arXiv subjects

Dmitri Bykov

Publications and source records attributed to Dmitri Bykov.

At least 37 records · Page 2Linked to original sources

arXiv2020

The $\mathsf{CP^{n-1}}$-model with fermions: a new look

We elaborate the formulation of the $\mathsf{CP^{n-1}}$ sigma model with fermions as a gauged Gross-Neveu model. This approach allows to identify the super phase space of the model as a supersymplectic quotient. Potential chiral gauge anomalies are shown to receive contributions from bosons and fermions alike and are related to properties of this phase space. Along the way we demonstrate that the worldsheet supersymmetric model is a supersymplectic quotient of a model with target space supersymmetry. Possible generalizations to other quiver supervarieties are briefly discussed.

hep-th↗

arXiv2020

Quantum flag manifold $σ$-models and Hermitian Ricci flow

We show that flag manifold $σ$-models (including $\mathbb{CP}^{n-1}$, Grassmannian models as special cases) and their deformed versions may be cast in the form of gauged bosonic Thirring/Gross-Neveu-type systems. Quantum mechanically the gauging is violated by chiral anomalies, which may be cancelled by adding fermions. We conjecture that such models are integrable and check on some examples that the trigonometrically deformed geometries satisfy the generalized Ricci flow equations.

hep-th↗

arXiv2020

Deformed $σ$-models, Ricci flow and Toda field theories

It is shown that the Pohlmeyer map of a $σ$-model with a toric two-dimensional target space naturally leads to the `sausage' metric. We then elaborate the trigonometric deformation of the $\mathrm{CP}^{n-1}$-model, proving that its $T$-dual metric is Kähler and solves the Ricci flow equation. Finally, we discuss a relation between flag manifold $σ$-models and Toda field theories.

hep-th↗

arXiv2020

Higher spin sl_2 R-matrix from equivariant (co)homology

We compute the rational $\mathfrak{sl}_2$ $R$-matrix acting in the product of two spin-$\ell\over 2$ (${\ell \in \mathbb{N}}$) representations, using a method analogous to the one of Maulik and Okounkov, i.e., by studying the equivariant (co)homology of certain algebraic varieties. These varieties, first considered by Nekrasov and Shatashvili, are typically singular. They may be thought of as the higher spin generalizations of $A_1$ Nakajima quiver varieties (i.e., cotangent bundles of Grassmannians), the latter corresponding to $\ell=1$.

math-ph↗

arXiv2019

Flag manifold sigma-models and nilpotent orbits

In the present paper we study flag manifold sigma-models that admit a zero-curvature representation. It is shown that these models may be naturally considered as interacting (holomorphic and anti-holomorphic) $βγ$-systems. Besides, using the theory of nilpotent orbits of complex Lie groups, we establish a relation to the principal chiral model.

hep-th↗

arXiv2019

Ricci-flat metrics on vector bundles over flag manifolds

We construct explicit complete Ricci-flat metrics on the total spaces of certain vector bundles over flag manifolds of the group $SU(n)$, for all Kähler classes. These metrics are natural generalizations of the metrics of Candelas-de la Ossa on the conifold, Pando Zayas-Tseytlin on the canonical bundle over $\mathbb{CP}^1\times \mathbb{CP}^1$, as well as the metrics on canonical bundles over flag manifolds, recently constructed by van Coevering.

hep-th↗

arXiv2019

Flag manifold sigma-models: the 1/N-expansion and the anomaly two-form

We construct a gauged linear sigma-model representation and develop a 1/N-expansion for flag manifold sigma-models previously proposed by the author. Classically there exists a zero-curvature representation for the equations of motion of these models, which leads in particular to the existence of a conserved non-local charge. We show that at the quantum level this charge is no longer conserved and calculate explicitly the anomaly in its conservation law.

hep-th↗

arXiv2017

Ricci-flat metrics on the cone over $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$

We describe a framework for constructing the Ricci-flat metrics on the total space of the canonical bundle over $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ (the del Pezzo surface of rank one). We construct explicitly the first-order deformation of the so-called `orthotoric metric' on this manifold. We also show that the deformation of the corresponding conformal Killing-Yano form does not exist.

hep-th↗

arXiv2016

Complex structure-induced deformations of sigma-models

We describe a deformation of the principal chiral model (with an even-dimensional target space G) by a B-field proportional to the Kähler form on the target space. The equations of motion of the deformed model admit a zero-curvature representation. As a simplest example, we consider the case of G=S^1 x S^3. We also apply a variant of the construction to a deformation of the AdS_3 x S^3 x S^1 (super-)sigma-model.

hep-th↗

arXiv2016

Complex structures and zero-curvature equations for sigma-models

We construct zero-curvature representations for the equations of motion of a class of sigma-models with complex homogeneous target spaces, not necessarily symmetric. We show that in the symmetric case the proposed flat connection is gauge-equivalent to the conventional one.

hep-th↗

arXiv2015

Classical solutions of a flag manifold sigma-model

We study a sigma-model with target space the flag manifold U(3)/U(1)^3. A peculiarity of the model is that the complex structure on the target space enters explicitly in the action. We describe the classical solutions of the model for the case when the worldsheet is a sphere CP^1.

hep-th↗

arXiv2014

Integrable properties of sigma-models with non-symmetric target spaces

It is well-known that sigma-models with symmetric target spaces are classically integrable. At the example of the model with target space the flag manifold U(3)/U(1)^3 -- a non-symmetric space -- we show that the introduction of torsion allows to cast the equations of motion in the form of a zero-curvature condition for a one-parametric family of connections, which can be a sign of integrability of the theory. We also elaborate on geometric aspects of the proposed model.

hep-th↗

arXiv2014

Comments on the del Pezzo cone

We describe a framework for constructing the general Ricci-flat metric on the anticanonical cone over the del Pezzo surface of rank one.

hep-th↗

arXiv2013

The Kähler metric of a blow-up

After a review of the general properties of holomorphic spheres in complex surfaces we describe the local geometry in the vicinity of a CP^1 embedded with a negative normal bundle. As a by-product, we build (asymptotically locally hyperbolic) Kahler-Einstein metrics on the total spaces of the line bundles O(-m), m >= 3 over CP^1. We check that the behavior of the Kahler potential is compatible with the Chern-Weil formulas for the Euler characteristic and signature. We also describe two supersymmetric setups where relevant constructions arise.

hep-th↗

arXiv2012

The geometry of antiferromagnetic spin chains

We construct spin chains that describe relativistic sigma-models in the continuum limit, using symplectic geometry as a main tool. The target space can be an arbitrary complex flag manifold, and we find universal expressions for the metric and theta-term.

hep-th↗

arXiv2011

Haldane limits via Lagrangian embeddings

In the present paper we revisit the so-called Haldane limit, i.e. a particular continuum limit, which leads from a spin chain to a sigma model. We use the coherent state formulation of the path integral to reduce the problem to a semiclassical one, which leads us to the observation that the Haldane limit is closely related to a Lagrangian embedding into the classical phase space of the spin chain. Using this property, we find a spin chain whose limit produces a relativistic sigma model with target space the manifold of complete flags U(N)/U(1)^N. We discuss possible other future applications of Lagrangian/isotropic embeddings in this context.

hep-th↗

arXiv2010

The worldsheet low-energy limit of the AdS_4 x CP^3 superstring

We consider the AdS_4 x CP^3 IIA superstring sigma-model in the background of the "spinning string" classical solution, which possesses two Noether spins. In the limit when one of the spins is infinite there are massless excitations, which govern the infrared worldsheet properties of the model. We obtain a sigma-model of CP^3 with fermions, which describes the dynamics of these massless modes.

hep-th↗

arXiv2009

Off-shell symmetry algebra of the AdS_4 x CP^3 superstring

By direct calculation in classical theory we derive the central extension of the off-shell symmetry algebra for the string propagating in AdS_4 x CP^3. It turns out to be the same as in the case of the AdS_5 x S^5 string. We also elaborate on the kappa-symmetry gauge and explain, how it can be chosen in a way which does not break bosonic symmetries.

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