Laplacians on fuzzy Riemann surfaces

Adachi, Hiroyuki; Ishiki, Goro; Kanno, Satoshi; Matsumoto, Takaki

doi:10.1103/physrevd.103.126003

Cited by 5 publications

(5 citation statements)

References 44 publications

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“…This is explicitly shown for the case dim M = 2[13] and the proof can be easily generalized in the case of general Kähler manifold we are considering in this paper.…”

mentioning

confidence: 64%

“…In this paper, we investigate the Berezin-Toeplitz quantization of vector bundles over a general closed Kähler manifold. We show that the asymptotic properties of the Toeplitz operator given in [13,14] also exist in higher dimensional manifolds. We derive a large-p asymptotic expansion of the product T p (ϕ)T p (χ) for arbitrary sections of vector bundles (general fields) ϕ, χ, up to the second order in 1/p.…”

Section: Introductionmentioning

confidence: 86%

“…We can see that ∆ is a Hermitian operator which is positive semi-definite in terms of the Frobenius inner product. In [13], it is shown that the spectra of the Bochner and the matrix Laplacians agree in the large-p limit 4 .…”

Section: 21)mentioning

confidence: 99%

“…In this paper, we studied the Berezin-Toeplitz quantization of vector bundles over a general closed connected Kähler manifold, which is a continuation of our previous studies of two-dimensional cases [13,14]. In our formalism, we treated a vector bundle as a homomorphism bundle and treat its sections as some linear operator between suitable twisted spinor fields.…”

Section: Conclusion and Future Problemsmentioning

confidence: 99%

“…The Toeplitz operators for more general fields than functions were proposed in [8][9][10][11][12]. In more recent studies [13,14], it is shown that the Toeplitz operator of general fields on a closed Riemann surface enjoys beautiful properties, which are a natural generalization of (1.2).…”

Section: Introductionmentioning

confidence: 99%

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Vector bundles on fuzzy Kähler manifolds

Adachi¹,

Ishiki²,

Kanno³

2022

Preprint

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We propose a matrix regularization of vector bundles over a general closed Kähler manifold. This matrix regularization is given as a natural generalization of the Berezin-Toeplitz quantization and gives a map from sections of a vector bundle to matrices. We examine the asymptotic behaviors of the map in the large-N limit. For vector bundles with algebraic structure, we derive a beautiful correspondence of the algebra of sections and the algebra of corresponding matrices in the large-N limit. We give two explicit examples for monopole bundles over a complex projective space CP n and a torus T 2n .

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“…This is explicitly shown for the case dim M = 2[13] and the proof can be easily generalized in the case of general Kähler manifold we are considering in this paper.…”

mentioning

confidence: 64%

Section: Introductionmentioning

confidence: 86%

Section: 21)mentioning

confidence: 99%

Section: Conclusion and Future Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Vector bundles on fuzzy Kähler manifolds

Adachi¹,

Ishiki²,

Kanno³

2022

Preprint

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Matrix regularization for tensor fields

Adachi

Ishiki

Kanno

et al. 2022

Progress of Theoretical and Experimental Physics

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We propose a novel matrix regularization for tensor fields. In this regularization, tensor fields are described as rectangular matrices and both of area-preserving diffeomorphisms and local rotations of the orthonormal frame are realized as unitary similarity transformations of matrices in a unified way. We also show that the matrix commutator corresponds to the covariantized Poisson bracket for tensor fields in the large-N limit.

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