scholarly journals Deformed $$\sigma $$-models, Ricci flow and Toda field theories

2021 ◽  
Vol 111 (6) ◽  
Author(s):  
Dmitri Bykov ◽  
Dieter Lüst
Keyword(s):  
Sigma Models ◽  
Ricci Flow ◽  
Flag Manifold ◽  
Flow Equation ◽  
Target Space ◽  
Field Theories ◽  
Two Dimensional

AbstractIt is shown that the Pohlmeyer map of a $$\sigma $$ σ -model with a toric two-dimensional target space naturally leads to the ‘sausage’ metric. We then elaborate the trigonometric deformation of the $$\mathbb {CP}^{n-1}$$ 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 $$\sigma $$ σ -models and Toda field theories.

scholarly journals Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Athanasios Chatzistavrakidis ◽  
Grgur Šimunić
Keyword(s):  
Sigma Models ◽  
Sigma Model ◽  
Target Space ◽  
Two Dimensional ◽  
Courant Algebroids ◽  
Dirac Structures ◽  
Courant Algebroid ◽  
Abelian Case ◽  
Jacobi Manifolds ◽  
Jacobi Structure

Abstract 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.

Integrability in 2D fields theory/sigma-models

2019 ◽  
pp. 248-318
Author(s):  
Sergei L. Lukyanov ◽  
Alexander B. Zamolodchikov
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Sigma Models ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field ◽  
Zero Curvature ◽  
Basic Facts ◽  
Linear Sigma Models ◽  
General Aspects

This is a two-part course about the integrability of two-dimensional non-linear sigma models (2D NLSM). In the first part general aspects of classical integrability are discussed, based on the O(3) and O(4) sigma-models and the field theories related to them. The second part is devoted to the quantum 2D NLSM. Among the topics considered are: basic facts of conformal field theory, zero-curvature representations, integrals of motion, one-loop renormalizability of 2D NLSM, integrable structures in the so-called cigar and sausage models, and their RG flows. The text contains a large number of exercises of varying levels of difficulty.

scholarly journals Harmonic analysis of 2d CFT partition functions

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Nathan Benjamin ◽  
Scott Collier ◽  
A. Liam Fitzpatrick ◽  
Alexander Maloney ◽  
Eric Perlmutter
Keyword(s):  
Partition Function ◽  
Harmonic Analysis ◽  
Moduli Space ◽  
Target Space ◽  
Field Theories ◽  
Partition Functions ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Operator Spectrum

Abstract We apply the theory of harmonic analysis on the fundamental domain of SL(2, ℤ) to partition functions of two-dimensional conformal field theories. We decompose the partition function of c free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space ℍ/SL(2, ℤ), and of target space moduli space O(c, c; ℤ)\O(c, c; ℝ)/O(c)×O(c). This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to AdS3 gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.

scholarly journals On isometry anomalies in minimal 𝒩 = (0,1) and 𝒩 = (0,2) sigma models

2016 ◽  
Vol 31 (27) ◽  
pp. 1650147 ◽  
Author(s):  
Jin Chen ◽  
Xiaoyi Cui ◽  
Mikhail Shifman ◽  
Arkady Vainshtein
Keyword(s):  
Sigma Models ◽  
Local Level ◽  
Exceptional Case ◽  
Target Space ◽  
Two Dimensional ◽  
Fermion Loop ◽  
Pontryagin Class ◽  
Text Model ◽  
Anomaly Analysis ◽  
Minimal Formula

The two-dimensional minimal supersymmetric sigma models with homogeneous target spaces [Formula: see text] and chiral fermions of the same chirality are revisited. In particular, we look into the isometry anomalies in [Formula: see text] and [Formula: see text] models. These anomalies are generated by fermion loop diagrams which we explicitly calculate. In the case of [Formula: see text] sigma models the first Pontryagin class vanishes, so there is no global obstruction for the minimal [Formula: see text] supersymmetrization of these models. We show that at the local level isometries in these models can be made anomaly free by specifying the counterterms explicitly. Thus, there are no obstructions to quantizing the minimal [Formula: see text] models with the [Formula: see text] target space while preserving the isometries. This also includes [Formula: see text] (equivalent to [Formula: see text]) which is an exceptional case from the [Formula: see text] series. For other [Formula: see text] models, the isometry anomalies cannot be rescued even locally, this leads us to a discussion on the relation between the geometric and gauged formulations of the [Formula: see text] models to compare the original of different anomalies. A dual formalism of [Formula: see text] model is also given, in order to show the consistency of our isometry anomaly analysis in different formalisms. The concrete counterterms to be added, however, will be formalism dependent.

scholarly journals Sigma models on flags

2019 ◽  
Vol 6 (2) ◽  
Author(s):  
Kantaro Ohmori ◽  
Nathan Seiberg ◽  
Shu-Heng Shao
Keyword(s):  
Sigma Models ◽  
Flag Manifold ◽  
Target Space ◽  
Global Symmetry ◽  
Specific Focus ◽  
Non Linear ◽  
Linear Sigma Models ◽  
Special Case

We study (1+1)-dimensional non-linear sigma models whose target space is the flag manifold U(N)\over U(N_1)\times U(N_2)\cdots U(N_m), with a specific focus on the special case U(N)/U(1)^{N}U(N)/U(1)N. These generalize the well-known \mathbb{CP}^{N-1}ℂℙN−1 model. The general flag model exhibits several new elements that are not present in the special case of the \mathbb{CP}^{N-1}ℂℙN−1 model. It depends on more parameters, its global symmetry can be larger, and its ’t Hooft anomalies can be more subtle. Our discussion based on symmetry and anomaly suggests that for certain choices of the integers N_INI and for specific values of the parameters the model is gapless in the IR and is described by an SU(N)_1SU(N)1 WZW model. Some of the techniques we present can also be applied to other cases.

scholarly journals THE RADIAL PART OF THE ZERO-MODE HAMILTONIAN FOR SIGMA MODELS WITH GROUP TARGET SPACE

2004 ◽  
Vol 16 (05) ◽  
pp. 603-628 ◽  
Author(s):  
DOUG PICKRELL
Keyword(s):  
Sigma Models ◽  
Lie Group ◽  
Sigma Model ◽  
Zero Mode ◽  
Target Space ◽  
Two Dimensional ◽  
Radial Part ◽  
Group Target ◽  
Simply Connected ◽  
Connected Lie Group

In this note, we use geometric arguments to derive a possible form for the radial part of the "zero-mode Hamiltonian" for the two-dimensional sigma model with target space S3, or more generally a compact simply connected Lie group.

scholarly journals PSEUDODUALITY AND COMPLEX GEOMETRY IN SIGMA MODELS

2013 ◽  
Vol 10 (07) ◽  
pp. 1350034
Author(s):  
MUSTAFA SARISAMAN
Keyword(s):  
Sigma Models ◽  
Complex Geometry ◽  
Kähler Manifolds ◽  
Target Space ◽  
Kahler Manifolds ◽  
Two Dimensional ◽  
Additional Constraints

We study the pseudoduality transformations in two-dimensional N = (2, 2) sigma models on Kähler manifolds. We show that structures on the target space can be transformed into the pseudodual manifolds by means of (anti)holomorphic preserving mapping. This map requires that torsions related to individual spaces and riemann connection on pseudodual manifold must vanish. We also consider holomorphic isometries which puts additional constraints on the pseudoduality.

scholarly journals GENERALIZED MATHAI-QUILLEN TOPOLOGICAL SIGMA MODELS

1996 ◽  
Vol 11 (32n33) ◽  
pp. 2585-2600
Author(s):  
PABLO M. LLATAS
Keyword(s):  
Quantum Mechanics ◽  
Sigma Models ◽  
Topological Field Theories ◽  
Theoretical Approach ◽  
Sigma Model ◽  
Target Space ◽  
Field Theories ◽  
Internal Space ◽  
Topological Quantum ◽  
Topological Field

A simple field theoretical approach to Mathai-Quillen topological field theories of maps X : MI→MT from an internal space to a target space is presented. As an example of applications of our formalism we compute by applying our formulas the action and Q-variations of the fields of two well-known topological systems: topological quantum mechanics and type-A topological sigma model.

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