Nontoric topological string theory and anomalies

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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/

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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

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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

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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

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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

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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

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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?)

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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

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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

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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

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Toric diagrams setup
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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

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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

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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
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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 DE type current
  algebra and the CE 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

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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!

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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)

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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

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4.Sheaves of quantum automorphic functions
geometric Langlands program

• Whittaker standard special gemetric 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 Beilison-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.

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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