On the invariants of finite groups arising in a topological quantum field theory

• URL: http://arxiv.org/abs/2510.14971v2
• Date: Mon, 27 Oct 2025 22:04:50 GMT
• Title: On the invariants of finite groups arising in a topological quantum field theory
• Authors: Christopher A. Schroeder, Hung P. Tong-Viet,
• Abstract summary: finite group $G$ determines a family of numerical invariants associated with closed orientable surfaces.<n>These invariants can be viewed as natural extensions of the commuting probability $d(G)$.<n>By analyzing these higher-genus analogues, we establish new quantitative criteria relating the values of these invariants to key structural features of finite groups.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we investigate structural properties of finite groups that are detected by certain group invariants arising from Dijkgraaf--Witten theory, a topological quantum field theory, in one space and one time dimension. In this setting, each finite group $G$ determines a family of numerical invariants associated with closed orientable surfaces, expressed in terms of the degrees of the complex irreducible characters of $G$. These invariants can be viewed as natural extensions of the commuting probability $d(G)$, which measures the likelihood that two randomly chosen elements of $G$ commute and has been extensively studied in the literature. By analyzing these higher-genus analogues, we establish new quantitative criteria relating the values of these invariants to key structural features of finite groups, such as commutativity, nilpotency, supersolvability and solvability. Our results generalize several classical theorems concerning the commuting probability, thereby linking ideas from finite group theory and topological quantum field theory.

Related papers

• Counting gauge-invariant states with matter fields and finite gauge groups [51.56484100374058]
  We generalize the exact number of gauge-invariant states to include the case of scalar and fermionic matter.<n>Results are relevant for resource estimation and also as a crosscheck when working in a gauge-invariant basis.
  arXiv  Detail & Related papers  (2025-09-02T10:29:28Z)
• Predicting symmetries of quantum dynamics with optimal samples [41.42817348756889]
  Identifying symmetries in quantum dynamics is a crucial challenge with profound implications for quantum technologies.<n>We introduce a unified framework combining group representation theory and subgroup hypothesis testing to predict these symmetries with optimal efficiency.<n>We prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols.
  arXiv  Detail & Related papers  (2025-02-03T15:57:50Z)
• Abelian Spectral Topology of Multifold Exceptional Points [0.0]
  We revisit the classification of EP$n$s in generic systems and systems with local symmetries, generalize it in terms of more mathematically tractable (local) similarity relations, and extend it to include similarities as well as non-local symmetries.<n>Our work reveals the foundations on which the topological nature of EP$n$s resides, which will find great use in modern experiments within both classical and quantum physics.
  arXiv  Detail & Related papers  (2024-12-19T19:00:00Z)
• Topological nature of edge states for one-dimensional systems without symmetry protection [46.87902365052209]
  We numerically verify and analytically prove a winding number invariant that correctly predicts the number of edge states in one-dimensional, nearest-neighbor (between unit cells)<n>Our winding number is invariant under unitary or similarity transforms.
  arXiv  Detail & Related papers  (2024-12-13T19:44:54Z)
• Gaussian quantum Markov semigroups on finitely many modes admitting a normal invariant state [0.0]
  Gaussian quantum Markov semigroups (GQMSs) are of fundamental importance in modelling the evolution of several quantum systems.<n>We completely characterize those GQMSs that admit a normal invariant state and we provide a description of the set of normal invariant states.<n>We study the behavior of such semigroups for long times: firstly, we clarify the relationship between the decoherence-free subalgebra and the spectrum of $mathbfZ$.
  arXiv  Detail & Related papers  (2024-12-13T10:01:18Z)
• Constructive quantum mechanics based on finite groups [0.0]
  A formulation of quantum mechanics based on replacing the general unitary group by finite groups is considered. To solve problems arising in the context of this formulation, we use computer algebra and computational group theory methods.
  arXiv  Detail & Related papers  (2024-09-26T14:04:37Z)
• On reconstruction of states from evolution induced by quantum dynamical semigroups perturbed by covariant measures [50.24983453990065]
  We show the ability to restore states of quantum systems from evolution induced by quantum dynamical semigroups perturbed by covariant measures. Our procedure describes reconstruction of quantum states transmitted via quantum channels and as a particular example can be applied to reconstruction of photonic states transmitted via optical fibers.
  arXiv  Detail & Related papers  (2023-12-02T09:56:00Z)
• Equivariant simplicial distributions and quantum contextuality [0.0]
  We introduce an equivariant version of contextuality with respect to a symmetry group, which comes with natural applications to quantum theory. In the equivariant setting, we construct cohomology classes that can detect contextuality.
  arXiv  Detail & Related papers  (2023-10-27T13:25:46Z)
• Matter relative to quantum hypersurfaces [44.99833362998488]
  We extend the Page-Wootters formalism to quantum field theory. By treating hypersurfaces as quantum reference frames, we extend quantum frame transformations to changes between classical and nonclassical hypersurfaces.
  arXiv  Detail & Related papers  (2023-08-24T16:39:00Z)
• Covariant quantum field theory of tachyons [0.0]
  Three major misconceptions concerning quantized tachyon fields are shown to be a result of misrepresenting the Lorentz group in a too small Hilbert space. We establish an explicitly covariant framework that allows for the proper quantization of the tachyon fields eliminating all of these issues.
  arXiv  Detail & Related papers  (2023-08-01T11:00:14Z)
• Infinite Permutation Groups and the Origin of Quantum Mechanics [0.0]
  When the lattice is atomistic, it is isomorphic to the lattice of definably closed sets of a finitary relational structure in First Order Logic. We show that the automorphism group must belong to a family of permutation groups known as geometric Jordan groups. We then use the classification theorem for Jordan groups to argue that the combined requirements of probability and atomicism leave uncountably infinite Steiner 2-systems.
  arXiv  Detail & Related papers  (2023-07-24T18:00:16Z)
• General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory [44.99833362998488]
  We introduce quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple quantum numbers. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories.
  arXiv  Detail & Related papers  (2022-12-28T18:56:25Z)
• Equivariant Transduction through Invariant Alignment [71.45263447328374]
  We introduce a novel group-equivariant architecture that incorporates a group-in hard alignment mechanism. We find that our network's structure allows it to develop stronger equivariant properties than existing group-equivariant approaches. We additionally find that it outperforms previous group-equivariant networks empirically on the SCAN task.
  arXiv  Detail & Related papers  (2022-09-22T11:19:45Z)
• Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups [23.31108679980856]
  We prove that relative Hermitian modular categories give rise to modified Hermitian WRT-TQFTs. The Hermitian theory is applied to define new pseudo-Hermitian topological phases that can be considered as non-semisimple analogs of Levin-Wen models.
  arXiv  Detail & Related papers  (2022-08-30T23:26:31Z)
• Building Quantum Field Theories Out of Neurons [0.0]
  An approach to field theory is studied in which fields are comprised of $N$ constituent random neurons. Gaussianity is exhibited at large-$N$, potentially explaining a feature of field theories in Nature.
  arXiv  Detail & Related papers  (2021-12-08T19:05:36Z)
• Capacity of Group-invariant Linear Readouts from Equivariant Representations: How Many Objects can be Linearly Classified Under All Possible Views? [21.06669693699965]
  We find that the fraction of separable dichotomies is determined by the dimension of the space that is fixed by the group action. We show how this relation extends to operations such as convolutions, element-wise nonlinearities, and global and local pooling.
  arXiv  Detail & Related papers  (2021-10-14T15:46:53Z)
• Local unitary classes of states invariant under permutation subgroups [0.0]
  The study of entanglement properties of multi-qubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. We generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations.
  arXiv  Detail & Related papers  (2021-09-14T18:42:21Z)
• A refinement of Reznick's Positivstellensatz with applications to quantum information theory [72.8349503901712]
  In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares. Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
  arXiv  Detail & Related papers  (2019-09-04T11:46:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.