Volume 105B, number 2,3 PHYSICS LETTERS 1 October 1981 THE EQUIVALENCE OF Sp(2N) AND SO(-2N) GAUGE THEORIES R.L. MKRTCHYAN Landau Institute for Theoretical Physics, Academy of Sciences, Moscow, USSR Received 5 February 1981 It is proved that aU gauge invariant quantities in the Sp (2N) quantum gauge theory transform into those of SO (2N) after changing N--* -N, at fixed h = g2N. Matter multiplets transform into multiptets with transposed Young diagrams with nf nf(-1) d, where nf is the number of flavors, and d the number of squares in the Young diagram. In this letter we prove a simple relation between gauge theories, based on symplectic and orthogonal groups, in gauge invariant sectors. The following ex- act statement is claimed. Consider a gauge theory with the gauge group Sp(2N) and matter in multiplets with the Young dia- grams Mi, i = 1,2 ..... and with number of flavors n i. Then all gauge invariant expectation values in this the- ory coincide with the corresponding quantities in the SO(2N) gauge theory with the matter in transposed multiplets M/T , and with the number of flavors n i (- 1 ) di (d i is the number of squares in the Young diagram Mi), after changing N~ -N at fixed ~ = g2N, where g2 is the coupling constant, common for both theories. Notes: (i) As is well-known from the N expansion [1 ] the relevant coupling constant in nonabellian gauge theories is ~ = g2 N, its positiveness corresponds, for example, to the property of asymptotic freedom (see below). (ii) One may avoid the reversing of the sign of n i in the case of odd d i by introducing a subsidiary index (i.e. multiply all flavors), the trace of which is equal to 1 or -1. (iii) We shall see later that actually our statement holds for quantities invariant under global gauge trans- formations only. (iv) Naturally, there are ambiguities in changing the sign of N in quantities which are defined only for in- teger positive N. But in perturbation theory we give a 174 simple and natural definition: the group weight of each diagram may be represented by a polynomial of N(see below) and in this form they make sense for allN. (v) All representations of the group Sp(2N) are la- beled by Young diagrams, but SO(2N) has (spinor) representations, which do not correspond to any Young diagrams. So, more exactly one may speak about the embedding of the Sp(2N) theory in the SO (2N) one. (vi) The group weight of each diagram may be ex- pressed in terms of eigenvalues of the invariant (Casimir) operators on irreducible representations, hence this equivalence has to correspond to some rela- tions between the eigenvalues of the Casimir operators for the groups Sp(2N) and SO(2N). Actually, we proved that these eigenvalues for Sp(2N) transform to the eigenvalues for SO(2N) on a transposed repre- sentation under the change ofN ~ -N, but we do not continue this line here. (vii) Some relations between the groups Sp (2N) and SO(-2N) were known previously [2] in a purely group-theoretical framework, without any reference to gauge theories. In what follows we check the equivalence on some examples from perturbation theory, and describe gen- eral diagrammatic proofs. Then we carry out the most simple approach to the phenomena considered - the reformulation of the gauge invariant sector of the gauge theory in loop space, developed recently in the works of Makeenko and Migdal [3]. In the end of this letter we briefly discuss the case of matter in the defin- ing and adjoint representations, leaving aside the general case. 0 031-9163/81/0000-0000/$ 02.75 © 1981 North-Holland