Refining the Proof of Planar Equivalence

Abstract

We outline a full non-perturbative proof of planar (large-N) equivalence between bosonic correlators in a theory with Majorana fermions in the adjoint representation and one with Dirac fermions in the two-index (anti)symmetric representation. In a particular case (one flavor), this reduces to our previous result - planar equivalence between super-Yang--Mills theory and a non-supersymmetric ``orientifold field theory.'' The latter theory becomes one-flavor massless QCD at N=3.

Figures (6)

• Figure 1: One fermion loop (i.e. $\log \, \det \left( i\not\! \partial + \not\!\! A ^a \, T^a _r + J_\Psi\right)$) in the gluon field background (shown as shaded areas). The gluon fields "inside" and "outside" the loop do not communicate with each other at $N\to\infty$. This is indicated by distinct shadings. Averaging over the gluon field inside the loop is independent of averaging outside. Topologically, of course, the distinction between inside and outside is immaterial
• Figure 2: Example of fermion multiloops in the gluon field background, at $N\to\infty$. The background field outside loops 1 and 4 is the same. The background field inside loop 1 and outside loop 2 and 3 is the same. The background field inside loop 4 and outside loop 5 is the same.
• Figure 3: Two opposite-orientation contours in the sum (\ref{['wlineint']}).
• Figure 4: The 't Hooft double-line representation for $\langle W_{\rm adjoint} \rangle$. On the right we display a convenient graphic shorthand notation that we suggest to use in this problem. The black circle corresponds to $\langle {\rm Tr}\, U \rangle$, the white circle to $\langle {\rm Tr}\, U^\dagger \rangle$, while the segment connecting them indicates that both circles originate from one and the same fermion loop, see Fig. \ref{['Mone']}.
• Figure 5: A particular connected contribution to an expectation value $\langle W_1 W_2 W_3 W_4 W_5\rangle _c$ in ${\cal N}=1\ $ SYM theory. Dashed circles indicate averaging over a connected background field, for instance, the external lines of loop 1 and loop 4 are averaged over one and the same gluon field.