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Nontoric topological string theory and anomalies

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JPS 2006, Ehime University, 29 March. 1. Non-toric topological string theory and anomalies ... Unity of math and physics (dedicated to Gelfand and Gauss) ... – PowerPoint PPT presentation

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Title: Nontoric topological string theory and anomalies


1
Non-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/

2
Plan 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

3
1.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

4
Ziv 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

5
What 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

6
What 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

7
What 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?)

8
Digression 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

9
Dubrovins 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

10
2. 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

11
Toric diagrams setup
12
Gluing 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

13
Relation 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

14
Coordinate 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
15
3.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

16
Analyticial 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!

17
Whats 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)

18
Review 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

19
4.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.

20
5.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
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