Heterotic Sigma Model Research Articles

Heterotic integrable deformation of the principal chiral model

A novel classically integrable model is proposed. It is a deformation of the two-dimensional principal chiral model, embedded into a heterotic σ model by a particular heterotic gauge field. This is inspired by the bosonic part of the heterotic σ model and its recent Hamiltonian formulation in terms of O(d,d+n)-generalized geometry in Hatsuda [Gauged double field theory, current algebras and heterotic sigma models, ]. Classical integrability is shown by the construction of a Lax pair and a classical R matrix. The latter is almost of the canonical form with a twist function and solves the classical Yang-Baxter equation. Published by the American Physical Society 2024

Gauged double field theory, current algebras and heterotic sigma models

We study the O(D, D + n) generalized metric and the gauge symmetries in the gauged double field theory (DFT) in view of current algebras and sigma models. We show that the O(D, D + n) generalized metric in the gauged DFT is consistent with the heterotic sigma models at the leading order in the α′-corrections. We then study the non-Abelian gauge symmetries and current algebras of heterotic string theories. We show that the algebras exhibit the correct diffeomorphism, the B-field gauge transformations of the background fields together with the non-Abelian gauge transformations possibly with the appropriate local Lorentz transformations.

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.

Generalized supergravity equations and generalized Fradkin-Tseytlin counterterm

The generalized Fradkin-Tseytlin counterterm for the (type I) Green-Schwarz superstring is determined for background fields satisfying the generalized supergravity equations (GSE). For this purpose, we revisit the derivation of the GSE based upon the requirement of kappa-symmetry of the superstring action. Lifting the constraint of vanishing bosonic torsion components, we are able to make contact to several different torsion constraints used in the literature. It is argued that a natural geometric interpretation of the GSE vector field that generalizes the dilaton is as the torsion vector, which can combine with the dilatino spinor into the torsion supervector. To find the counterterm, we use old results for the one-loop effective action of the heterotic sigma model. The counterterm is covariant and involves the worldsheet torsion for vanishing curvature, but cannot be constructed as a local functional in terms of the worldsheet metric. It is shown that the Weyl anomaly cancels without imposing any further constraints on the background fields. In the case of ordinary supergravity, it reduces to the Fradkin-Tseytlin counterterm modulo an additional constraint.

Wilson loop of the heterotic sigma model and the sv-map

The single-valued projection (sv) is a relation between scattering amplitudes of gauge bosons in heterotic and open superstring theories. Recently we have studied sv from the aspect of nonlinear sigma models [1], where the gauge physics of open string sigma model is under the Wilson loop representation but the gauge physics of heterotic string sigma model is under the fermionic representation since the Wilson loop representation is absent in the heterotic case. There we showed that the sv comes from a sum of six radial orderings of heterotic vertices on the complex plane. In this paper, we propose a Wilson loop representation for the heterotic case and using the Wilson loop representation to show that sv comes from a sum of two opposite-directed contours of the heterotic sigma model. We firstly prove that the Wilson loop is the exact propagator of the fermion field that carry the gauge physics of the heterotic string in the fermionic representation. Then we construct the action of the heterotic string sigma model in terms of the Wilson loop, by exploring the geometry of the Wilson loop and by generalizing the nonabelian Stokes's theorem [2–4] to the fermionic case. After that, we compute some three loop and four loop diagrams as an example, to show how the sv for ζ2 and ζ3 arises from a sum of the contours of the Wilson loop. Finally we conjecture that this sum of contours of the Wilson loop is the mechanism behind the sv for general cases.

Sv-map between type I and heterotic sigma models

The scattering amplitudes of gauge bosons in heterotic and open superstring theories are related by the single-valued projection which yields heterotic amplitudes by selecting a subset of multiple zeta value coefficients in the α′ (string tension parameter) expansion of open string amplitudes. In the present work, we argue that this relation holds also at the level of low-energy expansions (or individual Feynman diagrams) of the respective effective actions, by investigating the beta functions of two-dimensional sigma models describing world-sheets of open and heterotic strings. We analyze the sigma model Feynman diagrams generating identical effective action terms in both theories and show that the heterotic coefficients are given by the single-valued projection of the open ones. The single-valued projection appears as a result of summing over all radial orderings of heterotic vertices on the complex plane representing string world-sheet.

Heterotic sigma models on T8 and the Borcherds automorphic form \u03a612

We consider the spectrum of BPS states of the heterotic sigma model with (0, 8) supersymmetry and T8 target, as well as its second-quantized counterpart. We show that the counting function for such states is intimately related to Borcherds’ automorphic form Φ12, a modular form which exhibits automorphy for O(2, 26; ℤ). We comment on possible implications for Umbral moonshine and theories of AdS3 gravity.

Novel branches of (0, 2) theories

We show that recently proposed linear sigma models with torsion can be obtained from unconventional branches of conventional gauge theories. This observation puts models with log interactions on firm footing. If non-anomalous multiplets are integrated out, the resulting low-energy theory involves log interactions of neutral fields. For these cases, we find a sigma model geometry which is both non-toric and includes brane sources. These are heterotic sigma models with branes. Surprisingly, there are massive models with compact complex non-Kahler target spaces, which include brane/anti-brane sources. The simplest conformal models describe wrapped heterotic NS5-branes. We present examples of both types.

Heterotic sigma models with N = 2 space-time supersymmetry

We study the non-linear sigma model realization of a heterotic vacuum with N=2 space-time supersymmetry. We examine the requirements of (0,2) + (0,4) world-sheet supersymmetry and show that a geometric vacuum must be described by a principal two-torus bundle over a K3 manifold.

A heterotic sigma model with novel target geometry

We construct a (1,2) heterotic sigma model whose target space geometry consists of a transitive Lie algebroid with complex structure on a Kaehler manifold. We show that, under certain geometrical and topological conditions, there are two distinguished topological half--twists of the heterotic sigma model leading to A and B type half--topological models. Each of these models is characterized by the usual topological BRST operator, stemming from the heterotic (0,2) supersymmetry, and a second BRST operator anticommuting with the former, originating from the (1,0) supersymmetry. These BRST operators combined in a certain way provide each half--topological model with two inequivalent BRST structures and, correspondingly, two distinct perturbative chiral algebras and chiral rings. The latter are studied in detail and characterized geometrically in terms of Lie algebroid cohomology in the quasiclassical limit.

Yang-Mills Chern-Simons corrections from the pure spinor superstring

Nilpotency of the pure spinor BRST operator in a curved background implies superspace equations of motion for the background. By computing one-loop corrections to nilpotency for the heterotic sigma model, the Yang-Mills Chern-Simons corrections to the background are derived.

Two-dimensional twisted sigma models, the mirror chiral de Rham complex, and twisted generalised mirror symmetry

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 a novel understanding of the sigma model one-loop beta function solely in terms of holomorphic data. At the $(2,2)$ locus, one can describe the resulting half-twisted variant of the topological B-model in terms of a $\it{mirror}$ "Chiral de Rham complex" (or CDR) defined by Malikov et al. in \cite{GMS1}. Via mirror symmetry, one can also derive various conjectural expressions relating the sheaf cohomology of the mirror CDR to that of the original CDR on pairs of Calabi-Yau mirror manifolds. An analysis of the half-twisted model on a non-K\"ahler group manifold with torsion also allows one to draw conclusions about the corresponding sheaves of CDR (and its mirror) that are consistent with mathematically established results by Ben-Bassat in \cite{ben} on the mirror symmetry of generalised complex manifolds. These conclusions therefore suggest an interesting relevance of the sheaf of CDR in the recent study of generalised mirror symmetry.

The Heterotic Supersymmetric Sigma Model in the Canonical Exterior Formalism

Starting from a classical 2D superconformal theory described by the Wess–Zumino–Witten action, the canonical exterior formalism on group manifold for the heterotic supersymmetric sigma model is constructed. The motion equations of the dynamical field and the constraints are found and analyzed from the geometric point of view. It can be seen how the use of the canonical exterior formalism is more adequate and simple because of its manifest covariance in all the steps. The relationship between the form brackets defined in the canonical exterior formalism and the Poisson-brackets is written. Later on, the Dirac-brackets are written by using the second class constraints provided by the canonical exterior formalism. As it can be seen the canonical exterior formalism allows to show how the canonical quantization of the heterotic supersymmetric sigma model is facilitated.

Geometry of theN=4,d=1nonlinear supermultiplet

We construct the general action for $N=4$, $d=1$ nonlinear supermultiplet including the most general interaction terms which depend on the arbitrary function $h$ obeying the Laplace equation on ${S}^{3}$. We find the bosonic field $B$ which depends on the components of nonlinear supermultiplet and transforms as a full time derivative under $N=4$ supersymmetry. The most general interaction is generated just by a Fayet-Iliopoulos term built from this auxiliary component. Being transformed through a full time derivative under $N=4$, $d=1$ supersymmetry, this auxiliary component $B$ may be dualized into a fourth scalar field giving rise to a four-dimensional $N=4$, $d=1$ sigma-model. We analyzed the geometry in the bosonic sector and find that it is not a hyper-K\"ahler one. With a particular choice of the target space metric $g$ the geometry in the bosonic sector coincides with the one which appears in heterotic (4, 0) sigma-model in $d=2$.

Two-dimensional twisted sigma models and the theory of chiral differential operators

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 obtain a novel understanding of the sigma model one-loop beta function solely in terms of holomorphic data. At the $(2,2)$ locus, where the obstruction vanishes for $\it{any}$ smooth manifold $X$, we obtain a purely mathematical description of the half-twisted variant of the topological A-model and (if $c_1(X) =0$) its elliptic genus. By studying the half-twisted $(2,2)$ model on $X=\mathbb {CP}^1$, one can show that a subset of the infinite-dimensional space of physical operators generates an underlying super-affine Lie algebra. Furthermore, on a non-K\ahler, parallelised, group manifold with torsion, we uncover a direct relationship between the modulus of the corresponding sheaves of chiral de Rham complex, and the level of the underlying WZW theory.

The geometry of sigma models with twisted supersymmetry

We investigate the relation between supersymmetry and geometry for two-dimensional sigma models with target spaces of arbitrary signature, and Lorentzian or Euclidean world-sheets. In particular, we consider twisted forms of the two-dimensional ( p, q) supersymmetry algebra. Superspace formulations of the ( p, q) heterotic sigma models with twisted or untwisted supersymmetry are given. For the twisted (2,1) and the pseudo-Kähler sigma models, we give extended superspace formulations.

HETEROTIC T-DUALITY AND THE RENORMALIZATION GROUP

We consider target space duality transformations for heterotic sigma models and strings away from renormalization group fixed points. By imposing certain consistency requirements between the T-duality symmetry and renormalization group flows, the one-loop gauge beta function is uniquely determined, without any diagram calculations. Classical T-duality symmetry is a valid quantum symmetry of the heterotic sigma model, severely constraining its renormalization flows at this one-loop order. The issue of heterotic anomalies and their cancellation is addressed from this duality constraining viewpoint.

The gauged (2, 1) heterotic sigma model

The geometry of (2, 1) supersymmetric sigma models with isometry symmetries is discussed. The gauging of such symmetries in superspace is then studied. We find that the coupling to the (2, 1) Yang-Mills supermultiplet can be achieved provided certain geometric conditions are satisfied. We construct the general gauged action, using an auxiliary vector to generate the full non-polynomial structure.

Geometry, isometries and gauging of (2, 1) heterotic sigma-models

The geometry of (2, 1) supersymmetric sigma-models is reviewed and the conditions under which they have isometry symmetries are analysed. Certain potentials are constructed that play an important role in the gauging of such symmetries. The gauged action is found for a special class of models.

A truly crazy idea about type-IIB supergravity and heterotic sigma-models

Within the confines of the standard NSR sigma-model approach, we construct an explicit and manifestly (1,0) heterotic sigma-model where the bosonic background fields are those of 10D, N = IIB supergravity. The novelty that permits this construction is to use the righton sector of the (1,0) NSR sigma-model to describe the currents of the positive chiral part of the SO(1,9) spin-bundle tensored with itself.