The modern version of conformal matrix model (CMM) describes conformal blocks in the Dijkgraaf-Vafa phase. Therefore it possesses a determinant representation and becomes a Toda chain τ -function only after a peculiar Fourier transform in internal dimensions. Moreover, in CMM Hirota equations arise in a peculiar discrete form (when the couplings of CMM are actually Miwa time-variables). Instead, this integrability property is actually independent of the measure in the original hypergeometric integral. To get hypergeometric functions, one needs to pick up a very special τ -function, satisfying an additional "string equation". Usually its role is played by the lowest L−1 Virasoro constraint, but, in the Miwa variables, it turns into a finite-difference equation with respect to the Miwa variables. One can get rid of these differences by rewriting the string equation in terms of some double ratios of the shifted τ -functions, and then these ratios satisfy more sophisticated equations equivalent to the discrete Painlevé equations by M. Jimbo and H. Sakai (q-PVI equation). They look much simpler in the q-deformed ("5d") matrix model, while in the "continuous" limit q −→ 1 to 4d one should consider the Miwa variables with non-unit multiplicities, what finally converts the simple discrete Painlevé q-PVI into sophisticated differential Painlevé VI equations, which will be considered elsewhere.
We briefly discuss the recent claims that the ordinary KP/Toda integrability, which is a characteristic property of ordinary eigenvalue matrix models, persists also for the Dijkgraaf-Vafa (DV) partition functions and for the refined topological vertex. We emphasize that in both cases what is meant is a particular representation of partition functions: a peculiar sum over all DV phases in the first case and hiding the deformation parameters in a sophisticated potential in the second case, i.e. essentially a reformulation of some questions in the new theory in the language of the old one. It is at best obscure if this treatment can be made consistent with the AGT relations and even with the quantization of the underlying integrable systems in the Nekrasov-Shatashvili limit, which seem to require a full-scale β-deformation of individual DV partition functions. Thus, it is unclear if the story of integrability is indeed closed by these recent considerations.1 When only one ǫ is non-vanishing, this corresponds to an ordinary quantization [16] of the underlying integrable system [17].
We study the six-dimensional pseudo-Riemannian spaces with two time-like coordinates that admit non-homothetic infinitesimal projective transformations. The metrics are manifestly obtained and the projective group properties are determined. We also find a generic definition of projective motion in the 6-dimensional rigid h-space.PACS : 02.40.Hw
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