Title: Nontoric topological string theory and anomalies
1Non-toric topological string theory and anomalies
- Makoto Sakurai,University of Tokyo (Hongo)
- makoto_at_hep-th.phys.s.u-tokyo.ac.jp
- To appear at hep-th
- http//www5f.biglobe.ne.jp/makotosakurai/
2Plan of talk
- 1.Review of toric topological strings and
topological vertex by topological chiral algebra - 2.Toric del Pezzo, anomalies / elliptic genus /
topological modular forms - 3.Non-toric del Pezzo surfaces and birationality
of open Gromov-Witten - 4.Sheaves of quantum automorphic functions
generalizing the Beilinson-Drinfeld 2003 chiral
algebra and Kontsevich-Manin 1994 /
Eguchi-Hori-Xiong - 5.Summary and future direction
31.Topological vertex by topological chiral
algebra(review)
- Eguchi-Hori-Xiong on closed GW, Eguchi-Kanno
Heisenberg representation of W-algebra / free
fermions of BPS open GW or B-/A-model for
Dijkgraaf-Vafa utilizing mirror symmetry (2003-)
of toric noncompact CY3 - We will extend the Heisenberg algebra by the
Beilinson-Drinfeld chiral algebra defined also
for non-Calabi-Yau complex n-folds / stacks - Sakurais interpretation of the q-Schur
function as the q-tensor category symmetry over
Ziv Rans space of finite points on worldsheets
2004 Fall JPS
4Ziv Ran space of factorization algebra
- However, worldsheet X is not necessarily affine,
typically ellpitic curve - Is 0 point fucntion lt1gt constant map?
- In addition,We have theD-module structureon
the moduli space (G-torsor) of 2d Yang-Mills /
Hitchin system or the jet scheme of the target
manifolds
5What was new in my previous work of 2004?
- Derived Tensor categories (Categories as object)
of GKZ (toric) / q-Kniznik-Zamolodchikov equation
rather than operators (noncommutative (Artin)
rings Eguchi-Kanno) are important namely, - 2-Category of ( category, functor between
objects) will be the true category in the DERIVED
Fukaya category definition of topological A-model
6What did I suggest in the previous works of 2005?
- To obtain A-model, we need hetetoric-B-model-A-mod
el duality - Not the A-I-B duality of E.Frenkel
7What was NOT perfect in my work?
- Could not define the topological A model
rigorously just a better definition - the Fukaya category of BPS stable D2 brane of
Calabi-Yau 3-folds does not have D4 branes
(co-isotropic) degrees of freedom
Kapustin-Orlov - We also do not yet understand the Karoubi
completion of A-model / Fano 3-folds of
differential category (in B-model?) in the
topological M-theory of stable 3-forms
(Kovalevs twisted connected sum?)
8Digression What were more technically obstacles?
- Fukaya category (special Lagrangian D-branes,
Maslov index) defined by the free loop space in
manifolds which behaves well in the
equivariant target space but, the Maslov class
is not easily computable without using instanton
calculus of quiver gauge theory Nakajima et.al. - We needed to use the formal loop space of disk
amplitude of B-branes Sakurai 2005 in the
birational mirror singular varieties to define
the path-integral the old work of Kontsevich on
motivic integration of local (hetero) / global
(B-model) loops / arcs - Geometric Langlands duality / Fourier-Mukai
transformation for D-modules this mathematics
will be discussed later in the Chapter 4
9Dubrovins Semisimplicity of Frobenius manifolds
- Manins proof in del Pezzo All genus closed GW
is detemined if we have low genus amplitudes and
Virasoro constraint could be modernized by
open-closed duality written by heterotic A/B
duality - We will have to understand Hirzebruch surface for
nodal curves which could not be in the Batyrev
ring in terms of q-D-modules of singular
worldsheet? to be examined in the future - However, may us restrict to genus 0 (rational
curve) and genus 1 elliptic curves in this talk
102. Toric del Pezzo surfaces and anomalies and
elliptic genus
- Extension of the works of Homological Mirror
Symmetrists Orlov Ueda - Picard groups of ADE current algebra are
important to determine the loop Sato Grassmannian
of local affine Kac-Moody group. Sakurai 2004
Fall - Sakurais non-flag variety by systematically
gluing WZW to determine gerbe F of Witten /
E.Frenkel-Losev-Nekrasov level 1 WZW ansatz can
determine the anomaly term of coordinate change
of beta-gamma half-twisted CFT of local
coordinates / tangent vectors computable
modular forms
11Toric diagrams setup
12Gluing of WZW or flag variety my previous
conjecture
- Sakurai 2004 Fall JPS WZW-flag variety gluing
conjecture by the WZW-Hitchin connection
correnspondence Laszlo was before Wittens
ansatz April - It turned out that it was already conjectured as
gerbes Gorbounov-Malikov-Schechtman 2000 for
elliptic genus but not in the sense of WZW model - It was Nikita A.Nekrasov 2005 Winter that
reviewed this from the viewpoint of explicit
Jacobian matrices g - They Losev-E.Frenkel-Nekrasov will conjecture
in the future work the coefficient as the level 1
WZW model
13Relation between Picard group and gauging of
isometry?
- My observation Picard groups (B-model) are the
universal object (G of loop group), written as
gauging of isometry (A-model) for group manifolds
(gauged WZW quantum groups) - However, dealing with non-WZW type from reductive
groups, and extended Dynkin diagrams
(non-toric) - the gluing of affine Lie algebras from
exceptional divisors - the dimension of the del Pezzo (complex 2) is not
G / Borel (flag) or G / Parabolic subgroup P
14Coordinate transformations for toric del Pezzo
diagram chasing
- We can take any U-V-W-U patches because del Pezzo
has no third sheaf cohomology - We can see non-vanishing gerbes but not the
relation to the exact non-zero elliptic genus
itself
T.Kimura 2003
153.Non-toric compact del Pezzo surfaces and
birationality of open Gromov-Witten
- Genus 0 closed part was done one decade ago by
Kontsevich-Manin J.Bryan-Leung,Pandharipande
et.al. but all genus open extension was not. - We borrowed in the last presentation JPS 2005
Fall the results of Auroux-Katzarkov-OrlovHori
, Walcher et.al. for dualizing A- / B-model. - Rather, we will now try the ADE type current
algebra and the ADE type dual Langlands groups
relatively trivial in the level of affine Hecke
algebra - However, analytically writing the global
automorphic sheaves (multi-valued hyperfunction)
is not easy Ch.4
16Analyticial results of past and my present
- Del Pezzo 9 forpseudo-modular formwhich was
done by the 12 nodal curve counting over
exceptional divisor of by Cremona relation of
Young tableau by J.Bryan-Leung genus 0
1997,Goettsche-Pandharipande 96 - However, this was andifferential method andon
the Fourier-Mukai transform of coherent sheaves
not D modules or S-duality. We need
Beilinson-Drinfeld chiral algebra of automorphic
sheaves!
17Whats different from other math-physicists?
- It is a refinement of the classic of
Cachazo-VafaEguchi-Sakai on the E-string / E
type Seiberg-Witten theory - However, we did not utilize the spectral curve /
meromorphic 2d Hitchin system explicitly
Donagi-Witten - We had a mysterious confusion on the symmetry
enhancement mechanism and real algebraic
geometry (algebraically closed?) in the work of
Vafa - Diaconescu et.al. is on the canonical
non-compact CY3-folds and different from ours
we would prefer Dolgachev surface / Enriques
3-folds Maulik-Pandharipande 2006 (non-toric)
18Review Generalized Homological Mirror symmetry
in derived category of Fano manifolds
- A-model (symplectic) differential graded
category of rational homotopy theory without
torsions, which includes the Floer cohomology as
the Homology theory - Do Leschetz fibration, Morse potential, and
vanishing cycles explain the open-closed duality
in the differential category? - B-model (algebro-geometry) Landau-Ginzburg
Potential as the critical point in phase other
than Calabi-Yau Hori-Vafa Auroux-Katzarkov-Orlo
v (non-toric Fano or ½ K3) from the viewpoints
of B-branes / exceptional sheaves
194.Sheaves of quantum automorphic functions
geometric Langlands program
- Whittaker standard special geometric function for
the differential equation is understood in the
language of automorphic sheaves for the q-KZ
equation Kazhdan,Etingof et.al. conjecture of
my talk - In the non-toric / non Gelfand-Kapranov-Zelevinsky
, non-Laumon-Lafforgues GLn type) - Genus 1 should demand the Beilinson-Bloch-Borel
zeta function / t Hooft regulator B-model (or
the analytic torsion A-model) - Delignes perverse sheaf was important to extend
the Poincare duality to singular variety /
D-modules.
205.Summary and future direction
- Summary
- Beilinson-Drinfeld chiral algebra had its origin
both in the Conformal Field Theory of Vafa-Witten
S-duality by non-toric q-KZ equation and Class
Field Theory of Galois / automorphic
representation sheaves of fundamental groups vs
Picard groups - Unity of math and physics (dedicated to Gelfand
and Gauss) - Genus 1 is also defined but more technically
difficult with higher K-theory - S-duality from Leibnitz algebraic analysis is
defined butexplicit Newtonian differential
analytical expressions are still elusive - Future
- Homological mirror symmetry and chiral algebra
for quintic Calabi-Yau compact 3-folds utilizing
BCOV conjecture - Deriving the melting crystals from Kashiwares
crystal basis - Higher Woodin inaccessible cardinal foundation or
Loeb-Wiener measure rather than just the
Brownian-matrix model CW complex (B-model) of
quantum gravity - Relation between Higher order logic and
n-category theory of functional integration
theory - Brown representability of moduli functor of stacks