We propose a physical interpretation of the chiral de Rham complex as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model. We show that the chiral de Rham complex on a Calabi-Yau manifold carries all information about the classical dynamics of the sigma model. Physically, this provides an operator realization of the non-linear sigma model. Mathematically, the idea suggests the use of Hamiltonian flow equations within the vertex algebra formalism with the possibility to incorporate both left and right moving sectors within one mathematical framework.
On any Calabi-Yau manifold X one can define a certain sheaf of chiral N=2 superconformal field theories, known as the chiral de Rham complex of X. It depends only on the complex structure of X, and its local structure is described by a simple free field theory. We show that the cohomology of this sheaf can be identified with the infinite-volume limit of the half-twisted sigma-model defined by E. Witten more than a decade ago. We also show that the correlators of the half-twisted model are independent of the Kahler moduli to all orders in worldsheet perturbation theory, and that the relation to…
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.
In this paper, we study the perturbative aspects of the half-twisted variant of Witten's topological A-model on a complex orbifold X/G, where G is an isometry group of X. The objective is to furnish a purely physical interpretation of the mathematical theory of the Chiral de Rham complex on orbifolds recently constructed by Frenkel and Szczesny in [Chiral de Rham complex and orbifolds, arXiv: math.AG/0307181]. In turn, one can obtain a novel understanding of the holomorphic (twisted) N=2 superconformal structure underlying the untwisted and twisted sectors of the quantum sigma model, purely in…
We show that the chiral de Rham complex of a generalized Calabi-Yau manifold carries N=2 supersymmetry. We discuss the corresponding topological twist for this N=2 algebra. We interpret this as an algebroid version of the super-Sugawara or Kac-Todorov construction.
We represent a generalization of Borisov's construction of chiral de Rham complex for the case of line bundle twisted chiral de Rham complex on Calabi-Yau hypersurface in projective space. We generalize the differential associated to the polytope $Δ$ of the projective space $\mathbb{P}^{d-1}$ by allowing nonzero modes for the screening currents forming this differential. It is shown that the numbers of screening current modes define the support function of toric divisor of a line bundle on $\mathbb{P}^{d-1}$ that twists the chiral de Rham complex on Calabi-Yau hypersurface.
The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold $Z$, and contains the ordinary de Rham complex at weight zero. Given a closed 3-form $H$ on $Z$, we construct the twisted chiral de Rham differential $D_H$, which coincides with the ordinary twisted differential in weight zero. We show that its cohomology vanishes in positive weight and coincides with the ordinary twisted cohomology in weight zero. As a consequence, we propose that in a background flux, Ramond-Ramond fields can be interpreted as …
In this note we calculate elliptic genus in various examples of twisted chiral de Rham complex on two dimensional toric compact manifolds and Calabi-Yau hypersurfaces in toric manifolds. At first the elliptic genus is calculated for the line bundle twisted chiral de Rham complex on a compact smooth toric manifold and K3 hypersurface in \mathbb{P}^{3} . Then we twist chiral de Rham complex by sheaves localized on positive codimension submanifolds in \mathbb{P}^{2} and calculate in each case the elliptic genus. In the last example the elliptic genus of chiral de Rham complex on \mathbb{P}^{2} tw…
In this paper, we study the perturbative aspects of the half-twisted variant of Witten's topological A-model coupled to a non-dynamical gauge field with Kahler target space X being a G-manifold. Our main objective is to furnish a purely physical interpretation of the equivariant cohomology of the chiral de Rham complex, recently constructed by Lian and Linshaw in [arXiv:math/0501084], called the "chiral equivariant cohomology". In doing so, one finds that key mathematical results such as the vanishing in the chiral equivariant cohomology of positive weight classes, lend themselves to straightf…
We propose a localization formula for the chiral de Rham complex generalizing the well-known localization procedure in topological theories. Our formula takes into account the contribution due to the massive modes. The key to achieve this is to view the non-linear beta-gamma system as a gauge theory. For abelian gauge groups we are in the realm of toric geometry. Including the b-c system, the formula reproduces the known results for the elliptic genus of toric varieties. We compute the partition function of several models.
In this paper, we study the perturbative aspects of a "B-twisted" two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal E$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted as an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on $X$. In addition, one can also obtain …
Interpreting the chiral de Rham complex (CDR) as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model, we suggest a setup for the study of CDR on manifolds with special holonomy. We show how to systematically construct global sections of CDR from differential forms, and investigate the algebra of the sections corresponding to the covariantly constant forms associated with the special holonomy. As a concrete example, we construct two commuting copies of the Odake algebra (an extension of the N=2 superconformal algebra) on the space of global sections of CDR of a Calabi…
Let $Z$ be a principal circle bundle over a base manifold $M$ equipped with an integral closed $3$-form $H$ called the flux. Let $\widehat{Z}$ be the T-dual circle bundle over $M$ with flux $\widehat{H}$. Han and Mathai recently constructed the $\mathbb{Z}_2$-graded space of exotic differential forms $\mathcal{A}^{\bar{k}}(\widehat{Z})$. It has an additional $\mathbb{Z}$-grading such that the degree zero component coincides with the space of invariant twisted differential forms $Ω^{\bar{k}}(\widehat{Z}, \widehat{H})^{\widehat{\mathbb{T}}}$, and it admits a differential that extends the twisted…
We construct a sheaf of N=2 vertex algebras naturally associated to any Poisson manifold. The relation of this sheaf to the chiral de Rham complex is discussed. We reprove the result about the existence of two commuting N = 2 superconformal structures on the space of sections of the chiral de Rham complex of a Calabi-Yau manifold, but now calculated in a manifest N=2 formalism. We discuss how the semi-classical limit of this sheaf of N=2 vertex algebras is related to the classical supersymmetric non-linear sigma model.
We construct an embedding of two commuting copies of the N=2 superconformal vertex algebra in the space of global sections of the twisted chiral-anti-chiral de Rham complex of a generalized Calabi-Yau metric manifold, including the case when there is a non-trivial H-flux and non-vanishing dilaton. The 4 corresponding BRST charges are well defined on any generalized Kahler manifold. This allows one to consider the half-twisted model defining thus the chiral de Rham complex of a generalized Kahler manifold. The classical limit of this result allows one to recover the celebrated generalized Kahle…
We use path integral methods and topological quantum field theory techniques to investigate a generic classical Hamiltonian system. In particular, we show that Floer's instanton equation is related to a functional Euler character in the quantum cohomology defined by the topological nonlinear $σ$--model. This relation is an infinite dimensional analog of the relation between Poincaré--Hopf and Gauss--Bonnet--Chern formulæ$~$ in classical Morse theory, and can also be viewed as a loop space generalization of the Lefschetz fixed point theorem. By applying localization techniques to path integrals…
We extend our analysis in [arXiv:0801.4782] and show that the chiral algebras of (0,2) sigma models are totally trivialized by worldsheet instantons for all complete flag manifolds of compact semisimple Lie groups. Consequently, supersymmetry is spontaneously broken. Our results verify Stolz's idea that there are no harmonic spinors on the loop spaces of these flag manifolds. Moreover, they also imply that the kernels of certain twisted Dirac operators on these spaces will be empty under a "quantum" deformation of their geometries.
We review the enveloping algebra of the 10 dimensional chiral sigma matrices. To facilitate the computation of the product of several chiral sigma matrices we have developed a symbolic program. Using this program one can reduce the multiplication of the sigma matrices down to linear combinations of irreducilbe elements. We are able to quickly derive several identities that are not restricted to traces. A copy of the program written in the Mathematica language is provided for the community.
With the linear sigma model, we have studied Bose-Einstein condensation and the chiral phase transition in the chiral limit for an interacting pion system. A $μ-T$ phase diagram including these two phenomena is presented. It is found that the phase plane has been divided into three areas: the Bose-Einstein condensation area, the chiral symmetry broken phase area and the chiral symmetry restored phase area. Bose-Einstein condensation can happen either from the chiral symmetry broken phase or from the restored phase. We show that the onset of the chiral phase transition is restricted in the area…
We study d=2, N=(2,2) non-linear sigma-models in (2,2) superspace. By analyzing the most general constraints on a superfield, we show that through an appropriate choice of coordinates, there are no other superfields than chiral, twisted chiral and semi-chiral ones. We study the resulting sigma-models and we speculate on the possibility that all (2,2) non-linear sigma-models can be described using these fields. We apply the results to two examples: the SU(2) x U(1) and the SU(2) x SU(2) WZW model. Pending upon the choice of complex structures, the former can be described in terms of either one …
Starting with the projective-superspace off-shell formulation for four-dimensional N = 2 supersymmetric sigma-models on cotangent bundles of arbitrary Hermitian symmetric spaces, their on-shell description in terms of N = 1 chiral superfields is developed. In particular, we derive a universal representation for the hyperkaehler potential in terms of the curvature of the symmetric base space. Within the tangent-bundle formulation for such sigma-models, completed recently in arXiv:0709.2633 and realized in terms of N = 1 chiral and complex linear superfields, we give a new universal formula for …
We explore the nonperturbative aspects of the chiral algebras of N = (0,2) sigma models, which perturbatively are intimately related to the theory of chiral differential operators (CDOs). The grading by charge and scaling dimension is anomalous if the first Chern class of the target space is nonzero. This has some nontrivial consequences for the chiral algebra. As an example, we study the case where the target space is CP^1, and show that worldsheet instantons trivialize the chiral algebra entirely. Consequently, supersymmetry is spontaneously broken in this model. We then turn to a closer loo…
We compare Chiral Perturbation Theory (ChPT) and the Linear Sigma Model (LSM) as realizations of low energy QCD for light mesons in a chirally imbalanced medium. The relations between the low-energy constants of the Chiral Lagrangian and the corresponding constants of the Linear Sigma Model are established as well as the expressions for the decay constant of the $π$-meson in the medium and the mass of the $a_0$. In the large $N_c$ count taken from QCD the correspondence of ChPT and LSM is remarkably good and give a solid ground for search of chiral imbalance manifestation in pion physics. A po…
This is a review of the relation between supersymmetric non-linear sigma models and target space geometry. In particular, we report on the derivation of generalized Kähler geometry from sigma models with additional spinorial superfields. Some of the results reviewed are: Generalized complex geometry from sigma models in the Lagrangian formulation; Coordinatization of generalized Kähler geometry in terms of chiral, twisted chiral and semi-chiral superfields; Generalized Kähler geometry from sigma models in the Hamiltonian formulation.
CERN measurements of pi(-)p->pi(-)pi(+)n on polarized target at 17.2 GeV/c enable experimental determination of partial wave production amplitudes below 1080 MeV. The measured S-wave transversity amplitudes provide evidence for a narrow scalar resonance sigma(770). The assumption of analyticity of production amplitudes in dipion mass allows to determine S-wave helicity amplitudes S_0 and S_1. The amplitude S_1 is related to pi(-)pi(+)->pi(-)pi(+) scattering. There are four "down" solutions (1, 1bar), (2, 1bar), (1, 2bar) and (2, 2bar) selected by unitarity in pipi scattering. Ellis-Lanik relat…
A multi-parameter integrable deformation of the principal chiral model is presented. The Yang-Baxter and bi-Yang-Baxter sigma-models, the principal chiral model plus a Wess-Zumino term and the TsT transformation of the principal chiral model are all recovered when the appropriate deformation parameters vanish. When the Lie group is SU(2), we show that this four-parameter integrable deformation of the SU(2) principal chiral model corresponds to the Lukyanov model.
We consider a general N=(2,2) non-linear sigma-model in (2,2) superspace. Depending on the details of the complex structures involved, an off-shell description can be given in terms of chiral, twisted chiral and semi-chiral superfields. Using superspace techniques, we derive the conditions the potential has to satisfy in order to be ultra-violet finite at one loop. We pay particular attention to the effects due to the presence of semi-chiral superfields. A complete description of N=(2,2) strings follows from this.
In this paper, we study the perturbative aspects of a twisted version of the two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal E$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted in terms of an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on $X$. One can also …
We develop the duality transformation rules in two-dimensional theories in the superfield formalism. Even if the chiral superfield which we dualize involves an F-term, we can dualize it by virtue of the property of chiral superfields. We apply the duality transformation rule of the neutral chiral superfield to the ${\cal N}=(4,4)$ gauged linear sigma model for five-branes. We also investigate the duality transformation rule of the charged chiral superfield in the ${\cal N} = (4,4)$ gauged linear sigma model for the $A_1$-type ALE space. In both cases we obtain the dual Lagrangians in the super…
We compute the free energy in the presence of a chemical potential coupled to a conserved charge in the effective SU(N)xSU(N) scalar field theory to third order for asymmetric volumes in general d-dimensions, using dimensional regularization. We also compute the mass gap in a finite box with periodic boundary conditions.
It is shown how derived brackets naturally arise in sigma-models via Poisson- or antibracket, generalizing a recent observation by Alekseev and Strobl. On the way to a precise formulation of this relation, an explicit coordinate expression for the derived bracket is obtained. The generalized Nijenhuis tensor of generalized complex geometry is shown to coincide up to a de-Rham closed term with the derived bracket of the structure with itself, and a new coordinate expression for this tensor is presented. The insight is applied to two known two-dimensional sigma models in a background with genera…
In this note, we construct new representations of D=2, N=4 supersymmetry which do not involve chiral or twisted chiral multiplets. These multiplets may make it possible to circumvent no-go theorems about N=4 superspace formulations of WZWN-models.
The Supersymmetric Dual Sigma Model (SDSM) is a local field theory introduced to be nonlocally equivalent to the Supersymmetric Chiral nonlinear sigma-Model (SCM), this dual equivalence being proven by explicit canonical transformation in tangent space. This model is here reconstructed in superspace and identified as a chiral-entwined supersymmetrization of the Dual Sigma Model (DSM). This analysis sheds light on the Boson-Fermion Symphysis of the dual transition, and on the new geometry of the DSM.
Non-linear sigma models with extended supersymmetry have constrained target space geometries, and can serve as effective tools for investigating and constructing new geometries. Analyzing the geometrical and topological properties of sigma models is necessary to understand the underlying structures of string theory. The most general two-dimensional sigma model with manifest N=(2,2) supersymmetry can be parametrized by chiral, twisted chiral and semichiral superfields. In the research presented in this thesis, N=(4,4) (twisted) supersymmetry is constructed for a semichiral sigma model. It is fo…
In four-dimensional N=1 Minkowski superspace, general nonlinear sigma models with four-dimensional target spaces may be realised in term of CCL (chiral and complex linear) dynamical variables which consist of a chiral scalar, a complex linear scalar and their conjugate superfields. Here we introduce CCL sigma models that are invariant under U(1) "duality rotations" exchanging the dynamical variables and their equations of motion. The Lagrangians of such sigma models prove to obey a partial differential equation that is analogous to the self-duality equation obeyed by U(1) duality invariant mod…
We study the mechanism of topological superconductivity in a hierarchical chain of chiral non-linear sigma-models (models of current algebra) in one, two, and three spatial dimensions. The models have roots in the 1D Peierls-Frohlich model and illustrate how the 1D Frohlich's ideal conductivity extends to a genuine superconductivity in dimensions higher than one. The mechanism is based on the fact that a point-like topological soliton carries an electric charge. We discuss a flux quantization mechanism and show that it is essentially a generalization of the persistent current phenomenon, known…
It is commonly asserted that there is a \hat G\times G centreless Kac-Moody extension of the manifest G\times G global symmetry of the two-dimensional principal chiral model (PCM) for the group manifold G. Here, we show that the symmetry is in fact larger, namely \hat G\times \hat G, the full centreless Kac-Moody extension of the entire manifest G\times G global symmetry. Extending previous results in the literature, we also obtain an explicit realisation of the Virasoro-like symmetry of the PCM, generated by K_n=L_{n+1} - L_{n-1} for both positive and negative n. We show that these generators…
In this paper we discuss symmetry breakdown in NJL model at low N_c. In particular we propose a modified NJL model that displays a symmetry breakdown and also at finite temperatures under certain conditions the chiral fluctuations in this model give rise to a phase analogous to pseudogap phase of strong-coupling and low carrier density superconductors.
Certain features associated with the symmetry reduction of the vacuum Einstein equations by two commuting, space-like Killing vector fields are studied. In particular, the discussion encompasses the equations for the Gowdy $T^3$ cosmology and cylindrical gravitational waves. We first point out a relation between the $SL(2,R)$ (or SO(3)) $σ$ and principal chiral models, and then show that the reduced Einstein equations can be obtained from a dimensional reduction of the standard SL(2,R) sigma-model in three dimensions. The reduced equations can also be derived from the action of a `generalized'…
We consider vector and axialvector mesons in the framework of a gauged linear sigma model with chiral $U(N_f)_R \times U(N_f)_L$ symmetry. For $N_f=2$, we investigate the behavior of the chiral condensate and the meson masses as a function of temperature by solving a system of coupled Dyson-Schwinger equations derived via the 2PI formalism in double-bubble approximation. We find that the inclusion of vector and axialvector mesons tends to sharpen the chiral transition. Within our approximation scheme, the mass of the $ρ$ meson increases by about 100 MeV towards the chiral transition.
We introduce a sigma model lagrangian generalising a number of new and old models which can be thought of as chiral, including the Schild string, ambitwistor strings, and the recently introduced tensionless AdS twistor strings. This "chiral sigma model" describes maps from a $p$-brane worldvolume into a symplectic space and is manifestly diffeomorphism- and Weyl-invariant despite the absence of a worldvolume metric. Construction of the Batalin-Vilkovisky master action leads to a BRST operator under which the gauge-fixed action is BRST-exact; we discuss whether this implies that the chiral bran…
