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Andrei Tyurin

Publications and source records attributed to Andrei Tyurin.

10 recordsLinked to original sources
arXiv2001

Delzant models of moduli spaces

For every genus g, we construct a smooth, complete, rational polarized algebraic variety DM_g together with a normal crossing divisor D = sum D_i, such that for every moduli space M_C(2,0) of semistable topologically trivial vector bundles of rank 2 on an algebraic curve C of genus g there exists a holomorphic isomorphism f: M_C(2,0) minus K_2 -> DM_g minus D, where K_2 is the Kummer variety of the Jacobian of C, sending the polarization of DM_g to the theta divisor of the moduli space. This isomorphism induces isomorphisms of the spaces of conformal blocks.

math.AG↗
arXiv2001

Lattice gauge theories and the Florentino conjecture

We study the relation between the space of representation classes of the fundamental group of a Riemann surface and gauge theory on trivalent graphs. We construct a partial gauge fixing in the latter gauge theory. As an application we get a proof of a conjecture of Florentino.

math.DG↗
arXiv2000

Three mathematical faces of SU(2) - spin networks

Spin networks are at the core of quantum gravity. Our aim is to plug the mathematical community at large into the procedures turn to create a finite quantum theory of general relativity. For this, because of the different cultural backgraund, we would like to change the tack: to relate discrete (combinatorial) objects to the standard "contineous" geometry.

math.DG↗
arXiv1999

Complexification of Bohr-Sommerfeld conditions

The complex version of Bohr-Sommerfeld conditions is proposed. The BPU-construction (see [D.Borthwick, T. Paul and A. Uribe, Legendrian distributions with applications to the non-vanishing of Poincaré series of large weight, Invent. math, 122 (1995), 359-402, preprint hep-th/9406036] or [Andrei Tyurin, On Bohr-Sommerfeld bases, math.AG/9909084]) is generalized to this complexification. The new feature of this generalization is a spectral curve. The geometry of such curves is investigated.

math.AG↗
arXiv1999

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 Poincaré 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]).

math.AG↗
arXiv1999

Quantization and ``theta functions''

Geometric Quantization links holomorphic geometry with real geometry, a relation that is a prototype for the modern development of mirror symmetry. We show how to use this treatment to construct a special basis in every space of conformal blocks. This is a direct generalization of the basis of theta functions with characteristics in every complete linear system on an Abelian variety (see Mumford's "Tata lectures on theta" cite(Mumford)). The same construction generalizes the classical theory of theta functions to vector bundles of higher rank on Abelian varieties and K3 surfaces. We also discuss the geometry behind these constructions.

math.AG↗
arXiv1999

Geometric quantization and mirror symmetry

After the appearance of my preprint [T3] (Special Lagrangian geometry and slightly deformed algebraic geometry (spLag and sdAG), Warwick preprint 22/1998, alg-geom/9806006, 54 pp.). I received an e-mail from Cumrun Vafa, who recognized that the subject is closely related to that of his preprint [V] (Extending mirror conjecture to Calabi-Yau with bundles, hep-th/9804131, 7 pp.). This text started out as an e-mail ``reply'' to his letter. All the constructions we propose have well known ``spectral curve'' prototypes (see for example Friedman and other [FMW], Bershadsky and other [BJPS] and a number of others). Roughly speaking, our constructions are the spectral curve construction plus the phase geometry described in [T3]. So this text should really come before [T3], as motivation for the development of the geometry of the phase map in [T3].

math.AG↗
arXiv1998

Special Langrangian geometry and slightly deformed algebraic geometry (spLag and sdAG)

The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries are parallel to classical gauge theories and their complexifications. This article explains this parallelism, so that the appearance of invariants of new type in complexified gauge theory (see Donaldson--Thomas [D-T] and Thomas [T]) can be accompanied by analogous invariants in the theory of special Lagrangian cycles, for which the development is at present much more modest than in gauge theory. We discuss related geometric constructions, arising from mirror symmetry and symplectic geometry.

math.AG↗
arXiv1994

Spin canonical invariants of 4-manifolds and algebraic surfaces

The paper is a colloquial-style discussion of invariants of algebraic surfaces analogous to the Donaldson polynomials, arising from moduli spaces of ``jumping'' Yang--Mills instantons, or moduli spaces of jumping vector bundles. The invariants have the following applications: (1) to the Van de Ven conjecture that the Kodaira dimension is a diffeomorphism invariant; (2) to proving that algebraic surfaces with $p_g > 0$ have a proper sublattice of $H^2(X,\Z)$ invariant under diffeomorphism; (3) to proving the same result as (2) for surfaces with $p_g = 0$, in particular the Barlow surface.

alg-geom↗