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.
of Research Unit
Location of the Research
Scientific Sectors interested
in the Research
of the Research Unit
|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 T-duality, 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 so-called symmetry enhancement mechanism. The study of Calabi-Yau 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 so-called D-branes, 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 so-called
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 Calabi-Yau manifold X, are equivalent -- from the viewpoint of the associated conformal
quantum field theory -- to models of a different type compactified on another Calabi-Yau 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 high-energy behavior of the theory compactified on Y by
studying perturbatively the low-energy limit of the theory compactified on X. One can also study mirror symmetry by
exploiting techniques from noncommutative geometry; indeed, the so-called 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 (1997-2000)
 C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez e J.M. Munoz Porras. Mirror symmetry on K3 surfaces via
Fourier-Mukai transform. Commun. Math. Phys. 195 (1998) 79-93.
 C. Bartocci, U. Bruzzo, G. Sanguinetti. Categorial mirror symmetry for K3 surfaces. Commun. Math. Phys. 206
 U. Bruzzo e G. Sanguinetti. Mirror symmetry on K3 surfaces as a hyper-kaehler rotation. Lett. Math. Phys. 45
 E. Aldrovandi e L. A. Takhtajan. Generating Functional in CFT and Effective Action for Two-Dimensional Quantum
Gravity on Higher Genus Riemann Surfaces. Commun. Math. Phys. 188 (1997) 29-67.
 G. Landi, F. Lizzi e R. Szabo. String Geometry and the Noncommutative Torus. Commun. Math. Phys. 206 (1999)
 C. Bartocci, U. Bruzzo e D. Hernandez Ruiperez. A Fourier-Mukai transform for stable bundles on K3 surfaces. J.
reine angew. Math. 486 (1997) 1-16.
 C. Bartocci, U. Bruzzo D. Hernandez Ruiperez. Existence of mu-stable vector bundles on K3 surfaces and the
Fourier-Mukai transform. In Lect. Notes Pure Appl. Math. 200, M. Dekker Publisher, 1998. pp. 245--258.
 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.
 C. Bartocci, U. Bruzzo e D. Hernandez Ruiperez. A hyperkaehler Fourier transform. Diff. Geom. Appl. 8 (1998)
 G. Landi. An Introduction to Noncommutative Spaces and Their Geometries. Lecture Notes in Physics, m51
(Springer-Verlag, Berlin-Heidelberg, 1997).
 G. Landi e C. Rovelli. General Relativity in Terms of Dirac Eigenvalues Phys. Rev. Lett. 78 (1997) 3051.
 G. Landi e C. Rovelli. Gravity from Dirac Eigenvalues. Mod. Phys. Lett. A13 (1998) 479.
 A.P. Balachandran, G. Bimonte, G. Landi, F. Lizzi e P. Teotonio-Sobrinho. Lattice Gauge Fields and
Noncommutative Geometry. J. Geom. Phys. 24 (1998) 353.
 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
 L. Dabrowski, P.M.Hajac e P. Siniscalco. Explicit Hopf-Galois Description of Induced Frobenius Homomorphisms,
in: "Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions", D. Kastler et al. (Nova
Science Publisher, Commack NY, 1999.
 L. Dabrowski, F. Nesti e P. Siniscalco. A Finite Quantum Symmetry of M(3,C). Int. J. Mod. Phys. A 13 (1998)
 E. Ercolessi, G. Landi e P. Teotonio-Sobrinho. K-theory of Noncommutative Lattices. K-Theory 18 (1999)
 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. 85-113.
 T. Krajewski . Classification of finite spectral triples, J. Geom. Phys. 28 (1997) 1-30.
 T. Krajewski. Constraints of the scalar potential from spectral action principle, to appear in J. Math. Phys. (2000).
 T. Krajewski e R. Wulkenhaar. On Kreimer's Hopf algebra structure of Feynman graphs, Eur. Phys. J. C7 (1998)
 T. Krajewski e R. Wulkenhaar . Perturbative quantum gauge fields on the noncommutative torus, to appear in Int.
J. Mod. Phys. A (2000).
 G. Bonelli, L. Bonora, F. Nesti e A. Tomasiello. Matrix string theory and its moduli space. Nucl. Phys. B554 (1999)
 G. Bonelli, L. Bonora, F. Nesti e A. Tomasiello. Heterotic string theory and Riemann surfaces. Nucl. Phys. B564
 M.Boggi. Compactification of configurations of points on P^1 and
quadratic transformations of projective space. Ind. Math. 1999.
 M.Boggi e M.Pikaart. Galois covers of moduli of curves. Comp. Math. 120 (2000) 171-191.
 G. Panati e A. Teta. The Flux-Across-Surfaces 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
pair of Calabi-Yau 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 Calabi-Yau manifold X of complex dimension n is the moduli space
of fibrations of X in (real) n-dimensional 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 Calabi-Yau manifolds of complex dimension 4, in particular in the hyperkaehler case (here the
Strominger-Yau-Zaslow construction implies that the Calabi-Yau manifold is an algebraically completely integrable
-- study of the transformation pattern of D-branes under mirror symmetry. Here the aim is to extend to the 3- and
4-dimensional 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 Fourier-Mukai
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 Morita-equivalent noncommutative tori give rise to isomorphic BPS spectra and equivalent field
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 quasi-conformal 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 ind-Teichmueller spaces and the descent of the
variational complex with respect to the Deligne complex.