Exact results and holography of Wilson loops in $ \mathcal{N} = 2 $ superconformal (quiver) gauge theories

Abstract

Using localization, matrix model and saddle-point techniques, we determine exact behavior of circular Wilson loop in N = 2 superconformal (quiver) gauge theories in the large number limit of colors. Focusing at planar and large 't Hooft couling limits, we compare its asymptotic behavior with well-known exponential growth of Wilson loop in N = 4 super Yang-Mills theory with respect to 't Hooft coupling. For theory with gauge group SU(N) coupled to 2N fundamental hypermultiplets, we find that Wilson loop exhibits non-exponential growth -at most, it can grow as a power of 't Hooft coupling. For theory with gauge group SU(N) × SU(N) and bifundamental hypermultiplets, there are two Wilson loops associated with two gauge groups.We find Wilson loop in untwisted sector grows exponentially large as in N = 4 super Yang-Mills theory. We then find Wilson loop in twisted sector exhibits non-analytic behavior with respect to difference of the two 't Hooft coupling constants. By letting one gauge coupling constant hierarchically larger/smaller than the other, we show that Wilson loops in the second type theory interpolate to Wilson loops in the first type theory. We infer implications of these findings from holographic dual description in terms of minimal surface of dual string worldsheet.We suggest intuitive interpretation that in both classes of theory holographic dual background must involve string scale geometry even at planar and large 't Hooft coupling limit and that new results found in the gauge theory side are attributable to worldsheet instantons and infinite resummation therein. Our interpretation also indicates that holographic dual of these gauge theories is provided by certain non-critical string theories. Reduction to One-Matrix ModelThe work [6] provided a proof for the conjecture [4,5] that the evaluation of the half-BPS Wilson loop in N = 4 super Yang-Mills theory [2, 3] is reduced to a related problem in a Gaussian Hermitian one-matrix model. In this section, we show that the similar reduction also works for N = 2 superconformal gauge theories of general quiver type. The resulting matrix model is, however, not Gaussian but includes non-trivial vertices due to nontrivial one-loop determinant.

Rigid holography and six-dimensional N = 2 0 $$ \mathcal{N}=\left(2,0\right) $$ theories on AdS5 × S $$ \mathbb{S} $$ 1

“…(The situation for K = 2 was previously studied in [19] and also in [20].) First, we can identify what this limit corresponds to in the space of exactly marginal deformations that we described above.…”

“…The previous exact computations of Wilson loop operators [19] indicated hierarchically separated BPS string tensions in the singular limit. 28 Extensions to 't Hooft loops and some correlation functions are straightforward, and the correlation functions of loop operators should be compared with those of the BPS strings in the six-dimensional N = (2, 0) theory on AdS 5 × S 1 .…”

“…19 The discussion above suggests that these currents should map by the AdS/CFT correspondence to the A K−1 theory on AdS 5 × S 1 . Naively, this implies that this bulk field theory must contain, at the origin of its 'moduli space', massless high-spin gauge fields, including a massless graviton (corresponding to the energy-momentum tensor of the SU(K) gauge theory, which is separately conserved in this limit).…”

“…The hierarchical behavior of the Wilson and 't Hooft loop operators as exact marginal deformations are turned on away from the orbifold point was first initiated in [19].…”

SUSY gauge theories on squashed three-spheres

“…In particular, in [3] it has been argued that the 3D partition function can be used to determine the superconformal R-symmetry at the infrared fixed points, in a similar spirit to a-maximization in 4D gauge theories. It has also been attracting wide attention from various other fields, for example Wilson loops [5,6], 3D dualities [2], large-N duality of topological string [7] and the ABJM theory at strong coupling [8,9,10]. The localization technique can also be applied to the theory on S 2 × S 1 and allows one to compute the superconformal index.…”