17 research outputs found
Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants
Inspired by the work of Alday, Gaiotto and Tachikawa (AGT), we saw the
revival of Poincar{\'{e}}'s uniformization problem and Fuchsian equations
obtained thereof.
Three distinguished aspects are possessed by Fuchsian equations. First, they
are available via imposing a classical Liouville limit on level-two null-vector
conditions. Second, they fall into some A_1-type integrable systems. Third, the
stress-tensor present there (in terms of the Q-form) manifests itself as a kind
of one-dimensional "curve".
Thereby, a contact with the recently proposed Nekrasov-Shatashvili limit was
soon made on the one hand, whilst the seemingly mysterious derivation of
Seiberg-Witten prepotentials from integrable models become resolved on the
other hand. Moreover, AGT conjecture can just be regarded as a quantum version
of the previous Poincar{\'{e}}'s approach.
Equipped with these observations, we examined relations between spheric and
toric (classical) conformal blocks via Calogero-Moser/Heun duality. Besides, as
Sutherland model is also obtainable from Calogero-Moser by pinching tori at one
point, we tried to understand its eigenstates from the viewpoint of toric
diagrams with possibly many surface operators (toric branes) inserted. A
picture called "bubbling pants" then emerged and reproduced well-known results
of the non-critical self-dual c=1 string theory under a "blown-down" limit.Comment: 17 pages, 4 figures; v2: corrections and references added; v3:
Section 2.4.1 newly added thanks to JHEP referee advice. That classical
four-point spheric conformal blocks reproducing known SW prepotentials is
demonstrated via more examples, to appear in JHEP; v4: TexStyle changed onl
Hamiltonian structure of 2+1 dimensional gravity
A summary is given of some results and perspectives of the hamiltonian ADM
approach to 2+1 dimensional gravity. After recalling the classical results for
closed universes in absence of matter we go over the the case in which matter
is present in the form of point spinless particles. Here the maximally slicing
gauge proves most effective by relating 2+1 dimensional gravity to the Riemann-
Hilbert problem. It is possible to solve the gravitational field in terms of
the particle degrees of freedom thus reaching a reduced dynamics which involves
only the particle positions and momenta. Such a dynamics is proven to be
hamiltonian and the hamiltonian is given by the boundary term in the
gravitational action. As an illustration the two body hamiltonian is used to
provide the canonical quantization of the two particle system.Comment: 13 pages,2 figures,latex, Plenary talk at SIGRAV2000 Conferenc
Classical conformal blocks from TBA for the elliptic Calogero-Moser system
The so-called Poghossian identities connecting the toric and spherical
blocks, the AGT relation on the torus and the Nekrasov-Shatashvili formula for
the elliptic Calogero-Moser Yang's (eCMY) functional are used to derive certain
expressions for the classical 4-point block on the sphere. The main motivation
for this line of research is the longstanding open problem of uniformization of
the 4-punctured Riemann sphere, where the 4-point classical block plays a
crucial role. It is found that the obtained representation for certain 4-point
classical blocks implies the relation between the accessory parameter of the
Fuchsian uniformization of the 4-punctured sphere and the eCMY functional.
Additionally, a relation between the 4-point classical block and the ,
twisted superpotential is found and further used to re-derive the
instanton sector of the Seiberg-Witten prepotential of the , supersymmetric gauge theory from the classical block.Comment: 25 pages, no figures, latex+JHEP3, published versio
Accessory parameters for Liouville theory on the torus
We give an implicit equation for the accessory parameter on the torus which
is the necessary and sufficient condition to obtain the monodromy of the
conformal factor. It is shown that the perturbative series for the accessory
parameter in the coupling constant converges in a finite disk and give a
rigorous lower bound for the radius of convergence. We work out explicitly the
perturbative result to second order in the coupling for the accessory parameter
and to third order for the one-point function. Modular invariance is discussed
and exploited. At the non perturbative level it is shown that the accessory
parameter is a continuous function of the coupling in the whole physical region
and that it is analytic except at most a finite number of points. We also prove
that the accessory parameter as a function of the modulus of the torus is
continuous and real-analytic except at most for a zero measure set. Three
soluble cases in which the solution can be expressed in terms of hypergeometric
functions are explicitly treated.Comment: 30 pages, LaTex; typos corrected, discussion of eq.(74) improve
Symplectic geometry of the moduli space of projective structures in homological coordinates
We study the symplectic geometry of the space of linear differential equations with holomorphic coefficients of the form on Riemann surfaces of genus g. This space coincides with the moduli space of projective connections which is an affine bundle modelled on the cotangent bundle . We show that for several choices of the origin, or base, holomorphic projective connection (such as Bergman, Wirtinger or Schottky) the canonical Poisson structure on induces the Goldman bracket on the monodromy character variety. These different choices give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations; we find the corresponding generating functions. Combined with a prior theorem of Kawai, our results show the symplectic equivalence between the embeddings of induced by the Bers and Bergman projective connections into the space of projective structures. The main technical tools are variational formulas with respect to homological Darboux coordinates on the space of holomorphic quadratic differentials. In particular, we get a new system of differential equations for the Prym matrix of the canonical two-sheeted covering of a Riemann surface defined by a quadratic differential