Exact Lagrangian submanifolds in simply-connected cotangent bundles

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler, using a different approach.Comment: 28 pages, 3 figures. Version 2 -- derivation and discussion of the spectral sequence considerably expanded. Other minor change

On the full, strongly exceptional collections on toric varieties with Picard number three

We investigate full strongly exceptional collections on smooth, com- plete toric varieties. We obtain explicit results for a large family of varieties with Picard number three, containing many of the families already known. We also describe the relations between the collections and the split of the push forward of the trivial line bundle by the toric Frobenius morphism

On Bohr-Sommerfeld bases

This paper combines algebraic and Lagrangian geometry to construct a special basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We use the method of [D. Borthwick, T. Paul and A. Uribe, Legendrian distributions with applications to the non-vanishing of Poincar\'e series of large weight, Invent. math, 122 (1995), 359-402, preprint hep-th/9406036], whereby every vector of a BS basis is defined by some half-weighted Legendrian distribution coming from a Bohr-Sommerfeld fibre of a real polarization of the underlying symplectic manifold. The advantage of BS bases (compared to bases of theta functions in [A. Tyurin, Quantization and ``theta functions'', Jussieu preprint 216 (Apr 1999), e-print math.AG/9904046, 32pp.]) is that we can use information from the skillful analysis of the asymptotics of quantum states. This gives that Bohr-Sommerfeld bases are unitary quasi-classically. Thus we can apply these bases to compare the Hitchin connection with the KZ connection defined by the monodromy of the Knizhnik-Zamolodchikov equation in combinatorial theory (see, for example, [T. Kohno, Topological invariants for 3-manifolds using representations of mapping class group I, Topology 31 (1992), 203-230; II, Contemp. math 175} (1994), 193-217]).Comment: 43 pages, uses: latex2e with amsmath,amsfonts,theore

On Pure Spinor Superfield Formalism

We show that a certain superfield formalism can be used to find an off-shell supersymmetric description for some supersymmetric field theories where conventional superfield formalism does not work. This "new" formalism contains even auxiliary variables in addition to conventional odd super-coordinates. The idea of this construction is similar to the pure spinor formalism developed by N.Berkovits. It is demonstrated that using this formalism it is possible to prove that the certain Chern-Simons-like (Witten's OSFT-like) theory can be considered as an off-shell version for some on-shell supersymmetric field theories. We use the simplest non-trivial model found in [2] to illustrate the power of this pure spinor superfield formalism. Then we redo all the calculations for the case of 10-dimensional Super-Yang-Mills theory. The construction of off-shell description for this theory is more subtle in comparison with the model of [2] and requires additional Z_2 projection. We discover experimentally (through a direct explicit calculation) a non-trivial Z_2 duality at the level of Feynman diagrams. The nature of this duality requires a better investigation

Syzygy algebras for the Segre embeddings

We describe the syzygy spaces for the Segre embedding $\mathbb{P}(U)\times\mathbb{P}(V)\subset\mathbb{P}(U\otimes V)$ in terms of representations of ${\rm GL}(U)\times {\rm GL}(V)$ and construct the minimal resolutions of the sheaves $\mathscr{O}_{\mathbb{P}(U)\times\mathbb{P}(V)}(a,b)$ in $D(\mathbb{P}(U\otimes V))$ for $a\geqslant-\dim(U)$ and $b\geqslant-\dim(V)$. Also we prove some property of multiplication on syzygy spaces of the Segre embedding.Comment: 17 pages, 11 picture

Seiberg Duality is an Exceptional Mutation

The low energy gauge theory living on D-branes probing a del Pezzo singularity of a non-compact Calabi-Yau manifold is not unique. In fact there is a large equivalence class of such gauge theories related by Seiberg duality. As a step toward characterizing this class, we show that Seiberg duality can be defined consistently as an admissible mutation of a strongly exceptional collection of coherent sheaves.Comment: 32 pages, 4 figures; v2 refs added, "orbifold point" discussion refined; v3 version to appear in JHEP, discussion of torsion sheaves improve

Multiloop Superstring Amplitudes from Non-Minimal Pure Spinor Formalism

Using the non-minimal version of the pure spinor formalism, manifestly super-Poincare covariant superstring scattering amplitudes can be computed as in topological string theory without the need of picture-changing operators. The only subtlety comes from regularizing the functional integral over the pure spinor ghosts. In this paper, it is shown how to regularize this functional integral in a BRST-invariant manner, allowing the computation of arbitrary multiloop amplitudes. The regularization method simplifies for scattering amplitudes which contribute to ten-dimensional F-terms, i.e. terms in the ten-dimensional superspace action which do not involve integration over the maximum number of $\theta$'s.Comment: 23 pages harvmac, added acknowledgemen

Mutation in triangulated categories and rigid Cohen-Macaulay modules

We introduce the notion of mutation of $n$-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules over certain Veronese subrings.Comment: 52 pages. To appear in Invent. Mat