MINISTERO DELL'ISTRUZIONE, DELL'UNIVERSITÀ E DELLA RICERCA

CO.FIN.
GEOMETRIC METHODS IN THE THEORY OF NONLINEAR WAVES
AND THEIR APPLICATIONS.
[general project]

research program 2006-2008

RESEARCH UNIT: PAVIA

PEOPLE

CARFORA Mario
MARZUOLI Annalisa
GILLI Valeria
DAPPIAGGI Claudio

Full Professor of Mathematical Physics (University of Pavia).
Vice-Director of the Department of Nuclear and Theoretical Physics of the University of Pavia.

1997-2001:Associate Professor of Theoretical Physics (Univ. Pavia).
1992-1997: Associate Professor of Mathematical Physics (SISSA-ISAS, Trieste).
1985-1992: Researcher (Univ.Pavia).
1981-1985: Researcher (Univ. of Rome "La Sapienza").
1981: Ph.D. in Physics (Univ. of Texas at Dallas, supervisor W. Rindler).
1977: Laurea in Physics ("La Sapienza", Rome).

Honours:

Winner of the Bruno Finzi Prize for Mathematical Physics of the Istituto Lombardo di Scienze (1997);
2nd Prize of The Gravity Research Foundation (with C. Rovelli and A. Bareira, 1996).
Member of the Scientific Committee of the National Group of Mathematical Physics (GNFM) of the National Institute of Advanced Mathematics (INDAM).
Member of the Scientific Committee of SIGRAV (Italian Society for Relativity and Gravitational Physics).

Local coordinator (Pavia) of the theoretical group INFN (Italian National Institute for Nuclear Physics) (1987-1993);
Local coordinator (Pavia) of the European HCM project Constrained Dynamical Systems (1993-96);
Member of the Scientific Committee of the SIGRAV Graduate School in Contemporary Relativity and Gravitational Physics.
Member of the Scientific Committee of the meeting "Problemi Attuali di Fisica Teorica" (Vietri, 2001, 2004, 2005, 2006); Chair of the Quantum Gravity workshop.
President (together with Bruno Bertotti) of the Scientific Committee of the International Conference "Spacetime in Action: one hundred years of Relativity" (Pavia March 29- April 2), framed within the UNESCO World Year of Physics 2005 Programme.
Chairman of the Classical and Quantum Gravity workshop of the Italian Meeting on General Relativity and Gravitational Physics (2004, 2006).

Moduli spaces, integrable structures and evolutive PDEs: applications in Ricci flow theory and quantum algorithms.

The physics of two-dimensional quantum gravity and string theory has provided a number of deep and not completely understood connections between Riemann moduli space theory, piecewise-linear geometry, and the theory of integrable systems, (e.g., [1], [2], [3], [4], [5], [6]). Such a wide spectrum of applications has a two-fold origin. On the mathematical side it is deeply related to the fact that the moduli space admits natural (semi-simplicial) decompositions which are in a one-to-one correspondence with classes of suitably decorated graphs. On the physical side it is connected to the observation that conformal field theory (CFT) and Topological quantum field theory, ubiquitous in many of the physical systems mentioned above, are based on algebraic structures parametrized by the moduli space of genus g Riemann surfaces with N punctures M(g;N). This parametrization is consistent with the operation of sewing any two such surfaces together, provided that we match the complex structure in the overlap and keep track of which puncture is ingoing and which is outgoing. This sewing gives to M(g;N) and its cell decomposition the structure of an operad. In such a setting a CFT leads to a natural algebra, over the cell decomposition of Riemann moduli space, which can be related to the algebra of physical space of states of the theory and to their dynamics. Both such descriptions are associated with the existence of an underlying integrable structure described by nonlinear integrable models of dispersionless type satisfying the constraints represented by the string equations : KdV, KP and Toda hierarchies for the simplicial models, and the so-called Ground Ring of Operators for the CFT living on the Riemann surface [7]. Most likely, at the root of the connection between the CFT framework, the cellular structure of M(g,N), and the underlying integrabilty there is the common operadic structure mentioned above. Such a correspondence has provided one of the most successful tool for a rigorous mathematical interpretation of ideas in topological quantum field theory.

Recently, new applications of these ideas have made their impact in several guises also in Ricci flow theory and even in quantum computing. Ricci flow [8] is an evolutionary system of PDEs of geometric origin defined by deforming a given Riemannian metric in the direction of its Ricci tensor. It arises in a variety of non-linear problems in physics, typically as the renormalization group flow for non-linear sigma-model [9], wheras in Riemannian geometry has been introduced by R. Hamilton [10] in his attempt to a proof of the Thurston geometrization conjecture. In this latter connection, it has led to ground breaking results owing to the recent seminal work of G. Perelman [11], [12], [13]. In dimension two, the Ricci flow resembles a (non-linear) heat flow equation on a surface due to the (weakly) parabolic character of the metric deformation in terms of the flow parameter t. Thus, it exhibits a dissipative behavior in t, but,
as shown by I. Bakas [14], it possesses an underlying infinite dimensional Lie algebras structure which gives rise to a flow admitting a Toda field theoretic interpretation and turns out to be integrable in space. Since the two-dimensional Ricci flow provides an alternative proof of the standard uniformization theorem for surfaces one is led to discuss to what extent the Ricci flow considered as a dynamical system on the Riemann moduli space M(g) is connected to the above mentioned integrable hierarchies. This analysis relies on the study of the trajectories of the flow in regimes where the deformation parameter tends to certain limiting values and diffeomorphism solitons appear.
We believe that characterizing this connection could be an important step forward in the study of the relation between
the evolutive PDEs describing the two-dimensional Ricci flow and the dynamics of the integrable hierarchies characterizing the geometry of the moduli spaces M(g,N). The project will address some of the issues of interest in this area, with particular emphasis on the roles of Perelman's shrinker entropy functionals [11], [15] and the construction of explicit examples in terms of the Egorof metrics which naturally appear in the study of Gromov-Witten invariants.


By exploiting techniques from hyperbolic geometry and the theory of geometric structures is also possible to extend the connection between the operad structure of M(g;N) and integrable systems to some aspects of Chern-Simons and Wess-Zumino-Witten (WZW) theory (for this latter see e.g. [16]). This remark suggests that integrable hierarchies may also have a relevant role in the development of quantum algorithms based on the recoupling theory of angular momenta.
In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing, besides the standard Boolean quantum circuit model based on qubits and Boolean gates. The approach of the spin-network quantum simulator proposed in [17] relies on the (re)coupling theory of SU(2) angular momenta (Racah-Wigner tensor category) and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum circuit model.
Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables.
When working with purely discrete unitary gates, the computational space of the simulator is naturally modelled as a family of finite-states and discrete-time quantum automata which in turn represent discrete versions of topological models of quantum computation based on modular functors of SU(2) Chern--Simons theory at some fixed level k [18]. Note that the discretized quantum theory underlying such computational scheme actually belongs to the class of SU(2) state sum models introduced by Turaev and Viro [19] in 1992 and widely used in 3-dimensional simplicial quantum gravity models and in Conformal Field Theory.

Preliminary results [20] suggest that in some case (the computation of knots polynomials) the growth rate of the time complexity function in terms of the integer k can be estimated to be (polynomially) bounded by resorting to the interplay between Chern--Simons theory and the cell decomposition of the moduli space M(g;N). This remarks suggests to use the techniques and ideas from the theory of integrable hierarchies and simplicial quantum gravity in order to get a deeper understanding of the time complexity function.
It is such characterization that the Pavia group will focus on under the present proposal.

[1] J. Ambjorn, B. Durhuus, T. Jo'nsson "Quantization of Geometry" Cambridge University Press (1997), see also J. Ambjorn, M. Carfora, A. Marzuoli, "The geometry of dynamical triangulations" Lecture Notes in Physics m50, Springer (1997).

[2] M. Kontsevitch, "Intersection theory on the moduli space of curves and the matrix Airy function", Commun. Math. Phys. 147 (1992) 1­23; E. Witten, "Two-dimensional gravity and intersection theory on moduli space", Surveys in Diff. Geom. 1 (1991), 243-310.

[3] R. C. Penner, "Perturbative series and the moduli space of punctured surfaces", J. Diff. Geom. 27 (1988), 35-53.

[4] B. Dubrovin, Y. Zhang "Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants"
math.DG/0108160.

[5] A. Okounkov, R. Pandharipande, "Gromov-Witten theory, Hurwitz numbers, and matrix models I", (2001) math.ag/0101147.

[6] D. Gaiotto and L. Rastelli, "A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model" hep-th/0312196 (2003);
see also R. Gopakumar, "From free fields to ADS -II", hep-th/0402063 (2004).

[7] E. Witten "Ground Ring of two dimensional String Theory", Nucl. Phys. B 373, 187 (1992) hep-th/9108004; see also
I. Kostov, "Boundary ground ring and disc correlation functions in Liouville quantum gravity", Proceedings of the Conference "Lie theory and its applications in Physics" (June, 2003) hep-th/0402098.

[8] B. Chow, D. Knopf "The Ricci flow: an introduction" Mathematical Surveys and Monographs, Vol. 110, American Math.Soc. (2004).

[9] D. H. Friedan, "Nonlinear models in 2+epsilon dimensions", Ann. Physics 163 (1985), no. 2, 318--419.

[10] R. Hamilton "Three-manifolds with positive Ricci curvature" J. Differential Geom. 17 (1982), 255-306.

[11] G. Perelman "The entropy formula for the Ricci flow and its geometric applications" math.DG/0211159.

[12] G. Perelman "Ricci flow with surgery on Three-Manifolds" math.DG/0303109.

[13] G. Perelman "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" math.DG/0307245.

[14] I. Bakas "The algebraic structure of geometric flows in two dimensions", arXiv:hep-th/0507284.

[15] M. Carfora, "Fokker-Planck dynamics and entropies for the
renormalized Ricci flow" arXiv:math.DG/0507309 v2.

[16] M. Carfora, "Discretized Gravity and the SU(2) WZW model" Class. Quant. Gravity 21 (2004) S109-S126.

[17] A. Marzuoli, M. Rasetti "Computing spin networks" Annals of Phys. 318 (2005) 345-407.

[18] M. H. Freedman, A. Kitaev, M. Larsen, Z. Wang "Topological Quantum Computation" Bull. Amer. Math. Soc. 40 (2002) 31-38.

[19] V.G. Turaev, O. Ya. Viro, "State sum invariants of 3-manifolds and quantum 6j-symbols" Topology 31 (1992) 865-902.

[20] S. Garnerone, A. Marzuoli, M. Rasetti "Quantum automata, braid group and link polynomials" ArXiv:quant-ph/0601169.

Under the present proposal, the research programme of the Pavia group will focus on the following lines:

Geometry of the moduli space M(g,N) of genus g Riemann surfaces with N punctures and its operad structure. Cellular decompositions of M(g,N): Ribbon graphs associated with the singular Euclidean structures (defined by random Regge triangulations), and ribbon graphs induced by the horocyclic parametrization defined by R. Penner. Connections between such two descriptions. Dispersionless integrable hierarchies and the geometry of these cellular decompositions of M(g,N). Evolutive PDEs of geometrical type connected with the two-dimensional Ricci flow, interpretation as a dynamical system on the moduli space M(g,N) and connections with the integrable hierarchies describing the topology of M(g,N).
Three-dimensional hyperbolic geometry and two-dimensional singular Euclidean structures. Chern-Simons theory and cellular decomposition of the moduli space M(g,N). Applications to problems in quantum computing knots invariants. This latter line of research is prompted by the geometrical analysis briefly discussed above and stems from observing that metrically triangulated surfaces can be naturally associated with three-dimensional hyperbolic manifolds triangulated by means of ideal hyperbolic tetrahedra (this being a consequence of the Ahlfors-Bers theorem according to which the 3D hyperbolic structure in a manifold is characterized by the conformal structure on the boundary of the manifold).

The Pavia team and the other teams partecipating in the proposal each have experience and high level of expertise in these areas. As far as the planned timetable is concerned, the program of research under this Project is divided into two phases. In the first year we will be concentrating in assessing the issue of a proper characterization of the connections among the cellular geometry of M(g,N), defined either by singular Euclidean structures or by the Penner construction, three-dimensional hyperbolic geometry, and the integrable hierarchies governing the topology of M(g,N). In the second year, the main effort of the team will be directed in studying the applications of the model in discussing the evolutive PDEs associated with the two-dimensional Ricci flow, and the interaction between the cellular geometry of M(g,N) and problems of quantum computations of knots invariants.


We believe that an important aspect of our research programme is the allocation of funds for a two-years postdoctoral position that will offer the opportunity to young researchers, interested into such a lively and fascinating field, to interact with our team and with the other teams partecipating in the proposal.