The Affine Connection of Supersymmetric SO(N)/Sp(N) Theories

Abstract

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.

8 pages. v2 derivation of the critical exponents for the operator relations of SO(N)/Sp(N), comments and refs. added

References in corpus (5)

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• On the Matter of the Dijkgraaf--Vafa Conjecture
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Cited by in corpus (6)

• On Low Rank Classical Groups in String Theory, Gauge Theory and Matrix Models
• Supersymmetric Gauge Theories with Flavors and Matrix Models
• More on N=1 Matrix Model Curve for Arbitrary N
• Konishi Anomalies and Curves without Adjoints
• Effective superpotential for U(N) with antisymmetric matter
• Dual Interpretations of Seiberg-Witten and Dijkgraaf-Vafa curves