Noncommutative Wess–Zumino–Witten actions and their Seiberg–Witten invariance

We analyze the noncommutative two-dimensional Wess–Zumino–Witten model and its properties under Seiberg–Witten transformations in the operator formulation. We prove that the model is invariant under such transformations even for the noncritical (non-chiral) case, in which the coefficients of the kinetic and Wess–Zumino terms are not related. The pure Wess–Zumino term represents a singular case in which this transformation fails to reach a commutative limit. We also discuss potential implications of this result for bosonization.     

Fredholm Property of Operators from 2D String Field Theory

In a recent study of Landau-Ginzburg model of string field theory by Gaiotto, Moore and Witten, there appears a type of perturbed Cauchy-Riemann equation, i.e. the ζ -instanton equation. Solutions of ζ -instanton equation have degenerate asymptotics. This degeneracy is a severe restriction for obtaining the Fredholm property and constructing relevant homology theory. In this article, we study the Fredholm property of a sort of differential operators with degenerate asymptotics. As an application, we verify certain Fredholm property of the linearized operator of ζ -instanton equations.     

Non-commutative Lorentz symmetry and the origin of the Seiberg–Witten map

We show that the non-commutative Yang–Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The non-commutative Yang–Mills action is invariant under combined conformal transformations of the Yang–Mills field and of the non-commutativity parameter . The Seiberg–Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action. Received: 6 November 2001 / Published online: 5 April 2002     

Moyal Deformation,Seiberg–Witten Maps,and Integrable Models

A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg–Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg–Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.     

Renormalisability of the matter determinants in noncommutative gauge theory in the enveloping-algebra formalism

We consider noncommutative gauge theory defined by means of Seiberg–Witten maps for an arbitrary semisimple gauge group. We compute the one-loop UV divergent matter contributions to the gauge field effective action to all orders in the noncommutative parameters θ. We do this for Dirac fermions and complex scalars carrying arbitrary representations of the gauge group. We use path-integral methods in the framework of dimensional regularisation and consider arbitrary invertible Seiberg–Witten maps that are linear in the matter fields. Surprisingly, it turns out that the UV divergent parts of the matter contributions are proportional to the noncommutative Yang–Mills action where traces are taken over the representation of the matter fields; this result supports the need to include such traces in the classical action of the gauge sector of the noncommutative theory.     

Construction of singular vertex operators as degenerate primary conformal fields

A general construction of null fields is given through the use of a vertex operator representation of conformal quantum fields. Singular vertex operators are constructed as possessing the conformal properties of a degenerate primary field. A systematic approach to obtaining the partial differential equations for correlation functions is suggested.     

On the Riemannian geometry of Seiberg–Witten moduli spaces

We construct a natural L²$L^2$-metric on the perturbed Seiberg–Witten moduli spaces _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ of a compact 4-manifold M$M$, and we study the resulting Riemannian geometry of _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$. We derive a formula which expresses the sectional curvature of _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case M$M$ is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U(1)$U\left(1\right)$ bundle P→_Mμ+$\mathfrak{P}\to {\mathfrak{M}}_{{\mu }^{+}}$ such that the bundle projection becomes a Riemannian submersion. On a Kähler surface M$M$, the L²$L^2$-metric on _Mμ+${\mathfrak{M}}_{{\mu }^{+}}$ coincides with the natural Kähler metric on moduli spaces of vortices.     

Theory of extreme correlations using canonical Fermions and path integrals

The  t$t$–J$J$  model is studied using a novel and rigorous mapping of the Gutzwiller projected electrons, in terms of canonical electrons. The mapping has considerable similarity to the Dyson–Maleev transformation relating spin operators to canonical Bosons. This representation gives rise to a non Hermitian quantum theory, characterized by minimal redundancies. A path integral representation of the canonical theory is given. Using it, the salient results of the extremely correlated Fermi liquid (ECFL) theory, including the previously found Schwinger equations of motion, are easily rederived. Further, a transparent physical interpretation of the previously introduced auxiliary Greens function and the ‘caparison factor’, is obtained.     

Infinite conformal symmetry in two-dimensional quantum field theory

We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of Virasoro algebra, and that the correlation functions are built up of the “conformal blocks” which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the degenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations.     

On a Selberg–Schur Integral

A generalization of Selberg’s beta integral involving Schur polynomials associated with partitions with entries not greater than 2 is explicitly computed. The complex version of this integral is given after proving a general statement concerning the complex extensions of Selberg–Schur integrals. All these results have interesting applications in both mathematics and physics, particularly in conformal field theory, because the conformal blocks for the Wess–Zumino–Novikov–Witten model with underlying affine structure can be obtained by analytical continuation of these integrals.      

Avoidance of a Landau pole by flat contributions in QED

We consider massless Quantum Electrodynamics in the momentum scheme and carry forward an approach based on Dyson–Schwinger equations to approximate both the β$\beta$-function and the renormalized photon self-energy (Yeats, 2011). Starting from the Callan–Symanzik equation, we derive a renormalization group (RG) recursion identity which implies a non-linear ODE for the anomalous dimension and extract a sufficient but not necessary criterion for the existence of a Landau pole. This criterion implies a necessary condition for QED to have no such pole. Solving the differential equation exactly for a toy model case, we integrate the corresponding RG equation for the running coupling and find that even though the β$\beta$-function entails a Landau pole it exhibits a flat contribution capable of decreasing its growth, in other cases possibly to the extent that such a pole is avoided altogether. Finally, by applying the recursion identity, we compute the photon propagator and investigate the effect of flat contributions on both spacelike and timelike photons.     

Noncommutative theories, the axial anomaly, Fujikawa's method and the Atiyah–Singer index

Fujikawa's method is employed to compute at first order in the noncommutative parameter the U(1)_A anomaly for noncommutative SU(N). We consider the most general Seiberg–Witten map which commutes with hermiticity and complex conjugation and a noncommutative matrix parameter, θ^μν, which is of “magnetic” type. Our results for SU(N) can be readily generalized to cover the case of general nonsemisimple gauge groups when the symmetric Seiberg–Witten map is used. Connection with the Atiyah–Singer index theorem is also made.     

A first-order system approach for diffusion equation. II: Unification of advection and diffusion

In this paper, we unify advection and diffusion into a single hyperbolic system by extending the first-order system approach introduced for the diffusion equation [J. Comput. Phys., 227 (2007) 315–352] to the advection–diffusion equation. Specifically, we construct a unified hyperbolic advection–diffusion system by expressing the diffusion term as a first-order hyperbolic system and simply adding the advection term to it. Naturally then, we develop upwind schemes for this entire   system; there is thus no need to develop two different schemes, i.e., advection and diffusion schemes. We show that numerical schemes constructed in this way can be automatically uniformly accurate, allow O(h)$O(h)$ time step, and compute the solution gradients (viscous stresses/heat fluxes for the Navier–Stokes equations) simultaneously to the same order of accuracy as the main variable, for all Reynolds numbers. We present numerical results for boundary-layer type problems on non-uniform grids in one dimension and irregular triangular grids in two dimensions to demonstrate various remarkable advantages of the proposed approach. In particular, we show that the schemes solving the first-order advection–diffusion system give a tremendous speed-up in CPU time over traditional scalar schemes despite the additional cost of carrying extra variables and solving equations for them. We conclude the paper with discussions on further developments to come.     

Quantum corrections to Bekenstein–Hawking black hole entropy and gravity partition functions

Algebraic aspects of the computation of partition functions for quantum gravity and black holes in _AdS3${\mathit{AdS}}_3$ are discussed. We compute the sub-leading quantum corrections to the Bekenstein–Hawking entropy. It is shown that the quantum corrections to the classical result can be included systematically by making use of the comparison with conformal field theory partition functions, via the _AdS3/_CFT2${\mathit{AdS}}_3/{\mathit{CFT}}_2$ correspondence. This leads to a better understanding of the role of modular and spectral functions, from the point of view of the representation theory of infinite-dimensional Lie algebras. Besides, the sum of known quantum contributions to the partition function can be presented in a closed form, involving the Patterson–Selberg spectral function. These contributions can be reproduced in a holomorphically factorized theory whose partition functions are associated with the formal characters of the Virasoro modules. We propose a spectral function formulation for quantum corrections to the elliptic genus from supergravity states.     

String modular phases in Calabi–Yau families

We investigate the structure of singular Calabi–Yau varieties in moduli spaces that contain a Brieskorn–Pham point. Our main tool is a construction of families of deformed motives over the parameter space. We analyze these motives for general fibers and explicitly compute the L$L$-series for singular fibers for several families. We find that the resulting motivic L$L$-functions agree with the L$L$-series of modular forms whose weight depends both on the rank of the motive and the degree of the degeneration of the variety. Surprisingly, these motivic L$L$-functions are identical in several cases to L$L$-series derived from weighted Fermat hypersurfaces. This shows that singular Calabi–Yau spaces of non-conifold type can admit a string worldsheet interpretation, much like rational theories, and that the corresponding irrational conformal field theories inherit information from the Gepner conformal field theory of the weighted Fermat fiber of the family. These results suggest that phase transitions via non-conifold configurations are physically plausible. In the case of severe degenerations we find a dimensional transmutation of the motives. This suggests further that singular configurations with non-conifold singularities may facilitate transitions between Calabi–Yau varieties of different dimensions.     

Universal R operator with Jordanian deformation of conformal symmetry

The Jordanian deformation of sl(2) bialgebra structure is studied in view of physical applications to breaking of conformal symmetry in the high energy asymptotics of scattering. Representations are formulated in terms of polynomials, generators in terms of differential operators. The deformed R operator with generic representations is analyzed in spectral and integral forms.     

Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions

Employing the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the XXX spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated T–Q$T\text{–}Q$ relation and the Bethe ansatz equations are derived.