Algebraic Geometry and Topological Strings Seminar 

I will discuss joint work with Jim Bryan in which we show how to modify the definition of the small quantum cohomology of an orbifold to take into account divisorial twisted sectors. This quantum cohomology is conjecturally very closely related to the quantum cohomology of any crepant resolution of singularities. In particular, we can directly verify this conjecture in a strong form in the example where the orbifold is the n'th symmetric product of the complex plane, with crepant resolution given by the Hilbert scheme of points.

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REFERENCES:

J. Bryan, R. Pandharipande, The local Gromov-Witten theory of curves, math.AG/0411037.
A. Okounkov, R. Pandharipande, The quantum cohomology of the Hilbert scheme of points in the plane, math.AG/0411210.

We explain that certain Gopakumar-Vafa invariants (and, conjecturally, hence Gromow-Witten invariants) for Calabi-Yau manifolds that admit a K3 fibration can be collected in a generating function. This function is in general an automorphic form determined by the fibration. In the class of K3 fibrations in toric varieties in which the Picard lattice of the fiber has rank one, we show how this automorphic form can be determined from the topology of the fibration.

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REFERENCES:

R. Gopakumar, C. Vafa, M-theory and topological strings,II, hep-th/9812127.
S. Katz, A. Klemm, C. Vafa, M-Theory, Topological Strings and Spinning Black Holes, hep-th/9910181.
J. Harvey, G. Moore, Algebras, BPS States, and Strings, hep-th/9510182.
M. Marino, G. Moore, Counting higher genus curves in a Calabi-Yau manifold, hep-th/9808131.
A. Klemm, M. Kreuzer, E. Riegler, E. Scheidegger, Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections, hep-th/0410018.
E. Scheidegger,Topological Strings on K3 fibrations, I, to appear.

We present Golyshev's modularity conjecture that states that the counting equations for Fano threefolds with Picard group $Z$ (which codes their Gromov-Witten invariants) are modular. To check it we find Gromov-Witten infariants of them. For this we generalise Givental's Theorem (in the Fano case) that gives us Gromov-Witten invariants for complete intersections in toric varieties with non-negative canonical bundle.
REFERENCES:

• V. Golyshev, The geometricity problem and modularity of some Riemann-Roch variations, Dokl. Akad. Nauk 386 (2002) 583-588.
• V. Przyjalkowski, Quantum cohomology of smooth complete intersections in weighted projective spaces and singular toric varieties, math.AG/0507232.
• A. Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), 141-175, alg-geom/9701016

In this lecture I will give an overview of ``String topology''. This is a purely topological theory first introduced by Chas and Sullivan, that has developed into a vast topological field theory of structure on the homology of loop spaces of manifolds, and in spaces of paths with boundary values in D-branes. I will describe a Morse theoretic viewpoint of string topology. This involves representing moduli space of Riemann surfaces by a category of ribbon graphs. Using this together with analytic work of Salaman and Weber, we relate string topology operations on the loop space LM, with the Gromov Witten theory of the cotangent bundle T*M.

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REFERENCES:

M. Chas and D. Sullivan, String Topology, math.GT/9911159. To appear in Ann. of Math.
R.L. Cohen, Morse theory, graphs, and string topology, math.GT/0411272. To appear in Proc. SMS/NATO Adv. study inst. on Morse theoretic methods in nonlinear analysis and symplectic topology, Kluwer press, 2005.
R.L. Cohen and V. Godin, A polarized view of string topology, Topology, Geometry, and Quantum Field Theory, London Math. Soc. Lecture Notes, vol. 308 (2004) 127-154.
R.L. Cohen and J.D.S. Jones, A homotopy theoretic realization of string topology, Math. Annalen, 324 (2002) 773-798.
R.L. Cohen and A. Voronov, Notes on String Topology, math.GT/0503625. To appear in CRM Lecture Notes from summer school on string topology and Hochschild homology, Almeria Spain, 2005.
D. Sullivan, Open and closed string field theory interpreted in classical algebraic topology, Topology, geometry and quantum field theory, London Math. Soc. Lecture Notes, 308, Cambridge Univ. Press, Cambridge, 2004, math.QA/0302332, p. 344-357.

We start with a brief introduction into knot homology theories and categorification of polynomial knot invariants. Of particular interest are homology theories of Ozsvath-Szabo-Rasmussen and Khovanov-Rozansky which provide a homological lift of the Alexander polynomial and the quantum sl(N) invariant, respectively. Motivated by the ideas from physics, we then present a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. We also describe the geometric meaning of the new knot invariants in terms of the enumerative geometry of Riemann surfaces with boundaries in a certain Calabi-Yau three-fold.

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REFERENCES:

S. Gukov, A. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, hep-th/0412243.
N.M. Dunfield, S. Gukov and J. Rasmussen, The Superpolynomial for Knot Homologies, math.gt/0505662.

Finding relations in the tautological rings of the moduli spaces of stable curves has been an important question. In this talk, I will propose a conjectural framework of deriving all/some tautological equations. These conjectures are inspired by a study of Givental's axiomatic Gromov-Witten theory, for which some background will be provided.

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REFERENCES:

Y.P. Lee, Witten's conjecture, Virasoro conjecture, and invariance of tautological equations, updated version available at http://www.math.utah.edu/~yplee/research/ite3.pdf.

Motivated by the AdS/CFT and black hole microstate counting, we discuss quantization of infinitely-dimensional families of regular supergravity solutions with Anti-de-Sitter or flat asymptotics. We demonstrate the general methods on two recent examples of such families - the "bubbling AdS" family of Lin-Lunin-Maldacena and the "D1-D5 with angular momentum" family of Lunin-Maldacena-Maoz.

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REFERENCES:

L.Maoz, V.S.Rychkov, Geometry Quantization from Supergravity: the Case of 'Bubbling Ads', hep-th/0508059.
L.Grant, L.Maoz, J.Marsano, K.Papadodimas, V.S.Rychkov, Minisuperspace Quantization of 'Bubbling Ads' and Free Fermion Droplets, hep-th/0505079.
H.Lin, O.Lunin, J.Maldacena, Bubbling Ads Space and 1/2 BPS Geometries, hep-th/0409174.
O.Lunin, J.Maldacena, L.Maoz, Gravity Solutions for the D1-D5 System with Angular Momentum, hep-th/0212210.

In this talk, we will explain how to give an explicit closed-formula of all-genus Gromov-Witten theory for certain noncompact surface geometries.

In this talk I will consider non-linear gauged sigma-models with Kahler domain and target. For a special choice of potential these models admit Bogomolny (or self-duality) equations, the so-called symplectic vortex equations. I will then describe a topological field theory that studies the moduli space of solutions of these equations. The correlation functions of the theory are closely related to the recently introduced Hamiltonian Gromov-Witten invariants.

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REFERENCES:

J.M.Baptista, "A Topological Gauged Sigma Model", hep-th/0502152.
K.Cieliebak, A.R.Gaio, I.Mundet i Riera, D.Salamon, "The Symplectic Vortex Equations and Invariants of Hamiltonian Group Actions", math.SG/0111176.

I will discuss some recent attempts to understand the meaning of nonperturbative topological strings---one approach via counting of BPS microstates, and in particular its extension to open topological strings, and another via topological M-theory.

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REFERENCES:

M.Aganagic, A.Neitzke, C.Vafa, "BPS Microstates and the Open Topological String Wave Function", hep-th/0504054.
R.Dijkgraaf, S.Gukov, A.Neitzke, C.Vafa, "Topological M-theory as Unification of Form Theories of Gravity", hep-th/0411073.

We take a look at the nontorsion case, which correspond to infinite dimensional projective bundles. We cast this in the realm of C*-algebras and groupoids, using Morita equivalences and Brauer groups, with reference to the general analysis of the interplay between groupoids and algebras as a central theme of the noncommutative geometry.

1. M. Atiyah, G. Segal, Twisted K-theory, math.KT/0407054.
2. A. Connes, Noncommutative Geometry, Academic Press, 1994.
3. P. Donavan and M. Karoubi, Graded Brauer groups and K-theory with local coefficients, Publ. Math IHES 38 (1970) 5-25.
4. N.P. Landsman, Mathematical topics between classical and quantum mechanics. Springer Monographs in Mathematics. Springer-Verlag, New York, 1998
5. J. Rosenberg, Continuous-trace algebras from the bundle theoretic point of view. J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368-381.
6. I. Raeburn, D.P. Williams, Morita equivalence and continuous-trace C*-algebras, Mathematical Surveys and Monographs, 60. American Mathematical Society, Providence, RI, 1998. xiv+327 pp.
7. A. Kumjian, P. S. Muhly, J. N. Renault, D. P. Williams, The Brauer Group of a Locally Compact Groupoid, funct-an/9706004.

I will explain the present status of the conjectural correspondence between descendents in Gromov-Witten and Donaldson-Thomas theories, in particular, the appearance of the Frankenstein series $F(q)=\sum _{a,b}{a}^2{q}^{ab}$.