On the perturbative chiral ring for marginally deformed Script N=4 SYM theories

Abstract

For N = 1 SU(N) SYM theories obtained as marginal deformations of the N = 4 parent theory we study perturbatively some sectors of the chiral ring in the weak coupling regime and for finite N. By exploiting the relation between the definition of chiral ring and the effective superpotential we develop a procedure which allows us to easily determine protected chiral operators up to n loops once the superpotential has been computed up to (n−1) order. In particular, for the Lunin-Maldacena β-deformed theory we determine the quantum structure of a large class of operators up to three loops. We extend our procedure to more general Leigh-Strassler deformations whose chiral ring is not fully understood yet and determine the weight-two and weight-three sectors up to two loops. We use our results to infer general properties of the chiral ring.

On the all-order perturbative finiteness of the deformed N=4 SYM theory

“…When restricted to three loops, the general formula exactly reproduces the result of our graph calculation. Also, we can immediately explain the observation of [8] that α is not corrected at two loopsit simply follows from the fact that γ Φ has no two-loop contribution. In Section 3.2.4 we generalize the construction of O F to the most general deformed theory which is made finite by a single condition [1,16].…”

“…Only the triple chiral vertex receives at order g 5 a finite non-planar correction from the first (super)diagram in Figure 3. It affects both the [ , ] ω and [ , ] Ω structures in the effective superpotential W eff [8]. In particular, the correction to the [ , ] ω structure is…”

“…The absence of a two-loop correction in (63) (independently noticed in [8]) is due to the specific form (52) of the two-loop contribution which allows us to rewrite the finiteness condition up to three loops in the form (49). …”

Marginal deformations and quasi-Hopf algebras

“…One could also ask similar questions regarding the spectrum, and for instance revisit studies of chiral primaries in the Leigh-Strassler theories making use of the discrete symmetries (e.g. [55,56]) from the quantum algebra perspective. Note that in recent work [57,46,58], it was established that a different Hopf algebra, the Yangian, acts as a symmetry on the action of N = 4 SYM as well as that of the β-deformed theory [45], and its presence indeed led to Slavnov-Taylor relations between correlation functions.…”

Wrapping effects in supersymmetric gauge theories

“…The β-deformation breaks the SU (4) R-symmetry of the N = 4 theory to U (1) R . The other symmetries that are still present after the deformation are the Z 3 associated to cyclic permutations of the scalar superfields, and two non-R-symmetry U (1) factors [79]. The Feynman rules for the propagators and the vertices with vectors remain the same as for the undeformed theory, whereas the contributions for chiral and antichiral vertices get an additional factor of q or q, depending on the order of the fields…”