Makoto Sakurai,

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Beilinson-Drinfeld chiral algebras for del Pezzo
surfaces

• Makoto Sakurai,
• Department of Physics,
• The University of Tokyo

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Why del Pezzo for chiral algebras?

• It describes Fano varieties which are recently
  well-understood by the Homological Mirror
  Symmetry with superpotentials of
  Auroux-Katzarkov-Orlov.
• The Fourier-Mukai transformation between DX
  modules and OX modules, which were studied by
  Edward Frenkel and Gaitsgory, are little
  understood in explicit examples.
• We are interested in the case of non-toric
  B-model (e.g. higher blowup del Pezzo surfaces),
  which cannot be described by the topological
  vertex of Dijkgraaf-Vafa and Okounkov et.al.
• Therefore we will use the coordinate free
  formalism of either the topological M-theory by
  generalized complex manifolds of Hitchin or the
  formal loop / arc space methods of
  Kapranov-Vasserot.

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Table of contents

• Review on the Operator-Product-Expansion of
  chiral algebras by generalized complex structures
  and Courant brackets.
• Detailed computations of 2nd Chern character in
  the case of 1,2,3 point blowup and comparison
  between Nekrasov and Witten.
• Conjectural extensions to (open) Gromov-Witten
  invariants (derived Fukaya category) and
  non-toric arc space.

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1.Beilinson-Drinfeld chiral algebras definition
and convention
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Useful Operator-Product-Expansions
For smooth complex manifolds X, and its open
neighborhood U
Inner product and Lie bracket
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and, its generalization by topological M-theory
of generalized complex manifolds
Generalized complex structures with the metric g
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Generalization of OPE preserving chiral
differential operators of Malikov-Schechtman

• Nekrasovs ansatz of coordinate changes in terms
  of the above ß? CFT taking ?s as the target
  space affine coordinates.

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OPE Preserving equations with respect to the
generalized complex structure
With the generalized complex structures
It is easy to compute the symmetric part sab, and
the antisymmetric part µ is obtained by the
Maurer-Cartan equation
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2.Gerbes of chiral differential operators

• If we have the 3 affine patches U1, U2, U3, then
  we can have the chain of coordinate changes
  U1?U2?U3?U1

This formula can be obtained from a careful
treatment of the cross terms and normal
orderings, which cancel the singular parts.
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Comparison of Nekarsovs Ansatz with Wittens CP2

• Let us take Ui(i1,2,3 mod 3) are the standard
  affine coordinate defined by the projective
  coordinates ?i?0. And v and w are the
  inhomogeneous coordinates v ?1/?0,w ?2/?0
• Take the ansatz of Witten, which assume that the
  Pontrjagin anomaly comes from the generator F
  dv dw / v w ?ij fij d?i? d?j

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Wittens computation in terms of
Malikov-Schechtmans OPE preserving Ansatz
In agreement with Nekrasov, only the symmetric
part s is alive. But be aware that the summation
of ss is not vanishing and there is still an
extra term comes from the Taylor expansion of OPE
in normal ordered g ß
In the step of 1 coordinate change
However, the computation is not so
straightforward. It is because there is a cross
term between (old) ß ? in new ß and the
cancellation of singular terms in the normal
ordering terms is nontrivial.
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1st,2nd, and 3rd steps of Wittens calculation
more detailed demonstration
More on 1st step compared with Nekrasov Bij part
The result of 2nd step in short
The computation of 3rd step in detail
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Direct comparison of Nekrasovs anomaly 2-form
and Wittens computation
There is a good agreement between Wittens OPE
preserving computation or the pole anallysis of
the generator AND the anomaly 2-form of Nerakov,
which was derived from the generalized complex
structure ansatz
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In the case of 1,2,3 point blowups of CP2,we have
4,5,6 affine coordinates
Toric diagrams for lower blowups of CP2 and
their affine coordinate changes
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2nd Chern character is 3(n-1) (n0,1,2,3 blowups)
in consistence with Grothendieck-Hirzebruch
Riemann-Roch theorem
For example, n1 The obstruction vanishes
Because of the two 1 and two -1 contribution to
d log x ? d log y cancel
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3.Future works for the derived Fukaya category
and geometric Langlands

• The generalized complex manifolds are useful in a
  coordinate free manner, but it still depends on
  the choise of Cech coverings. How can we compute
  in a coordinate free manner?
• We prefer the formal loop space method of
  Kapranov-Vasserot. For example, we can conclude
  that the genus 0 small quantum cohomology of
  toric Fano varieties.In terms of A 2nd
  homologyclass and A its dominant cone
• In the work of J.Bryan-Leung (1997), we get the
  pseudo-modular form of generatingfunction for
  closedGromov-Witten invariantsin ½ K3 (9 points
  blowups)by curve countingmethods and
  combinatorics.We hope to prove the modularityof
  non-toric compact del Pezzo surfaces