Z THEORY

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Z THEORY

• Nikita Nekrasov
• IHES/ITEP
• Nagoya, 9 December 2004

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Based on the work done in collaboration with

• A.Losev,
• A.Marshakov,
• D.Maulik,
• A.Okounkov,
• H.Ooguri,
• R.Pandharipande,
• C.Vafa

2002-2004
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Z THEORY

• Interplay between
• (non-perturbative) topological strings
• and
• topological gauge theory
• Other names
• mathematical M-theory,
• topological M-theory,
• m/f-theory

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TOPOLOGICAL STRINGS

• Special amplitudes in Type II superstring
  compactifications on Calabi-Yau threefolds
• Simplified string theories,
• interesting on their own
• Mathematically better understood
• Come in several variants
• A, B, (C), open, closed,

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PERTURBATIVE vs NONPERTURBATIVE

• Usual string expansion
• perturbation theory in the string coupling

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NONPERTURBATIVE EFFECTS

• In field theory from space-time Lagrangian
• In string theory need something else
• Known sources of nonpert effects
• D-branes and NS-branes
• This lecture will not mention NS branes, except
  for fundamental strings

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A model
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A model
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D-branes in A-model
!
L
Sum over Lagrangian submanifolds in X, whose
homology classes belong to a Lagrangian
sublattice in the middle dim homology
Subtleties in integration over the moduli of
Lagrangain submanifolds. In the simplest cases
reduces to the study of Chern-Simons gauge
theory on L
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ALL GENUS A STRING

• Theory of Kahler gravity
• Only a few terms in the large volume expansion
  are known
• For toric varieties one can write down a
  functional which will reproduce localization
  diagrams could be a useful hint

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B STORY

• Genus zero part
• classical theory of variations of Hodge
  structure (for Calabi-Yaus)
• Generalizations Saitos theory of primitive
  form, Oscillating integrals singularity theory
  noncommutative geometrygerbes.

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D-branes in B model

• Derived category of the category of coherent
  sheaves
• Main examples
• holomorphic bundles
• ideal sheaves of curves and points
• D-brane charge the element of K(X).
• Chern character in cohomology H(X)

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All genus B closed string
KODAIRA-SPENCER THEORY OF GRAVITY
CUBIC FIELD THEORY () NON-LOCAL (mildly /-
) BACKGROUND DEPENDENT (-) NO IDEA ABOUT THE
NON-PERTURBATIVE COMPLETION
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B open string field theory

• HOLOMORPHIC CHERN-SIMONS

W holomorphic (3,0) form on the
Calabi-Yau X
THIS ACTION IS NEVER GAUGE INVARIANT NEED TO
COUPLE B TO B
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B open string field theory

• CHERN-SIMONS

F closed 3 form on the Calabi-Yau X
THIS ACTION IS GAUGE INVARIANT
GENERALIZE TO SUPERCONNECTIONS TO GET THE OBJECTS
IN DERIVED CATEGORY
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HITCHINS GRAVITY IN 6d

• Replace Kodaira-Spencer Lagrangian which
  describes deformations of ( X, W )

BY LAGRANGIAN FOR F
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HITCHINS GRAVITY IN 6d
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HITCHINS GRAVITY IN 6d
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NAÏVE EXPECTATION

• Full string partition function
• Perturbative disconnected partition function
• X
• D-brane partition function

Z (full) Z(closed) X Z (open) ???
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D-brane partition function

• Sum over (all?) D-brane charges
• Integrate (what?) over the moduli space (?) of
  D-branes with these charges
• ?
• ? ? ? ?
• ? ?
  ?

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Particular case of B-model D-brane counting
problem

• Donaldson-Thomas theory

Counting ideal sheaves torsion free sheaves of
rank one with trivial determinant
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LOCALIZATION IN THE TORIC CASE

• Sum over torus-invariant ideals melting crystals

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Monomial ideals in 2d
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DUALITIES IN TOPOLOGICAL STRINGS

• T-duality (mirror symmetry)
• S-duality

INSPIRED BY THE PHYSICAL SUPERSTRING DUALITIES
HINTS FOR THE EXISTENCE OF HIGHER DIMENSIONAL
THEORY
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T-DUALITY

• CLOSED/OPEN TYPE A TOPOLOGICAL STRING ON
  
• CLOSED/OPEN TYPE B TOPOLOGICAL
• STRING ON

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T-DUALITY

• Complex structure moduli of
• Complexified Kahler moduli of

AND VICE VERSA
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S-DUALITY

• OPEN CLOSED TYPE A
• STRING ON X
• OPEN CLOSED TYPE B
• STRING ON X

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GW DT correspondence

• Choice of the lattice L in the K(X)
• Ch(L)

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Describing curves using their equations

• ENUMERATIVE PROBLEM
• Virtual fundamental cycle in the Hilbert scheme
  of curves and points
• For CY threefold expected dim 0
• Generating function of integrals of 1

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GW DT correspondence degree zero

• Partition function
• sum over finite codimension monomial ideals in
  Cx,y,z
• sum over 3d partitions
• a power of MacMahon function

COINCIDES WITH DT EXPRESSION AS AN ASYMPTOTIC
SERIES
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QUANTUM FOAM

• The Donaldson-Thomas partition function
• can be interpreted as the partition function
• of Kahler gravity theory
• Important lesson metric only exists in the
  asymptotic expansion in string coupling constant.
  
• In the DT expansion
• ideal sheaves (gauge theory)

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ON TO SEVEN DIMENSIONS

• DT THEORY HAS A NATURAL
• K-THEORETIC GENERALIZATION
• CORRESPONDS TO THE GAUGE THEORY ON

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DONALDSON-WITTEN

• FOUR DIMENSIONAL GAUGE THEORY
• Gauge group G (A, B, C, D, E, F, G - type)
• Z - INSTANTON PARTITION FUNCTION
• GEOMETRY EMERGING FROM GAUGE THEORY (DIFFERENT
  FROM AdS/CFT) Seiberg-Witten curves, as limit
  shapes

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DW GW correspondence

• Gauge group G corresponds to GW theory on

GEOMETRIC ENGINEERING OF 4d GAUGE THEORIES
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INSTANTON partition function
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DW GW correspondence
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DW GW correspondence

• In the G SU(N) case the instanton partition
  function can be evaluated explicitly (random 2d
  partitions)
• Admits a generalization (higher Casimirs Chern
  classes of the universal bundle)
• The generalization is non-trivial for N1
  (Hilbert scheme of points on the plane)
• Maps to GW theory of the projective line

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RANDOM PARTITIONS

• Fixed point formula for Z, for GSU(N)
• The sum over N-tuples of partitions
• The sum has a saddle point limit shape
• It gives a geometric object Seiberg-Witten
  curve the mirror to

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WHAT IS Z THEORY?

• Dualities Unification of t and s moduli
• (complex and Kahler) suggest a
• theory of closed 3-form in 7-dimensions, or
  some chiral theory in 8d
• Candidates on the market
• 3-form Chern-Simons in 7d
• coupled to topological gauge theory
• Hitchins theory of gravity in 7d coupled
• to the theory of associative cycles
• ? ? ?? ?

TO BE CONTINUED.........
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FOR BETTER TIMES..

• MAKE THEM SPECIAL HOLONOMY SPACETIMES..