Nekrasov Functions and Exact Bohr-Sommerfeld Integrals

In the case of SU(2), associated by the AGT relation to the 2d Liouville theory, the Seiberg-Witten prepotential is constructed from the Bohr-Sommerfeld periods of 1d sine-Gordon model. If the same construction is literally applied to monodromies of exact wave functions, the prepotential turns into the one-parametric Nekrasov prepotential F(a,\epsilon_1) with the other epsilon parameter vanishing, \epsilon_2=0, and \epsilon_1 playing the role of the Planck constant in the sine-Gordon Shroedinger equation, \hbar=\epsilon_1. This seems to be in accordance with the recent claim in arXiv:0908.4052 and poses a problem of describing the full Nekrasov function as a seemingly straightforward double-parametric quantization of sine-Gordon model. This also provides a new link between the Liouville and sine-Gordon theories.Comment: 10 page

On AGT description of N=2 SCFT with N_f=4

We consider Alday-Gaiotto-Tachikawa (AGT) realization of the Nekrasov partition function of N=2 SCFT. We focus our attention on the SU(2) theory with N_f=4 flavor symmetry, whose partition function, according to AGT, is given by the Liouville four-point function on the sphere. The gauge theory with N_f=4 is known to exhibit SO(8) symmetry. We explain how the Weyl symmetry transformations of SO(8) flavor symmetry are realized in the Liouville theory picture. This is associated to functional properties of the Liouville four-point function that are a priori unexpected. In turn, this can be thought of as a non-trivial consistency check of AGT conjecture. We also make some comments on elementary surface operators and WZW theory.Comment: 18 pages. v2, a misinterpretation in the gauge theory side has been corrected; title and introduction were changed accordingl

Non-Perturbative Topological Strings And Conformal Blocks

We give a non-perturbative completion of a class of closed topological string theories in terms of building blocks of dual open strings. In the specific case where the open string is given by a matrix model these blocks correspond to a choice of integration contour. We then apply this definition to the AGT setup where the dual matrix model has logarithmic potential and is conjecturally equivalent to Liouville conformal field theory. By studying the natural contours of these matrix integrals and their monodromy properties, we propose a precise map between topological string blocks and Liouville conformal blocks. Remarkably, this description makes use of the light-cone diagrams of closed string field theory, where the critical points of the matrix potential correspond to string interaction points.Comment: 36 page

The matrix model version of AGT conjecture and CIV-DV prepotential

Recently exact formulas were provided for partition function of conformal (multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted as Dotsenko-Fateev correlator of screenings and analytically continued in the number of screening insertions, represents generic Virasoro conformal blocks. Actually these formulas describe the lowest terms of the q_a-expansion, where q_a parameterize the shape of the Penner potential, and are exact in the filling numbers N_a. At the same time, the older theory of CIV-DV prepotential, straightforwardly extended to arbitrary beta and to non-polynomial potentials, provides an alternative expansion: in powers of N_a and exact in q_a. We check that the two expansions coincide in the overlapping region, i.e. for the lowest terms of expansions in both q_a and N_a. This coincidence is somewhat non-trivial, since the two methods use different integration contours: integrals in one case are of the B-function (Euler-Selberg) type, while in the other case they are Gaussian integrals.Comment: 27 pages, 1 figur

Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions

We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (\beta-ensemble), dS = ydz + O(\epsilon^2) = \sum_p \epsilon^{2p} \rho_\beta^{(p|1)}(z), where \epsilon and \beta are related to the LNS parameters \epsilon_1 and \epsilon_2. This provides explicit formulas for conformal blocks in terms of analytically continued contour integrals and resolves the old puzzle of the free-field description of generic conformal blocks through the Dotsenko-Fateev integrals. Most important, this completes the GKMMM description of SW theory in terms of integrability theory with the help of exact BS integrals, and provides an extended manifestation of the basic principle which states that the effective actions are the tau-functions of integrable hierarchies.Comment: 14 page

Roles for the Conserved Spc105p/Kre28p Complex in Kinetochore-Microtubule Binding and the Spindle Assembly Checkpoint

Kinetochores attach sister chromatids to microtubules of the mitotic spindle and orchestrate chromosome disjunction at anaphase. Although S. cerevisiae has the simplest known kinetochores, they nonetheless contain approximately 70 subunits that assemble on centromeric DNA in a hierarchical manner. Developing an accurate picture of the DNA-binding, linker and microtubule-binding layers of kinetochores, including the functions of individual proteins in these layers, is a key challenge in the field of yeast chromosome segregation. Moreover, comparison of orthologous proteins in yeast and humans promises to extend insight obtained from the study of simple fungal kinetochores to complex animal cell kinetochores.We show that S. cerevisiae Spc105p forms a heterotrimeric complex with Kre28p, the likely orthologue of the metazoan kinetochore protein Zwint-1. Through systematic analysis of interdependencies among kinetochore complexes, focused on Spc105p/Kre28p, we develop a comprehensive picture of the assembly hierarchy of budding yeast kinetochores. We find Spc105p/Kre28p to comprise the third linker complex that, along with the Ndc80 and MIND linker complexes, is responsible for bridging between centromeric heterochromatin and kinetochore MAPs and motors. Like the Ndc80 complex, Spc105p/Kre28p is also essential for kinetochore binding by components of the spindle assembly checkpoint. Moreover, these functions are conserved in human cells.Spc105p/Kre28p is the last of the core linker complexes to be analyzed in yeast and we show it to be required for kinetochore binding by a discrete subset of kMAPs (Bim1p, Bik1p, Slk19p) and motors (Cin8p, Kar3p), all of which are nonessential. Strikingly, dissociation of these proteins from kinetochores prevents bipolar attachment, even though the Ndc80 and DASH complexes, the two best-studied kMAPs, are still present. The failure of Spc105 deficient kinetochores to bind correctly to spindle microtubules and to recruit checkpoint proteins in yeast and human cells explains the observed severity of missegregation phenotypes