Pants decompositions of random surfaces

Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus $g$ for which any pants decomposition requires curves of total length at least $g^{7/6 - \epsilon}$. Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil-Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.Comment: 16 pages, 4 figure

Embeddings of algebras in derived categories of surfaces

By a result of Orlov there always exists an embedding of the derived category of a finite-dimensional algebra of finite global dimension into the derived category of a high-dimensional smooth projective variety. In this article we give some restrictions on those algebras whose derived categories can be embedded into the bounded derived category of a smooth projective surface. This is then applied to obtain explicit results for hereditary algebras.Comment: 13 pages; revised versio

Matrix Model for Discretized Moduli Space

We study the algebraic geometrical background of the Penner--Kontsevich matrix model with the potential N\alpha \tr {\bigl(- \fr 12 \L X\L X +\log (1-X)+X\bigr)}. We show that this model describes intersection indices of linear bundles on the discretized moduli space right in the same fashion as the Kontsevich model is related to intersection indices (cohomological classes) on the Riemann surfaces of arbitrary genera. The special role of the logarithmic potential originated from the Penner matrix model is demonstrated. The boundary effects which was unessential in the case of the Kontsevich model are now relevant, and intersection indices on the discretized moduli space of genus $g$ are expressed through Kontsevich's indices of the genus $g$ and of the lower genera

Microlocal sheaves and quiver varieties

We relate Nakajima Quiver Varieties (or, rather, their multiplicative version) with moduli spaces of perverse sheaves. More precisely, we consider a generalization of the concept of perverse sheaves: microlocal sheaves on a nodal curve X. They are defined as perverse sheaves on normalization of X with a Fourier transform condition near each node and form an abelian category M(X). One has a similar triangulated category DM(X) of microlocal complexes. For a compact X we show that DM(X) is Calabi-Yau of dimension 2. In the case when all components of X are rational, M(X) is equivalent to the category of representations of the multiplicative pre-projective algebra associated to the intersection graph of X. Quiver varieties in the proper sense are obtained as moduli spaces of microlocal sheaves with a framing of vanishing cycles at singular points. The case when components of X have higher genus, leads to interesting generalizations of preprojective algebras and quiver varieties. We analyze them from the point of view of pseudo-Hamiltonian reduction and group-valued moment maps.Comment: 49 page

Nonperturbative Model Of Liouville Gravity

We obtain nonperturbative results in the framework of continuous Liouville theory. In particular, we express the specific heat ${\cal Z}$ of pure gravity in terms of an expansion of integrals on moduli spaces of punctured Riemann spheres. The integrands are written in terms of the Liouville action. We show that ${\cal Z}$ satisfies the Painlev\'e I.Comment: 11 pages, LaTex fil