10 research outputs found
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