1. Z THEORY
Nikita Nekrasov
IHES/ITEP
Nagoya, 9 December 2004

2. Based on the work done in collaboration with:
A.Losev,
A.Marshakov,
D.Maulik,
A.Okounkov,
H.Ooguri,
R.Pandharipande,
C.Vafa

4. (non-perturbative) topological strings topological gauge theory
Z THEORY
Interplay between
(non-perturbative) topological strings
and
topological gauge theory
Other names:
mathematical M-theory,
topological M-theory,
m/f-theory

5. 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,…

6. PERTURBATIVE vs NONPERTURBATIVE
Usual string expansion:
perturbation theory in the string coupling

7. 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

8. A model

9. A model

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

11. 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

12. B STORY Genus zero part =
classical theory of variations of Hodge structure (for Calabi-Yau’s)
Generalizations: Saito’s theory of primitive form, Oscillating integrals – singularity theory; noncommutative geometry;gerbes.

13. 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)

14. 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

15. 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*

16. 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

17. HITCHIN’S GRAVITY IN 6d BY LAGRANGIAN FOR F
Replace Kodaira-Spencer Lagrangian which describes deformations of ( X, W )
BY LAGRANGIAN FOR F

18. HITCHIN’S GRAVITY IN 6d

19. HITCHIN’S GRAVITY IN 6d

20. NAÏVE EXPECTATION Full string partition function =
Perturbative disconnected partition function X
D-brane partition function
Z (full) = Z(closed) X Z (open) ???

21. D-brane partition function
Sum over (all?) D-brane charges
Integrate (what?) over the moduli space (?) of D-branes with these charges ?
? ? ? ?
? ? ?

22. Particular case of B-model D-brane counting problem
Donaldson-Thomas theory
Counting ideal sheaves:
torsion free sheaves of rank one with trivial determinant

23. LOCALIZATION IN THE TORIC CASE
Sum over torus-invariant ideals: melting crystals

24. Monomial ideals in 2d

25. DUALITIES IN TOPOLOGICAL STRINGS
T-duality (mirror symmetry)
S-duality
INSPIRED BY
THE PHYSICAL SUPERSTRING DUALITIES
HINTS FOR THE EXISTENCE OF
HIGHER DIMENSIONAL THEORY

26. T-DUALITY CLOSED/OPEN TYPE A TOPOLOGICAL STRING ON =
CLOSED/OPEN TYPE B TOPOLOGICAL
STRING ON

27. T-DUALITY Complex structure moduli of = Complexified Kahler moduli of
AND VICE VERSA

28. S-DUALITY
OPEN + CLOSED TYPE A
STRING ON X =
OPEN + CLOSED TYPE B

29. Choice of the lattice L in the K(X):
GW – DT correspondence
Choice of the lattice L in the K(X):
Ch(L)

30. 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

31. GW – DT correspondence: degree zero
Partition function =
sum over finite codimension monomial ideals in C[x,y,z] =
sum over 3d partitions =
a power of MacMahon function:
COINCIDES WITH DT EXPRESSION AS AN ASYMPTOTIC SERIES

32. 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)

33. K-THEORETIC GENERALIZATION
ON TO SEVEN DIMENSIONS
DT THEORY HAS A NATURAL
K-THEORETIC GENERALIZATION
CORRESPONDS TO THE GAUGE THEORY ON

34. 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

35. GEOMETRIC ENGINEERING OF 4d GAUGE THEORIES
DW – GW correspondence
Gauge group G corresponds to GW theory on
GEOMETRIC ENGINEERING OF 4d GAUGE THEORIES

36. INSTANTON partition function

37. DW – GW correspondence

38. 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 N=1 (Hilbert scheme of points on the plane)
Maps to GW theory of the projective line

39. RANDOM PARTITIONS Fixed point formula for Z, for G=SU(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

40. WHAT IS Z THEORY? TO BE CONTINUED.........
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;
Hitchin’s theory of gravity in 7d coupled
to the theory of associative cycles;
? ? ?? ?
TO BE CONTINUED

41. MAKE THEM SPECIAL HOLONOMY SPACETIMES…..
FOR BETTER TIMES…..
MAKE THEM SPECIAL HOLONOMY SPACETIMES…..