Singularities, Integrability, Symmetries (Scientific Coordinator: Francesco Calogero  Universita' La Sapienza  Roma) 
Local Research Unit: Symmetries, singularities and integrability issues in string theory and supersymmetric field theories. 

Scientific Coordinator
of Research Unit
Ugo Bruzzo Location of the Research
Unit
Scientific Sectors interested
in the Research
Key Words


Human Resources 

University Personnel
of the Research Unit


Scientific Background 

String theory provides a striking example of the tight relations which
occur among integrable systems, symmetries and
singularities. According to recent studies, one can associate algebraically integrable systems to the moduli spaces of string theory compactifications. String theory models display symmetries of different kinds: the nonlinear sigma models obtained from the compactification procedure possess the usual local gauge symmetries, supersymmetries and conformal symmetries; but new symmetries arise, which are of a purely quantum and nonperturbative nature. The main examples are Tduality, mirror symmetry, and others. Singularity theory turns out to be of great importance in string theory: the compactification spaces, as a result of precise physical requirements, are fibrations whose fibers may display quite complicated singularities. The latters are in turn related to the symmetries of the theory, as it is for instance shown by the socalled symmetry enhancement mechanism. The study of CalabiYau varieties with orbifold singularities is one of the techniques commonly employed in analysing the string theory models. The quantum spectrum of string theory models is in part described by the socalled Dbranes, which may be regarded as topological defects of the compactification spaces. As a result, a better understanding of the relations among integrability, symmetries and singularities, along the lines which inspire the national research project "SINTESI", appears to be of the utmost importance to allow a more comprehensive approach to the study of string theory. The main threads of the research activity of the SISSA local unit, which are tightly interrelated, are two: the socalled mirror symmetry, and the rigorous formulation of the string theory action functional. Starting with the early '90s, remarkable nonpertubative symmetries have been discovered in string theory, which contain the germs of very important and interesting mathematical questions. One can indeed see that some string theory models, compactified on a CalabiYau manifold X, are equivalent  from the viewpoint of the associated conformal quantum field theory  to models of a different type compactified on another CalabiYau manifold Y (the mirror dual to X). This duality is quite important from the physical viewpoint because the identification of the two theories inverts the energy range, so that one can gain information about the highenergy behavior of the theory compactified on Y by studying perturbatively the lowenergy limit of the theory compactified on X. One can also study mirror symmetry by exploiting techniques from noncommutative geometry; indeed, the socalled Morita equivalence seems to correspond to mirror symmetry between string theory models compactified on different noncommutative tori. The SISSA local unit is formed by researchers who are active in this area (cf. Bibliography). In particular they have recently organized a series of Workshops and Schools on topics related to the algebraic geometry of strings and integrable systems (cf. the curriculum of the scientific coordinator of the local unit). ReferencesBIBLIOGRAFIA RECENTE SCELTA DELL'UNITA' LOCALE (19972000)[1] C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez e J.M. Munoz Porras. Mirror symmetry on K3 surfaces via FourierMukai transform. Commun. Math. Phys. 195 (1998) 7993. [2] C. Bartocci, U. Bruzzo, G. Sanguinetti. Categorial mirror symmetry for K3 surfaces. Commun. Math. Phys. 206 (1999) 265272. [3] U. Bruzzo e G. Sanguinetti. Mirror symmetry on K3 surfaces as a hyperkaehler rotation. Lett. Math. Phys. 45 (1998) 295301. [4] E. Aldrovandi e L. A. Takhtajan. Generating Functional in CFT and Effective Action for TwoDimensional Quantum Gravity on Higher Genus Riemann Surfaces. Commun. Math. Phys. 188 (1997) 2967. [5] G. Landi, F. Lizzi e R. Szabo. String Geometry and the Noncommutative Torus. Commun. Math. Phys. 206 (1999) 603637. [6] C. Bartocci, U. Bruzzo e D. Hernandez Ruiperez. A FourierMukai transform for stable bundles on K3 surfaces. J. reine angew. Math. 486 (1997) 116. [7] C. Bartocci, U. Bruzzo D. Hernandez Ruiperez. Existence of mustable vector bundles on K3 surfaces and the FourierMukai transform. In Lect. Notes Pure Appl. Math. 200, M. Dekker Publisher, 1998. pp. 245258. [8] C. Bartocci, U. Bruzzo e D. Hernandez Ruiperez. Moduli of reflexive K3 surfaces. In "Complex Analysis and Geometry", a cura di V. Ancona et al. Pitman Research Notes in Mathematics Series, Longman, Harlwow, UK, 1997. [9] C. Bartocci, U. Bruzzo e D. Hernandez Ruiperez. A hyperkaehler Fourier transform. Diff. Geom. Appl. 8 (1998) 239249. [10] G. Landi. An Introduction to Noncommutative Spaces and Their Geometries. Lecture Notes in Physics, m51 (SpringerVerlag, BerlinHeidelberg, 1997). [11] G. Landi e C. Rovelli. General Relativity in Terms of Dirac Eigenvalues Phys. Rev. Lett. 78 (1997) 3051. [12] G. Landi e C. Rovelli. Gravity from Dirac Eigenvalues. Mod. Phys. Lett. A13 (1998) 479. [13] A.P. Balachandran, G. Bimonte, G. Landi, F. Lizzi e P. TeotonioSobrinho. Lattice Gauge Fields and Noncommutative Geometry. J. Geom. Phys. 24 (1998) 353. [14] L. Dabrowski, F. Nesti e P. Siniscalco On the Drinfeld twist for U_h sl(2) Proc. Conv. Naz. Relativita' Generale, Roma 1996. World Sci. 1997, Singapore [15] L. Dabrowski, P.M.Hajac e P. Siniscalco. Explicit HopfGalois Description of Induced Frobenius Homomorphisms, in: "Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions", D. Kastler et al. (Nova Science Publisher, Commack NY, 1999. [16] L. Dabrowski, F. Nesti e P. Siniscalco. A Finite Quantum Symmetry of M(3,C). Int. J. Mod. Phys. A 13 (1998) 41474162. [17] E. Ercolessi, G. Landi e P. TeotonioSobrinho. Ktheory of Noncommutative Lattices. KTheory 18 (1999) 339362. [18] G. Landi e F. Lizzi. Projective Systems of Noncommutative Lattices as a Pregeometric Substratum, in: "Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions", D. Kastler et al. (Nova Science Publisher, Commack NY, 1999), pp. 85113. [19] T. Krajewski . Classification of finite spectral triples, J. Geom. Phys. 28 (1997) 130. [20] T. Krajewski. Constraints of the scalar potential from spectral action principle, to appear in J. Math. Phys. (2000). [21] T. Krajewski e R. Wulkenhaar. On Kreimer's Hopf algebra structure of Feynman graphs, Eur. Phys. J. C7 (1998) 699708. [22] T. Krajewski e R. Wulkenhaar . Perturbative quantum gauge fields on the noncommutative torus, to appear in Int. J. Mod. Phys. A (2000). [23] G. Bonelli, L. Bonora, F. Nesti e A. Tomasiello. Matrix string theory and its moduli space. Nucl. Phys. B554 (1999) 103135. [24] G. Bonelli, L. Bonora, F. Nesti e A. Tomasiello. Heterotic string theory and Riemann surfaces. Nucl. Phys. B564 (2000) 86102. [25] M.Boggi. Compactification of configurations of points on P^1 and quadratic transformations of projective space. Ind. Math. 1999. [26] M.Boggi e M.Pikaart. Galois covers of moduli of curves. Comp. Math. 120 (2000) 171191. [27] G. Panati e A. Teta. The FluxAcrossSurfaces Theorem for a Point Interaction Hamiltonian", in H. Holden et al., Infinite Dimensional (Stochastic) Analysis and Quantum Physics, CMS Proceeding series, Leipzig, 1999. 

Research Program Description 

Part of the activity of the SISSA unit
will deal with the problem of giving a good definition of the notion of
"mirror
pair of CalabiYau manifolds". An important proposal has been advanced, on the basis of physical considerations, by Strominger, Yau e Zaslow; their conjectural definition of "mirror dual" is given in terms of special Lagrangian geometry. The basic idea is that the mirror Y of a CalabiYau manifold X of complex dimension n is the moduli space of fibrations of X in (real) ndimensional special Lagrangian tori equipped with a flat line bundle. Specifically, the following questions will be examined:  characterization of mirror symmetry, firstly at the topological level, and subsequently at the level of complex structures, for CalabiYau manifolds of complex dimension 4, in particular in the hyperkaehler case (here the StromingerYauZaslow construction implies that the CalabiYau manifold is an algebraically completely integrable system).  study of the transformation pattern of Dbranes under mirror symmetry. Here the aim is to extend to the 3 and 4dimensional case some results already obtained for complex dimension 2. In particular the n=3 case, for which complex geometry techniques are of no use, requires the development of a relative transform of the FourierMukai type for Lagrangian fibrations with singular fibres.  study of the relation between mirror symmetry and Morita equivalence, in view of the fact that string theory models compactified on Moritaequivalent noncommutative tori give rise to isomorphic BPS spectra and equivalent field theories. As for the research on the structure of the action functional in string theory, the aim is to improve some results about the expression of the Polyakov and Liouville actions in terms of quasiconformal maps between high genus Riemann surfaces, providing a description of the action as a Deligne cohomology class with respect to good covers. "Bad" covers will be also considered, together with the links with the indTeichmueller spaces and the descent of the variational complex with respect to the Deligne complex. 