<i>X</i> <sub>2</sub> series of universal quantum dimensions

Avetisyan, M. Y.; Mkrtchyan, R. L.

doi:10.1088/1751-8121/ab5f4d

Cited by 9 publications

(9 citation statements)

References 34 publications

(97 reference statements)

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“…In the present paper we study a feature of quantum dimensions, which has been noticed in [8], and was later called linear resolvability [9], LR.…”

Section: Introductionmentioning

confidence: 91%

“…Subsequently a number of universal formulae, particularly for dimensions and quantum dimensions, have been derived in [6,7,8,9], as well as Diophantine classification of simple Lie algebras [10], based on the universal quantum dimension of the adjoint representation (1) [11,12], has been worked out. For example, the quantum dimension of the adjoint representation of each of the simple Lie algebra is given by the following function:…”

Section: Introductionmentioning

confidence: 99%

“…As the simplest example to demonstrate this phenomenon [8], consider the following expression for universal dimension of one of the irreducible representations, appearing in the decomposition of the symmetric square of the adjoint representation. It was denoted as Y 2 (β) by Vogel (see Appendix A), and we refer to its dimension similarly as Y 2 (β):…”

Section: Introductionmentioning

confidence: 99%

“…It was proved in [8] that the universal formula of quantum dimensions of Cartan products of arbitrary powers of the adjoint and X 2 representations 2 X(x, k, n) [8,9] is LR at the points from the Vogel's table.…”

Section: Introductionmentioning

confidence: 99%

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On linear resolvability of universal quantum dimensions

Avetisyan¹,

Mkrtchyan²

2021

Preprint

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In his study of finite (Vassiliev's) knot invariants Vogel introduced the so-called universal parameters, belonging to projective plane, which particularly parameterize the simple Lie algebras by the Vogel's table. Subsequently a number of quantities, such as some universal knot invariants, (quantum) dimensions of simple Lie algebras, etc., have been represented in terms of these parameters, i.e. in the universal form. We prove that at the points from the Vogel's table all known universal quantum dimension formulae are linearly resolvable, i.e. yield finite answers even if these points are singular, provided one restricts them to the appropriate lines. We show, that the same phenomenon takes place for another three distinguished points in Vogel's plane -E 7 1 2 , X1, and X2. We also examine the same formulae on linear resolvability at the remaining 48 distinguished points in Vogel's plane, which correspond to the so-called Y -objects, and discover that there are three points among them, which are regular for all known quantum dimension formulae. Two of them happen to be sharing a remarkable similarity with the simple Lie algebras, namely, the universal formulae yield integer-valued outputs (dimensions) at those points in the classical limit.

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“…In the present paper we study a feature of quantum dimensions, which has been noticed in [8], and was later called linear resolvability [9], LR.…”

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On linear resolvability of universal quantum dimensions

Avetisyan¹,

Mkrtchyan²

2021

Preprint

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“…A number of universal formulae have been derived both in the scope of the representation theory and in physical gauge theories, constructed on simple Lie algebras: [1][2][3][4][5][6][7]. Let's present an example [2]:…”

Section: Introductionmentioning

confidence: 99%

Universal dimensions of simple Lie algebras and configurations of points and lines

Avetisyan¹

2021

Proceedings of RDP Online Workshop "Recent Advances in Mathematical Physics" — PoS(Regio2020)

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Exceptionally simple integrated correlators in $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory

Dorigoni,

Vallarino

2023

J. High Energ. Phys.

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Supersymmetric localisation has led to several modern developments in the study of integrated correlators in $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills (SYM) theory. In particular, exact results have been derived for certain integrated four-point functions of superconformal primary operators in the stress tensor multiplet which are valid for all classical gauge groups, SU(N), SO(N), and USp(2N), and for all values of the complex coupling, τ = θ/(2π) + 4πi/$$ {g}_{YM}^2 $$ g YM 2 . In this work we extend this analysis and provide a unified two-dimensional lattice sum representation valid for all simple gauge groups, in particular for the exceptional series Er (with r = 6, 7, 8), F4 and G2. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality which for the cases of F4 and G2 is given by particular Fuchsian groups. We show that the perturbation expansion of these integrated correlators is universal in the sense that it can be written as a single function of three parameters, called Vogel parameters, and a suitable ’t Hooft-like coupling. To obtain the perturbative expansion for the integrated correlator with a given gauge group we simply need substituting in this single universal expression specific values for the Vogel parameters. At the non-perturbative level we conjecture a formula for the one-instanton Nekrasov partition function valid for all simple gauge groups and for general Ω-deformation background. We check that our expression reduces in various limits to known results and that it produces, via supersymmetric localisation, the same one-instanton contribution to the integrated correlator as the one derived from the lattice sum representation. Finally, we consider the action of the hyperbolic Laplace operator with respect to τ on the integrated correlators with exceptional gauge groups and derive inhomogeneous Laplace equations very similar to the ones previously obtained for classical gauge groups.

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X ₂ series of universal quantum dimensions

Cited by 9 publications

References 34 publications

On linear resolvability of universal quantum dimensions

On linear resolvability of universal quantum dimensions

Universal dimensions of simple Lie algebras and configurations of points and lines

Exceptionally simple integrated correlators in $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory

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X 2 series of universal quantum dimensions

Cited by 9 publications

References 34 publications

On linear resolvability of universal quantum dimensions

On linear resolvability of universal quantum dimensions

Universal dimensions of simple Lie algebras and configurations of points and lines

Exceptionally simple integrated correlators in $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory

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X ₂ series of universal quantum dimensions