Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M ₂(ℂ)=ℂℤ₂·ℂℤ₂. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct q-monopoles on all the Podleś quantum spheres S ² _q,s . Received: 25 September 1998 / Accepted: 23 February 2000     

A solution for galactic disks with Yukawian gravitational potential

We present a new solution for the rotation curves of galactic disks with gravitational potential of the Yukawa type. We follow the technique employed by Toomre in 1963 in the study of galactic disks in the Newtonian theory. This new solution allows an easy comparison between the Newtonian solution and the Yukawian one. Therefore, constraints on the parameters of theories of gravitation can be imposed, which in the weak field limit reduce to Yukawian potentials. We then apply our formulae to the study of rotation curves for a zero-thickness exponential disk and compare it with the Newtonian case studied by Freeman in 1970. As an application of the mathematical tool developed here, we show that in any theory of gravity with a massive graviton (this means a gravitational potential of the Yukawa type), a strong limit can be imposed on the mass (m _g) of this particle. For example, in order to obtain a galactic disk with a scale length of b∼ 10 kpc, we should have a massive graviton of m _g ≪ 10⁻⁵⁹g. This result is much more restrictive than those inferred from solar system observations.     

The one-loop Green’s functions of dimensionally reduced gauge theories

We apply the dimensional regularization technique as well as that by dimensional reduction to the calculation of the regularized one-loop Green’s functions ind ₀-dimensional Yang-Mills theory with real massless scalars and spinors in arbitrary (real) representations of a gauge groupG. As a particular example, the super-symmetrically regularized one-loop Green’s functions of theN=4 supersymmetric Yang-Mills model are derived.     

Krein Regularization of <Emphasis Type="Italic">λϕ</Emphasis><Superscript>4</Superscript>

We calculate the four-point function in λϕ ⁴ theory by using Krein regularization and compare our result, which is finite, with the usual result in λϕ ⁴ theory. The effective coupling constant (λ _μ ) is also calculated in this method.     

A Condensed Matter Interpretation of SM Fermions and Gauge Fields

We present the bundle (Aff(3)⊗ℂ⊗Λ)(ℝ³), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each (ℂ⊗Λ)(ℝ³) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space (Aff(3)⊗ℂ)(ℤ³). This space allows a simple physical interpretation as a phase space of a lattice of cells. We find the SM SU(3) _c ×SU(2) _L ×U(1) _Y action on (Aff(3)⊗ℂ⊗Λ)(ℝ³) to be a maximal anomaly-free gauge action preserving E(3) symmetry and symplectic structure, which can be constructed using two simple types of gauge-like lattice fields: Wilson gauge fields and correction terms for lattice deformations. The lattice fermion fields we propose to quantize as low energy states of a canonical quantum theory with ℤ₂-degenerated vacuum state. We construct anticommuting fermion operators for the resulting ℤ₂-valued (spin) field theory. A metric theory of gravity compatible with this model is presented too.     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

A purely algebraic construction of a gauge and renormalization group invariant scalar glueball operator

This paper presents a complete algebraic proof of the renormalizability of the gauge invariant d=4 operator F _{μ ν} ²(x) to all orders of perturbation theory in pure Yang–Mills gauge theory, whereby working in the Landau gauge. This renormalization is far from being trivial as mixing occurs with other d=4 gauge variant operators, which we identify explicitly. We determine the mixing matrix Z to all orders in perturbation theory by using only algebraic arguments and consequently we can uncover a renormalization group invariant by using the anomalous dimension matrix Γ derived from Z. We also present a future plan for calculating the mass of the lightest scalar glueball with the help of the framework we have set up.     

Quasiclassical Lian-Zuckerman Homotopy Algebras,Courant Algebroids and Gauge Theory

We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to “light modes”, i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the A _∞-subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the category of VOA with a formal parameter to the category of A _∞-algebras. The associated generalized Maurer-Cartan equation gives the analogue of the Yang-Mills equation for a wide class of VOAs. Applying this construction to an example of VOA generated by β - γ systems, we find a remarkable relation between the Courant algebroid and the homotopy algebra of the Yang-Mills theory.     

Superbosonization of Invariant Random Matrix Ensembles

‘Superbosonization’ is a new variant of the method of commuting and anti-commuting variables as used in studying random matrix models of disordered and chaotic quantum systems. We here give a concise mathematical exposition of the key formulas of superbosonization. Conceived by analogy with the bosonization technique for Dirac fermions, the new method differs from the traditional one in that the superbosonization field is dual to the usual Hubbard-Stratonovich field. The present paper addresses invariant random matrix ensembles with symmetry group U _n , O _n , or USp _n , giving precise definitions and conditions of validity in each case. The method is illustrated at the example of Wegner’s n-orbital model. Superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non-Gaussian and/or non-invariant type.     

Neutron-neutron scattering length from the reaction γd → πnn employing chiral perturbation theory

We discuss the possibility to extract the neutron-neutron scattering length a_nn from experimental spectra on the reaction γd → π⁺ nn . The transition operator is calculated to high accuracy from chiral perturbation theory. We argue that for properly chosen kinematics, the theoretical uncertainty of the method can be as low as 0.1 fm.     

On Quasi-Hopf Superalgebras

In this work we investigate several important aspects of the structure theory of the recently introduced quasi-Hopf superalgebras (QHSAs), which play a fundamental role in knot theory and integrable systems. In particular we introduce the opposite structure and prove in detail (for the graded case) Drinfeld's result that the coproduct induced on a QHSA is obtained from the coproduct Δ by twisting. The corresponding “Drinfeld twist'”F _D is explicitly constructed, as well as its inverse, and we investigate the complete QHSA associated with . We give a universal proof that the coassociator and canonical elements correspond to twisting the original coassociator Φ=Φ₁₂₃ and canonical elements α,β with the Drinfeld twist F _D . Moreover in the quasi-triangular case, it is shown algebraically that the R-matrix corresponds to twisting the original R-matrix R with F _D . This has important consequences in knot theory, which will be investigated elsewhere. Received: 14 December 1998 / Accepted: 29 January 2000     

Classes on Compactifications of the Moduli Space of Curves Through Solutions to the Quantum Master Equation

In this paper we describe a construction which produces classes in compactifications of the moduli space of curves. This construction extends a construction of Kontsevich which produces classes in the open moduli space from the initial data of a cyclic A _∞-algebra. The initial data for our construction are what we call a ‘quantum A _∞-algebra’, which arises as a type of deformation of a cyclic A _∞-algebra. The deformation theory for these structures is described explicitly. We construct a family of examples of quantum A _∞-algebras which extend a family of cyclic A _∞-algebras, introduced by Kontsevich, which are known to produce all the kappa classes using his construction.      

Time evolution of <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">μν</Emphasis></Subscript> and the cosmological constant problem

We study the cosmic time evolution of an effective quantum field theory energy-momentum tensor T _μν and show that, as a consequence of the effective nature of the theory, T _μν is such that the vacuum energy decreases with time. We find that the zero point energy at present time is washed out by the cosmological evolution. The implications of this finding for the cosmological constant problem are investigated.     

Vertex Operators – From a Toy Model to Lattice Algebras

Within the framework of the discrete Wess–Zumino–Novikov–Witten theory we analyze the structure of vertex operators on a lattice. In particular, the lattice analogues of operator product expansions and braid relations are discussed. As the main physical application, a rigorous construction for the discrete counterpart g _n $ of the group valued field g(x) is provided. We study several automorphisms of the lattice algebras including discretizations of the evolution in the WZNW model. Our analysis is based on the theory of modular Hopf algebras and its formulation in terms of universal elements. Algebras of vertex operators and their structure constants are obtained for the deformed universal enveloping algebras . Throughout the whole paper, the abelian WZNW model is used as a simple example to illustrate the steps of our construction. Received: 16 December 1996 / Accepted: 5 May 1997     

The K ℓ3 scalar form factors in the standard model

We discuss the predictions of the standard model for the scalar form factors of K _ℓ3 decays. Our analysis is based on the results of chiral perturbation theory, large N _c estimates of low-energy couplings and dispersive methods. It includes a discussion of isospin-violating effects of strong and electromagnetic origin. This work was supported in part by EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”.     

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

In this paper we further develop the theory of α-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two “chiral” induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU(n WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices, i.e. on the “physical spectrum” of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU _{( n )} and many conformal inclusions, covering in particular all type I modular invariants of SU(2) and SU(3), and we conjecture that it holds also for any other conformal inclusion of SU _{( n )} as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion SU(3)₅⊂SU(6)₁ is computed for the first time. Received: 24 December 1998 / Accepted: 22 February 1999     

Weak Solutions of General Systems of Hyperbolic Conservation Laws

 In this paper, we establish the existence theory for general system of hyperbolic conservation laws and obtain the uniform L ₁ boundness for the solutions. The existence theory generalizes the classical Glimm theory for systems, for which each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of Lax. We construct the solutions by the Glimm scheme through the wave tracing method. One of the key elements is a new way of measuring the potential interaction of the waves of the same characteristic family involving the angle between waves. A new analysis is introduced to verify the consistency of the wave tracing procedure. The entropy functional is used to study the L ₁ boundedness. Received: 16 October 2001 / Accepted: 8 May 2002 Published online: 4 September 2002 RID="★" ID="★" The research was supported in part by NSF Grant DMS-9803323. RID="★★" ID="★★" The research was supported in part by the RGC Competitive Earmarked Research Grant CityU 1032/98P.     

Properties of synchrotron radiation

Muon capture on hydrogen gives a unique possibility for a measurement of the pseudo-scalar form factor g _p (q _c ² = -0.88 m _μ ² ) of the nucleonic weak current, thus providing a sensitive test of the QCD chiral symmetry perturbation theory which predicts the value of this form factor with a precision of Δg _p /g _p ≃ 2%. For adequate comparison with theory, the muon capture rate Λ_c should be measured with a precision of ΔΛ_c/Λ_c ≤ 1%, that is an order of magnitude better than the precision of the present world data. We report on the project of an experiment designed to provide the required precision. Also, we present the final result of our previous experiment on a high precision measurement of the μ³He capture rate and compare this result with the PCAC prediction. This revised version was published online in August 2006 with corrections to the Cover Date.     

Theory of melting,vacancy model

A theory of melting based on vacancy model is formulated. The polymer solution theory is used for derivation of the melting equation for a two-species model of melting solid. Under simplifying assumptions the analysis leads to a simple correlation betweenT _m and 〈v〉, the average energy of interaction between the vibrating atoms. Pseudopotential method is used for calculating 〈v〉 for the alkali metals lithium, sodium, potassium and rubidium at temperatureT _m. The calculated values ofT _m〈v〉 are in accord with those expected from our model. Application to the high pressure melting curves of solids is also discussed.     

Inequalities for a unified family of Voigt functions in several variables

The classical Voigt functions occur frequently in a wide variety of problems in astrophysical spectroscopy, emission, absorption and transfer of radiation in heated atmosphere, and plasma dispersion, and indeed also in the theory of neutron reactions. Here, in the present paper, by applying several known upper bounds for the first-kind Bessel function J _ν (x) given recently by (for example) Landau, Olenko and Krasikov, sharp bounding inequalities are obtained for the unified multivariable Voigt function V _μ,ν (x; y) in terms of the confluent Fox-Wright function ₁ψ₀ and its incomplete variant ₁ψ₀. Connections of the unified multivariable Voigt function V _μ,ν (x; y) with other unifications and generalizations of the classical Voigt function are also briefly pointed out.