Paolo Soriani

A search for non-perturbative dualities of local N = 2 Yang - Mills theories from Calabi - Yau 3-folds

The generalization of the rigid special geometry of the vector multiplet quantum moduli space to the case of supergravity is discussed through the notion of a dynamical Calabi - Yau 3-fold. Duality symmetries of this manifold are connected with the analogous dualities associated with the dynamical Riemann surface of the rigid theory. N = 2 rigid gauge theories are reviewed in a framework ready for comparison with the local case. As a byproduct we give in general the full duality group (quantum monodromy) for an arbitrary rigid SU(r+1) gauge theory, extending previous explicit constructions for the r = 1,2 cases. In the coupling to gravity, R-symmetry and monodromy groups of the dynamical Riemann surface, whose structure we discuss in detail, are embedded into the symplectic duality group associated with the moduli space of the dynamical Calabi - Yau 3-fold.

R Symmetry and the Topological Twist of N = 2 Effective Supergravities of Heterotic Strings

We discuss R symmetries in locally supersymmetric N = 2 gauge theories coupled to hypermultiplets which can be thought of as effective theories of heterotic superstring models. In this type of supergravities a suitable R symmetry exists and can be used to topologically twist the theory: the vector multiplet containing the dilaton–axion field has different R charge assignments with respect to the other vector multiplets. Correspondingly a system of coupled instanton equations emerges, mixing gravitational and Yang–Mills instantons with triholomorphic hyperinstantons and axion instantons. For the tree level classical special manifolds ST(n) = SU(1,1)/U(1) × SO(2,n)/[SO(2) × SO(n)], R symmetry with the specified properties is a continuous symmetry, but for the quantum-corrected manifolds a discrete R group of electric–magnetic duality rotations is sufficient and we argue that it exists.

Constrained topological gravity from twisted N=2 Liouville theory

In this paper we show that there exists a new class of topological field theories whose correlators are intersection numbers of cohomology classes in a constrained moduli space. Our specific example is a formulation of 2D topological gravity. The constrained moduli-space is the Poincare dual of the top Chern class of the bundle ehol → Mg, whose sections are the holomorphic differentials. Its complex dimensions is 2g − 3, rather than 3g − 3. We derive our model by performing the A-topological twist of N=2 supergravity, that we identify with N=2 Liouville theory, whose rheonomic construction is also presented. The peculiar field theoretical mechanism, rooted in BRST cohomology, that is responsible for the constraint on moduli space is discussed, the key point being the fact that the graviphoton becomes a Lagrange multiplier after twist. The relation with conformal field theories is also explored. Our formulation of N=2 Liouville theory leads to a representation of the N=2 superconformal algebra with c=cLiouville+cghost=6−6, instead of the value c=3−9 that is obtained by untwisting the Verlinde and Verlinde formulation of topological gravity. The different central charge is the shadow, in conformal field theory, of the constraint on moduli space. Our representation of the c=6, N=2 algebra can be split into the direct sum of a minimal model with c=32 and a “maximal” model with c=92. Considerations on the matter coupling of constrained topological gravity are also presented. Their study requires the analysis of both the A-twist and the B-twist of N=2 matter coupled supergravity, that we postpone to future work.

From N=2 supergravity to constrained moduli spaces

In this talk we review some results concerning a mechanism for reducing the moduli space of a topological field theory to a proper submanifold of the ordinary moduli space. Such mechanism is explicitly realized in the example of constrained topological gravity, obtained by topologically twisting the N=2 Liouville theory.In this talk we review some results concerning a mechanism for reducing the moduli space of a topological field theory to a proper submanifold of the ordinary moduli space. Such mechanism is explicitly realized in the example of constrained topological gravity, obtained by topologically twisting the N=2 Liouville theory.