Twisted N=2 Supersymmetry with Central Charge and Equivariant Cohomology

We present an equivariant extension of the Thom form with respect to a vector field action, in the framework of the Mathai-Quillen formalism. The associated Topological Quantum Field Theories correspond to twisted N=2 supersymmetric theories with a central charge. We analyze in detail two different cases: topological sigma models and non-abelian monopoles on four-manifolds. Received: 2 April 1996 / Accepted: 15 July 1996     

Topological massive sigma models

In this paper we construct topological sigma models which include a potential and are related to twisted massive supersymmetric sigma models. Contrary to a previous construction these models have no central charge and do not require the manifold to admit a Killing vector. We use the topological massive sigma model constructed here to simplify the calculation of the observables. Lastly it is noted that this model can be viewed as interpolating between topological massless sigma models and topological Landau-Ginzburg models.     

Chiral Nonlinear sigma models as models for topological superconductivity

We study the mechanism of topological superconductivity in a hierarchical chain of chiral nonlinear sigma models (models of current algebra) in one, two, and three spatial dimensions. The models illustrate how the 1D Fr?hlich's ideal conductivity extends to a genuine superconductivity in dimensions higher than one. The mechanism is based on the fact that a pointlike topological soliton carries an electric charge. We discuss a flux quantization mechanism and show that it is essentially a generalization of the persistent current phenomenon, known in quantum wires. We also discuss why the superconducting state is stable in the presence of a weak disorder.     

Topological field theories,Nicolai maps and BRST quantization

We establish a connection between topological field theories, Nicolai maps, BRST quantization and Langevin equations. In particular we show that there is a one-to-one correspondence between global unbroken supersymmetric theories which admit a Nicolai map and theories which arise as the BRST quantization of the square of the Langevin equation, setting the random field to zero. As such they are topological in nature. As an example we consider the topological quantum field theory of Witten in the Labastida-Pernici form and show that it is the first example of a theory admitting a complete Nicolai map in four dimensions. We also consider the topological sigma models of Witten and show that they too arise from the BRST quantization of the square of the Langevin equation.     

BRST model for equivariant cohomology and representatives for the equivariant thom class

In this paper the BRST formalism for topological field theories is studied in a mathematical setting. The BRST operator is obtained as a member of a one parameter family of operators connecting the Weil model and the Cartan model for equivariant cohomology. Furthermore, the BRST operator is identified as the sum of an equivariant derivation and its Fourier transform. Using this, the Mathai-Quillen representative for the Thom class of associated vector bundles is obtained as the Fourier transform of a simple BRST closed element.Supported by the SV FOM/SMC Mathematical Physics, The Netherlands     

Generalized Kähler Geometry from Supersymmetric Sigma Models

We give a physical derivation of generalized Kähler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri (Generalized complex geometry, DPhil thesis, Oxford University, 2004) regarding the equivalence between generalized Kähler geometry and the bi-hermitean geometry of Gates et al. (Nucl Phys B248:157, 1984). When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.     

The existence of Solitons in Gauged Sigma Models with Broken Symmetry: Some Remarks

We prove the existence of multisolitons in the gauged self-dual Maxwell and Chern–Simons sigma models with broken symmetry. In the context of topological solutions, the existence proofs rely on established results on earlier models. In the context of nontopological solutions with radial symmetry, the proof is by adapting a shooting argument.     

Classical solutions of higher-dimensional nonlinear sigma models on spheres

We construct several classical solutions of higher-dimensional nonlinear sigma models on spheres. These solutions are characterized by typical topological maps, in particular, four famous Hopf maps and the universal maps of theK-theory.Dedicated to the late Professor Shichiro Oka.     

The Mathai-Quillen formalism and topological field theory

These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the Mathai-Quillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the Atiyah-Jeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections.     

Dirac Sigma Models

We introduce a new topological sigma model, whose fields are bundle maps from the tangent bundle of a 2-dimensional world-sheet to a Dirac subbundle of an exact Courant algebroid over a target manifold. It generalizes simultaneously the (twisted) Poisson sigma model as well as the G/G-WZW model. The equations of motion are satisfied, iff the corresponding classical field is a Lie algebroid morphism. The Dirac Sigma Model has an inherently topological part as well as a kinetic term which uses a metric on worldsheet and target. The latter contribution serves as a kind of regulator for the theory, while at least classically the gauge invariant content turns out to be independent of any additional structure. In the (twisted) Poisson case one may drop the kinetic term altogether, obtaining the WZ-Poisson sigma model; in general, however, it is compulsory for establishing the morphism property.     

Quantum criticality and minimal conductivity in graphene with long-range disorder

We consider the conductivity sigma of graphene with negligible intervalley scattering at half filling. We derive the effective field theory, which, for the case of a potential disorder, is a symplectic-class sigma model including a topological term with theta=pi. As a consequence, the system is at a quantum critical point with a universal value of the conductivity of the order of e(2)/h. When the effective time-reversal symmetry is broken, the symmetry class becomes unitary, and sigma acquires the value characteristic for the quantum Hall transition.     

Topologically Stable Lumps in SO(d) Gauged O((d+1) Sigma Models in d Dimensions: d =2,3,4

The gauge-invariant topological charge is defined for, and the inequalities supplying the lower bound on the action of an SO(4) gauged O(5) sigma model in four dimensions are established. The consistency of the solution with finiteness of the action and with topological stability is briefly verified for a particular dynamical example. Against the background of the topologically stable finite energy solitons of SO(d) gauged O(d+1) sigma models in d dimensions already known for d=2 and for d=3, the present example can be viewed as a demonstration by induction for the existence of such solitons in the case of arbitrary k.     

Meron solutions of the sigma model and singular harmonic maps

We study the meron solutions of theO(3) sigma model in two-dimensions using the singular harmonic maps. We find new solutions of the topological charge (the Brouwer degree) zero having one dimensional singularities of the fold and collapse types. In general any singularity of a solution of the sigma model is locally equivalent with one of the four types of singular points of a harmonic map.     

Quantized anomalous Hall effect in two-dimensional ferromagnets: quantum Hall effect in metals

We study the effect of disorder on the anomalous Hall effect (AHE) in two-dimensional ferromagnets. The topological nature of the AHE leads to the integer quantum Hall effect from a metal, i.e., the quantization of sigma(xy) induced by the localization except for the few extended states carrying Chern numbers. Extensive numerical study on a model reveals that Pruisken's two-parameter scaling theory holds even when the system has no gap with the overlapping multibands and without the uniform magnetic field. Therefore, the condition for the quantized AHE is given only by the Hall conductivity sigma(xy) without the quantum correction, i.e., /sigma(xy)/>e(2)/(2h).     

Bogomol'nyi solitons and Hermitian symmetric spaces

We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chern-Simons (CS) gauge field and we study self-dual solitons. When the target space is given by a Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol'nyi bounded from below by a topological charge. The Bogomol'nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the CP(N − 1) case is explored. We also discuss the self-dual solitons in the case of noncompact SU(1, 1) and present a detailed analysis for the rotationally symmetric solutions.     

Hofstadter butterfly and integer quantum hall effect in three dimensions

For a three-dimensional (3D) lattice in magnetic fields we have shown that the hopping along the third direction, which normally smears out the Landau quantization gaps, can rather give rise to a Hofstadter's butterfly specific to 3D when a criterion is fulfilled by anisotropic (quasi-one-dimensional) systems. In 3D the angle of the magnetic field plays the role of the field intensity in 2D, so that the butterfly can occur in much smaller fields. We have also calculated the Hall conductivity in terms of the topological invariant in the Kohmoto-Halperin-Wu formula, and each of sigma(xy),sigma(zx) is found to be quantized.     

Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization

We discuss, for a two-dimensional Dirac Hamiltonian with a random scalar potential, the presence of a Z2 topological term in the nonlinear sigma model encoding the physics of Anderson localization in the symplectic symmetry class. The Z2 topological term realizes the sign of the Pfaffian of a family of Dirac operators. We compute the corresponding global anomaly, i.e., the change in the sign of the Pfaffian by studying a spectral flow numerically. This Z2 topological effect can be relevant to graphene when the impurity potential is long ranged and, also, to the two-dimensional boundaries of a three-dimensional lattice model of Z2 topological insulators in the symplectic symmetry class.     

GLSMs for partial flag manifolds

In this paper we outline some aspects of nonabelian gauged linear sigma models. First, we review how partial flag manifolds (generalizing Grassmannians) are described physically by nonabelian gauged linear sigma models, paying attention to realizations of tangent bundles and other aspects pertinent to (0, 2) models. Second, we review constructions of Calabi–Yau complete intersections within such flag manifolds, and properties of the gauged linear sigma models. We discuss a number of examples of nonabelian GLSMs in which the Kähler phases are not birational, and in which at least one phase is realized in some fashion other than as a complete intersection, extending previous work of Hori–Tong. We also review an example of an abelian GLSM exhibiting the same phenomenon. We tentatively identify the mathematical relationship between such non-birational phases, as examples of Kuznetsov’s homological projective duality. Finally, we discuss linear sigma model moduli spaces in these gauged linear sigma models. We argue that the moduli spaces being realized physically by these GLSMs are precisely Quot and hyperquot schemes, as one would expect mathematically.     

Vector and Spinor Decomposition of SU（2） Gauge Potential, Their Equivalence, and Knot Structure in SU（2） Chern-Simons Theory

In this paper, spinor and vector decompositions of SU（2） gauge potential are presented and their equivalence is constructed using a simply proposal. We also obtain the action of Faddeev nonlinear 0（3） sigma model from the SU（2） mass/ve gauge field theory, which is proposed according to the gauge invariant principle. At last, the knot structure in SU（2） Chern-Simons filed theory is discussed in terms of the Φ-mapping topological current theory, The topological charge of the knot is characterized by the Hopf indices and the Brouwer degrees of Φ-mapping.