ON THE TWISTOR AND SPINOR GEOMETRY OF LOOP SPACES

Mauro Spera
Dipartimento di Informatica 
Universitˆ degli Studi di Verona
Ca' Vignal 2, Strada Le Grazie 15, 37134 Verona, Italia

In this talk I report on recent joint work with Tilmann WURZBACHER, L.M.A.M. UniversitŽ Paul Verlaine - Metz et C.N.R.S. France.

Loop (and, more generally, mapping) spaces provide a natural arena for infinite dimensional differential geometry and
topology - viewed either  per se or as a tool for finite dimensional geometry  - and reveal subtle and fascinating structures, often arising from theoretical and mathematical physics, such as the notion of string structure, the loop space analogue of the notion of spin structure (see e.g. [K],[BMcL]).
In finite dimensions, spin structures and bundles of spinors turn out to be closely related to twistor spaces (compare e.g. [D-V]). 
In the paper [SW3], motivated by the finite dimensional portrait, we introduced two natural "twistor spaces" associated to the loop space of a Riemannian spin manifold and showed their equivalence, at least in the Kaehlerian case.
They both arise as isotropic Grassmannian fibrations (parametrizing
complex structures on a real Hilbert space), the first one coming from
looping the frame bundle of M and Fourier mode decomposition of the Hilbert space L^2(S^1, C^d) (see e.g. [PS, SW1, Wu2]), the second one from the analogous decomposition relative to the Atiyah operator family, induced by the Levi-Civita connection, on the complexified tangent bundle of the loop space ([A],[Wu1]).
In the case of loop spaces, the obstruction to the existence of a string structure is a Dixmier-Douady class which can be interpreted in terms of gerbes, via an infinite dimensional version of the 
Gotay-Lashof-Sniatycki-Weinstein construction given in [GLSW] and [B].
The above results, besides having some interest in themselves, provide
possibly useful steps towards the rigorous construction of a
Dirac-Ramond operator on loop spaces, beyond the flat case (see e.g. [SW2]
and references therein), in the spirit of the "twistorial"
reinterpretation of the ordinary finite dimensional Dirac operator
outlined in [D-V]. 


References

[A] M. F. Atiyah,  Circular symmetry and stationary phase approximation, 
in: Colloque en l'honneur de L. Schwartz, Vol.I, AstŽrisque Vol. 131, SMF (1985) 43-59.

[B] J. L. Brylinski,  Loop spaces, characteristic classes and geometric quantization, Birkha"user, Basel, 1993.

[BMcL] J. L. Brylinski and D. Maclaughlin, The geometry of degree-four characteristic classes and of line bundles on loop spaces I,  Duke Math. J. 75 (1994), 603-638.

[D-V] M. Dubois-Violette,  Structures complexes au-dessus des variŽtŽs, applications, in: Mathematics and Physics (Paris, 1979-1982), Prog. Math. 37, Birkha"user, Boston, MA, (1983) 1-42.

[GLSW] M. Gotay, R. Lashof, J. Sniatycki and A. Weinstein,  Closed forms on symplectic fibre bundles, Comm. Math. Helv. 58 (1983), 617-621.

[K] T. Killingback,  World-sheet anomalies and loop geometry,
Nucl. Phys. B 288 (1987), 578-588. 
  
[PS] A. Pressley and G. Segal,  Loop groups, Oxford University Press, Oxford, 1986.   

[SW1] M. Spera and T. Wurzbacher, Determinants, Pfaffians, and quasi-free
representations of the CAR algebra, Rev.Math.Phys. 10 (1998), 705-721.

[SW2]  M. Spera and T. Wurzbacher, The Dirac-Ramond operator on loops in flat space, J. Funct. Anal. 197 (2003), 110-139.

[SW3]  M. Spera and T. Wurzbacher, Twistor spaces and spinor
over loop spaces, Math. Ann. (to appear).

[SW4]  M. Spera and T. Wurzbacher, Good coverings and smooth partitions of unity for mapping spaces, (in final preparation).

[Wu1]  T. Wurzbacher, Symplectic geometry of the loop space of a Riemannian
manifold,  J. Geom. Phys. 16 (1995), 345-384.

[Wu2] T. Wurzbacher, Fermionic Second Quantization,
in: Infinite dimensional Kaehler manifolds (Oberwolfach, 1995), 
DMV Sem., 31, Birkha"user, Basel, (2001)  287-375.