf {M}}_{(k_{+},k_{-})}(Sigma ). On Sigma =S^2, rigorous estimates of g_{L^2} close to D are obtained which imply that {textsf {M}}_{(1,1)}(S^2) has finite volume and is geodesically incomplete. On Sigma ={mathbb {R}}^2, careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for g_{L^2} in the limits of small and large separation. All these results make use of a localization formula, expressing g_{L^2} in terms of data at the (anti)vortex positions, which is established for general {textsf {M}}_{(k_{+},k_{-})}(Sigma ). For arbitrary compact Sigma , a natural compactification of the space {{textsf {M}}}_{(k_{+},k_{-})}(Sigma ) is proposed in terms of a certain limit of gauged linear sigma models, leading to formulae for its volume and total scalar curvature. The volume formula agrees with the result established for mathrm{Vol}(mathsf{M}_{(1,1)}(S^2)), and allows for a detailed study of the thermodynamics of vortex-antivortex gas mixtures. It is found that the equation of state is independent of the genus of Sigma , and that the entropy of mixing is always positive.

Disk, interval, point: on constructions of quantum field theories with holomorphic action functionals

Bosonic quantum field theories with holomorphic action functionals are realized by two types of constructions involving supersymmetric quantum field theories, compactified on an interval in one type and compactified on a disk and deformed in the other. We establish the equivalence between the two types of constructions by reducing the disk to the interval and the interval to a point. As examples, we discuss constructions of zero-dimensional gauged sigma model, gauged quantum mechanics, gauged symplectic bosons in two dimensions, and Chern-Simons theory and its higher-dimensional variants.

Skyrmions and spin waves in frustrated ferromagnets at low applied magnetic field

A continuum model of frustrated ferromagnets is analyzed in detail in the regime of low applied magnetic field, $H_0<1/4$, where the ground state is a spatially varying conical spiral. By changing variables to a corotating spin field, the model is reformulated as a gauged sigma model in a fixed background gauge, allowing the construction of stable isolated Skyrmions, and stable multi-Skyrmion clusters, which approach the conical ground state at spatial infinity. Owing to the spatial anisotropy induced by the ground state, these Skyrmions exhibit only discrete symmetries, and are of neither N\'eel nor Bloch type. These Skyrmions are continuously connected to the more familar solutions in the high field regime ($H_0>1/4$), acquiring axial symmetry in the limit $H_0\rightarrow 1/4$. The propagation of small amplitude spin waves through the conical ground state is also analyzed and is found to depend strongly on both $H_0$ and propagation direction relative to the ground state. In contrast to spin waves in the high field regime ($H_0>1/4$) there is no spectral gap: waves may propagate with any angular frequency.

Asymptotic behaviors of governing equation of gauged sigma model for Heisenberg ferromagnet

In this note, we study weak solutions of equation (0.1)Δu=4eu1+eu−4π∑i=1Nδpi+4π∑j=1MδqjinR2,where {δpi}i=1N (resp. {δqj}j=1M ) are Dirac masses concentrated at the points pi,i=1,…,N, (resp. qj,j=1,…,M) and N−M>1. Eq. (0.1) represents a governing equation of gauged sigma model for Heisenberg ferromagnet. We show that it has a sequence of solutions uβ having behaviors as −βln|x|+O(1) at infinity with a free parameter β∈(2,2(N−M)), and our concern in this paper is to study the asymptotic behavior of bβ as β approaching the extremal values 2 and 2(N−M).

Momentum Sections in Hamiltonian Mechanics and Sigma Models

We show a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic manifold by considering gauged sigma models on a higher dimensional manifold.

Gauged sigma models and magnetic Skyrmions

We define a gauged non-linear sigma model for a 2-sphere valued field and a SU(2)SU(2) connection on an arbitrary Riemann surface whose energy functional reduces to that for critically coupled magnetic skyrmions in the plane, with arbitrary Dzyaloshinskii-Moriya interaction, for a suitably chosen gauge field. We use the interplay of unitary and holomorphic structures to derive a general solution of the first order Bogomol’nyi equation of the model for any given connection. We illustrate this formula with examples, and also point out applications to the study of impurities.

Categorical Equivalence and the Renormalization Group

AbstractIn this article we review how categorical equivalences are realized by renormalization group flow in physical realizations of stacks, derived categories, and derived schemes. We begin by reviewing the physical realization of sigma models on stacks, as (universality classes of) gauged sigma models, and look in particular at properties of sigma models on gerbes (equivalently, sigma models with restrictions on nonperturbative sectors), and ‘decomposition,’ in which two‐dimensional sigma models on gerbes decompose into disjoint unions of ordinary theories. We also discuss stack structures on examples of moduli spaces of SCFTs, focusing on elliptic curves, and implications of subtleties there for string dualities in other dimensions. In the second part of this article, we review the physical realization of derived categories in terms of renormalization group flow (time evolution) of combinations of D‐branes, antibranes, and tachyons. In the third part of this article, we review how Landau–Ginzburg models provide a physical realization of derived schemes, and also outline an example of a derived structure on a moduli spaces of SCFTs.

Chern-Simons gauged sigma model into &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbb{H}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and its self-dual equations

We propose the Chern-Simons gauged sigma model from $\mathbb{R}^{1+2}$ into the hyperbolic plane $\mathbb{H}^2$. We seek a static configuration of this model and derive self-dual equations. We also establish some existence results for solutions of the self-dual equations under appropriate boundary conditions near $\infty$

Algebraic formulation of higher gauge theory

In this paper, we present a purely algebraic formulation of higher gauge theory and gauged sigma models based on the abstract theory of graded commutative algebras and their morphisms. The formulation incorporates naturally Becchi - Rouet -Stora - Tyutin (BRST) symmetry and is also suitable for Alexandrov - Kontsevich - Schwartz-Zaboronsky (AKSZ) type constructions. It is also shown that for a full-fledged Batalin-Vilkovisky formulation including ghost degrees of freedom, higher gauge and gauged sigma model fields must be viewed as internal smooth functions on the shifted tangent bundle of a space-time manifold valued in a shifted L∞-algebroid encoding symmetry. The relationship to other formulations where the L∞-algebroid arises from a higher Lie groupoid by Lie differentiation is highlighted.

Higher groupoid bundles, higher spaces, and self‐dual tensor field equations

We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a self‐contained review on simplicial sets as models of (∞, 1)‐categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Ševera, that maps higher groupoids to ‐algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain six‐dimensional superconformal field theories via a Penrose–Ward transform of higher groupoid bundles over a twistor space. This construction reduces the search for non‐Abelian self‐dual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.

2d gauge theories and generalized geometry

We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra $\mathfrak{g}$ leads naturally to the appearance of the "generalized tangent bundle" $\mathbb{T}M \equiv TM \oplus T^*M$ by means of composite fields. Gauge transformations of the composite fields follow the Courant bracket, closing upon the choice of a Dirac structure $D \subset \mathbb{T}M$ (or, more generally, the choide of a "small Dirac-Rinehart sheaf" $\cal{D}$), in which the fields as well as the symmetry parameters are to take values. In these new variables, the gauge theory takes the form of a (non-topological) Dirac sigma model, which is applicable in a more general context and proves to be universal in two space-time dimensions: A gauging of $\mathfrak{g}$ of a standard sigma model with Wess-Zumino term exists, \emph{iff} there is a prolongation of the rigid symmetry to a Lie algebroid morphism from the action Lie algebroid $M \times \mathfrak{g}\to M$ into $D\to M$ (or the algebraic analogue of the morphism in the case of $\cal{D}$). The gauged sigma model results from a pullback by this morphism from the Dirac sigma model, which proves to be universal in two-spacetime dimensions in this sense.

Rigidly supersymmetric gauge theories on curved superspace

In this note we construct rigidly supersymmetric gauged sigma models and gauge theories on certain Einstein four-manifolds, and discuss constraints on these theories. In work elsewhere, it was recently shown that on some nontrivial Einstein four-manifolds such as AdS4, N = 1 rigidly supersymmetric sigma models are constrained to have target spaces with exact Kähler forms. Similarly, in gauged sigma models and gauge theories, we find that supersymmetry imposes constraints on Fayet-Iliopoulos parameters, which have the effect of enforcing that Kähler forms on quotient spaces be exact. We discuss the ‘background principle’ in this context. We also discuss general aspects of universality classes of gauged sigma models, as encoded by stacks, and also discuss affine bundle structures implicit in these constructions. In an appendix, we discuss how anomalies in four-dimensional gauge theories, such as those which play an important role in our analysis, can be recast in the language of stacks.

Quasi-topological gauged sigma models, the geometric Langlands program, and knots

We construct and study a closed, two-dimensional, quasi-topological (0, 2) gauged sigma model with target space a smooth G-manifold, where G is any compact and connected Lie group. When the target space is a flag manifold of simple G, and the gauge group is a Cartan subgroup thereof, the perturbative model describes, purely physically, the recently formulated mathematical theory of “Twisted Chiral Differential Operators”. This paves the way, via a generalized T -duality, for a natural physical interpretation of the geometric Langlands correspondence for simply-connected, simple, complex Lie groups. In particular, the Hecke eigensheaves and Hecke operators can be described in terms of the correlation functions of certain operators that underlie the infinite-dimensional chiral algebra of the flag manifold model. Nevertheless, nonperturbative worldsheet twisted-instantons can, in some situations, trivialize the chiral algebra completely. This leads to a spontaneous breaking of supersymmetry whilst implying certain delicate conditions for the existence of BeilinsonDrinfeld D-modules. Via supersymmetric gauged quantum mechanics on loop space, these conditions can be understood to be intimately related to a conjecture by Hohn-Stolz [1] regarding the vanishing of the Witten genus on string manifolds with positive Ricci curvature. An interesting connection to Chern-Simons theory is also uncovered, whence we would be able to (i) relate general knot invariants of three-manifolds and Khovanov homology to “quantum” ramified D-modules and Lagrangian intersection Floer homology; (ii) furnish physical proofs of mathematical conjectures by Seidel-Smith [2] and Gaitsgory [3, 4] which relate knots to symplectic geometry and Langlands duality, respectively. ∗E-mail: tan@ias.edu ar X iv :1 11 1. 06 91 v1 [ he pth ] 2 N ov 2 01 1