Abstract
We study chiral rings of 4d mathcal{N} = 1 supersymmetric gauge theories via the notion of K-stability. We show that when using Hilbert series to perform the computations of Futaki invariants, it is not enough to only include the test symmetry information in the former’s denominator. We discuss a way to modify the numerator so that K-stability can be correctly determined, and a rescaling method is also applied to simplify the calculations involving test configurations. All of these are illustrated with a host of examples, by considering vacuum moduli spaces of various theories. Using Gröbner basis and plethystic techniques, many non-complete intersections can also be addressed, thus expanding the list of known theories in the literature.
Highlights
Original ring.1 It was argued in [1] that this is equivalent to the concept of K-stability.2 In [2, 3], for a polarized ring with symmetry/Reeb vector field ζ, K-stability is determined via perturbing the ring by a test symmetry η for some symmetry η and small
We study chiral rings of 4d N = 1 supersymmetric gauge theories via the notion of K-stability
We show that when using Hilbert series to perform the computations of Futaki invariants, it is not enough to only include the test symmetry information in the former’s denominator
Summary
We shall focus on the chiral rings of (3+1)-dimensional SCFT [9,10,11] for whose supersymmetry we will write in N = 1 language. The above should be compared and contrasted with the calculation of the classical vacuum moduli space (VMS), which is the GIT quotient of J by the complexified gauge group [13]. This is done by considering the minimal set of gauge invariant operators (GIOs) Gj in the theory, each being a single-trace operator, and a polynomial in the φi. In [17], it was shown that all the classical VMSs are affine Calabi-Yau (Gorenstein) singularities
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