International Journal of Modern Physics A

Within the framework of the Exact Renormalization Group, a manifestly gauge invariant calculus is constructed for SU(N) Yang–Mills. The methodology is comprehensively illustrated with a proof, to all orders in perturbation theory, that the β function has no explicit dependence on either the seed action or details of the covariantization of the cutoff. The cancellation of these nonuniversal contributions is done in an entirely diagrammatic fashion.

In this paper we study the phenomenon of UV/IR mixing in noncommutative field theories from the point of view of worldsheet open–closed duality in string theory. New infrared divergences in noncommutative field theories arise as a result of integrating over high momentum modes in the loops. These are believed to come from integrating out additional bulk closed string modes. We analyze this issue in detail for the bosonic theory and further for the supersymmetric theory on the C²/Z₂ orbifold. We elucidate on the exact role played by the constant background B-field in this correspondence.

It is argued that there are strong similarities between the infrared physics of N = 2 supersymmetric Yang–Mills and that of the quantum Hall effect, both systems exhibit a hierarchy of vacua with a subgroup of the modular group mapping between them. The coupling flow for pure SU(2) N = 2 supersymmetric Yang–Mills in four dimensions is reexamined and an earlier suggestion in the literature, that was singular at strong coupling, is modified to a form that is well behaved at both weak and strong coupling and describes the crossover in an analytic fashion. Similarities between the phase diagram and the flow of SUSY Yang–Mills and that of the quantum Hall effect are then described, with the Hall conductivity in the latter playing the role of the θ-parameter in the former. Hall plateaux, with odd denominator filling fractions, are analogous to fixed points at strong coupling in N = 2 SUSY Yang–Mills, where the massless degrees of freedom carry an odd monopole charge.

In the framework of rainbow approximation of the Dyson–Schwinger equation approach, it is shown that there exist two qualitatively distinct solutions of the Dyson–Schwinger equation for the quark propagator in the case of nonzero current quark mass. One solution corresponds to the "Nambu–Goldstone" phase and the other one corresponds to the "Wigner" phase in the chiral limit.

We examine D-branes on AdS₃, and find a 3-brane wrapping the entire AdS₃, in addition to 1-branes and instantonic 2-branes previously discussed in the literature. The 3-brane is found using a construction of Maldacena, Moore, and Seiberg. We show that all these branes satisfy Cardy's condition and extract the open string spectrum on them.

We set up a reality condition for Darboux transformations of time-dependent Schrödinger equations (TDSE) with position-dependent (effective) mass. The reality condition guarantees the potential in the transformed TDSE to be real-valued and hence physical. This paper is a sequel of our former one.³⁶

Forces defined in the framework of optical reference geometry are introduced in the case of stationary and axially symmetric Kerr black hole and naked-singularity space–times with a repulsive cosmological constant. Properties of the forces acting on test particles moving along circular orbits in the equatorial plane are discussed, whereas it is shown where the gravitational force vanishes and changes its orientation and where the centrifugal force vanishes and changes its orientation independently of the velocity of test particles related to the optical geometry; the Coriolis force does not vanish for the velocity being nonzero. The space–times are classified according to the number of circular orbits where the gravitational and centrifugal forces vanish.

We find exact solutions describing Ricci flows of four-dimensional pp-waves nonlinearly deformed by two-/three-dimensional solitons. Such solutions are parametrized by five-dimensional metrics with generic off-diagonal terms and connections with nontrivial torsion which can be related, for instance, to antisymmetric tensor sources in string gravity. There are defined nontrivial limits to four-dimensional configurations and the Einstein gravity.

From the ADHM construction on noncommutative vector space, we investigate different U(1) instanton solutions tied by partial isometry transformations. We recast these solutions under a form of vector fields in noncommutative vector space which makes possible the calculus of their fluxes through fuzzy spheres. For this end, we establish the noncommutative analog of Gauss theorem from which we show that the flux of the U(1) instantons through fuzzy spheres does not depend on the radius of these spheres and it is invariant under partial isometry transformations.

We suggest an infinite-dimensional extension of gauge transformations which includes non-Abelian tensor gauge fields. In this extension of the Yang–Mills theory the vector gauge boson becomes a member of a bigger family of gauge bosons of arbitrarily large integer spins. The invariant Lagrangian does not contain higher derivatives of tensor gauge fields and all interactions take place through three- and four-particle exchanges with dimensionless coupling constant.

In recent papers we suggested an infinite-dimensional extension of gauge transformations which includes non-Abelian tensor gauge fields. Extended gauge transformations of non-Abelian tensor gauge fields form a new large group which has a natural geometrical interpretation in terms of extended current algebra associated with compact Lie group. We shall demonstrate that one can construct two infinite series of gauge invariant quadratic forms, so that a linear combination of them comprises the general Lagrangian. The general Lagrangian exhibits enhanced local gauge invariance with double number of gauge parameters and allows to eliminate all negative norm states of the nonsymmetric second-rank tensor gauge field. Therefore it describes two polarizations of helicity-two massless charged tensor gauge boson and the helicity-zero Kalb–Ramond field. The geometrical interpretation of the enhanced gauge symmetry with double number of gauge parameters is not yet known.

Bekenstein–Hawking formalism of black hole thermodynamics should be modified to incorporate quantum gravitational effects. Generalized Uncertainty Principle (GUP) provides a suitable framework to perform such modifications. In this paper, we consider a general form of GUP to find black hole thermodynamics in a model universe with large extra dimensions. We will show that black holes radiate mainly in the four-dimensional brane. Existence of black holes remnants as a possible candidate for dark matter is discussed.

Mehta has shown that eigenvectors of the finite Fourier transform with the matrix , 0 ≤ j, k ≤ N-1, can be defined in terms of the classical Hermite functions as , where . We argue that the finite Fourier transform does actually govern also some q-extensions of Mehta's eigenvectors , associated with certain well-known orthogonal q-polynomial families. For the pairs of the continuous q-Hermite and q^-1-Hermite polynomials, the Rogers–Szegő and Stieltjes–Wigert polynomials, and the discrete q-Hermite polynomials of types I and II such links are explicitly derived.

In the limit when the base q → 1 these q-extensions coincide with Mehta's eigenvectors , whereas in the continuous limit (i.e. when the parameter N → ∞) they correspond to the classical Fourier integral transforms between the above-mentioned pairs of q-polynomial families.