Title: Duality between open GromovWitten invariants and BeilinsonDrinfeld chiral algebras
1Duality between open Gromov-Witten invariants and
Beilinson-Drinfeld chiral algebras
Makoto Sakurai, Univesity of Tokyo, School of
Science, Department of Physics, Eguchi
Lab(makoto_at_hep-th.phys.s.u-tokyo.ac.jp) http//www
5f.biglobe.ne.jp/makotosakurai/
2Table of contents
- Motivation and backgrounds
- General theory
- Explicit calculation chiral algebras for del
Pezzo surfaces - Extension of topological M-theory G2 holonomy
construction - Conclusion and future direction
3a)Motivation 1String / Gauge duality
- Better understanding on the Chern-Simons /
Seiberg-Witten duality - Worldsheet instanton for (0,2) heterotic sigma
model / Beilinson-Drinfeld chiral algebra
4Motivation 2Infinite analysis and String theory
- Folklore Infinite Heisenberg algebra is the
representation of loop groups LG Hitchin-Segal /
Segal-Pressley 80s Counterpart of anomaly
cancellation / elliptic genus W
- Study on the loop groups inspired Malliavin
stochastic analysis - Loop space Morse-Floer in the Atiyah-Bott-Witten
localization theorem of A-model (symplectic)
What is the infinite algebraic geometry?
5Motivation 3 Geometry and Arithmetic
- Quantum integrability of topological vertex vs
Virasoro conjecture - Loop spaces and motives K-theory of
infinite-dimensional sheaves D not clear its
physical meaning - Mirror symmetry, quantum geometric Langlands
(when the target is a gaug e gruop), and S-duality
6Why Beilinson-Drinfeld chiral algebras?
- Short history of Beilinson-Drinfeld (80s-)
- Malikov-SchechtmanMS et.al. coordinate
dependent, mimicking chiral rings - Kapranov-Vaserot AK A)Coordinate independent
loops / motives B)Quantum cohomology for toric
Fanos - Beilinson-Drinfeld BD Sheaf-theoretical beyond
Cech,tensor categories, and mirror symmetry of
D-modules / geometric Langlands Still difficult
to do things from first principle - Representation theory of affine Kac-Moodys not
essential for SCFT loop spaces are intrinsic
define topological observables - Whats new in my work S Sakruai
- Unify stringy topological invariants by infinite
algebro-geometry - Chiral algebras for higher non-toric del Pezzo
surfaces
7b)General theoryWarmup by G/B and definitions
and reviews MS
- G/B by loop groups LG. HQ quantum cohomology
ring - , calculation by affine covers and loop
space Exceptional locus by the toric action
,where ?0M
is the loop space that respects the complex
structure
8Definition of Hitchin system / 2d Yang-Mills
theory (after Hitchin)
- Let P be a principal G-bundle over a Riemann
surface S, which satisfies self-duality
equations - It is also descibed as the representation of
fundamental group p1(S) in the gauge group G - Affine curve S is the WZW model (flag manifolds)
L
9Disk amplitude and 2 dim YM / SUSY Poisson
sigma-model AKMSSS2 My interpretation
- M toric, L0M loop spaces as the boundary of
stable / holomorphic maps from D2 to M - It should be the supersymmetric sigma-model with
B-field / gerbes on Riemann surface, which
produces the q-deformation and the
infinite-dimensional sheaves - M not necessarily toric, L0M demands refined
motivic integrationD - 2D YM q-deformed of free fermion is the section
at affine coordinate / germ or curve (Laurant
expansion at a point)
10c)Explicit calculation of sheaves of chiral
primaries Sakurai new work
- Del Pezzo surfaces (k0,...,8)
- Toric del Pezzo for
- Non-toric for k gt 3
- Degree k surface inwith canonical sheaf
for
- Reproduces classics Eguchi-Hori-Xiong, but
different principles of loop spaces and
localization of loops (not virtual localization
of A-model) - Pull-buck of the homology classes of target space
M - Not previously done by mathematicians, bacause
their mathods were only in the toric Fano cases
11Future works in this directions
- Better definition of all-genus Gromov-Witten
invariants / the topological vertex DDDHP, but
we didnt yet derive from the first principle of
motives - Open-closed duality should explain why the
K-theory of Drinfeld D reproduces the chiral de
Rham complex (closed Gromov-Witten)SS2 - Algebro-geometric / categorical proof of
geometric transitions without using analytic
continuation SDDDHP - We couldnt calculate from coordinate-free (non
Cech) sheaf cohomology of Drinfeld, which
requires more algebro-geometry
12d)Extension of topological M-theoryS
- Towards the missing link between 2d Hitchin
systems ((0,2) heterotic) and the 7d Hitchin
functional (topological M-theory) From
Hitchin to Hitchin S2 Analogue of
mysterious duality? - The inconsistency betweenJoyces 7d G2 holonomy
manifoldsand the 7d SU(3) holonomy solutionsof
Hitchin flow equation - Kovalevs construction of G2 holonomy manifolds
from 2 Fano 3-folds should be useful it could be
the completion of the CY3 by the Landau-Ginzburg
phase S
13Adding Fano 3-folds to M-theory
- Twisted connected sum Kovalev of Fano 3-folds
Mukai produces strictly G2 holonomy
manifolds with asymptotically CY3 cylinder
boundaries which could be the initial conditions
for Hitchin flow equation, which should be
modified - It is also preferable from the LG / Fano B-model
duality Orlov 2005
14e)Conclusion and future direction
- Better understanding on the mathematical
principles of topological strings / M-theory
quantum Hitchin systems - Beautiful theory of infinite dimensional geometry
reproduces our past results However, - Explicit calculation was difficult to perform
more general target spaces are awaiting for our
challenges to ease its difficulty
15References
- ADE A.Adams,J.Distler,M.ErnebjergTopological
Heterotic Rings,hep-th/0506263 - AK S.Arkhipov and M.Kapranov Toric arc
schemes and quantum cohomology of toric
varieties, math.AG/0410054 - BD A.Beilinson and V.Drinfeld Chiral
algebras, AMS (2004) - D V.Drinfeld Infinite-dimensional vector
bundles in algebraic geometry (an introduction)
, math.AG/0309155 - DDDHPDiaconescu,Dijkgraaf,Donagi,Hofman,Pantev
Geometric transitions and integrable systems,
hep-th/0506196 - F Edward Frenkel Mirror symmetry in two
steps A-I-B, hep-th/0505131
16References 2
- L Y.Laszlo Hitchins and WZW connections are
the same, Journal of Differential Geometry, 49
(1998) 547-576 - MS F.Malikov, V.Schechtman "Deformations of
vertex algebras, quantum cohomology of toric
varieties, and elliptic genus", CMP 234 (2003),
no. 1, 77-100 - S Makoto Sakurai Moduli space of topological
M-theory and topological chiral algebras, to
appear - S2 Makoto Sakurai Presentations at the Japan
Physical Society, Sep 2004 Topological vertex
and geometric transition via Beilinson-Drinfeld
chiral algebras, Mar 2005 Mathematical
principles of topological strings / M-theory and
Hitchin systems - W Edward Witten Two-Dimensional Models With
(0,2) Supersymmety Perturbative Aspects,
hep-th/0504078