Quantum supergeometry and (co)bordism in quantum PDEs
by
Agostino Prastaro
University of Rome ''La Sapienza'', Italy

Oral Communication

Recently a new noncommutative geometry has been introduced by A.Prástaro [7, 8, 10] that allows us to describe noncommutative manifolds, (quantum manifolds), as geometric objects founded on pseudogroups. Such a geometric setting allows us to extend in a natural way to this new noncommutative framework, classic differential algebraic and homological objects. In this way we characterize quantum manifolds from the (co)homologic and (co)bordism point of view. This approach to noncommutative geometry differs from one introduced by A.Connes [1] as this last is founded in an algebraic geometric description of manifolds, that in the noncommutative context is not so natural to develop a geometric theory of partial differential equations (PDEs) and hence to treat quantum physics. (Actually such geometric algebraic approach is the uniquely adopted by the scientific community that works in noncommutative geometry. See e.g. also refs.[3, 4].) In fact, the approach by A.Connes has been first introduced to describe purely geometric situations, without any attention to solve physical problems. Really the most difficulties with such a theory are in their application to the theory of PDEs. In fact, it mediates in noncommutative contexts the commutative tools of the functional analysis, by means of some algebraic machinary. But in this way it is not able to build a geometric theory of PDEs - hence to obtain general theorems of existence of local and global solutions in quantum frameworks. (For example just at the beginning, starting with the definition of tangent space, the algebraic approach is very involved.)

The approach introduced by A.Prástaro is able to go on such difficulties, as its noncommutative manifolds are considered in a more geometric context and conceptually nearer to the point of view of the classic differential geometry. In fact, here quantum PDEs are seen as quantum submanifolds of suitable spaces (quantum derivative-jet spaces) and with a bit of more effort, it is possible to extend great results of the classic geometric theory of PDEs to such a category of quantum manifolds. In particular, we are able to obtain existence theorems of local solutions that extend ones obtained by D.Spencer [12] and H.Goldschmidt [2] on commutative manifolds and by us for supermanifolds [6]. Furthermore, we are able to extend also our more recent results on the integral (co)bordism groups in PDEs [7, 8, 10, 11] that are essential in order to characterize global solutions.

Aim of the present communication is to announce new results in this direction of the geometric theory of noncommutative PDEs. More precisely we aim to apply our approach to the study of a quantum PDE describing the quantum supergravity. For such an equation we obtain theorems of existence of local and global solutions. For the characterization of global solutions we utilize our recent theory on the integral (co)bordism groups in quantum PDEs, that allows us to calculate these groups also for quantum PDEs and that extend in this noncommutative framework the well known structures first introduced in algebraic topology by R.Thom and L.S.Pontrjagin [13, 5]. In this way we find global solutions having very sophisticate behaviours, like quantum tunnel effects and quantum black-holes.

REFERENCES

[1] CONNES, A. Noncommutative Geometry, Academic Press, San Diego, 1994.

[2] GOLDSCHMIDT, H., Integrability criteria for systems of non-linear partial differential equations, J. Differential Geometry 1(1967), 269-307.

[3] LYCHAGIN, V., Calculus and quantizations over Hopf algebras, Acta Appl. Math. 51(1998), 303-352.

[4] MANIN, Yu. Topics in Noncommutative Geometry, Princeton University Press, Princeton, N.J. 1991

[5] PONTRJAGIN, L.S., Smooth manifolds and their applications on homotopy theory, Amer. Math. soc. transl, 11(\1959), 1-114.

[6] PRASTARO, A., Geometry of super PDE's, Geometry in Partial Differential Equations, A.Prástaro & Th.M.Rassias (eds.), World Scientific, Singapore (1994), 259-315; Geometry of quantized super PDEìs, Amer. Math. Soc. Transl. 167)(2)(1995), 163-192; Quantum geometry of super PDE's, Rep. Math. Phys. 37(1)(1996), 23-140.

[7] PRASTARO, A., Geometry of PDEs and Mechanics, World Scientific, Singapore 1996, 760 pp.

[8] PRASTARO, A., (Co)bordism groups in PDEs and quantum PDEs, Rep. Math. Phys. 38(3)(1996), 443-455.

[9] PRASTARO, A., Quantum and integral (co)bordism groups in partial differential equations, Acta Appl. Math. 51(1998), 243-302; Quantum and integral bordism groups in the Navier-Stokes equation, New Developments in Differential Geometry, Budapest 1996, J.Szenthe (ed.), Kluwer Academic Publishers, Dordrecht 1998, 343-360; (Co)bordism groups in PDEs, Acta Appl. Math. 59(2)(1999), 111-202.

[10] PRASTARO, A., (Co)bordism groups in quantum PDEs, (to appear).

[11] PRASTARO & RASSIAS, Th.M., A geometric approach to a noncommutative generalized d'Alembert equation, C. R. Acad. Sc. Paris 330(I-7)(2000), 545-550.

[12] SPENCER, D., Determination of structures on manifolds defined by transitive continuous pseudogroups, I, II, III., Ann. Math. 75(1965), 1-114.

[13] THOM, R., Quelques propriétés globales des variétéa différentiables, Comm. Math. Helv. 28(1954), 17-86.

Date received: June 16, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-28.