arXiv subjects

S. A. Merkulov

Publications and source records attributed to S. A. Merkulov.

23 records · Page 2Linked to original sources

arXiv1998

Deformation quantization of the $n$-tuple point

Contrary to the classical methods of quantum mechanics, the deformation quantization can be carried out on phase spaces which are not even topological manifolds. In particular, the Moyal star product gives rise to a canonical functor $F$ from the category of affine analytic spaces to the category of associative (in general, non-commutative) $\C$-algebras. Curiously, if $X$ is the $n$-tuple point, $x^{n}=0$, then $F(X)$ is the algebra of $n\times n$ matrices.

math.QA↗

arXiv1998

The extended moduli space of special Lagrangian submanifolds

It is well known that the moduli space of all deformations of a compact special Lagrangian submanifold $X$ in a Calabi-Yau manifold $Y$ within the class of special Lagrangian submanifolds is isomorphic to the first de Rham cohomology group of $X$. Reinterpreting the embedding data $X\subset Y$ within the mathematical framework of the Batalin-Vilkovisky quantization, we find a natural deformation problem which extends the above moduli space to the full de Rham cohomology group of $X$.

math.AG↗

arXiv1998

Formality of canonical symplectic complexes and Frobenius manifolds

It is shown that the de Rham complex of a symplectic manifold $M$ satisfying the hard Lefschetz condition is formal. Moreover, it is shown that the differential Gerstenhaber-Batalin-Vilkoviski algebra associated to such a symplectic structure gives rise, along the lines explained in the papers of Barannikov and Kontsevich [alg-geom/9710032] and Manin [math/9801006], to the structure of a Frobenius manifold on the de Rham cohomology of $M$.

math.SG↗

arXiv1997

Semisimple Frobenius (super)manifolds and quantum cohomology of $P^r$

We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations. Finally, we calculate the Schlesinger initial conditions for solutions describing quantum cohomology of projective spaces.

alg-geom↗

arXiv1996

Exotic holonomies $\E_7^{(a)}$

It is proved that the Lie groups $\E_7^{(5)}$ and $\E^{(7)}_7$ represented in $\R^{56}$ and the Lie group $\E_7^{\C}$ represented in $\R^{112}$ occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connnections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension.

dg-ga↗