Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2003-07-29
JHEP 0310 (2003) 068
Physics
High Energy Physics
High Energy Physics - Theory
8 pages. v2 derivation of the critical exponents for the operator relations of SO(N)/Sp(N), comments and refs. added
Scientific paper
10.1088/1126-6708/2003/10/068
We study the covariance properties of the equations satisfied by the generating functions of the chiral operators R and T of supersymmetric SO(N)/Sp(N) theories with symmetric/antisymmetric tensors. It turns out that T is an affine connection. As such it cannot be integrated along cycles on Riemann surfaces. This explains the discrepancies observed by Kraus and Shigemori. Furthermore, by means of the polynomial defining the Riemann surface, seen as quadratic-differential, one can construct an affine connection that added to T leads to a new generating function which can be consistently integrated. Remarkably, thanks to an identity, the original equations are equivalent to equations involving only one-differentials. This provides a geometrical explanation of the map recently derived by Cachazo in the case of Sp(N) with antisymmetric tensor. Finally, we suggest a relation between the Riemann surfaces with rational periods which arise in studying the Laplacian on special Riemann surfaces and the integrality condition for the periods of T.
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