Abelian Lagrangian Algebraic Geometry (ALAG) is a geometric framework for understanding an interaction between the symplectic and algebraic geometries in the context of geometric quantization. It comes back to the last works of Andrey Tyurin (see [GT]–[T4]). I will give a review of the main concepts of this theory, recent developments, and open questions, focusing mostly on nonabelian theta-functions and generalized Knizhnik–Zamolodchikov equation.
Namely, let 2n-dimensional symplectic manifold (M, ) admit simultaneously a structure of real completely integrable system, i.e. lagrangian toric fibration
: M (1)
over a Delzant polytope ⊂Rn, and a structure of polarized Kähler manifold M1, where I is an integrable complex structure on TM. Assume for simplicity that there is a hermitian line bundle L on M that becomes holomorphic and ample on M1 and has c1(L)∈ H2 (M,Z) rationally proportional to the canonical class K1 of M1 and to the class [] provided by the symplectic structure.
For example, M can be a projective toric variety, or an algebraic abelian variety, or a moduli space of holomorphic vector bundles on a Riemann surface of genus g, e.t.c.
The real polarization (1) produces n-dimensional family of lagrangian cycles on M. ALAG explains that one should expect in this family a finite number of Bohr–Sommerfeld cycles (i.e. those S for which L|s has global covariantly constant trivializing section). This holds in all examples listed above.
Another important construction of ALAG (called BPU-mapping) attaches to each Bohr–Sommerfeld cycle a holomorphic section of L on M1 defined up to multiplication by a non-zero complex constant. In all examples above the sections coming from Bohr–Sommerfeld fibers of (1) form a basis of H0(M, L).
When the complex structure I varies along the moduli space M of polarized Kähler structures on M, the spaces H0(M, L) fill a holomorphic vector bundle M over M . Each real polarization (1) provides projective bundle ℙ(M ) with canonical flat connection whose horizontal constant sections are those coming from Bohr-Sommerfeld fibers of (1). This construction generalizes Knizhnik-Zamolodchikov equation and (usual) theta-functions, which appear when 𝑀 turns to Jacobian of an algebraic curve (i.e. to the moduli space of topologically trivial holomorphic line bundles on a Riemann surface).
Different real polarizations (1) of a fixed symplectic manifold M often form an interesting combinatorial structure. For example, when M is the moduli space of topologically trivial holomorphic vector bundles of rank 2 on a Riemann surface ∑ of genus g ≥ 2, then essentially different lagrangian fibrations (1) stay in bijection with «pants decompositions», i.e. scissions of ∑into pants by 3g − 3 simple pairwise non-isotopic loops. These decompositions form a famous graph whose edges are elementary regluings of pants. Each polarized Kähler structure I on M equips the edges of this graph with transition matrices between Bohr–Sommerfeld bases in H0(M, L) coming from polarizations (1) staying at the vertexes. I my talk I will try to explain the details of this picture as far as the time allows.
References
[GT] A. L. Gorodentsev, A. N. Tyurin. Abelian Lagrangian Algebraic Geometry. Izvestia RAN: Ser. Mat. 65:3 (2001) p. 15–50.
[T1] A. N. Tyurin. On Bohr-Sommerfeld bases. Univ. of Larwick preprint: Sept. 1999 / Publication 216.
[T2] A. N. Tyurin. Geometric quantization and mirror symmetry. arXiv:math.AG/9902027[math.AG].
[T3] A. N. Tyurin. Complexification of Bohr–Sommerfeld conditions. arXiv:math.AG/9909094[math.AG].
[T4] A. N. Tyurin. Qantization, Classical and Qantum Field Theory and Theta-Functions. arXiv:math/0210466[math.AG].