Roberto Zucchini - Academia.edu (original) (raw)

Papers by Roberto Zucchini

arXiv (Cornell University), Jun 29, 2019

The geometry of the total space of a principal bundle with regard to the action of the bundle's s... more The geometry of the total space of a principal bundle with regard to the action of the bundle's structure group is elegantly described by the bundle's operation, a collection of derivations consisting of the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and satisfying the six Cartan relations. Connections and gauge transformations are defined by the way they behave under the action of the operation's derivations. In the first paper of a series of two extending the ordinary theory, we constructed an operational total space theory of strict principal 2-bundles with reference to the action of the structure strict 2-group. Expressing this latter through a crossed module pE, Gq, the operation is based on the derived Lie group er1s¸G. In this paper, the second of the series, an original formulation of the theory of 2-connections and 1-and 2-gauge transformations of principal 2-bundles based on the operational framework is provided.

arXiv (Cornell University), Dec 13, 2011

In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict ... more In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict higher gauge theory, that avoids the subtleties of the relationship between Lie 2-groups and algebras by relying exclusively on the structure semistrict Lie 2-algebra v and its automorphism 2-group Aut(v). Gauge transformations are defined and shown to form a strict 2-group depending on v. Fields are v-valued and their global behaviour is controlled by appropriate gauge transformation gluing data organized as a strict 2-groupoid. In the second part, using the BV quantization method in the AKSZ geometrical version, we write down a 3dimensional semistrict higher BF gauge theory generalizing ordinary BF theory, carry out its gauge fixing and obtain as end result a semistrict higher topological gauge field theory of the Witten type. We also introduce a related 4-dimensional semistrict higher Chern-Simons gauge theory.

Nuclear Physics B, Feb 1, 1991

We study duality in simple two-dimensional orbifold models with metric and axionic backgrounds at... more We study duality in simple two-dimensional orbifold models with metric and axionic backgrounds at arbitrary orders in string perturbation theory. It is shown that duality involves certain linear relations among twist correlators at dual backgrounds which are independent of the perturbative order. Relying on methods of harmonic analysis, a nonperturbative representation (in the sense of string field theory) of a class of S-matrix elements is obtained in terms of a universal background-independent spectral distribution and non_holomomhic mndiOlar vertar functions of the background parameters. At the same time, we find new realizations of the Verlinde algebra and of hypergroups characterizing the S-matrix nonperturbatively. Various examples and calculations are worked out.

Journal of Mathematical Physics, Jun 1, 2017

In this paper, we present a purely algebraic formulation of higher gauge theory and gauged sigma ... more In this paper, we present a purely algebraic formulation of higher gauge theory and gauged sigma models based on the abstract theory of graded commutative algebras and their morphisms. The formulation incorporates naturally BRST symmetry and is also suitable for AKSZ type constructions. It is also shown that for a full-fledged BV formulation including ghost degrees of freedom, higher gauge and gauged sigma model fields must be viewed as internal smooth functions on the shifted tangent bundle of a space time manifold valued in a shifted L 8-algebroid encoding symmetry. The relationship to other formulations where the L 8-algebroid arises from a higher Lie groupoid by Lie differentiation is highlighted.

Journal of High Energy Physics, Oct 1, 2014

We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded ... more We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded in a semistrict Lie 2-algebra equipped with an invariant non singular bilinear form. We analyze the gauge invariance of the theory and show that action is invariant under a higher gauge transformation up to a higher winding number. We find that the theory admits two seemingly inequivalent canonical quantizations. The first is manifestly topological, it does not require a choice of any additional structure on the spacial 3-fold. The second, more akin to that of ordinary Chern-Simons theory, involves fixing a CR structure on the latter. Correspondingly, we obtain two sets of semistrict higher WZW Ward identities and we find the explicit expressions of two higher versions of the WZW action. We speculate that the model could be used to define 2-knot invariants of 4-folds.

Journal of Geometry and Physics, Sep 1, 2015

In this technical paper, we present a new formulation of higher parallel transport in strict high... more In this technical paper, we present a new formulation of higher parallel transport in strict higher gauge theory required for the rigorous construction of Wilson lines and surfaces. Our approach is based on an original notion of Lie crossed module cocycle and cocycle 1-and 2-gauge transformation with a non standard double category theoretic interpretation. We show its equivalence to earlier formulations.

Nuclear Physics B, Mar 1, 1986

The simple renormalization framework previously discussed by one of us is extended to treat the H... more The simple renormalization framework previously discussed by one of us is extended to treat the Higgs potential of the standard model. As an application, we obtain in a straightforward manner the one-loop correction delta(M) in the natural relation h(M) = &surd;1/2 GmumH2[1 + delta(M)], where mH is the physical mass of the Higgs boson, Gmu the accurately known mu decay coupling constant and h(M) the MS quartic Higgs coupling at mass scale M. The correction delta(M) contains contributions proportional to xi ≡ mH2/mZ2 and xi-1 and become large for xi>>1 or xi<<1. The dependence of h(M) on M and mH is analyzed by combining our one-loop results (which provide initial conditions) and approximate analytic solutions of relevant renormalization group equations. A parameter Mc, which roughly describes the mass scale at which perturbation theory breaks down and the possible onset of new physics, is discussed as a function of mH.

arXiv (Cornell University), May 24, 2019

It is a classic result that the geometry of the total space of a principal bundle with reference ... more It is a classic result that the geometry of the total space of a principal bundle with reference to the action of the bundle's structure group is codified in the bundle's operation, a collection of derivations comprising the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and obeying the six Cartan relations. In particular, connections and gauge transformations can be defined through the way they are acted upon by the operation's derivations. In this paper, the first of a series of two extending the ordinary theory, we construct an operational total space theory of strict principal 2-bundles with regard to the action of the structure strict 2-group. Expressing this latter via a crossed module pE, Gq, the operation is based on the derived Lie group er1s¸G. In the second paper, an original formulation of the theory of 2-connections and 1-and 2-gauge transformations based on the operational framework worked out here will be provided.

arXiv (Cornell University), Dec 13, 2011

We work out a formulation of higher gauge theory, whose symmetry is encoded in a semistrict Lie 2... more We work out a formulation of higher gauge theory, whose symmetry is encoded in a semistrict Lie 2-algebra v and which we call semistrict. We view v as a 2-term L-infinity algebra, a special case of strong homotopy Lie algebra generalizing an ordinary Lie algebra by allowing the Lie bracket to have a non trivial Jacobiator. Fields are v-valued and gauge transformations are special Aut(v)-valued maps organized as an ordinary group and acting on them. The global behaviour of fields is controlled by appropriate gauge transformation 1-cocycles. Using the BV quantization method in the AKSZ geometrical version, we write down a 3-dimensional semistrict higher BF gauge theory generalizing ordinary BF theory, carry out its gauge fixing and obtain as end result a semistrict higher topological gauge field theory of the Witten type. We also introduce a related 4-dimensional semistrict higher Chern--Simons gauge theory. We discuss merits and weaknesses of our formulation in relations to other approaches.

Fortschritte der Physik, May 14, 2019

Holonomy invariants in strict higher gauge theory have been studied in depth, aiming to applicati... more Holonomy invariants in strict higher gauge theory have been studied in depth, aiming to applications to higher Chern-Simons theory. For a flat 2-connection, the holonomy of surface knots of arbitrary genus has been defined and its covariance properties under 1-gauge transformation and change of base data have been determined. Using quandle theory, a definition of trace over a crossed module has been given that yields surface knot invariants upon application to 2-holonomies.

Journal of Geometry and Physics, Jun 1, 2003

Topological integrals appear frequently in Lagrangian field theories. On manifolds without bounda... more Topological integrals appear frequently in Lagrangian field theories. On manifolds without boundary, they can be treated in the framework of (absolute) (co)homology using the formalism of Cheeger-Simons differential characters. String and D-brane theory involve field theoretic models on worldvolumes with boundary. On manifolds with boundary, the proper treatment of topological integrals requires a generalization of the usual differential topological set up and leads naturally to relative (co)homology and relative Cheeger-Simons differential characters. In this paper, we present a construction of relative Cheeger-Simons differential characters which is computable in principle and which contains the ordinary Cheeger-Simons differential characters as a particular case.

Physics Letters B, May 1, 1991

We present a calculation of the effective action for induced conformal gravity on higher genus Ri... more We present a calculation of the effective action for induced conformal gravity on higher genus Riemann surfaces. Our expression, generalizing Polyakov's formula, depends holomorphically on the Beltrami differential and integrates the diffeomorphism anomaly.

International Journal of Geometric Methods in Modern Physics, Dec 1, 2004

We show that the global aspects of Abelian and center projection of a SU(2) gauge theory on an ar... more We show that the global aspects of Abelian and center projection of a SU(2) gauge theory on an arbitrary manifold are naturally described in terms of smooth Deligne cohomology. This is achieved through the introduction of a novel type of differential topological structure, called Cho structure. Half integral monopole charges appear naturally in this framework.

Communications in Mathematical Physics, Mar 1, 1993

We continue the model independent study of the Polyakov action on an arbitrary compact surface wi... more We continue the model independent study of the Polyakov action on an arbitrary compact surface without boundary of genus larger than 2 as the general solution of the relevant conformal Ward identity. A general formula for the Polyakov action and an explicit calculation of the energy-momentum tensor density is provided. It is further shown how Polyakov's SL(2,C) symmetry emerges in a curved base surface.

Physical Review Letters, Oct 20, 1986

An approximate analytic calculation of O(Zu2) corrections to Fermi decays is presented. When the ... more An approximate analytic calculation of O(Zu2) corrections to Fermi decays is presented. When the analysis of Koslowksy et a/. is modified to take into account the new results, it is found that each of the eight accurately studied s t values differs from the average by l~, thus significantly improving the comparison of experiments with conserved-vector-current predictions. The new & t values are lo~er than before, which also brings experiments into very good agreement with the three-generation standard model, at the level of its quantum corrections.

Annals of Physics, 1985

The unitary group describing the evolution of the quantum fluctuation around any classic' phase o... more The unitary group describing the evolution of the quantum fluctuation around any classic' phase orbit has a perturbative strongly asymptotic expansion in the small parameter h"'. Likewise, the mean values of the observables position and momentum in suitable timedependent states have an asymptotic expansion in h"', whose zeroth-order term satisfies the classic canonical equations. Similar expansions are found for the squared quantum dispersions. 'E' 1985 Academic Press, Inc. 1. INTRODUCTION When dealing with the Classical Limit two natural questions arise: first, understanding in what sense Classical Mechanics turns out to be the limit of Quantum Mechanics when ZI + 0 and putting the limit procedure on a firm mathematical basis, and second, expanding the quantum fluctuation around the solutions of the classical equations in a power series of the small parameter h, in order to obtain approximate expressions. Many authors have analyzed the first problem in the context of relativistic and non-relativistic particle and field theories [l-7]. A solution for the second problem is given by the famous WKB method [8]. However, there are alternative methods. J. Ginibre and G. Velo found an asymptotic expansion in h "* for the unitary group describing the quantum fluctuation around the solution of the classical field equations for non-relativistic bosons which in some cases is Bore1 summable [9]. The aim of this paper consists in investigating whether analogous results can be obtained in the context of a non-relativistic particle theory. The problem is partly simplified because the structure of the Hilbert space of the quantum states is simpler in a particle theory, and because the classical equations are local. However, the interaction is generally more complicated, so that a more complicated combinatorics is expected. Consider a quantum mechanical system with v degrees of freedom and with an analogous classical system. In Quantum Mechanics the states of the system are represented by rays of some Hilbert space P and the observables by selfadjoint operators in 5!. Conversely, in Classical Mechanics the states are points of a real 2vdimensional vector space 8, called phase space, and the observables real measurable functions defined on 8.

Physics Letters B, Mar 1, 1994

It is shown how the theory of classical W-algebras can be formulated on a higher genus Riemann su... more It is shown how the theory of classical W-algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. An intriguing relation between the theory of A 1 embeddings into simple Lie algebras and the holomorphic geometry of Riemann surfaces is exihibited.

Classical and Quantum Gravity, Oct 13, 1999

We determine from 11d supergravity the quadratic bulk action for the physical bosonic fields rele... more We determine from 11d supergravity the quadratic bulk action for the physical bosonic fields relevant for the computation of correlation functions of normalized chiral operators in D = 6, N = (0, 2) and D = 3, N = 8 supersymmetric CFT in the large N limit, as dictated by the AdS/CFT duality conjecture.

Journal of High Energy Physics, Dec 12, 2006

BiHermitian geometry, discovered long ago by Gates, Hull and Roček, is the most general sigma mod... more BiHermitian geometry, discovered long ago by Gates, Hull and Roček, is the most general sigma model target space geometry allowing for (2, 2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target space, we write down the explicit expression of the sigma model's action and BRST transformations and present a computation of the topological gauge fermion and the topological action.

Classical and Quantum Gravity, Feb 1, 1993

I show that the generalized Beltrami differentials and projective connections which appear natura... more I show that the generalized Beltrami differentials and projective connections which appear naturally in induced light cone W n gravity are geometrical fields parametrizing in one-to-one fashion generalized projective structures on a fixed base Riemann surface. I also show that W n symmetries are nothing but gauge transformations of the flat SL(n, C) vector bundles canonically associated to the generalized projective structures. This provides an original formulation of classical light cone W n geometry. From the knowledge of the symmetries, the full BRS algebra is derived. Inspired by the results of recent literature, I argue that quantum W n gravity may be formulated as an induced gauge theory of generalized projective connections. This leads to projective field theory. The possible anomalies arising at the quantum level are analyzed by solving Wess-Zumino consistency conditions. The implications for induced covariant W n gravity are briefly discussed. The results presented, valid for arbitrary n, reproduce those obtained for n = 2, 3 by different methods.

arXiv (Cornell University), Jun 29, 2019

The geometry of the total space of a principal bundle with regard to the action of the bundle's s... more The geometry of the total space of a principal bundle with regard to the action of the bundle's structure group is elegantly described by the bundle's operation, a collection of derivations consisting of the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and satisfying the six Cartan relations. Connections and gauge transformations are defined by the way they behave under the action of the operation's derivations. In the first paper of a series of two extending the ordinary theory, we constructed an operational total space theory of strict principal 2-bundles with reference to the action of the structure strict 2-group. Expressing this latter through a crossed module pE, Gq, the operation is based on the derived Lie group er1s¸G. In this paper, the second of the series, an original formulation of the theory of 2-connections and 1-and 2-gauge transformations of principal 2-bundles based on the operational framework is provided.

arXiv (Cornell University), Dec 13, 2011

In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict ... more In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict higher gauge theory, that avoids the subtleties of the relationship between Lie 2-groups and algebras by relying exclusively on the structure semistrict Lie 2-algebra v and its automorphism 2-group Aut(v). Gauge transformations are defined and shown to form a strict 2-group depending on v. Fields are v-valued and their global behaviour is controlled by appropriate gauge transformation gluing data organized as a strict 2-groupoid. In the second part, using the BV quantization method in the AKSZ geometrical version, we write down a 3dimensional semistrict higher BF gauge theory generalizing ordinary BF theory, carry out its gauge fixing and obtain as end result a semistrict higher topological gauge field theory of the Witten type. We also introduce a related 4-dimensional semistrict higher Chern-Simons gauge theory.

Nuclear Physics B, Feb 1, 1991

We study duality in simple two-dimensional orbifold models with metric and axionic backgrounds at... more We study duality in simple two-dimensional orbifold models with metric and axionic backgrounds at arbitrary orders in string perturbation theory. It is shown that duality involves certain linear relations among twist correlators at dual backgrounds which are independent of the perturbative order. Relying on methods of harmonic analysis, a nonperturbative representation (in the sense of string field theory) of a class of S-matrix elements is obtained in terms of a universal background-independent spectral distribution and non_holomomhic mndiOlar vertar functions of the background parameters. At the same time, we find new realizations of the Verlinde algebra and of hypergroups characterizing the S-matrix nonperturbatively. Various examples and calculations are worked out.

Journal of Mathematical Physics, Jun 1, 2017

In this paper, we present a purely algebraic formulation of higher gauge theory and gauged sigma ... more In this paper, we present a purely algebraic formulation of higher gauge theory and gauged sigma models based on the abstract theory of graded commutative algebras and their morphisms. The formulation incorporates naturally BRST symmetry and is also suitable for AKSZ type constructions. It is also shown that for a full-fledged BV formulation including ghost degrees of freedom, higher gauge and gauged sigma model fields must be viewed as internal smooth functions on the shifted tangent bundle of a space time manifold valued in a shifted L 8-algebroid encoding symmetry. The relationship to other formulations where the L 8-algebroid arises from a higher Lie groupoid by Lie differentiation is highlighted.

Journal of High Energy Physics, Oct 1, 2014

We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded ... more We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded in a semistrict Lie 2-algebra equipped with an invariant non singular bilinear form. We analyze the gauge invariance of the theory and show that action is invariant under a higher gauge transformation up to a higher winding number. We find that the theory admits two seemingly inequivalent canonical quantizations. The first is manifestly topological, it does not require a choice of any additional structure on the spacial 3-fold. The second, more akin to that of ordinary Chern-Simons theory, involves fixing a CR structure on the latter. Correspondingly, we obtain two sets of semistrict higher WZW Ward identities and we find the explicit expressions of two higher versions of the WZW action. We speculate that the model could be used to define 2-knot invariants of 4-folds.

Journal of Geometry and Physics, Sep 1, 2015

In this technical paper, we present a new formulation of higher parallel transport in strict high... more In this technical paper, we present a new formulation of higher parallel transport in strict higher gauge theory required for the rigorous construction of Wilson lines and surfaces. Our approach is based on an original notion of Lie crossed module cocycle and cocycle 1-and 2-gauge transformation with a non standard double category theoretic interpretation. We show its equivalence to earlier formulations.

Nuclear Physics B, Mar 1, 1986

The simple renormalization framework previously discussed by one of us is extended to treat the H... more The simple renormalization framework previously discussed by one of us is extended to treat the Higgs potential of the standard model. As an application, we obtain in a straightforward manner the one-loop correction delta(M) in the natural relation h(M) = &surd;1/2 GmumH2[1 + delta(M)], where mH is the physical mass of the Higgs boson, Gmu the accurately known mu decay coupling constant and h(M) the MS quartic Higgs coupling at mass scale M. The correction delta(M) contains contributions proportional to xi ≡ mH2/mZ2 and xi-1 and become large for xi>>1 or xi<<1. The dependence of h(M) on M and mH is analyzed by combining our one-loop results (which provide initial conditions) and approximate analytic solutions of relevant renormalization group equations. A parameter Mc, which roughly describes the mass scale at which perturbation theory breaks down and the possible onset of new physics, is discussed as a function of mH.

arXiv (Cornell University), May 24, 2019

It is a classic result that the geometry of the total space of a principal bundle with reference ... more It is a classic result that the geometry of the total space of a principal bundle with reference to the action of the bundle's structure group is codified in the bundle's operation, a collection of derivations comprising the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and obeying the six Cartan relations. In particular, connections and gauge transformations can be defined through the way they are acted upon by the operation's derivations. In this paper, the first of a series of two extending the ordinary theory, we construct an operational total space theory of strict principal 2-bundles with regard to the action of the structure strict 2-group. Expressing this latter via a crossed module pE, Gq, the operation is based on the derived Lie group er1s¸G. In the second paper, an original formulation of the theory of 2-connections and 1-and 2-gauge transformations based on the operational framework worked out here will be provided.

arXiv (Cornell University), Dec 13, 2011

We work out a formulation of higher gauge theory, whose symmetry is encoded in a semistrict Lie 2... more We work out a formulation of higher gauge theory, whose symmetry is encoded in a semistrict Lie 2-algebra v and which we call semistrict. We view v as a 2-term L-infinity algebra, a special case of strong homotopy Lie algebra generalizing an ordinary Lie algebra by allowing the Lie bracket to have a non trivial Jacobiator. Fields are v-valued and gauge transformations are special Aut(v)-valued maps organized as an ordinary group and acting on them. The global behaviour of fields is controlled by appropriate gauge transformation 1-cocycles. Using the BV quantization method in the AKSZ geometrical version, we write down a 3-dimensional semistrict higher BF gauge theory generalizing ordinary BF theory, carry out its gauge fixing and obtain as end result a semistrict higher topological gauge field theory of the Witten type. We also introduce a related 4-dimensional semistrict higher Chern--Simons gauge theory. We discuss merits and weaknesses of our formulation in relations to other approaches.

Fortschritte der Physik, May 14, 2019

Holonomy invariants in strict higher gauge theory have been studied in depth, aiming to applicati... more Holonomy invariants in strict higher gauge theory have been studied in depth, aiming to applications to higher Chern-Simons theory. For a flat 2-connection, the holonomy of surface knots of arbitrary genus has been defined and its covariance properties under 1-gauge transformation and change of base data have been determined. Using quandle theory, a definition of trace over a crossed module has been given that yields surface knot invariants upon application to 2-holonomies.

Journal of Geometry and Physics, Jun 1, 2003

Topological integrals appear frequently in Lagrangian field theories. On manifolds without bounda... more Topological integrals appear frequently in Lagrangian field theories. On manifolds without boundary, they can be treated in the framework of (absolute) (co)homology using the formalism of Cheeger-Simons differential characters. String and D-brane theory involve field theoretic models on worldvolumes with boundary. On manifolds with boundary, the proper treatment of topological integrals requires a generalization of the usual differential topological set up and leads naturally to relative (co)homology and relative Cheeger-Simons differential characters. In this paper, we present a construction of relative Cheeger-Simons differential characters which is computable in principle and which contains the ordinary Cheeger-Simons differential characters as a particular case.

Physics Letters B, May 1, 1991

We present a calculation of the effective action for induced conformal gravity on higher genus Ri... more We present a calculation of the effective action for induced conformal gravity on higher genus Riemann surfaces. Our expression, generalizing Polyakov's formula, depends holomorphically on the Beltrami differential and integrates the diffeomorphism anomaly.

International Journal of Geometric Methods in Modern Physics, Dec 1, 2004

We show that the global aspects of Abelian and center projection of a SU(2) gauge theory on an ar... more We show that the global aspects of Abelian and center projection of a SU(2) gauge theory on an arbitrary manifold are naturally described in terms of smooth Deligne cohomology. This is achieved through the introduction of a novel type of differential topological structure, called Cho structure. Half integral monopole charges appear naturally in this framework.

Communications in Mathematical Physics, Mar 1, 1993

We continue the model independent study of the Polyakov action on an arbitrary compact surface wi... more We continue the model independent study of the Polyakov action on an arbitrary compact surface without boundary of genus larger than 2 as the general solution of the relevant conformal Ward identity. A general formula for the Polyakov action and an explicit calculation of the energy-momentum tensor density is provided. It is further shown how Polyakov's SL(2,C) symmetry emerges in a curved base surface.

Physical Review Letters, Oct 20, 1986

An approximate analytic calculation of O(Zu2) corrections to Fermi decays is presented. When the ... more An approximate analytic calculation of O(Zu2) corrections to Fermi decays is presented. When the analysis of Koslowksy et a/. is modified to take into account the new results, it is found that each of the eight accurately studied s t values differs from the average by l~, thus significantly improving the comparison of experiments with conserved-vector-current predictions. The new & t values are lo~er than before, which also brings experiments into very good agreement with the three-generation standard model, at the level of its quantum corrections.

Annals of Physics, 1985

The unitary group describing the evolution of the quantum fluctuation around any classic' phase o... more The unitary group describing the evolution of the quantum fluctuation around any classic' phase orbit has a perturbative strongly asymptotic expansion in the small parameter h"'. Likewise, the mean values of the observables position and momentum in suitable timedependent states have an asymptotic expansion in h"', whose zeroth-order term satisfies the classic canonical equations. Similar expansions are found for the squared quantum dispersions. 'E' 1985 Academic Press, Inc. 1. INTRODUCTION When dealing with the Classical Limit two natural questions arise: first, understanding in what sense Classical Mechanics turns out to be the limit of Quantum Mechanics when ZI + 0 and putting the limit procedure on a firm mathematical basis, and second, expanding the quantum fluctuation around the solutions of the classical equations in a power series of the small parameter h, in order to obtain approximate expressions. Many authors have analyzed the first problem in the context of relativistic and non-relativistic particle and field theories [l-7]. A solution for the second problem is given by the famous WKB method [8]. However, there are alternative methods. J. Ginibre and G. Velo found an asymptotic expansion in h "* for the unitary group describing the quantum fluctuation around the solution of the classical field equations for non-relativistic bosons which in some cases is Bore1 summable [9]. The aim of this paper consists in investigating whether analogous results can be obtained in the context of a non-relativistic particle theory. The problem is partly simplified because the structure of the Hilbert space of the quantum states is simpler in a particle theory, and because the classical equations are local. However, the interaction is generally more complicated, so that a more complicated combinatorics is expected. Consider a quantum mechanical system with v degrees of freedom and with an analogous classical system. In Quantum Mechanics the states of the system are represented by rays of some Hilbert space P and the observables by selfadjoint operators in 5!. Conversely, in Classical Mechanics the states are points of a real 2vdimensional vector space 8, called phase space, and the observables real measurable functions defined on 8.

Physics Letters B, Mar 1, 1994

It is shown how the theory of classical W-algebras can be formulated on a higher genus Riemann su... more It is shown how the theory of classical W-algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. An intriguing relation between the theory of A 1 embeddings into simple Lie algebras and the holomorphic geometry of Riemann surfaces is exihibited.

Classical and Quantum Gravity, Oct 13, 1999

We determine from 11d supergravity the quadratic bulk action for the physical bosonic fields rele... more We determine from 11d supergravity the quadratic bulk action for the physical bosonic fields relevant for the computation of correlation functions of normalized chiral operators in D = 6, N = (0, 2) and D = 3, N = 8 supersymmetric CFT in the large N limit, as dictated by the AdS/CFT duality conjecture.

Journal of High Energy Physics, Dec 12, 2006

BiHermitian geometry, discovered long ago by Gates, Hull and Roček, is the most general sigma mod... more BiHermitian geometry, discovered long ago by Gates, Hull and Roček, is the most general sigma model target space geometry allowing for (2, 2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target space, we write down the explicit expression of the sigma model's action and BRST transformations and present a computation of the topological gauge fermion and the topological action.

Classical and Quantum Gravity, Feb 1, 1993

I show that the generalized Beltrami differentials and projective connections which appear natura... more I show that the generalized Beltrami differentials and projective connections which appear naturally in induced light cone W n gravity are geometrical fields parametrizing in one-to-one fashion generalized projective structures on a fixed base Riemann surface. I also show that W n symmetries are nothing but gauge transformations of the flat SL(n, C) vector bundles canonically associated to the generalized projective structures. This provides an original formulation of classical light cone W n geometry. From the knowledge of the symmetries, the full BRS algebra is derived. Inspired by the results of recent literature, I argue that quantum W n gravity may be formulated as an induced gauge theory of generalized projective connections. This leads to projective field theory. The possible anomalies arising at the quantum level are analyzed by solving Wess-Zumino consistency conditions. The implications for induced covariant W n gravity are briefly discussed. The results presented, valid for arbitrary n, reproduce those obtained for n = 2, 3 by different methods.