Two-dimensional Non-linear Sigma Models Research Articles

Marginal deformations of Calabi-Yau hypersurface hybrids with (2,2) supersymmetry

We study two-dimensional non-linear sigma models with (2,2) supersymmetry and a holomorphic superpotential that are believed to flow to unitary compact (2,2) superconformal theories with central charges cL = cR = 9. The SCFTs have a set of marginal deformations, and some of these can be realized as deformations of parameters of the UV theory, making it possible to apply techniques such as localization to probe the deformations of the SCFT in terms of a UV Lagrangian. In this work we describe the UV lifts of the remaining SCFT infinitesimal deformations, the so-called non-toric and non-polynomial deformations. Our UV theories naturally arise as geometric phases of gauged linear sigma models, and it may be possible to extend our results to find lifts of all SCFT deformations to the gauged linear sigma model.

On two-loop effective action of 2d sigma model

In this paper, we study two-loop contribution to the effective action of a two-dimensional non-linear sigma model. We derive a new formula, which can be applicable to a regularization of general type. As examples, we obtain known results for the dimensional regularization and investigate new types of a cutoff one. Also, we discuss non-local contributions and restrictions on the regularization.

Strings in bimetric spacetimes

We put forward a two-dimensional nonlinear sigma model that couples (bosonic) matter fields to topological Hořava gravity on a nonrelativistic worldsheet. In the target space, this sigma model describes classical strings propagating in a curved spacetime background, whose geometry is described by two distinct metric fields. We evaluate the renormalization group flows of this sigma model on a flat worldsheet and derive a set of beta-functionals for the bimetric fields. Imposing worldsheet Weyl invariance at the quantum level, we uncover a set of gravitational field equations that dictate the dynamics of the bimetric fields in the target space, where a unique massless spin-two excitation emerges. When the bimetric fields become identical, the sigma model gains an emergent Lorentz symmetry. In this single metric limit, the beta-functionals of the bimetric fields reduce to the Ricci flow equation that arises in bosonic string theory, and the bimetric gravitational field equations give rise to Einstein’s gravity.

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.

O(d,d) transformations preserve classical integrability

In this note, we study the action of O(d,d) transformations on the integrable structure of two-dimensional non-linear sigma models via the doubled formalism. We construct the Lax pairs associated with the O(d,d)-transformed model and find that they are in general non-local because they depend on the winding modes. We conclude that every O(d,d;R) deformation preserves integrability. As an application we compute the Lax pairs for continuous families of deformations, such as JJ¯ marginal deformations and TsT transformations of the three-sphere with H-flux.

Energy quantization for a nonlinear sigma model with critical gravitinos

We study some analytical and geometric properties of a two-dimensional nonlinear sigma model with gravitino which comes from supersymmetric string theory. When the action is critical w.r.t. variations of the various fields including the gravitino, there is a symmetric, traceless, and divergence-free energy-momentum tensor, which gives rise to a holomorphic quadratic differential. Using it we obtain a Pohozaev type identity and finally we can establish the energy identities along a weakly convergent sequence of fields with uniformly bounded energies.

Continuum limit of fishnet graphs and AdS sigma model

We consider the continuum limit of 4d planar fishnet diagrams using integrable spin chain methods borrowed from the mathcal{N} = 4 Super-Yang-Mills theory. These techniques give us control on the scaling dimensions of single-trace operators for all values of the coupling constant in the fishnet theory. We use them to study the thermodynamical limit of the BMN operator corresponding to the spin chain ferromagnetic vacuum. We find that its scaling dimension exhibits a critical behaviour when the coupling constant approaches Zamolodchikov’s critical coupling. Analysis close to that point suggests that the continuum limit of the fishnet graphs is controlled by the two-dimensional AdS5 non-linear sigma model. More generally, we present evidence that the fishnet diagrams define an integrable lattice regularization of the AdS5 model. A system of massless TBA equations is derived for the tachyon energy by dualizing the TBA equations of the weakly coupled planar mathcal{N} = 4 SYM theory.

Complex instantons in sigma models with chemical potential

We analyze two-dimensional nonlinear sigma models at nonzero chemical potentials, which are governed by a complex action. In the spirit of contour deformations (thimbles), we extend the fields into the complex plane, which allows us to incorporate the chemical potentials $\ensuremath{\mu}$ as twisted boundary conditions. We write down the equations of motion and find exact BPS-like solutions in terms of pairs of (anti)holomorphic functions, in particular generalizations of unit charge and fractional instantons to generic $\ensuremath{\mu}$. The decay of these solutions is controlled by the imaginary part of $\ensuremath{\mu}$ and a vanishing imaginary part causes jumps in the action. We analyze how the total charge is distributed into localized objects and to what extent these are characterized by topology.

Asymptotically free theory with scale invariant thermodynamics

A recently developed variational resummation technique, incorporating renormalization group properties consistently, has been shown to solve the scale dependence problem that plagues the evaluation of thermodynamical quantities, e.g., within the framework of approximations such as in the hard-thermal-loop resummed perturbation theory. This method is used in the present work to evaluate thermodynamical quantities within the two-dimensional nonlinear sigma model, which, apart from providing a technically simpler testing ground, shares some common features with Yang-Mills theories, like asymptotic freedom, trace anomaly and the nonperturbative generation of a mass gap. The present application confirms that nonperturbative results can be readily generated solely by considering the lowest-order (quasiparticle) contribution to the thermodynamic effective potential, when this quantity is required to be renormalization group invariant. We also show that when the next-to-leading correction from the method is accounted for, the results indicate convergence, apart from optimally preserving, within the approximations here considered, the sought-after scale invariance.

Regularity of Solutions of the Nonlinear Sigma Model with Gravitino

We propose a geometric setup to study analytic aspects of a variant of the super symmetric two-dimensional nonlinear sigma model. This functional extends the functional of Dirac-harmonic maps by gravitino fields. The system of Euler–Lagrange equations of the two-dimensional nonlinear sigma model with gravitino is calculated explicitly. The gravitino terms pose additional analytic difficulties to show smoothness of its weak solutions which are overcome using Rivière’s regularity theory and Riesz potential theory.

Coarse regularity of solutions to a nonlinear sigma-model with\xa0L^p\xa0gravitino

The regularity of weak solutions of a two-dimensional nonlinear sigma model with coarse gravitino is shown. Here the gravitino is only assumed to be in L^p for some p>4. The precise regularity results depend on the value of p.

Динамика взаимодействия блоховских доменных границ в двумерной нелинейной сигма-модели

Динамика взаимодействия блоховских доменных границ в двумерной нелинейной сигма-модели

Chiral operators in two-dimensional (0,2) theories and a test of triality

In this paper we compute spaces of chiral operators in general two-dimensional (0,2) nonlinear sigma models, both in theories twistable to the A/2 or B/2 model, as well as in non-twistable theories, and apply them to check recent duality conjectures. The fact that in a nonlinear sigma model, the Fock vacuum can act as a section of a line bundle on the target space plays a crucial role in our (0,2) computations, so we begin with a review of this property. We also take this opportunity to show how even in (2,2) theories, the Fock vacuum encodes in this way choices of target space spin structures, and discuss how such choices enter the A and B model topological field theories. We then compute chiral operators in general (0,2) nonlinear sigma models, and apply them to the recent Gadde-Gukov-Putrov triality proposal, which says that certain triples of (0,2) GLSMs should RG flow to nontrivial IR fixed points. We find that different UV theories in the same proposed universality class do not necessarily have the same space of chiral operators — but, the mismatched operators do not contribute to elliptic genera and are in non-integrable representations of the proposed IR affine symmetry groups, suggesting that the mismatched states become massive along RG flow. We find this state matching in examples not only of different geometric phases of the same GLSMs, but also in phases of different GLSMs, indirectly verifying the triality proposal, and giving a clean demonstration that (0,2) chiral rings are not topologically protected. We also check proposals for enhanced IR affine E 6 symmetries in one such model, verifying that (matching) chiral states in phases of corresponding GLSMs transform as 27’s, $$ \overline{\mathbf{27}}'\mathrm{s} $$ .

Two-Sphere Partition Functions and Gromov–Witten Invariants

Many N = (2,2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N = (2,2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories — recently computed via localization by Benini et al. and Doroud et al. — yields the exact Kahler potential on the quantum Kahler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kahler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in α ' . We compute these quantities for the quintic and for Rodland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P 7 , recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.

Schrödinger sigma models and Jordanian twists

We proceed to study the integrable structures of two-dimensional non-linear sigma models defined on three-dimensional Schrodinger spacetimes. We show that anisotropic Lax pairs are equivalent with isotropic Lax pairs with flat conserved currents under non-local gauge transformations. Then a quite non-trivial realization of the undeformed Yangian symmetry Y(sl(2)) is revealed after an appropriate gauge fixing, which is determined by comparing the gauge transformation to a quantum Jordanian twist. As a result, an exotic symmetry found in arXiv:1209.4147 may be interpreted as a Jordanian deformation of Y(sl(2)).

On a Semiclassical Limit of Loop Space Quantum Mechanics

Following earlier work, we view two-dimensional nonlinear sigma model as single particle quantum mechanics in the free loop space of the target space. In a natural semiclassical limit of this model, the wavefunction localizes on the submanifold of vanishing loops. One would expect that the semiclassical expansion should be related to the tubular expansion of the theory around the submanifold and effective dynamics on the submanifold is obtainable using Born-Oppenheimer approximation. We develop a framework to carry out such an analysis at the leading order. In particular, we show that the linearized tachyon effective equation is correctly reproduced up to divergent terms all proportional to the Ricci scalar. The steps are as follows: first we define a finite dimensional analogue of the loop space quantum mechanics (LSQM) where we discuss its tubular expansion and how that is related to a semiclassical expansion of the Hamiltonian. Then we study an explicit construction of the relevant tubular neighborhood in loop space using exponential maps. Such a tubular geometry is obtained from a Riemannian structure on the tangent bundle of target space which views the zero-section as a submanifold admitting a tubular neighborhood. Using this result and exploiting an analogy with the toy model, we arrive at the final result for LSQM.

Exotic symmetry and monodromy equivalence in Schrödinger sigma models

We consider the classical integrable structure of two-dimensional non-linear sigma models with target space three-dimensional Schrodinger spacetimes. There are the two descriptions to describe the classical dynamics: 1) the left description based on SL(2,R)_L and 2) the right description based on U(1)_R. We have shown the sl(2,R)_L Yangian and q-deformed Poincare algebras associated with them. We proceed to argue an infinite-dimensional extension of the q-deformed Poincare algebra. The corresponding charges are constructed by using a non-local map from the flat conserved currents related to the Yangian. The exotic tower structure of the charges is revealed by directly computing the classical Poisson brackets. Then the monodromy matrices in both descriptions are shown to be gauge-equivalent via the relation between the spectral parameters. We also give a simple Riemann sphere interpretation of this equivalence.

THE CLASSICAL EQUIVALENCE OF MONODROMY MATRICES IN SQUASHED SIGMA MODEL

We show Yangian and quantum affine symmetries are realized in two-dimensional non-linear sigma models with squashed spheres target spaces. According to these symmetries, there are two descriptions for the classical dynamics. In each description, a Lax pair and the corresponding monodromy matrix are constructed. Both Lax pairs reproduce the same equations of motion and hence they are equivalent. We show the gauge-equivalence by constructing a map between monodromy matrices with the relation of spectral parameters.

Emergent gravity in two dimensions

We explore models with emergent gravity and metric by means of numerical simulations. A particular type of two-dimensional non-linear sigma-model is regularized and discretized on a quadratic lattice. It is characterized by lattice diffeomorphism invariance which ensures in the continuum limit the symmetry of general coordinate transformations. We observe a collective order parameter with properties of a metric, showing Minkowski or Euclidean signature. The correlation functions of the metric reveal an interesting long-distance behavior with power-like decay. This universal critical behavior occurs without tuning of parameters and thus constitutes an example of “self-tuned criticality” for this type of sigma-models. We also find a non-vanishing expectation value of a “zweibein” related to the “internal” degrees of freedom of the scalar field, again with long-range correlations. The metric is well described as a composite of the zweibein. A scalar condensate breaks Euclidean rotation symmetry.

On the classical equivalence of monodromy matrices in squashed sigma model

We proceed to study the hybrid integrable structure in two-dimensional non-linear sigma models with target space three-dimensional squashed spheres. A quantum affine algebra and a pair of Yangian algebras are realized in the sigma models and, according to them, there are two descriptions to describe the classical dynamics 1) the trigonometric description and 2) the rational description, respectively. For every description, a Lax pair is constructed and the associated monodromy matrix is also constructed. In this paper we show the gauge-equivalence of the monodromy matrices in the trigonometric and rational description under a certain relation between spectral parameters and the rescalings of sl(2) generators.