[Paper Review] On the spectral and homological dimension of k-Minkowski space

Marco Matassa|arXiv (Cornell University)|Sep 4, 2013

Advanced Operator Algebra Research18 references3 citations

TL;DR

This paper extends a modular spectral triple construction for κ-Minkowski space to n-dimensions, using twisted and modular spectral triple frameworks to define a spectral dimension equal to the classical dimension. It shows that the twisted Hochschild homology recovers the full dimension n, avoiding the dimension drop seen in standard homology, with the twist linked to the inverse of the modular group of the KMS weight.

ABSTRACT

We extend the construction of a spectral triple for k-Minkowski space, previously given for the two-dimensional case, to the general n-dimensional case. This takes into account the modular group naturally arising from the symmetries of the geometry, and requires the use of notions that have been recently developed in the frameworks of twisted and modular spectral triples. First we compute the spectral dimension, using an appropriate weight, and show that in general it coincides with the classical one. We also study the classical limit and the analytic continuation of the associated zeta function. Then we compare this notion of dimension with the one coming from homology. To this end, we compute the twisted Hochschild dimension of the universal enveloping algebra underlying k-Minkowski space. The result is that twisting avoids the dimension drop, similarly to other examples coming from quantum groups. In particular, the simplest such twist is given by the inverse of the modular group mentioned above.

Motivation & Objective

• To generalize the modular spectral triple construction for κ-Minkowski space from 2D to n-dimensional spacetime.
• To compute the spectral dimension using a KMS weight and analyze its zeta function's analytic continuation and classical limit.
• To investigate the homological dimension via twisted Hochschild homology and compare it with the spectral dimension.
• To establish that the twist in homology theory restores the classical dimension, avoiding the dimension drop typical in quantum group examples.
• To identify the connection between the modular group of the KMS weight and the twist automorphism in the homology computation.

Proposed method

• Adapt the two-dimensional modular spectral triple construction to n-dimensions using the GNS construction from a KMS weight invariant under κ-Poincaré symmetries.
• Define a Dirac operator D such that the twisted commutator [D, a]σ is bounded, with σ an automorphism derived from symmetry and classical limit constraints.
• Compute the spectral dimension via the zeta function ζ(s) = Tr(|D|−s), showing it equals n and that the residue at s=n recovers the KMS weight ω up to a constant.
• Use the Poincaré–Birkhoff–Witt theorem to parametrize elements in the universal enveloping algebra U(ℊκ) and solve for invariance under the twist automorphism.
• Construct a cycle in twisted Hochschild homology using the differential δ and verify its closure under the twist, ensuring non-trivial n-dimensional homology.
• Identify the twist automorphism σ as the inverse of the modular group σiω, with the simplest case given by σ(x₁) = x₁ − i(n−1)λ, σ(xⱼ) = xⱼ for j > 1.

Experimental results

Research questions

• RQ1 Does the spectral dimension of n-dimensional κ-Minkowski space, defined via a modular spectral triple, coincide with the classical dimension?
• RQ2 How does the zeta function associated with the Dirac operator behave under analytic continuation, and what is its classical limit?
• RQ3 Does the twisted Hochschild homology of the universal enveloping algebra U(ℊκ) recover the full n-dimensional homological dimension, avoiding the dimension drop?
• RQ4 What is the role of the modular group of the KMS weight in determining the twist automorphism in the homology theory?
• RQ5 Can the orientation cycle in the twisted Hochschild complex be identified with the commutative volume form dx¹∧⋯∧dxⁿ?

Key findings

• The spectral dimension of n-dimensional κ-Minkowski space is equal to n, matching the classical dimension, as determined by the zeta function trace.
• The residue of the zeta function at s = n is proportional to the KMS weight ω, confirming consistency with the underlying geometry.
• The zeta function admits a meromorphic continuation to the complex plane with simple poles, including those from the commutative case and additional ones due to the deformation parameter.
• The twisted Hochschild homological dimension of U(ℊκ) is n, demonstrating that the twist restores the classical dimension and avoids the dimension drop.
• The twist automorphism is identified as the inverse of the modular group of the KMS weight, with the simplest case given by σ(x₁) = x₁ − i(n−1)λ.
• The canonical cycle in twisted Hochschild homology corresponds to the commutative volume form, with the same structure as in the classical case.