Sigma Model In Terms Research Articles

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.

On Yang–Baxter models, twist operators, and boundary conditions

We discuss homogeneous Yang–Baxter deformations of integrable sigma models in terms of twist operators. We show that the twist operators behave as the classical analogue of a Drinfeld twist, for all abelian and almost abelian deformations. We also use twist operators to rederive the well-known interpretation of TsT transformations—equivalent to abelian deformations—in terms of twisted boundary conditions. We discuss complications in extending this boundary condition picture to non-abelian deformations.

Nonabelian T-duals of the flat background

Classification of nonabelian T-duals of the flat matric in D=4 with respect to the four-dimensional continuous subgroups of the Poincare group is given. After dualizing the flat background we identify majority of dual models as conformal sigma models in plane- parallel wave backgrounds, most of them having torsion. We give their form in Brinkmann coordinates. Besides the pp-waves we find several diagonalizable curved metrics with nontrivial scalar curvature and torsion.Due to the nonabelian T-duality, we find general solution of classical field equations of all the sigma models in terms of d'Alembert solutions of wave equation.

Plane-parallel waves as duals of the flat background

We give a classification of non-Abelian T-duals (‘T’ standing for topological or toroidal) of the flat metric in D = 4 dimensions with respect to the four-dimensional continuous subgroups of the Poincaré group. After dualizing the flat background, we identify the majority of dual models as conformal sigma models in plane-parallel wave backgrounds, most of them having torsion. We give their form in Brinkmann coordinates. We find, besides the plane-parallel waves, several diagonalizable curved metrics with nontrivial scalar curvature and torsion. Using the non-Abelian T-duality, we find general solutions of the classical field equations for all the sigma models in terms of dʼAlembert solutions of the wave equation.

On the sigma-model of deformed special geometry

We discuss the deformed sigma-model that arises when considering four-dimensional N=2 abelian vector multiplets in the presence of an arbitrary chiral background field. In addition, we allow for a class of deformations of special geometry by non-holomorphic terms. We analyze the geometry of the sigma-model in terms of intrinsic torsion classes. We show that, generically, the deformed geometry is non-Kähler. We illustrate our findings with an example. We also express the deformed sigma-model in terms of the Hesse potential that underlies the real formulation of special geometry.

Topological D-branes and commutative algebra

We show that questions concerning the topological B-model on a Calabi–Yau manifold in the Landau–Ginzburg phase can be rephrased in the language of commutative algebra. This yields interesting and very practical methods for analyzing the model. We demonstrate how the relevant “Ext” groups and superpotentials can be computed efficiently by computer algebra packages such as Macaulay. This picture leads us to conjecture a general description of D-branes in linear sigma models in terms of triangulated categories. Each phase of the linear sigma model is associated with a different presentation of the category of D-branes.

Quantum phase transitions beyond the Landau paradigm in a Sp(4) spin system

We propose quantum phase transitions beyond the Landau's paradigm of Sp(4) spin Heisenberg models on the triangular and square lattices motivated by the exact $\text{Sp}(4)\ensuremath{\simeq}\text{SO}(5)$ symmetry of spin-3/2 fermionic cold atomic system with only $s$-wave scattering. On the triangular lattice, we study a phase transition between the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ spin ordered phase and a ${Z}_{2}$ spin liquid phase; this phase transition is described by an O(8) sigma model in terms of fractionalized spinon fields, with significant anomalous scaling dimensions of spin order parameters. On the square lattice, we propose a deconfined critical point between the Neel order and the valence bond solid (VBS) order, which is described by the CP(3) model, and the monopole effect of the compact U(1) gauge field is expected to be suppressed at the critical point.

TOPOLOGICAL OBJECTS IN THE O(3) NONLINEAR SIGMA MODEL

We study the topological defects in the nonlinear O(3) sigma model in terms of the decomposition of U(1) gauge potential. Time-dependent baby skyrmions are discussed in the (2+1)-dimensional spacetime with the CP1 field. Furthermore, we show that there are three kinds of topological defects–vortex lines, point defects and knot exist in the (3+1)-dimensional model, and their topological charges, locations and motions are determined by the ϕ-mapping topological current theory.

Path-integral over non-linearly realized groups and hierarchy solutions

The technical problem of deriving the full Green functions of the elementary pion fields of the nonlinear sigma model in terms of ancestor amplitudes involving only the flat connection and the nonlinear sigma model constraint is a very complex task. In this paper we solve this problem by integrating, order by order in the perturbative loop expansion, the local functional equation derived from the invariance of the SU(2) Haar measure under local left multiplication. This yields the perturbative definition of the path-integral over the non-linearly realized SU(2) group.

Weak Power-Counting Theorem for the Renormalization of the Nonlinear Sigma Model in Four Dimensions

The formulation of the non-linear sigma model in terms of flat connection allows the construction of a perturbative solution of a local functional equation encoding the underlying gauge symmetry. In this paper we discuss some properties of the solution at the one-loop level in D=4. We prove the validity of a weak power-counting theorem in the following form: although the number of divergent amplitudes is infinite only a finite number of divergent amplitudes have to be renormalized. The proof uses the linearized functional equation of which we provide the general solution in terms of local functionals. The counteterms are given in terms of linear combinations of these invariants and the coefficients are fixed by a finite number of divergent amplitudes. The latter contain only insertions of the composite operators $\phi_0$ (the constraint of the non-linear sigma model) and $F_\mu$ (the flat connection). These amplitudes are at the top of a hierarchy implicit in the functional equation. As an example we derive the counterterms for the four-point amplitudes.

Low-energy effective action of lightly doped two-legt−Jladders

We propose a low energy effective theory of lightly doped two-leg t-J ladders with the help of slave fermion technique. The continuum limit of this model consists of two kinds of Dirac fermions which are coupled to the O(3) non-linear sigma model in terms of the gauge coupling with opposite sign of "charges". In addition to the gauge interaction, there is another kind of attractive force between these Dirac fermions, which arises from the short-ranged antiferromagnetic order. We show that the latter is essential to determine the low energy properties of lightly doped two-leg t-J ladders. The effective Hamiltonian we obtain is a bosonic Gaussian model and the boson field basically describes the particle density fluctuation. We also find two types of gapped spin excitations. Finally, we discuss the possible instabilities: charge density wave (CDW) and singlet superconductivity (SC). We find that the SC instability dominates in our approximation. Our results indicate that lightly doped ladders fall into the universality class of Luther-Emery model.

Microscopic description of D-brane black holes

We discuss the derivation of the effective SCFT description of certain aspects of the near extremal 5-dimensional black hole, which is modelled by a collection of D1- and D5-branes, by studying the corresponding U( Q 1) × U( Q 5) gauge theory in 2-dimensions. We give an analysis of the Higgs branch moduli to extract the independent degrees of freedom in a unitary gauge and then indicate the relevance of the residual S( Q 1) × S( Q 5) Weyl invariance to extract the low-energy degrees of freedom. These are strings corresponding to the maximal cycle of the Weyl subgroup. The corresponding sigma model in terms of these strings is non-linear and non-local. We calculate the string tension. Assuming that long wavelength fluctuations are governed by a local SCFT with N=4 SUSY, we indicate how, under mild assumptions, the sigma model becomes a free SCFT with an internal symmetry group SO(4) and central charge c = 6. We also discuss the coupling of the SCFT to bulk matter. Our considerations imply that the calculation of the Hawking decay rate has a sound foundation in string theory and is valid in the black hole regime where gN ∼ 1.

Twistor spaces for hyper-Kähler manifolds with torsion

We construct the twistor space associated with an HKT manifold, that is, a hyper-Kähler manifold with torsion, a type of geometry that arises as the target space geometry in two-dimensional sigma models with (4,0) supersymmetry. We show that this twistor space has a natural complex structure and is a holomorphic fibre bundle over the complex projective line with fibre the associated HKT manifold. We also show how the metric and torsion of the HKT manifold can be determined from data on the twistor space by a reconstruction theorem. We give a geometric description of the sigma model (4,0) superfields as holomorphic maps (suitably understood) from a twistorial extension of (4,0) superspace (harmonic superspace) into the twistor space of the sigma model target manifold and write an action for the sigma model in terms of these (4,0) superfields.

Long-wavelength dynamics of the two-dimensional antiferromagnet

We perform the long-wavelength reduction of the two-dimensional quantum lattice rotator model. An intermediate scale is introduced and the short-wavelength antiferromagnetic spin fluctuations are integrated out. The saddle-point solution for the generalized continuous sigma-model in terms of parameters of the initial model is constructed.

Multiple mirror manifolds and topology change in string theory

We use mirror symmetry to establish the first concrete arena of spacetime topology change in string theory. In particular, we establish that the quantum theories based on certain nonlinear sigma models with topologically distinct target spaces can be smoothly connected even though classically a physical singularity would be encountered. We accomplish this by rephrasing the description of these nonlinear sigma models in terms of their mirror manifold partners - a description in which the full quantum theory can be described exactly using lowest order geometrical methods. We establish that, for the known class of mirror manifolds, the moduli space of the corresponding conformal field theory requires not just two but numerous topologically distinct Calabi-Yau manifolds for its geometric interpretation. A single family of continously connected conformal theories thereby probes a host of topologically distinct geometrical spaces giving rise to multiple mirror manifolds.