Chiral rings, Futaki invariants, plethystics, and Gröbner bases

Bao, Jiakang; He, Yang‐Hui; Xiao, Yan

doi:10.1007/jhep01(2021)203

Cited by 7 publications

(6 citation statements)

References 52 publications

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“…More precisely, the last equality should be K-semistability for product test configurations. We will not expound upon this here, and readers are referred to [64][65][66][67] for more details.…”

Section: Maximization Of Isoradial Mahler Measurementioning

confidence: 99%

Mahler Measure for a Quiver Symphony

Bao,

He,

Zahabi

2021

Preprint

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Adopting the Mahler measure from number theory, we introduce it to toric quiver gauge theories, and study some of its salient features and physical implications. We propose that the Mahler measure is a universal measure for the quiver, encoding its dynamics with the monotonic behaviour along a so-called Mahler flow including two special points at isoradial and tropical limits. Along the flow, the amoeba, from tropical geometry, provides geometric interpretations for the dynamics of the quiver. In the isoradial limit, the maximization of Mahler measure is shown to be equivalent to a-maximization. The Mahler measure and its derivative are closely related to the master space, leading to the property that the specular duals have the same functions as coefficients in their expansions, hinting the emergence of a free theory in the tropical limit. Moreover, they indicate the existence of phase transition. We also find that the Mahler measure should be invariant under Seiberg duality.

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Section: Maximization Of Isoradial Mahler Measurementioning

confidence: 99%

Mahler Measure for a Quiver Symphony

Bao,

He,

Zahabi

2021

Preprint

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“…P . Let (1) and (2) be the exponent vectors of 1 and 2 respectively. If W 1 and W 2 are equivalent, then there exists invertible such that…”

Section: Definitions 31 Pluriweighted-homogeneitymentioning

confidence: 99%

“…In addition to those theoretical properties which make it possible to obtain complexity bounds, this order usually gives the fastest Gröbner basis computation in practice. For some applications, having any Gröbner basis is enough, for instance for computing the Hilbert series of the ideal [2,30]. Furthermore, multistep strategies have been designed around this feature, combining those "easy" Gröbner basis computations with change of order algorithms in order to obtain a Gröbner basis for any wanted order.…”

Section: Introductionmentioning

confidence: 99%

On the computation of Gröbner bases for matrix-weighted homogeneous systems

Verron¹

2022

Preprint

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In this paper, we examine the structure of systems which are weighted homogeneous for several systems of weights, and how it impacts Gröbner basis computations. We present different ways to compute Gröbner bases for systems with this structure, either directly or by reducing to existing structures. We also present optimization techniques which are suitable for this structure.The most natural orderings to compute a Gröbner basis for systems with this structure are weighted orderings following the systems of weights, and we discuss the possibility to use the algorithms in order to directly compute a basis for such an order, regardless of the structure of the system.We discuss applicable notions of regularity which could be used to evaluate the complexity of the algorithm, and prove that they are generic if non-empty. Finally, we present experimental data from a prototype implementation of the algorithms in SageMath. CCS CONCEPTS• Computing methodologies → Algebraic algorithms.

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“…For it to be K-stable, F can be zero only when the norm vanishes. For the expression of Futaki invariants and the definition of norm, one is referred to [90,91]. More details of defining K-stability along with its calculations can also be found in [91].…”

Section: K-stability and Chiral Ringsmentioning

confidence: 99%

“…Indeed, for some theories (such as the worldvolume theory of D3s probing Gorenstein singularties), one can show that K-stability recovers the unitarity bounds or irrelevance of superpotential terms [89]. However, a counterexample was found in [91]. Therefore, it is natural to ask: Question 15.…”

Section: K-stability and Chiral Ringsmentioning

confidence: 99%

Some Open Questions in Quiver Gauge Theory

Bao,

Hanany,

et al. 2021

Preprint

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Quivers, gauge theories and singular geometries are of great interest in both mathematics and physics. In this note, we collect a few open questions which have arisen in various recent works at the intersection between gauge theories, representation theory, and algebraic geometry. The questions originate from the study of supersymmetric gauge theories in different dimensions with different supersymmetries. Although these constitute merely the tip of a vast iceberg, we hope this guide can give a hint of possible directions in future research.This is an invited contribution to a special volume of Proyecciones, E. Gasparim, Ed., and it is the hope that the questions are specific enough for research projects aimed at PhD students.

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Chiral rings, Futaki invariants, plethystics, and Gröbner bases

Cited by 7 publications

References 52 publications

Mahler Measure for a Quiver Symphony

Mahler Measure for a Quiver Symphony

On the computation of Gröbner bases for matrix-weighted homogeneous systems

Some Open Questions in Quiver Gauge Theory

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