General Sigma Model Research Articles

The Chiral Medium in a Generalized Sigma Model and the Chiral Perturbation Theory

The Chiral Medium in a Generalized Sigma Model and the Chiral Perturbation Theory

Duality, Generalized Global Symmetries and Jet Space Isometries

We revisit universal features of duality in linear and nonlinear relativistic scalar and Abelian 1-form theories with single or multiple fields, which exhibit ordinary or generalized global symmetries. We show that such global symmetries can be interpreted as generalized Killing isometries on a suitable, possibly graded, target space of fields or its jet space when the theory contains higher derivatives. This is realized via a generalized sigma model perspective motivated from the fact that higher spin particles can be Nambu–Goldstone bosons of spontaneously broken generalized global symmetries. We work out in detail the 2D examples of a compact scalar and the massless Heisenberg pion fireball model and the 4D examples of Maxwell, Born–Infeld, and ModMax electrodynamics. In all cases we identify the ’t Hooft anomaly that obstructs the simultaneous gauging of both global symmetries and confirm the anomaly matching under duality. These results readily generalize to higher gauge theories for p-forms. For multifield theories, we discuss the transformation of couplings under duality as two sets of Buscher rules for even or odd differential forms.

Generalized sigma model with dynamical antisymplectic potential and non-Abelian de Rham's differential

For topological sigma models, we propose that their local Lagragian density is allowed to depend non-linearly on the de Rham's "velocities" $D Z^{A}$. Then, by differentiating the Lagrangian density with respect to the latter de Rham's "velocities", we define a "dynamical" anti-symplectic potential, in terms of which a "dynamical" anti-symplectic metric is defined, as well. We define the local and the functional antibracket via the dynamical anti-symplectic metric. Finally, we show that the generalized action of the sigma model satisfies the functional master equation, as required.

Spontaneous violation of spatial parity in dense baryonic matter in a generalized U(2)L×U(2)R sigma model

Based on a generalized U(2)L×U(2)R sigma model describing ground and excited states of scalar and pseudoscalar meson singlets and triplets, we investigate the conditions for spatial parity violation for dense baryonic matter in the hadron phase. We introduce’ t Hooft-type vertices into the model for violating the U(1)A symmetry of quantum chromodynamics by the anomaly. At large values of the chemical potential, broken-parity states can appear in the model, which corresponds to condensation of pseudoscalar fields. We give the conditions on the parameters of the generalized sigma model that determine the transition to the phase with spontaneously broken parity. For different signs of the constants of two’ t Hooft-type vertices, either pseudoscalar isotriplets or isosinglets undergo condensation.

The common origin of linear and nonlinear chiral multiplets in [formula omitted] mechanics

Elaborating on previous work [F. Delduc, E. Ivanov, Nucl. Phys. B 753 (2006) 211, hep-th/0605211; F. Delduc, E. Ivanov, Nucl. Phys. B 770 (2007) 179, hep-th/0611247], we show how the linear and nonlinear chiral multiplets of N = 4 supersymmetric mechanics with the off-shell content ( 2 , 4 , 2 ) can be obtained by gauging three distinct two-parameter isometries of the “root” ( 4 , 4 , 0 ) multiplet actions. In particular, two different gauge groups, one Abelian and one non-Abelian, lead, albeit in a disguised form in the second case, to the same (unique) nonlinear chiral multiplet. This provides an evidence that no other nonlinear chiral N = 4 multiplets exist. General sigma model type actions are discussed, together with the restricted potential terms coming from the Fayet–Iliopoulos terms associated with Abelian gauge superfields. As in our previous work, we use the manifestly supersymmetric language of N = 4 , d = 1 harmonic superspace. A novel point is the necessity to use in parallel the λ and τ gauge frames, with the “bridges” between these two frames playing a crucial role. It is the N = 4 harmonic analyticity which, though being non-manifest in the τ frame, gives rise to both linear and nonlinear chirality constraints.

BiHermitian supersymmetric quantum mechanics

BiHermitian geometry, discovered long ago by Gates, Hull and Roček, is the most general sigma model target space geometry allowing for (2, 2) world sheet supersymmetry. In this paper, we work out supersymmetric quantum mechanics for a biHermitian target space. We display the full supersymmetry of the model and illustrate in detail its quantization procedure. Finally, we show that the quantized model reproduces the Hodge theory for compact twisted generalized Kähler manifolds recently developed by Gualtieri in [33]. This allows us to recover and put in a broader context the results on the biHermitian topological sigma models obtained by Kapustin and Li in [9].

The biHermitian topological sigma model

BiHermitian geometry, discovered long ago by Gates, Hull and Roceck, is the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target space, we write down the explicit expression of the sigma model's action and BRST transformations and present a computation of the topological gauge fermion and the topological action.

A topological sigma model of biKähler geometry

BiKaehler geometry is characterized by a Riemannian metric g_{ab} and two covariantly constant generally non commuting complex structures K_+^a_b, K_-^a_b, with respect to which g_{ab} is Hermitian. It is a particular case of the biHermitian geometry of Gates, Hull and Roceck, the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. We present a sigma model for biKaehler geometry that is topological in the following sense: i) the action is invariant under a fermionic symmetry delta; ii) delta is nilpotent on shell; iii) the action is delta--exact on shell up to a topological term; iv) the resulting field theory depends only on a subset of the target space geometrical data. The biKaehler sigma model is obtainable by gauge fixing the Hitchin model with generalized Kaehler target space. It further contains the customary A topological sigma model as a particular case. However, it is not seemingly related to the (2,2) supersymmetric biKaehler sigma model by twisting in general.

Boundary integrability of nonlinear sigma models

We describe recent work on the classical integrability of the principal chiral model and general sigma models with boundaries in (compact) symmetric spaces.

Circular and Folded Multi-Spin Strings in Spin Chain Sigma Models

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 S5 spins (J1,J2) in the AdS5 ? S5 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 AdS5 spin S and one S5 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 (S1,S2,J).

Diverse N=(4,4) Twisted Multiplets in the N=(2,2) Superspace

We describe four different types of the ${\cal N} = (4,4)$ twisted supermultiplets in two-dimensional ${\cal N} = (2,2)$ superspace ${\bf R}^{1,1|2,2}$. All these multiplets are presented by a pair of chiral and twisted chiral superfields and differ in the transformation properties under an extra hidden ${\cal N} = (2,2)$ supersymmetry. The sigma model ${\cal N} = (2,2)$ superfield Lagrangians for each type of the ${\cal N} = (4,4)$ twisted supermultiplets are real functions subjected to some differential constraints implied by the hidden supersymmetry. We prove that the general sigma model action, with all types of ${\cal N} = (4,4)$ twisted multiplets originally included, is reduced to a sum of sigma model actions for separate types. An interaction between the multiplets of different sorts is possible only through the appropriate mass terms, and only for those multiplets which belong to the same `self-dual' pair.

Diversity of off-shell twisted(4,4)multiplets inSU(2)×SU(2)harmonic superspace

We elaborate on four different types of twisted ${\cal N}=(4,4)$ supermultiplets in the $SU(2) \times SU(2)$, 2D harmonic superspace. In the conventional ${\cal N}=(4,4)$, 2D superspace they are described by the superfields $\hat q^{i a}$, $\hat q^{\und i a}$, $\hat q^{i \und a}$, $\hat q^{\und i \und a}$ subjected to proper differential constraints, $(i, \und i, a, \und a)$ being the doublet indices of four groups SU(2) which form the full R-symmetry group $SO(4)_L\times SO(4)_R$ of ${\cal N}=(4,4)$ supersymmetry. We construct the torsionful off--shell sigma model actions for each type of these multiplets, as well as the corresponding invariant mass terms, in an analytic subspace of the $SU(2) \times SU(2)$ harmonic superspace. As an instructive example, ${\cal N}=(4,4)$ superconformal extension of the $SU(2) \times U(1)$ WZNW sigma model action and its massive deformation are presented for the multiplet $\hat q^{i \und a}$ . We prove that ${\cal N}=(4,4)$ supersymmetry requires the general sigma model action of pair of different multiplets to split into a sum of sigma model actions of each multiplet. This phenomenon also persists if a larger number of non-equivalent multiplets are simultaneously included. We show that different multiplets may interact with each other only through mixed mass terms which can be set up for multiplets belonging to ``self-dual'' pairs $(\hat q^{i a}, \hat q^{\und i \und a})$ and $(\hat q^{\und i a}, \hat q^{i \und a})$ . The multiplets from different pairs cannot interact at all. For a ``self-dual'' pair of the twisted multiplets we give the most general form of the on-shell scalar potential.

The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps

The most fruitful approach to studying low energy soliton dynamics in field theories of Bogomol’nyi type is the geodesic approximation of Manton. In the case of vortices and monopoles, Stuart has obtained rigorous estimates of the errors in this approximation, and hence proved that it is valid in the low speed regime. His method employs energy estimates which rely on a key coercivity property of the Hessian of the energy functional of the theory under consideration. In this article we prove an analogous coercivity property for the Hessian of the energy functional of a general sigma model with compact Kähler domain and target. We go on to prove a continuity property for our result, and show that, for the CP1 model on S2, the Hessian fails to be globally coercive in the degree 1 sector. We present numerical evidence which suggests that the Hessian is globally coercive in a certain equivariance class of the degree n sector for n⩾2. We also prove that, within the geodesic approximation, a single CP1 lump moving on S2 does not generically travel on a great circle.

Tduality and conformal invariance at two loops

We show that the conformal invariance conditions for a general sigma-model with torsion are invariant under T-duality through two loops.

String field equations from the generalized sigma model II

We improve and extend a method introduced in an earlier paper for deriving string field equations. The idea is to impose conformal invariance on a generalized sigma model, using a background field method that ensures covariance under very general non-local coordinate transformations. The method is used to derive the free string equations, as well as the interacting equations for the graviton-dilaton system. The full interacting string field equations derived by this method should be manifestly background independent.

String field equations from the generalized sigma model

We propose a new approach for deriving the string field equations from a general sigma model on the world-sheet. This approach leads to an equation which combines some of the attractive features of both the renormalization group method and the covariant beta function treatment of the massless excitations. It has the advantage of being covariant under a very general set of both local and non-local transformations in the field space. We apply it to the tachyon, massless and first massive level, and show that the resulting field equations reproduce the correct spectrum of a left-right symmetric closed bosonic string.

T-duality for open strings

The T-duality transformations between open and closed superstrings in different D-manifolds are generalized to curved backgrounds with commuting isometries. We address some global aspects like the occurrence of orientifold boundaries in general sigma models, higher genus worldsheets, and the case of non-compact isometries. The various world-volume effective actions are shown to transform properly under T-duality. We also include a brief discussion of the canonical transformations of boundary states in the operator formalism.

SIGMA MODELS WITH (2, 2) WORLD SHEET SUPERSYMMETRY

We propose a new mechanism for constructing (2, 2) superspace actions for WZWN sigma models. We use the already known semi-chiral superfields, but we choose the Lagrangian in a special way, so that it has a gauge invariance capable of eliminating half of the initially propagating degrees of freedom. Thus we avoid the doubling of the target space dimension. The nonlinear differential conditions on the Lagrangian are derived and it is shown that this type of action corresponds to the general (2, 2) sigma model.

STATICITY, CIRCULARITY, AND THE FIRST LAW OF BLACK HOLE PHYSICS

We give a brief summary of some of our ongoing work on stationary black holes for non-linear matter models, such as gauge field theories (including Higgs fields) or general sigma models. Our main result is a generalization of the Bardeen-Carter-Hawking formula for the variation of the mass of stationary black holes, which involves only global quantities and boundary integrals. Results and open questions related to the staticity problem for non-rotating black holes, and to the circularity and Frobenius conditions for rotating ones, are briefly discussed.

GAUGED HETEROTIC SIGMA-MODELS

The gauging of isometries in general sigma-models which include fermionic terms which represent the interaction of strings with background Yang-Mills fields is considered. Gauging is possible only if certain obstructions are absent. The quantum gauge anomaly is discussed, and then the (1, 0) supersymmetric generalization of the gauge action given.