New strings for old Veneziano amplitudes: I. Analytical treatment

The bosonic string theory evolved as an attempt to find a physical/quantum mechanical model capable of reproducing Euler’s beta function (Veneziano amplitude) and its multidimensional analogue. The multidimensional analogue of beta function was studied mathematically for some time from different angles by mathematicians such as Selberg, Weil and Deligne, among many others. The results of their studies apparently were not taken into account in physics literature on string theory. In a recent publication [IJMPA 19 (2004) 1655] an attempt was made to restore the missing links. The results of this publication are incomplete, however, since no attempts were made at reproduction of known spectra of both open an closed bosonic strings or at restoration of the underlying model(s) reproducing such spectra. Nevertheless, as discussed in this publication the existing mathematical interpretation of the multidimensional analogue of Euler’s beta function as one of the periods associated with the corresponding differential form “living” on the Fermat-type (hyper)surfaces, happens to be crucial for restoration of the quantum/statistical mechanical model reproducing such generalized beta function. Unlike the traditional formulations, this model is supersymmetric. Details leading to restoration of this model will be presented in the forthcoming Parts 2–4 of our work. They are devoted, respectively, to the group-theoretic, symplectic and combinatorial treatments of this model. In this paper the discussion is restricted mainly to the study of analytical properties of the multiparticle Veneziano and Veneziano-like (tachyon-free) amplitudes. In the last case, we demonstrate that the Veneziano-like amplitudes alone (with parameters adjusted accordingly) are capable of reproducing known spectra of both open and closed bosonic strings. The choice of parameters is subject to some constraints dictated by the mathematical interpretation of these amplitudes as periods of Fermat-type (hyper)surfaces considered as complex manifolds of Hodge-type.     

On the meissner and abelian higgs effects

We provide an axiomatic Euclidean field-theoretic treatment of the Meissner and Abelian Higgs effects, which extends our previous method to a fully quantum theoretic model of a gauge field, with U(1) symmetry, coupled to a scalar field. Our main results are that the former effect implies the latter one, and ensues from the condition of off-diagonal long-range order.     

Adèlic formulas for string amplitudes in fields of algebraic numbers

On the basis of the analysis on adèle groups (Tate's formula) for any field of algebraic numbers, a regularization of infinite adèlic products of gamma and beta functions of local fields is proposed. The formulas obtained are applied to representations of the four-point crossing symmetric Veneziano and Virasoro-Shapiro amplitudes through regularized adèlic products of the corresponding string (open and closed, resp.) amplitudes.     

A note on gauge symmetry breaking

We show that the breaking of Abelian gauge symmetry implies the existence of dipole singularities in the correlation functions of the (Abelian) Higgs model. We also show that the noninvariance of the Wightman functions does not preclude the implementability of the global gauge symmetry. An explicit example of gauge symmetry breaking (Ferrari's model) is discussed.     

Are vortex numbers preserved?

We study noncommutative vortex solutions that minimize the action functional of the Abelian Higgs model in 2-dimensional noncommutative Euclidean space. We first consider vortex solutions which are deformed from solutions defined on commutative Euclidean space to the noncommutative one. We construct solutions whose vortex numbers are unchanged under the noncommutative deformation. Another class of noncommutative vortex solutions via a Fock space representation is also studied.     

A fresh look at generalized Veneziano amplitudes

The dual resonance model, which was a precursor of string theory was based upon the idea that two-particle scattering amplitudes should be expressible equivalently as a sum of contributions of an infinite number of s channel poles each corresponding to a finite number of particles with definite spin, or as a similar sum of t channel poles. The famous example of Veneziano [Nuovo Cimento A 57 (1968) 190] satisfies all these requirements, and is additionally ghost free. We recall other trajectories which provide solutions to the duality constraints, e.g. the general Mobiüs trajectories and the logarithmic trajectories, which were thought to be lacking this last feature. We however demonstrate, partly empirically, the existence of a regime within a particular deformation of the Veneziano amplitude for logarithmic trajectories for which the 4-point amplitude remains ghost free.     

A model of the T-dependent pseudogap and its competition with superconductivity in copper oxides

Results for pseudogaps are obtained from a band model, where the stability of the gap depends on the amplitudes of vibrational displacements, or magnetic moments, and their coupling to electrons. A one-particle gap is favored by normal thermal excitations of phonons or spin waves. Another gap can be generated by spontaneous waves at lower temperature, if the electronic energy gain overcomes the elastic/magnetic energy needed for increased amplitudes of the oscillations. This state is characterized by charge or spin density waves. The pseudogap has many features in common with the superconducting gap, and the model lends support to the interpretation that the pseudogap is a precursor of, and competes with, superconducting pairing.     

Asymptotic freedom for quantum chromodynamics with top yukawa and Higgs interactions

Asymptotic freedom of QCD is extended to the enlarged system including the top Yukawa and Higgs interactions. Necessary and sufficient conditions for asymptotic freedom are given. By expansion with respect to powers of, all couplings are determined which are compatible with asymptotic freedom. It is found that the Higgs coupling is a function of the top coupling and that both couplings have upper bounds which correspond to the nontrivial case of reduction. The ultraviolet behavior of the coupling is controlled in all orders of the expansion.Work supported in part by the NSF: PHY-91-23780.     

Kirillov’s character formula,the holomorphic Peter–Weyl theorem,and the Blattner–Kostant–Sternberg pairing

By means of the orbit method we show that, for a compact Lie group, the Blattner–Kostant–Sternberg pairing map, with the constants being appropriately fixed, is unitary. Along the way we establish a holomorphic Peter–Weyl theorem for the complexification of a compact Lie group. Among our crucial tools is Kirillov’s character formula. The basic observation is that the Weyl vector is lurking behind the Kirillov character formula, as well as behind the requisite half-form correction on which the Blatter–Kostant–Sternberg-pairing for the compact Lie group relies, and thus produces the appropriate shift which, in turn, controls the unitarity of the BKS-pairing map. Our methods are independent of heat kernel harmonic analysis, which is used by B. C. Hall to obtain a number of these results [B.C. Hall, The Segal–Bargmann Coherent State Transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103–151; B.C. Hall, Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type, Comm. Math. Phys. 226 (2002) 233–268, quant.ph/0012015].     

An interplay between static potential and Reggeon trajectory for QCD string

I consider two cases where QCD string is described by an effective theory of long strings: the static potential and meson scattering amplitudes in the Regge regime. I show how they can be solved in the mean-field approximation, justified by the large number of space–time dimensions, and argue that it turns out to be exact. I compare contributions from QCD string and perturbative QCD and discuss experimental consequences for the scattering amplitudes.     

Sp(2) renormalization

The renormalization of general gauge theories on flat and curved space–time backgrounds is considered within the Sp(2)-covariant quantization method. We assume the existence of a gauge-invariant and diffeomorphism invariant regularization. Using the Sp(2)-covariant formalism one can show that the theory possesses gauge-invariant and diffeomorphism invariant renormalizability to all orders in the loop expansion and the extended BRST-symmetry after renormalization is preserved. The advantage of the Sp(2) method compared to the standard Batalin–Vilkovisky approach is that, in reducible theories, the structure of ghosts and ghosts for ghosts and auxiliary fields is described in terms of irreducible representations of the Sp(2) group. This makes the presentation of solutions to the master equations in more simple and systematic way because they are Sp(2)-scalars.     

Dual Superconductivity in Abelian Higgs Model of QCD

The study of generalized field associated with Abelian dyons has been undertaken and it has been demonstrated that topologically, a non-Abelian gauge theory is equivalent to a set of Abelian gauge theories supplemented by dyons which undergo condensation leading to confinement and consequently to superconducting model of QCD vacuum, where the Higgs field plays the role of a regulator only. Constructing the effective action for dyonic field in Abelian projection of QCD, it has been demonstrated that any charge (electrical or magnetic) of dyon screens its own direct potential to which it minimally couples and anti-screens the dual potential leading to dual superconductivity in accordance with generalized Meissner effect. In this Abelian projection of QCD an Abelian Higgs model (AHM) has been successfully constructed and it has been shown to incorporate dual superconductivity and confinement as the consequence of dyonic condensation. It has been demonstrated that in AHM t’ Hooft loop creates the string (AHM-string) around which the monopole current under London limit leads to vanishing coherence length in the chromo-magnetic superconductor. It has also been shown that in London limit the squared density of monopole current around AHM-string has a maximum at the distance of the order of penetration length.     

Regge theory and statistical mechanics

An interesting connection between the Regge theory of scattering, the Veneziano amplitude, the Lee–Yang theorems in statistical mechanics and nonextensive Renyi entropy is addressed. In this scheme the standard entropy and the Renyi entropy appear to be different limits of a unique mathematical object. This framework sheds light on the physical origin of nonextensivity. A non-trivial application to spin glass theory is shortly outlined.     

Functional integral via functional equation

Discretization ofp-adic Grassmann-valued -model leads to a hierarchical model with the Hamtilonian given by a nontrivial functional integral over the Grassmann variables. Using renormalization group arguments, we reduce the calculation of this integral to a functional equation. The problem of the convergence of the perturbation expansion of this integral, realized as a small-divisors problem, is investigated.     

The existence of generalised self-dual Chern-Simons vortices

We establish the existence and uniqueness of radially symmetric self-dual topological vortices in thep=2 members of the hierarchies of (generalised) Chern-Simons Higgs and Abelian Higgs models. We also obtain all possible symmetric nontopological vortices in the Chern-Simons model characterised by an additional parameter governing the decay rates of the fields.Supported in part by CEC under grant HCM-ERBCHRXCT930362.Supported in part by NSF under grant DMS-9400243.     

AdS/QCD and Applications of Light-Front Holography

Light-front holography leads to a rigorous connection between hadronic amplitudes in a higher dimensional anti-de Sitter(AdS) space and frame-independent light-front wavefunctions of hadrons in(3 + 1)-dimensional physical space-time,thus providing a compelling physical interpretation of the AdS/CFT correspondence principle and AdS/QCD,a useful framework which describes the correspondence between theories in a modified AdS 5 background and confining field theories in physical space-time.To a first semiclassical approximation,where quantum loops and quark masses are not included,this approach leads to a single-variable light-front Schro¨dinger equation which determines the eigenspectrum and the light-front wavefunctions of hadrons for general spin and orbital angular momentum.The coordinate z in AdS space is uniquely identified with a Lorentz-invariant coordinate ζ which measures the separation of the constituents within a hadron at equal light-front time.The internal structure of hadrons is explicitly introduced and the angular momentum of the constituents plays a key role.We give an overview of the light-front holographic approach to strongly coupled QCD.In particular,we study the photon-to-meson transition form factors(TFFs) FMγ(Q 2) for γ→ M using light-front holographic methods.The results for the TFFs for the η and η ' mesons are also presented.Some novel features of QCD are discussed,including the consequences of confinement for quark and gluon condensates.A method for computing the hadronization of quark and gluon jets at the amplitude level is outlined.     

Towards physically motivated proofs of the Poincaré and geometrization conjectures

Although the Poincaré and the geometrization conjectures were recently proved by Perelman, the proof relies heavily on properties of the Ricci flow previously investigated in great detail by Hamilton. Physical realization of such a flow can be found, for instance, in the work by Friedan [D. Friedan, Nonlinear models in 2+ε$2+\epsilon$ dimensions, Ann. Phys. 163 (1985) 318–419]. In his work the renormalization group flow for a nonlinear sigma model in 2+ε$2+\epsilon$ dimensions was obtained and studied. For ε=0$\epsilon =0$, by approximating the β$\beta$-function for such a flow by the lowest order terms in the sigma model coupling constant, the equations for Ricci flow are obtained. In view of such an approximation, the existence of this type of flow in Nature is questionable. In this work, we find totally independent justification for the existence of Ricci flows in Nature. This is achieved by developing a new formalism extending the results of two-dimensional conformal field theories (CFT’s) to three and higher dimensions. Equations describing critical dynamics of these CFT’s are examples of the Yamabe and Ricci flows realizable in Nature. Although in the original works by Perelman some physically motivated arguments can be found, their role in his proof remain rather obscure. In this paper, steps are made toward making these arguments more explicit, thus creating an opportunity for developing alternative, more physically motivated, proofs of the Poincaré and geometrization conjectures.     

Quantum ball as an example of quantum Kähler manifold

Using the notion of the coherent state map we present a construction of the C^∗$C^{\ast }$-algebra which is considered as a quantum version of the unit ball in C^N${\mathbb{C}}^N$. The structure of the quantum ball as well as its physical interpretation is investigated. Applying the reduction procedure to the quantum ball we obtain the Heisenberg–Weyl algebra.