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Z THEORY

Z THEORY. Nikita Nekrasov IHES/ITEP Nagoya, 9 December 2004. Based on the work done in collaboration with:. A.Losev, A.Marshakov, D.Maulik, A.Okounkov, H.Ooguri, R.Pandharipande, C.Vafa. 2002-2004. Z THEORY. Interplay between (non-perturbative) topological strings and

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Z THEORY

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  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 2002-2004

  3. Z THEORY Interplay between (non-perturbative) topological strings and topological gauge theory Other names: mathematicalM-theory, topologicalM-theory, m/f-theory

  4. 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,…

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

  6. 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

  7. A model

  8. A model

  9. 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

  10. 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

  11. 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.

  12. 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)

  13. 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

  14. 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*

  15. 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

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

  17. HITCHIN’S GRAVITY IN 6d

  18. HITCHIN’S GRAVITY IN 6d

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

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

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

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

  23. Monomial ideals in 2d

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

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

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

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

  28. GW – DT correspondence Choice of the latticeLin the K(X): Ch(L)

  29. Describing curves using their equations ENUMERATIVE PROBLEM • Virtual fundamental cycle in the Hilbert scheme of curves and points • For CY threefold: expecteddim = 0 • Generating function of integrals of 1

  30. 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

  31. 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)

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

  33. 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

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

  35. INSTANTON partition function

  36. DW – GW correspondence

  37. 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

  38. 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

  39. 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; Hitchin’s theory of gravity in 7d coupled to the theory of associative cycles; ? ? ?? ? TO BE CONTINUED.........

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

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