arXiv subjects

R. L. Mkrtchyan

Publications and source records attributed to R. L. Mkrtchyan.

At least 19 recordsLinked to original sources

arXiv2026

Uniqueness of universal quantum dimensions

We prove, under some assumptions, the uniqueness of universal quantum dimension formulae. We first show that the quantum dimensions of classical algebras have a specific form that can be represented universally, in Vogel's sense, as the product/ratio of q-dimension factors with arguments $aα+bβ+cγ$ where $c=0, 1, 2$ and $a, b$ are integers. On this basis, we derive simple uniqueness criteria, according to which all known universal quantum dimensions are unique.

math-ph↗

arXiv2026

Universal quantum dimensions: $γ$-dependent factors

We develop an approach for computing the universal, in Vogel's sense, quantum dimensions. In a previous work a method for calculating the $γ$-independent part of these dimensions for universal multiplets with nonzero associated members for classical algebras was proposed. In the present paper, this approach is extended to the $γ$-dependent factors. In particular, we derive these contributions for the $E$ universal multiplet, which appears in the universal decomposition of the fourth power of the adjoint representation, thereby obtaining its universal quantum dimension for the first time. The analysis requires knowledge of the quantum dimensions of the classical and $E_8$ members of the multiplet; for the remaining exceptional algebras, the corresponding quantum dimensions are recovered automatically, providing an additional consistency check of the approach. We also partially extend a previously conjectured relation between the $sl$ and $so, sp$ members - formulated in terms of vertical and horizontal sums of Young diagrams - to the exceptional case. Finally, we present an explicit algorithm implementing the proposed method, which enables the derivation of universal formulae for higher-dimensional representations. The universal quantum dimensions provide, in some cases, a unified description of Chern-Simons observables, including knot invariants and partition functions, in a form valid simultaneously for all gauge algebras.

math.QA↗

arXiv2026

Universal Quantum Dimensions: I. $γ$-Independent Factors

We propose a method for computing universal (in Vogel's sense) quantum dimension formulae for universal multiplets whose associated $sl$, $so$, and $sp$ representations are nonzero. The method uses the relation between $sl$ and $so$ representations given by the vertical-sum operation, and the dual relation between $sl$ and $sp$ representations given by the horizontal-sum operation on the corresponding Young diagrams. The usual quantum dimensions of these three representations, together with subtleties related to the invariance of universal formulae under automorphisms of the $sl$ Dynkin diagram, allow one to determine the $γ$-independent factors of a universal quantum dimension (note that $γ$ is the only parameter for classical algebras, depending on their rank). Using this approach, we compute the $γ$-independent factors for (known) adjoints' universal quantum dimension, and also obtain such a factor in one new case. We discuss how to extend this approach to the $γ$-dependent factors in the quantum dimension formulae, and other issues. This is another instance in which calculations purely within the classical algebras predict the answers for the exceptional cases, due to the hidden universality structure of the theory of simple Lie algebras.

math-ph↗

arXiv2025

The Casimir eigenvalues on $ad^{\otimes k}$ of SU(N) are linear on N

We consider eigenvalues of the Casimir operator on the naturally defined \textit{stable sequences} of representations of $su(N)$ algebra and prove that eigenvalues are linear over $N$ iff $λ_1+2λ_2+...+kλ_k=λ_{N-1}+2λ_{N-2}+...+kλ_{N-k}$, where $λ_i$ are Dynkin labels, and $λ_i=0$ for $k<i<N-k$, with fixed $k$. These representations are exactly those which appear in the decomposition of $ad(su(N))^{\otimes k}$, therefore this linearity admits the presentation of eigenvalues in the universal, in Vogel's sense, form, and supports the hypothesis of universal decomposition of $ad^{\otimes k}$ into Casimir eigenspaces.

math-ph↗

arXiv2025

On the universal Casimir spectrum

We conjecture the connection between $su$ and $so$ members of universal, in Vogel's sense, multiplets. The key element is the notion of the {\it vertical componentwise sum} $\oplus_v$ of Young diagrams. Representations in the decomposition of the power of the adjoint representation of $su(N)$ algebra can be parameterized by a couple of $N$-independent Young diagrams $λ$ and $τ$, with equal area. We assume that the $so(N)$ member of the universal (Casimir) multiplet of a given $su(N)$ representation is the $so$ representation with $λ\oplus_v τ$ Young diagram. This allows one to obtain the universal form of the Casimir eigenvalue on that multiplet. Conjecture is checked for all known cases: universal decompositions of powers of adjoint up to fourth, and series of universal representations. On this basis we suggest the set of universal Casimirs for fifth power of adjoint. We also conjecture that vertical sum operation is a kind of the (dual version of the) folding map of Dynkin diagrams. This will hopefully explain the intrinsic symmetry of universal formulae with respect to the automorphisms of Dynkin diagrams.

hep-th↗

arXiv2025

$N \leftrightarrow -N$ duality of SU(N) for stable sequences of representations

We generalize $N \leftrightarrow -N$ duality of dimension formulae of $SU(N)$ representations on a (class of) representations with $N$-dependent Young diagrams (which include the adjoint representation), and on eigenvalues of the Casimir operator for those representations. We discuss the consequences for the hypothesis of universal decomposition of powers of the adjoint representation into Casimir subspaces.

math-ph↗

arXiv2023

Refined $E_n$ Chern-Simons theory

The partition function of refined Chern-Simons theory on 3d sphere for the exceptional $E_n$ gauge algebras is presented in terms of multiple sine functions. Gopakumar-Vafa (BPS) approximation is calculated and presented in the form of some refined topological string partition function.

hep-th↗

arXiv2023

Two-fold refinement of non simply laced Chern-Simons theories

Inspired by the two-parameter Macdonald-Cherednik deformation of the formulae for non simply laced simple Lie algebras, we propose a two-fold refinement of the partition function of the corresponding Chern-Simons theory on $S^3$. It is based on a two-fold refinement of the Kac-Peterson formula for the volume of the fundamental domain of the coroot lattice of non simply laced Lie algebras. We further derive explicit integral representations of the two-fold refined Chern-Simons partition functions. We also present the corresponding generalized universal-like expressions for them. With these formulae in hand, one can try to investigate a possible duality of the corresponding Chern-Simons theories with hypothetical two-fold refined topological string theories.

hep-th↗

arXiv2022

On refined Chern-Simons / topological string duality for classical gauge groups

We present the partition function of the refined Chern-Simons theory on $S^3$ with arbitrary A,B,C,D gauge algebra in terms of multiple sine functions. For B and C cases this representation is novel. It allows us to conjecture duality to some refined and orientifolded versions of the topological string on the resolved conifold, and carry out the detailed identification of different contributions. The free energies for D and C algebras possess the usual halved contribution from the A theory, i.e. orientable surfaces, and contributions of non-orientable surfaces with one cross-cup, with opposite signs, similar as for the non-refined theories. However, in the refined case, both theories possess in addition a non-zero contribution of orientable surfaces with two cross-cups. In particular, we observe a trebling of the Kähler parameter, in the sense of a refinement and world-sheet (i.e. the number of cross-cups) dependent quantum shift. For B algebra the contribution of Klein bottles is zero, as is the case in the non-refined theory, and the one-cross-cup terms differ from the D and C cases. For the (refined) constant maps terms of these theories we suggest a modular-invariant representation, which leads to natural topological string interpretation. We also calculate some non-perturbative corrections.

hep-th↗

arXiv2021

On partition functions of refined Chern-Simons theories on $S^3$

We present a new expression for the partition function of refined Chern-Simons theory on $S^3$ with arbitrary gauge group, which is explicitly equal to $1$, when the coupling constant is zero. Using this form of partition function we show that the previously known Krefl-Schwarz representation of partition function of refined Chern-Simons theory on $S^3$ can be generalized to all simply-laced algebras. For all non-simply-laced gauge algebras, we derive similar representations of that partition function, which makes it possible to transform it into a product of multiple sine functions aiming at the further establishment of duality with refined topological strings.

hep-th↗

arXiv2021

Uniqueness of universal dimensions and configurations of points and lines

The problem of uniqueness of universal formulae for (quantum) dimensions of simple Lie algebras is investigated. We present generic functions, which multiplied by a universal (quantum) dimension formula, preserve both its structure and its values at the points from Vogel's table. Connection of some of these functions with geometrical configurations, such as the famous Pappus-Brianchon-Pascal $(9_3)_1$ configuration of points and lines, is established. Particularly, the appropriate realizable configuration $(144_336_{12})$ (yet to be found) will provide a symmetric non-uniqueness factor for any universal dimension formula.

math.QA↗

arXiv2021

On linear resolvability of universal quantum dimensions

In his study of finite (Vassiliev's) knot invariants,Vogel introduced the so-called universal parameters, belonging to the 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 - $E_{7\frac{1}{2}}, X_1,$ and $X_2$. 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.

math-ph↗

arXiv2020

Chern-Simons theory with the exceptional gauge group as a refined topological string

We present the partition function of Chern-Simons theory with the exceptional gauge group on three-sphere in the form of a partition function of the refined closed topological string with relation $2τ=g_s(1-b) $ between single Kähler parameter $τ$, string coupling constant $g_s$ and refinement parameter $b$, where $b=\frac{5}{3},\frac{5}{2},3,4,6$ for $G_2, F_4, E_6, E_7, E_8$, respectively. The non-zero BPS invariants $N^d_{J_L,J_R}$ ($d$ - degree) are $N^2_{0,\frac{1}{2}}=1, N^{11}_{0,1}=1$. Besides these terms, partition function of Chern-Simons theory contains term corresponding to the refined constant maps of string theory. Derivation is based on the universal (in Vogel's sense) form of a Chern-Simons partition function on three-sphere, restricted to exceptional line $Exc$ with Vogel's parameters satisfying $γ=2(α+β)$. This line contains points, corresponding to the all exceptional groups. The same results are obtained for $F$ line $γ=α+β$ (containing $SU(4), SO(10)$ and $E_6$ groups), with the non-zero $N^2_{0,\frac{1}{2}}=1, N^{7}_{0,1}=1$. In both cases refinement parameter $b$ ($=-ε_2/ε_1$ in terms of Nekrasov's parameters) is given in terms of universal parameters, restricted to the line, by $b=-β/α$.

hep-th↗

arXiv2019

On universal quantum dimensions of certain two-parameter series of representations

We present the universal, in Vogel's sense, expression for the quantum dimension of Cartan product of an arbitrary number of adjoint and $X_2$ representations of simple Lie algebras. The same formula mysteriously gives quantum dimensions of some other representations of the same Lie algebra under permutations of universal parameters. We list these representations for exceptional algebras and stable versions for classical algebras, when the rank of the classical algebra is sufficiently large w.r.t. the powers of representations. We show that universal formulae can have singularities on Vogel's plane for some algebras and that they give correct answers when restricted on appropriate lines on Vogel's plane. We note that the same irreducible representation can have several universal formulae for its (quantum) dimension, and discuss the implication of this phenomena on the Cohen - de Man method of calculation of universal formulae.

math-ph↗

arXiv2018

$X_2$ series of universal quantum dimensions

The antisymmetric square of the adjoint representation of any simple Lie algebra is equal to the sum of adjoint and $X_2$ representations. We present universal formulae for quantum dimensions of an arbitrary Cartan power of $X_2$. They are analyzed for singular cases and permuted universal Vogel's parameters. $X_2$ has been the only representation in the decomposition of the square of the adjoint with unknown universal series. Application to universal knot polynomials is discussed.

hep-th↗

arXiv2017

Partition function of Chern-Simons theory as renormalized q-dimension

We calculate $q$-dimension of $k$-th Cartan power of fundamental representation $Λ_0$, corresponding to affine root of affine simply laced Kac-Moody algebras, and show that in the limit $q\rightarrow 1 $, and with natural renormalization, it is equal to universal partition function of Chern-Simons theory on three-dimensional sphere.

hep-th↗

arXiv2017

On Universal Quantum Dimensions

We represent in the universal form restricted one-instanton partition function of supersymmetric Yang-Mills theory. It is based on the derivation of universal expressions for quantum dimensions (universal characters) of Cartan powers of adjoint and some other series of irreps of simple Lie algebras. These formulae also provide a proof of formulae for universal quantum dimensions for low-dimensional representations, needed in derivation of universal knot polynomials (i.e. colored Wilson averages of Chern-Simons theory on 3d sphere). As a check of the (complicated) formulae for universal quantum dimensions we prove numerically Deligne's hypothesis on universal characters for symmetric cube of adjoint representation.

math.RT↗

arXiv2016

Diophantine equations, Platonic solids, McKay correspondence, equivelar maps and Vogel's universality

We notice that one of the Diophantine equations, $knm=2kn+2km+2nm$, arising in the universality originated Diophantine classification of simple Lie algebras, has interesting interpretations for two different sets of signs of variables. In both cases it describes "regular polyhedrons" with $k$ edges in each vertex, $n$ edges of each face, with total number of edges $|m|$, and Euler characteristics $χ=\pm 2$. In the case of negative $m$ this equation corresponds to $χ=2$ and describes true regular polyhedrons, Platonic solids. The case with positive $m$ corresponds to Euler characteristic $χ=-2$ and describes the so called equivelar maps (charts) on the surface of genus $2$. In the former case there are two routes from Platonic solids to simple Lie algebras - abovementioned Diophantine classification and McKay correspondence. We compare them for all solutions of this type, and find coincidence in the case of icosahedron (dodecahedron), corresponding to $E_8$ algebra. In the case of positive $k$, $n$ and $m$ we obtain in this way the interpretation of (some of) the mysterious solutions (Y-objects), appearing in the Diophantine classification and having some similarities with simple Lie algebras.

math-ph↗