Abstract
It was known that quantum curves and super Chern-Simons matrix models correspond to each other. From the viewpoint of symmetry, the algebraic curve of genus one, called the del Pezzo curve, enjoys symmetry of the exceptional algebra, while the super Chern-Simons matrix model is described by the free energy of topological strings on the del Pezzo background with the symmetry broken. We study the symmetry breaking of the quantum cousin of the algebraic curve and reproduce the results in the super Chern-Simons matrix model.
Highlights
JHEP01(2019)210 structure of the correspondence is much clearer, though, besides the difficulty in the interpretation in terms of the M2-branes, it was difficult to compute directly the kernels of the spectral operators for general parameters of the curves until recently with the important progress in [39, 40]
We find that the breaking patterns are completely consistent with the previous results in [31] from the superconformal ChernSimons matrix models
We again generate all elements of the E7 Weyl group by using a computer, we find that the answer is 3840 elements generated by s4, s5, s6, s1s7s6s5s4s3s4s5s6s7s1, s2, (5.21)
Summary
The main purpose of this section is to explain our motivation of studying quantum curves It was proposed [5, 19, 20] that the N = 6 superconformal Chern-Simons theory with gauge group U(N1)k×U(N2)−k (with the subscripts k, −k denoting the ChernSimons levels) and two pairs of bifundamental matters describes the worldvolume theory of min(N1, N2) M2-branes and |N2 − N1| fractional M2-branes on the target space C4/Zk. With the localization techniques [41], the partition function, as well as the one-point functions (and hopefully the two-point functions [42]) of the half-BPS Wilson loop in the N = 6 superconformal Chern-Simons theory on S3, which is originally defined with the infinitedimensional path integral, reduces to a finite-dimensional matrix integration. Since the (1, 1, 1, 1) model without rank deformations corresponds to (MI, MII) = (k/2, k/2), the instanton exponent is given by d · T = d μeff + πi k dI dII.
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