Searcharxiv⌕ Search

arXiv subjects

R. Trinchero

Publications and source records attributed to R. Trinchero.

13 recordsLinked to original sources
arXiv2013

QCD condensates and holographic Wilson loops for asymptotically AdS spaces

The minimization of the Nambu-Goto action for a surface whose contour defines a circular Wilson loop of radius a placed at a finite value of the coordinate orthogonal to the boundary is considered. This is done for asymptotically AdS spaces. The condensates of even dimension $n=2$ through $10$ are calculated in terms of the coefficient of $a^{n}$ in the expansion of the on-shell subtracted Nambu-Goto action for small $a$ The subtraction employed is such that it presents no conflict with conformal invariance in the AdS case and need not introduce an additional infrared scale for the case of confining geometries. It is shown that the UV value of the condensates is universal in the sense that they only depends on the first coefficients of the difference with the AdS case.

hep-ph↗
arXiv2012

Scalar field on non-integer dimensional spaces

Deformations of the canonical spectral triples over the n-dimensional torus are considered. These deformations have a discrete dimension spectrum consisting of non-integer values less than n. The differential algebra corresponding to these spectral triples is studied. No junk forms appear for non-vanishing deformation parameter. The action of a scalar field in these spaces is considered, leading to non-trivial extra structure compared to the integer dimensional cases, which does not involve a loss of covariance. One-loop contributions are computed leading to finite results for non-vanishing deformation.

math-ph↗
arXiv2010

Paths on graphs and associated quantum groupoids

Given any simple biorientable graph it is shown that there exists a weak {*}-Hopf algebra constructed on the vector space of graded endomorphisms of essential paths on the graph. This construction is based on a direct sum decomposition of the space of paths into orthogonal subspaces one of which is the space of essential paths. Two simple examples are worked out with certain detail, the ADE graph $A_{3}$ and the affine graph $A_{[2]}$. For the first example the weak {*}-Hopf algebra coincides with the so called double triangle algebra. No use is made of Ocneanu's cell calculus.

math.QA↗
arXiv2008

Examples of non integer dimensional geometries

Two examples of spectral triples with non-integer dimension spectrum are considered. These triples involve commutative C*-algebras. The first example has complex dimension spectrum and trivial differential algebra. The other is a parameter dependent deformation of the canonical spectral triple over S1. Its dimension spectrum includes real non-integer values. It has a non-trivial differential algebra and in contrast with the one dimensional case there are no junk forms for a non-vanishing deformation parameter. The distance on this space depends non-trivially on this parameter.

math-ph↗
arXiv1999

Associated quantum vector bundles and symplectic structure on a quantum plane

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bundles without requiring the use of a (bicovariant) differential structure over H. Moreover, if H is coquasitriangular, it coacts naturally on the associated bundle, and the differential structure is covariant. We apply this construction to the case of the finite quotient of the SL_q(2) function Hopf algebra at a root of unity (q^3=1) as the structure group, and a reduced 2-dimensional quantum plane as both the "base manifold" and fibre, getting an algebra which generalizes the notion of classical phase space for this quantum space. We also build explicitly a differential complex for this phase space algebra, and find that levels 0 and 2 support a (co)representation of the quantum symplectic group. On this phase space we define vector fields, and with the help of the Sp_q structure we introduce a symplectic form relating 1-forms to vector fields. This leads naturally to the introduction of Poisson brackets, a necessary step to do "classical" mechanics on a quantum space, the quantum plane.

math-ph↗
arXiv1999

Hopf stars, twisted Hopf stars and scalar products on quantum spaces

The properties of Hopf star operations and twisted Hopf stars operations on quantum groups are discussed in relation with the theory of representations (star representations). Invariant Hermitian sesquilinear forms (scalar products) on modules or module-algebras are then defined and analyzed. Particular attention is paid to scalar products that can be associated with the Killing form (when it exists) or with the left (or right) invariant integrals on the quantum group. Our results are systematically illustrated in the case of a family of non semi-simple and finite dimensional quantum groups that are obtained as Hopf quotients of the quantum enveloping algebra U_q(sl(2,C)), q being an N-th root of unity. Many explicit results concerning the case N=3 are given. We also mention several physical motivations for the present work: conformal field theory, spin chains, integrable models, generalized Yang-Mills theory with quantum group action and the search for finite quantum groups symmetries in particle physics.

math-ph↗
arXiv1999

Differential calculus and connections on a quantum plane at a cubic root of unity

We consider the algebra of N x N matrices as a reduced quantum plane on which a finite-dimensional quantum group H acts. This quantum group is a quotient of U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take N=3; in that case \dim(H) = 27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess-Zumino complex. The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

math-ph↗
arXiv1998

Geometry of the reduced quantum plane

We consider the space M of NxN matrices as a reduced quantum plane and discuss its geometry under the action and coaction of finite dimensional quantum groups (a quotient of U_q(SL(2)), q being an N-th root of unity, and its dual). We also introduce a differential calculus for M: a quotient of the Wess Zumino complex. We shall restrict ourselves to the case N odd and often choose the particular value N=3. The present paper (to appear in the proceedings of the conference "Quantum Groups and Fundamental Physical Applications", Palerme, December 1997) is essentially a short version of math-ph/9807012.

math-ph↗
arXiv1998

Non-Commutative Corepresentations of Quantum Groups

We consider a twisted version of quantum groups corepresentations. This generalization amounts to include in the theory the case where quantum space coordinates and its endomorphism matrix entries belong to a non-commutative quadratic algebra.

math.QA↗
arXiv1998

Finite dimensional quantum group covariant differential calculus on a complex matrix algebra

Using the fact that the algebra M(3,C) of 3 x 3 complex matrices can be taken as a reduced quantum plane, we build a differential calculus Omega(S) on the quantum space S defined by the algebra C^\infty(M) \otimes M(3,C), where M is a space-time manifold. This calculus is covariant under the action and coaction of finite dimensional dual quantum groups. We study the star structures on these quantum groups and the compatible one in M(3,C). This leads to an invariant scalar product on the later space. We analyse the differential algebra Omega(M(3,C)) in terms of quantum group representations, and consider in particular the space of one-forms on S since its elements can be considered as generalized gauge fields.

math.QA↗
arXiv1996

Quantum mechanics over a q-deformed (0+1)-dimensional superspace

We built up a explicit realization of (0+1)-dimensional q-deformed superspace coordinates as operators on standard superspace. A q-generalization of supersymmetric transformations is obtained, enabling us to introduce scalar superfields and a q-supersymmetric action. We consider a functional integral based on this action. Integration is implemented, at the level of the coordinates and at the level of the fields, as traces over the corresponding representation spaces. Evaluation of these traces lead us to standard functional integrals. The generation of a mass term for the fermion field leads, at this level, to an explicitely broken version of supersymmetric quantum mechanics.

hep-th↗
arXiv1996

On Quantum Groups Co-Representations

We carry out a generalization of quantum group co-representations in order to encode in this structure those cases where non-commutativity between endomorphism matrix entries and quantum space coordinates happens.

q-alg↗