Pith. sign in

REVIEW 4 major objections 3 minor 83 references

Quantization of Algebraic Varieties Defined by Casimir Polynomials via Matrix Regularization: Fuzzy $S^7$ and Beyond

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that algebraic varieties defined by a single Casimir polynomial of a compact semisimple Lie algebra admit a weak matrix regularization whose classical limit is the variety, and constructs fuzzy $S^7$ as the working example.

desk verdict A correct but carefully scoped existence theorem for quantizing Casimir level sets; the title overstates what 'fuzzy S^7' means. read the letter →

arxiv 2608.04717 v1 pith:6O7VQXY6 submitted 2026-08-05 hep-th math-phmath.MP

Akifumi Sako This is my paper · ORCID
classification hep-thmath-phmath.MP MSC 81R6053D1722E46
keywords matrixregularizationweakfuzzyS^7CasimirpolynomialscoadjointorbitsLie-Poissonalgebrareduciblerepresentationsdenseorbitfilling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quantizing an algebraic variety means approximating its algebra of functions by matrix algebras, with Poisson brackets turning into commutators. This paper proposes a way to do this for varieties defined by fixing one Casimir polynomial of a compact semisimple Lie algebra, such as the seven-sphere $S^7$ inside $\mathfrak{su}(3)^*$. The key move is to quantize the variety piece by piece, using irreducible representations for each coadjoint orbit inside it, and then glue the pieces into a reducible representation; as the representations grow, the orbits fill the variety densely and the matrix algebras converge to the original variety in the classical limit. The construction is carried out explicitly for fuzzy $S^7$, and a general theorem asserts the same works for any compact semisimple Lie algebra. If correct, this gives a systematic way to produce fuzzy versions of many odd-dimensional spaces that ordinary symplectic quantization cannot directly handle.

What carries the argument

The central object is the weak matrix regularization of a Lie–Poisson algebra, specifically its reducible-representation form $q^R_{A/I,\mu} := \oplus_{a=1}^{m_\mu} q^a_{A/I,\mu}$, built by direct sum over quantizations $q^a$ associated with irreducible representations. Each irreducible block has its own Planck constant $\hbar_a(\mu)$, chosen so that the defining Casimir equation $C_k(x)=\lambda_k$ holds as an operator identity on that block; the direct sum then quantizes the whole variety. The mechanism that carries the argument is convergence in the sense of relations: the kernel of an irreducible quantization, generated by representation-dependent Casimir relations, converges as $\dim V_\mu \to \infty$ to the defining relations of the coadjoint orbit selected by the sequence of highest weights. Dense filling of the variety follows because the remaining Casimir values are continuous functions of the highest-weight direction, so letting the weight ratios range over a dense set makes the associated orbits fill $S^7$ (or $S_k$) in the classical limit.

What would settle it

Take the explicit sequence $q^R_{A/I,k} = \oplus_{p+q=k} q_{A/I,(p,q)}$ in (5.29) and compute, for increasing $k$, the operator norm of the commutator defect $[q^R(f), q^R(g)] - \widehat{\hbar} \, q^R(\{f,g\})$ for a fixed pair of low-degree polynomials $f,g$; a bound that grows without limit as $k$ increases would falsify the uniform error estimate that Theorem 2.11 asserts. Equivalently, check whether the kernels of the block quantizations converge in the Hausdorff metric to the ideal generated by $C_2(x)-r^2$ and $C_3(x)-r^3R$ for every admissible $R$ in the interval $[-\sqrt{3}/3, \sqrt{3}/3]$.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is Theorem 5.2: there is a weak matrix regularization of $S^7$ built from a sequence of reducible representations whose classical limit, in the sense of weak matrix regularization, is $S^7$. More generally, Theorem 6.3 states that for any compact semisimple Lie algebra and any admissible nonzero value of one of its Casimir polynomials, the algebraic variety $S_k = \{x : C_k(x) = \lambda_k\}$ admits such a weak matrix regularization. The construction proceeds by decomposing $S_k$ into coadjoint orbits, choosing for each orbit a sequence of irreducible representations that quantizes it in the sense of relations, and then taking direct sums of those representations with block-dependent Planck constants rescaled so that the fixed Casimir equation holds on each block; the classical limit recovers $S_k$ because the chosen orbits become dense in it. This is a dense-orbit quantization scheme, distinct from standard fuzzy-space constructions that quantize a single symplectic leaf.

Load-bearing premise

The construction depends on Theorem 2.11, stated without proof, that the block-diagonal direct sum of these quantizations is still a weak matrix regularization with error terms controlled by the largest block Planck constant; if that control fails as the number of blocks grows, the dense orbit filling would not produce a genuine quantization.

Editorial extensions

If this is right

  • Fuzzy $S^7$ is realized as a genuine weak matrix regularization, not merely as an informal union of fuzzy spheres.
  • For any compact semisimple Lie algebra, every admissible single-Casimir level set $S_k$ acquires a fuzzy-space quantization, including odd-dimensional spheres such as $S^{n^2-2}$ from type $A_{n-1}$ and $S^{n(2n+1)-1}$ from type $B_n$.
  • The same construction applies to algebraic varieties defined by Casimir polynomials of degree greater than two, not only quadratic ones.
  • The method provides a systematic way to build matrix configurations whose classical geometries are prescribed Casimir varieties, relevant to matrix models whose classical solutions are Lie algebra elements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The dense-filling construction suggests a general recipe: any Poisson variety that decomposes into a family of quantizable symplectic leaves, with the leaves dense in a suitable limit, could be quantized by this direct-sum method; the paper only needs the leaf family to be coadjoint orbits.
  • If Theorem 2.11's uniform error estimates hold, the classical limit should be witnessed by a natural filtration of the matrix algebras by PBW degree, and making such a filtration explicit would strengthen the weak regularization into a sharper correspondence.
  • The method may extend to noncompact Lie algebras if the orbit method and admissible representations are replaced by suitable fall-off conditions, though the dense filling of unbounded orbits would require a different limiting argument.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The paper studies quantization of algebraic varieties defined by fixing one Casimir polynomial of a compact semisimple Lie algebra, using the framework of weak matrix regularization of Lie-Poisson algebras. It reviews the construction for irreducible representations, shows that suitable sequences of representations yield, in a certain 'sense of relations,' classical limits that are coadjoint orbits, and then forms reducible representations by direct sums so that the corresponding coadjoint orbits become dense in the desired variety. The main concrete result is Theorem 5.2, which claims a weak matrix regularization of S^7 built from the reducible sequence (5.29), and the general version is Theorem 6.3 for any admissible single-Casimir level set S_k in a compact semisimple Lie algebra.

Significance. The paper contains several correct and useful ingredients: the su(3) Casimir eigenvalue formulas (B.5)-(B.7), the range computation for the normalized cubic Casimir (Proposition 3.2), the density argument for the lattice directions p/q in (5.31)-(5.32), and the general reduction to dominant Weyl chamber elements in Theorem 6.3. These calculations are explicit and mostly checkable. The proposed 'dense-orbit' construction is an interesting way to approximate a Casimir level set by a growing family of coadjoint-orbit quantizations, and the paper is honest in Section 7 about the open problem of strengthening the convergence. The central advertised claim, however, goes beyond what is proved, because the classical limit is defined in a weaker sense than a genuine matrix regularization of the function algebra of the variety.

major comments (4)
  1. [§5.1, Definition 5.1, Theorem 5.2] The notion of 'classical limit in the sense of weak matrix regularization' is strictly weaker than what is normally meant by a matrix regularization of S^7 and does not justify the title's 'fuzzy S^7'. For the explicit sequence (5.29), End(V_k) is isomorphic to the direct sum of End(V_(p,q)) over p+q=k, and every quantization map q^R_{A/I,k} is block diagonal. The center of the generated matrix algebra therefore contains the k+1 block projectors, whereas the commutative C*-algebra C(S^7) has trivial center. No mechanism is provided for off-block matrix elements to appear, so there is no continuity imposed between neighboring coadjoint orbits. The paper itself acknowledges after Eq. (5.19) that the asymptotic algebra homomorphism is not asserted to be surjective degree by degree, and Section 7 lists 'a more precise analysis of convergence in the sense of relations' as open. Thus the proven result is the existence of a dense family of coadjoint-orbit quantizations, not a quantization of functions on S^7. The theorem should either be strengthened by proving genuine convergence of the quantized algebras to C(S^7), or the abstract and title claims should be reformulated to state the dense-orbit result precisely.
  2. [§2.3, Theorem 2.11] Theorem 2.11 is stated without proof and is load-bearing for both Theorem 5.2 and Theorem 6.3. It is not a trivial consequence of Theorem 2.8 when the number m_μ of blocks grows with the sequence, because the block-dependent scaling factors r_a(μ) enter the error term in (2.19) and must be controlled against max_a |ℏ_a(μ)| and the PBW filtration. A complete proof, or a reference to a proof with explicit hypotheses, is needed, including the case m_μ→∞ where individual block dimensions grow at different rates. Without this estimate, the statement that q^R_{A/I,μ} is a weak matrix regularization is not established.
  3. [§6, Theorem 6.3(iii)] The diagonal selection argument in part (iii) of the proof is only sketched. For each j and each a=1,...,m_j, the proof chooses a 'sufficiently large member' of the sequence corresponding to O_a, but it must be shown that these choices can be made simultaneously so that each selected block satisfies the required approximation within the same j, and that the finite unions ∪_{a≤m_j} O_a become dense in S_k as j→∞. This requires a quantitative version of 'converges in the sense of relations' with an error bound that decreases with the representation dimension. As written, the density statement does not follow formally from the preceding qualitative convergence statements.
  4. [§5.1, Eqs. (5.11)-(5.14)] The passage from the remainder r_{C_3} to the asymptotic orbit relation C_3 = r^3 R uses an ℏ-correction term ℏ D(e^{(p,q)}). Although the correction is said to be O(ℏ), the estimate needed to show that it vanishes uniformly along the chosen subsequence is not written out. The needed bound involves the PBW filtration and the same uniformity issue as Theorem 2.11, so it should be made explicit in order to establish the 'convergence in the sense of relations' of each irreducible block.
minor comments (3)
  1. [§5.1, Eq. (5.11)] The notation 'r C_3(x)' in Eq. (5.11) is ambiguous: it appears to denote the remainder r_{C_3}(x), but the subscript is not typeset, making it easy to confuse with the radius r. Please use a clear notation such as r_{C_3}(x).
  2. [§2.3 and §5] The symbol ℏ(μ) is used both for the global sequence and for the block parameters ℏ_a(μ)=ℏ(μ) r_a(μ). This overloaded notation becomes confusing in the su(3) construction, where pairs (p_n,q_n) are also indexed. A separate symbol for the global scale and for each block scale would improve readability.
  3. [§7] The statement that the same construction yields fuzzy S^{n^2-2} for A_{n-1} and analogous spheres for other compact simple Lie algebras is not demonstrated; the necessary analogue of the density argument for the relevant lattice directions should at least be sketched or deferred to a future paper.

Circularity Check

2 steps flagged · score 6.0 of 10

The 'classical limit is S^7' result is effectively Definition 5.1 applied to a direct-sum construction; the underlying reducible-representation regularization is also imported from the author's own [34,35].

  1. self definitional [Definition 5.1; Theorem 5.2; Section 5.1 after (5.19); Section 7]
    "Definition 5.1. Let M be an algebraic variety that admits a decomposition into a disjoint union M=`_i M_i. Suppose that the matrix regularizations q^a_{A/I,\mu} introduced in Definition 2.10, each associated with an irreducible component of the representation, converge to the spaces M_{i_a}_\mu in the sense of relations. Suppose further that the resulting sequence of spaces M_{i_a}_\mu fills M densely. ... Then we say that q^R_{A/I,\mu}:A_g/I(C)\to End(V_\mu) in Definition 2.10 has M as its classical limit in the sense of weak matrix regularization. ... Theorem 5.2."

    The theorem is just the definition applied to the constructed direct sum: the proof checks that each irreducible block converges to a coadjoint orbit and that the p/q directions fill the cubic-Casimir interval, then invokes Definition 5.1. No independent convergence to C(S^7) is established. The paper itself concedes that the asymptotic algebra homomorphism 'need not be surjective onto the coordinate ring of the coadjoint orbit' and lists 'a more precise analysis of convergence in the sense of relations' as open. Thus the advertised conclusion 'classical limit is S^7' is stipulated by the paper's own weakened classical-limit notion rather than derived from a standard matrix-regularization criterion.

  2. self citation load bearing [Section 2.3, Theorem 2.11; relied on by (5.29) and Theorem 6.3]
    "Finally, we obtain the following. Theorem 2.11. q^R_{A/I,\mu} is a weak matrix regularization, i.e., for any [f],[g]\in A_g/I(C), there exists P=... such that [q^R_{A/I,\mu}([f]), q^R_{A/I,\mu}([g])] = \hat{\hbar}(\mu)q^R_{A/I,\mu}(\{[f],[g]\}) + (\hat{\hbar}(\mu))^2P."

    This theorem is the only statement that the reducible-representation map used in the explicit fuzzy S^7 sequence (5.29) and in the general Theorem 6.3 is a weak matrix regularization. It is stated without proof in the present paper; the surrounding text says that the precise definition is given in [34] and the remark routes the construction through [35], where [34] is the author's own paper with J. Gohara and [35] is Gohara's doctoral thesis. The product and commutator estimates controlling the direct sum as the number of blocks grows are therefore load-bearing premises taken from the author's own prior work rather than established in this paper.

full rationale

The asymptotic Casimir-limit arguments are not circular: Proposition 3.1 and the su(3) range of R follow from the Weyl dimension formula, the Harish-Chandra isomorphism, and elementary limits, and the lattice-direction gap estimate in Section 5 is a standard independent calculation. I am not claiming that the prior papers [34,35] are wrong. The circularity lies in the packaging: 'classical limit in the sense of weak matrix regularization' is defined to be precisely the dense-filling property that the reducible construction achieves, and then Theorems 5.2 and 6.3 assert that property as the main result. The paper's own caveats, that the asymptotic algebra homomorphism need not be surjective degree by degree and that a more precise convergence-in-relations analysis remains open, confirm that this is not convergence to C(S^7) in the ordinary sense. In addition, the key estimate that makes the reducible direct sum a weak matrix regularization is imported from the author's own previous work via Theorem 2.11 without proof here. Together these make the advertised 'fuzzy S^7' partially definitional and partially self-citational, although genuine independent content remains in the orbit decomposition and density proofs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The central claim rests on the imported weak matrix regularization theorems [34,35] and on standard Lie theory facts. There are no fitted data or hidden constants; the only chosen inputs are the radius r, the asymptotic direction t, and the block count m_j, all construction parameters. The invented notions of convergence are weaker than standard fuzzy-space convergence, which is the main reason the headline 'fuzzy S^7' should be read with caution.

free parameters (3)
  • radius r = r > 0
    Defines the level set C_2(x)=r^2. Chosen by hand as the radius of fuzzy S^7; not fitted to data.
  • asymptotic direction t = lim_{n→∞} p_n/q_n = t in [0,∞]
    Selects which coadjoint orbit a sequence of irreducible quantizations converges to. The construction uses density of rational directions to cover S^7.
  • block count m_j = increasing sequence m_j → ∞
    In Theorem 6.3, m_j is the number of orbits included at step j; chosen to make the union denser as j grows.
assumptions (5)
  • domain assumption Theorems 2.5, 2.6, 2.8, 2.11: q_{A/I,mu} and q^R_{A/I,mu} are weak matrix regularizations with the stated product and commutator estimates.
    Stated without proof and cited to [34] and [35]. These imported results carry the quantization step of the whole paper.
  • standard math Chevalley restriction theorem and Harish-Chandra isomorphism relate invariant polynomial values on the Lie algebra to Casimir eigenvalues on highest-weight modules.
    Used in Proposition 3.1 and Theorem 6.3 to compute limits of rescaled Casimir eigenvalues.
  • standard math Weyl dimension formula and the asymptotic dim V_mu ~ ||mu||^{#Phi+} for regular directions.
    Proposition 3.1 uses this to show ℏ(mu)||mu|| → alpha and that the sequence is a weak matrix regularization.
  • domain assumption The reduced Groebner basis of the principal ideal (C_k(x)-lambda_k) is the polynomial itself, or at least the remainder map R_G is well-defined and compatible with q_mu.
    The paper restricts to single-Casimir ideals and uses R_G throughout; no proof is given that remainders behave as needed for the reducible construction.
  • domain assumption For the chosen sequences, the PBW-degree bound n_mu ≥ deg r_f eventually holds for all f used in the constructions.
    Needed so that q_{A/I,mu}([f]) is defined without truncation; the paper states this can be arranged by taking dim V_mu large.
invented entities (2)
  • classical limit in the sense of relations
    purpose: Defines convergence of a matrix regularization sequence to an algebraic variety when the kernel relations of the quantization maps converge to the defining ideal of the variety, without requiring degreewise isomorphism of function algebras.
    Introduced in Section 5. The authors explicitly state that phi_mu need not become surjective degree by degree, so this notion is weaker than standard convergence of matrix algebras to C(M).
  • classical limit in the sense of weak matrix regularization
    purpose: Aggregates per-orbit relation-limits by requiring the corresponding coadjoint orbits to fill the target variety densely; used to state Theorems 5.2 and 6.3.
    Definition 5.1. Because the quantization is a direct sum over orbits and the filling is only dense, this is a paper-specific convergence concept without an external falsifiable handle.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantization of Algebraic Varieties Defined by Casimir Polynomials via Matrix Regularization: Fuzzy $S^7$ and Beyond." pith.science (2026). https://pith.science/paper/6O7VQXY6

@misc{pith2026260804717,
  author       = {Pith},
  title        = {Pith review of: Quantization of Algebraic Varieties Defined by Casimir Polynomials via Matrix Regularization: Fuzzy $S^7$ and Beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6O7VQXY6}},
  note         = {Machine review of arXiv:2608.04717}
}
abstract

We study the quantization of algebraic varieties defined by equations involving Casimir polynomials of compact semisimple Lie algebras. The Casimir polynomials belong to the Poisson center of the corresponding Lie-Poisson algebra. For this purpose, we employ a recently developed matrix regularization of Lie-Poisson algebras. In particular, using its formulation based on reducible representations, we construct quantizations of these algebraic varieties through their decomposition into coadjoint orbits, including singular orbits. As a concrete example, we present the construction of fuzzy $S^7$ in detail.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

83 extracted references · 66 canonical work pages

  1. [1]

    Lie Groups and Geometric Aspects of Isometric Actions,

    M. M. Alexandrino and R. G. Bettiol, “Lie Groups and Geometric Aspects of Isometric Actions,” Springer, Cham (2015)

    2015

  2. [2]

    Fuzzy CP2

    G. Alexanian, A.P. Balachandran, G. Immirzi and B. Ydri, “FuzzyCP 2,” Journal of Geometry and Physics,42, (2002), 28-53. [arXiv:hep-th/0103023[hep- th]]

    work page Pith review arXiv 2002

  3. [3]

    Finite N Matrix Models of Noncommutative Gauge Theory

    J. Ambjorn, Y. M. Makeenko, J. Nishimura and R. J. Szabo, “Finite N matrix models of noncommutative gauge theory,” Journal of High Energy Physics,11 (1999), 029 [arXiv:hep-th/9911041 [hep-th]]

    work page Pith review arXiv 1999

  4. [4]

    Nonperturbative Dynamics of Noncommutative Gauge Theory

    J. Ambjorn, Y. M. Makeenko, J. Nishimura and R. J. Szabo, “Nonperturbative dynamics of noncommutative gauge theory,” Physics Letters B,480(2000), 399- 408 [arXiv:hep-th/0002158 [hep-th]]

    work page Pith review arXiv 2000

  5. [5]

    Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory

    J. Ambjorn, Y. M. Makeenko, J. Nishimura and R. J. Szabo, “Lattice gauge fields and discrete noncommutative Yang-Mills theory,” Journal of High Energy Physics, 05(2000), 023 [arXiv:hep-th/0004147 [hep-th]]

    work page Pith review arXiv 2000

  6. [6]

    Progress in the numerical stud- ies of the type IIB matrix model,

    K. N. Anagnostopoulos, T. Azuma, K. Hatakeyama, M. Hirasawa, Y. Ito, J. Nishimura, S. K. Papadoudis and A. Tsuchiya, “Progress in the numerical stud- ies of the type IIB matrix model,” The European Physical Journal Special Topics, 232, no.23-24, (2023), 3681-3695. [arXiv:2210.17537 [hep-th]]

    arXiv 2023

  7. [7]

    The emergence of (3+1)-dimensional expanding spacetime from complex Langevin simulations of the Lorentzian type IIB matrix model with de- formations,

    K. N. Anagnostopoulos, T. Azuma, M. Hirasawa, J. Nishimura, S. Papadoudis and A. Tsuchiya, “The emergence of (3+1)-dimensional expanding spacetime from complex Langevin simulations of the Lorentzian type IIB matrix model with de- formations,” [arXiv:2604.19836 [hep-th]]

    arXiv

  8. [8]

    Noncommutative Yang-Mills in IIB Matrix Model

    H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, “Noncommu- tative Yang-Mills in IIB matrix model,” Nuclear Physics B,565(2000), 176-192 doi:10.1016/S0550-3213(99)00633-1 [arXiv:hep-th/9908141 [hep-th]]

    work page Pith review arXiv 2000

Show all 83 references
  1. [9]

    Noncommuta- tive Riemann Surfaces by Embeddings inR 3,

    J. Arnlind, M. Bordemann, L. Hofer, J. Hoppe and H. Shimada, “Noncommuta- tive Riemann Surfaces by Embeddings inR 3,” Communications in Mathematical Physics,288(2009), 403-429. [arXiv:0711.2588 [math-ph]]

    2009 arXiv

  2. [10]

    Fuzzy Riemann surfaces,

    J. Arnlind, M. Bordemann, L. Hofer, J. Hoppe and H. Shimada, “Fuzzy Riemann surfaces,” Journal of High Energy Physics,2009.06, (2009), 047

    2009

  3. [11]

    Noncommutative Minimal Surfaces,

    J. Arnlind, J. Choe and J. Hoppe, “Noncommutative Minimal Surfaces,” Letters in Mathematical Physics,106, (2016), 1109-1129. [arXiv:1301.0757 [math.QA]]

    2016 arXiv

  4. [12]

    More Membrane Matrix Model Solutions, and Minimal Surfaces inS 7,

    J. Arnlind and J. Hoppe, “More Membrane Matrix Model Solutions, and Minimal Surfaces inS 7,” (2003). [arXiv:hep-th/0312062 [hep-th]]

    2003 arXiv

  5. [13]

    Discrete Minimal Surface Algebras

    J. Arnlind and J. Hoppe, “Discrete Minimal Surface Algebras” Symmetry, Inte- grability and Geometry: Methods and Applications (SIGMA) Contribution to the Special Issue “Noncommutative Spaces and Fields,” 6 (2010), 042, 18pp. [arXiv:0903.5237[math.QA]]

    2010 arXiv

  6. [14]

    Multi-linear Formulation of Differential Geometry and Matrix Regularizations,

    J. Arnlind, J. Hoppe and G. Huisken, “Multi-linear Formulation of Differential Geometry and Matrix Regularizations,” Journal of Differential Geometry,91.1, (2012), 1-39. 33

    2012

  7. [15]

    Quantum Minimal Surfaces,

    J. Arnlind, J. Hoppe and M. Kontsevich, “Quantum Minimal Surfaces,” (2019), [arXiv:1903.10792 [math-ph]]

    2019 arXiv

  8. [16]

    Commuta- tive geometry for non-commutative D-branes by tachyon condensation,

    T. Asakawa, G. Ishiki, T. Matsumoto, S. Matsuura, and H. Muraki, “Commuta- tive geometry for non-commutative D-branes by tachyon condensation,” Progress of Theoretical and Experimental Physics, (2018), 063B04

    2018

  9. [17]

    Defining the Type IIB Matrix Model without Breaking Lorentz Symmetry,

    Y. Asano, J. Nishimura, W. Piensuk and N. Yamamori, “Defining the Type IIB Matrix Model without Breaking Lorentz Symmetry,” Physical Review Letters,134 no.4, (2025), 041603. [arXiv:2404.14045 [hep-th]]

    2025 arXiv

  10. [18]

    Path integral for the closed superstring and the matrix model,

    Y. Asano, “Path integral for the closed superstring and the matrix model,” [arXiv:2604.25052 [hep-th]]

    arXiv

  11. [19]

    Dynamical aspects of the fuzzy CP 2 in the large N reduced model with a cubic term,

    T. Azuma, S. Bal, K. Nagao and J. Nishimura, “Dynamical aspects of the fuzzy CP 2 in the large N reduced model with a cubic term,” Journal of High Energy Physics,05, (2006), 061. [arXiv:hep-th/0405277 [hep-th]]

    2006 arXiv

  12. [20]

    M Theory As A Matrix Model: A Conjecture,

    T. Banks, W. Fischler, S. Shenker and L. Susskind, “M Theory As A Matrix Model: A Conjecture,” Physical Review D,55, (1997), 5112-5128

    1997

  13. [21]

    Fuzzy torus in matrix model,

    S. Bal, M. Hanada, H. Kawai and F. Kubo, “Fuzzy torus in matrix model,” Nuclear Physics B,727, (2005), 196-217. [arXiv:hep-th/0412303 [hep-th]]

    2005 arXiv

  14. [22]

    Lectures on fuzzy and fuzzy SUSY physics,

    A. P. Balachandran, S. K¨ urk¸ c¨ uo˘ glu, and S. Vaidya, “Lectures on fuzzy and fuzzy SUSY physics,” World Scientific, (2007)

    2007

  15. [23]

    Fuzzy complex projective spaces and their star products,

    A. P. Balachandran, B. P. Dolan, J. H. Lee, X. Martin and D. O’Connor, “Fuzzy complex projective spaces and their star products,” Journal of Geometry and Physics,43, (2002), 184-204. [arXiv:hep-th/0107099 [hep-th]]

    2002 arXiv

  16. [24]

    Matrix embeddings on flatR 3 and the geometry of membranes,

    D. Berenstein and E. Dzienkowski, “Matrix embeddings on flatR 3 and the geometry of membranes,” Physical Review D,86, (2012), 086001

    2012

  17. [25]

    gl(∞) and geomet- ric quantization,

    M. Bordemann, J. Hoppe, P. Schaller and M. Schlichenmaier, “gl(∞) and geomet- ric quantization,” Communications in Mathematical Physics,138no. 2, (1991), 209-244

    1991

  18. [26]

    Toeplitz quantization of K¨ ahler manifolds andgl(N),N→ ∞limits,

    M. Bordemann, E. Meinrenken and M. Schlichenmaier, “Toeplitz quantization of K¨ ahler manifolds andgl(N),N→ ∞limits,” Communications in Mathematical Physics,165.2, (1994), 281-296

    1994

  19. [27]

    Inequivalence between the Euclidean and Lorentzian Versions of the Type IIB Matrix Model from Lefschetz Thimble Calculations,

    C. Y. Chou, J. Nishimura and A. Tripathi, “Inequivalence between the Euclidean and Lorentzian Versions of the Type IIB Matrix Model from Lefschetz Thimble Calculations,” Physical Review Letters,134(2025), 211601. [arXiv:2501.17798 [hep-th]]

    2025 arXiv

  20. [28]

    Scaling limits of the fuzzy sphere at one loop,

    C. S. Chu, J. Madore and H. Steinacker, “Scaling limits of the fuzzy sphere at one loop,” Journal of High Energy Physics,08, (2001), 038. [arXiv:hep-th/0106205 [hep-th]]

    2001 arXiv

  21. [29]

    Noncommutative geometry and matrix theory: Compactification on Tori,

    A. Connes, M.R. Douglas and A. Schwarz, “Noncommutative geometry and matrix theory: Compactification on Tori,” Journal of High Energy Physics,1998.02, (1998), 003

    1998

  22. [30]

    On the Quantum Mechanics of Superme- mbranes,

    B. de Wit, J. Hoppe and H. Nicolai, “On the Quantum Mechanics of Superme- mbranes,” Nuclear Physics B,305.4, (1988), 545-581. 34

    1988

  23. [31]

    Abstract Algebra,

    D. Dummit and R. Foote, “Abstract Algebra,” 3rd Edition, John Wiley and Sons, Inc. (2004)

    2004

  24. [32]

    Categorical Perspective on Quantization of Poisson Algebra,

    J. Gohara, Y. Hirota and A. Sako, “Categorical Perspective on Quantization of Poisson Algebra,” Journal of Mathematical Physics,61no.7, (2020), 073506. [arXiv:1907.08665 [math-ph]]

    2020 arXiv

  25. [33]

    A Generalization of the Quantization of Poisson Manifolds,

    J. Gohara, Y. Hirota and A. Sako, “A Generalization of the Quantization of Poisson Manifolds,” Geometry, Integrability and Quantization,21, (2020), 138-148

    2020

  26. [34]

    Quantization of Lie–Poisson algebra and Lie algebra solutions of mass-deformed type IIB matrix model,

    J. Gohara and A. Sako, “Quantization of Lie–Poisson algebra and Lie algebra solutions of mass-deformed type IIB matrix model,” J. Math. Phys.67(2026) no.2, 022301 doi:10.1063/5.0278248 [arXiv:2503.24060 [hep-th]]

    2026

  27. [35]

    Generalization of Matrix Regularization and a Unified Framework for Quantization,

    J. Gohara , “Generalization of Matrix Regularization and a Unified Framework for Quantization,” Doctoral thesis in Tokyo University of Science (2026) available at https://tus.repo.nii.ac.jp/records/2000602

    2026

  28. [36]

    The Twisted Eguchi-Kawai Model: A Re- duced Model for Large N Lattice Gauge Theory,

    A. Gonzalez-Arroyo and M. Okawa, “The Twisted Eguchi-Kawai Model: A Re- duced Model for Large N Lattice Gauge Theory,” Physical Review D,27(1983), 23972411

    1983

  29. [37]

    Connections, Curvature, and Coho- mology volume III

    W. Greub, S. Halperin, and R. Vanstone, “ Connections, Curvature, and Coho- mology volume III” Academic Press (1976)

    1976

  30. [38]

    Finite gauge theory on fuzzyCP 2,

    H. Grosse and H. Steinacker, “Finite gauge theory on fuzzyCP 2,” Nuclear Physics B,707, (2005), 145-198. [arXiv:hep-th/0407089 [hep-th]]

    2005 arXiv

  31. [39]

    Towards a nonperturbative covariant regularization in 4-D quantum field theory,

    H. Grosse and A. Strohmaier, “Towards a nonperturbative covariant regularization in 4-D quantum field theory,” Letters in Mathematical Physics,48, (1999), 163-

    1999

  32. [40]

    Coherent state induced star product onR 3 λ and the fuzzy sphere,

    A. B. Hammou, M. Lagraa and M. M. Sheikh-Jabbari, “Coherent state induced star product onR 3 λ and the fuzzy sphere,” Physical Review D66(2002), 025025 [arXiv:hep-th/0110291 [hep-th]]

    2002 arXiv

  33. [41]

    Introduction to Lie Algebras and Representation Theory

    J. E. Humphreys, “Introduction to Lie Algebras and Representation Theory.” Springer, GTM 9, (1972)

    1972

  34. [42]

    Non-Compact Hopf Maps and Fuzzy Ultra-Hyperboloids,

    K. Hasebe, “Non-Compact Hopf Maps and Fuzzy Ultra-Hyperboloids,” Nuclear Physics B,865, (2012), 148-199. [arXiv:1207.1968 [hep-th]]

    2012 arXiv

  35. [43]

    Super Landau Model and Howe Duality: From Supermonopole Har- monics to Quantum Matrix Geometry,

    K. Hasebe, “Super Landau Model and Howe Duality: From Supermonopole Har- monics to Quantum Matrix Geometry,” [arXiv:2604.24112 [hep-th]]

    arXiv

  36. [44]

    Quantum theory of a massless relativistic surface and a two-dimensional bound state problem,

    J. Hoppe, “Quantum theory of a massless relativistic surface and a two-dimensional bound state problem,” Soryushiron Kenkyu Electronics,80.3, (1989), 145-202

    1989

  37. [45]

    A large-Nreduced model as superstring,

    N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, “A large-Nreduced model as superstring,” Nuclear Physics B,498.1-2, (1997), 467-491

    1997

  38. [46]

    K¨ ahler structure in the commutative limit of matrix geometry,

    G. Ishiki, T. Matsumoto, and H. Muraki, “K¨ ahler structure in the commutative limit of matrix geometry,” Journal of High Energy Physics,08, (2016), 042

    2016

  39. [47]

    Information metric, Berry connection, and Berezin-Toeplitz quantization for matrix geometry,

    G. Ishiki, T. Matsumoto, and H. Muraki, “Information metric, Berry connection, and Berezin-Toeplitz quantization for matrix geometry,” Physical Review D,98, (2018), 026002

    2018

  40. [48]

    2D fuzzy Anti-de Sitter space from matrix models,

    D. Jurman and H. Steinacker, “2D fuzzy Anti-de Sitter space from matrix models,” Journal of High Energy Physics,01, (2014), 100. [arXiv:1309.1598 [hep-th]]. 35

    2014 arXiv

  41. [49]

    Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis,

    D. Jurman, “Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis,” Fortschritte der Physik,67no.4, (2019), 1800061. [arXiv:1710.01491 [math-ph]]

    2019 arXiv

  42. [50]

    Description of curved spacetimes by finite- size matrices in the type IIB matrix model,

    K. Hattori, T. Seko and A. Tsuchiya, “Description of curved spacetimes by finite- size matrices in the type IIB matrix model,” Progress of Theoretical and Experi- mental Physics 2026, Issue 8, 083B01(2026), ptag120. [arXiv:2606.24577 [hep-th]]

    2026 arXiv

  43. [51]

    Expanding universe as a classical solution in the Lorentzian matrix model for nonperturbative superstring theory,

    S. W. Kim, J. Nishimura and A. Tsuchiya, “Expanding universe as a classical solution in the Lorentzian matrix model for nonperturbative superstring theory,” Physical Review D,86(2012), 027901 [arXiv:1110.4803 [hep-th]]

    2012 arXiv

  44. [52]

    Expanding (3+1)-dimensional uni- verse from a Lorentzian matrix model for superstring theory in (9+1)-dimensions,

    S. W. Kim, J. Nishimura and A. Tsuchiya, “Expanding (3+1)-dimensional uni- verse from a Lorentzian matrix model for superstring theory in (9+1)-dimensions,” Physical Review Letters,108, (2012), 011601. [arXiv:1108.1540 [hep-th]]

    2012 arXiv

  45. [53]

    Late time behaviors of the expanding universe in the IIB matrix model,

    S. W. Kim, J. Nishimura and A. Tsuchiya, “Late time behaviors of the expanding universe in the IIB matrix model,” Journal of High Energy Physics,10, (2012),

    2012

  46. [54]

    Lectures on the Orbit Method,

    A. A. Kirillov, “Lectures on the Orbit Method,” Graduate Studies in Mathematics, vol. 64, American Mathematical Society, 2004

    2004

  47. [55]

    Quantum Riemann Surfaces I. The Unit Disc,

    S. Klimek and A. Lesniewski, “Quantum Riemann Surfaces I. The Unit Disc,” Communications in Mathematical Physics,146, (1992), 103-122

    1992

  48. [56]

    Lie Group Representations on Polynomial Rings,

    B. Kostant, “Lie Group Representations on Polynomial Rings,” American Journal of Mathematics, vol.85no.3, (1963), 327-404

    1963

  49. [57]

    Quantization and Unitary Representations,

    B. Kostant, “Quantization and Unitary Representations,” Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics, vol. 170, Springer, Berlin, (1970) 87–208,

    1970

  50. [58]

    Effective mass and symmetry breaking in the Ishibashi-Kawai- Kitazawa-Tsuchiya matrix model from compactification,

    S. Laliberte, “Effective mass and symmetry breaking in the Ishibashi-Kawai- Kitazawa-Tsuchiya matrix model from compactification,” Physical Review D,110 no.2, (2024), 026024. [arXiv:2401.16401 [hep-th]]

    2024 arXiv

  51. [59]

    Theorie der transformationsgruppen,

    S. Lie, “Theorie der transformationsgruppen,” (Zweiter Abschnitt, unter mitwirkung von Prof. Dr. Friedrich Engel), Teubner, Leipzig, 1890

  52. [60]

    Deformation Quantization of Nonregular Orbits of Compact Lie Groups,

    M. A. Lled´ o, “Deformation Quantization of Nonregular Orbits of Compact Lie Groups,” Lett. Math. Phys.58, 57–67 (2001); arXiv:math/0105191. doi:10.1023/A:1012560617508

    2001 arXiv

  53. [61]

    Poisson-Lie groups, dressing transformations and Bruhat decompositions,

    J.H. Lu and A. Weinstein, “Poisson-Lie groups, dressing transformations and Bruhat decompositions,” Journal of Differential Geometry, vol.31, (1990), 501- 526

    1990

  54. [62]

    The fuzzy sphere,

    J. Madore, “The fuzzy sphere,” Classical and Quantum Gravity,9.1, (1992), 69-87

    1992

  55. [63]

    An Introduction to Noncommutative Differential Geometry and its Physical Applications,

    J. Madore, “ An Introduction to Noncommutative Differential Geometry and its Physical Applications,” London Mathematical Society Lecture Note Series, Cam- bridge University Press, (1999)

    1999

  56. [64]

    On brane solutions in M(atrix) theory,

    V. P. Nair and S. Randjbar-Daemi, “On brane solutions in M(atrix) theory,” Nu- clear Physics B,533, (1998), 333-347. [arXiv:hep-th/9802187 [hep-th]]

    1998 arXiv

  57. [65]

    Deformation Quantization of Polynomial Poisson Algebras,

    M. Penkava and P. Vanhaecke, “Deformation Quantization of Polynomial Poisson Algebras,” Journal of Algebra,2271, (2000), 365-393. [arXiv:math/9804022] 36

    2000 arXiv

  58. [66]

    Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures,

    M. Rieffel, “Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures,” American Journal of Mathematics,112no. 4, (1990), 657-685

    1990

  59. [67]

    Dirac Operators for Matrix Algebras Converging to Coad- joint Orbits,

    M. A. Rieffel, “Dirac Operators for Matrix Algebras Converging to Coad- joint Orbits,” Communications in Mathematical Physics,401(2023), 1951-2009. [arXiv:2108.01136 [math.OA]]

    2023 arXiv

  60. [68]

    Category of quantizations and inverse problem,

    A. Sako, “Category of quantizations and inverse problem,” Nuclear Physics B, 989, (2023), 116146. [arXiv:2205.09019 [math-ph]]

    2023 arXiv

  61. [69]

    Lie Algebra and Quantization in Quantum World,

    A. Sako, “Lie Algebra and Quantization in Quantum World,” Lie Theory and its Applications in Physics, Varna, Bulgaria, June 2023, Conference proceedings, Springer Proceedings in Mathematics and Statistics, 473, (2025) 557-566

    2025

  62. [70]

    Measuring finite Quantum Geometries via Quasi-Coherent States,

    L. Schneiderbauer and H. Steinacker, “Measuring finite Quantum Geometries via Quasi-Coherent States,” Journal of Physics A,49no. 28, (2016), 285301. [arXiv:1601.08007 [hep-th]]

    2016 arXiv

  63. [71]

    Membrane topology and matrix regularization,

    H. Shimada, “Membrane topology and matrix regularization,” Nuclear Physics B,685, (2004), 297-320

    2004

  64. [72]

    Structure of Dynamical Systems: A Symplectic View of Physics,

    J.-M. Souriau, “Structure of Dynamical Systems: A Symplectic View of Physics,” Translated from the French by C. H. Cushman-de Vries, Progress in Mathematics, vol. 149, Birkh¨ auser, Boston, (1997)

    1997

  65. [73]

    The fuzzy 4-hyperboloidH 4 n and higher- spin in Yang–Mills matrix models,

    M. Sperling and H. C. Steinacker, “The fuzzy 4-hyperboloidH 4 n and higher- spin in Yang–Mills matrix models,” Nuclear Physics B,941, (2019), 680-743. [arXiv:1806.05907 [hep-th]]

    2019 arXiv

  66. [74]

    Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model,

    M. Sperling and H. C. Steinacker, “Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model,” Journal of High Energy Physics,07, (2019), 010. [arXiv:1901.03522 [hep-th]]

    2019 arXiv

  67. [75]

    Non-commutative geometry and matrix models,

    H. Steinacker, “Non-commutative geometry and matrix models,” Proceedings of Science,QGQGS2011, (2011), 004. [arXiv:1109.5521 [hep-th]]

    2011 arXiv

  68. [76]

    Quantum (Matrix) Geometry and Quasi-Coherent States,

    H. C. Steinacker, “Quantum (Matrix) Geometry and Quasi-Coherent States,” Journal of Physics A: Mathematical and Theoretical,54(2021), 055401. doi:10.1088/1751-8121/abd735 [arXiv:2009.03400 [hep-th]]

    2021 arXiv

  69. [77]

    Quantum Geometry, Matrix Theory, and Gravity,

    H. C. Steinacker, “Quantum Geometry, Matrix Theory, and Gravity,” Cambridge University Press, (2024)

    2024

  70. [78]

    Noncommutative field theories onR 3 λ: Toward UV/IR mixing freedom,

    P. Vitale and J. C. Wallet, “Noncommutative field theories onR 3 λ: Toward UV/IR mixing freedom,” JHEP04(2013), 115 [arXiv:1212.5131 [hep-th]]

    2013 arXiv

  71. [79]

    Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory,

    B. Ydri, “Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory,” [arXiv:1708.00734 [hep-th]]

    arXiv

  72. [80]

    The local structure of Poisson manifolds,

    A. Weinstein, “The local structure of Poisson manifolds,” Journal of Differential Geometry,18(3), (1983), 523-557

    1983

  73. [81]

    Geometric Quantization,

    N. M. J. Woodhouse, “Geometric Quantization,” 2nd ed., Oxford Mathematical Monographs, Clarendon Press, Oxford, (1991). 37

    1991

  74. [147]

    [arXiv:1208.0711 [hep-th]]

    arXiv

  75. [179]

    [arXiv:hep-th/9902138 [hep-th]]

    arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.