"There is geometry in the humming of the strings, there is music in the spacing of the spheres."
— Pythagoras
"There is geometry in the humming of the strings, there is music in the spacing of the spheres."
— Pythagoras
The following list of works can also be found on Google Scholar, ResearchGate, Semantic Scholar, Academia.edu or zbMATH Open.
Publications
C. J. D. Kemp, Relations between multiple zeta values and delta values from Drinfeld’s associator series, J. Number Theory 292, 314-339 (2027) [arXiv:2504.16747 [math.NT]].
Abstract: It is shown that novel relations between multiple zeta values and single-variable multiple polylogarithms at 1/2 (delta values) can be derived by comparing two distinct, yet a priori equal, formulae for Drinfeld's Knizhnik-Zamolodchikov series. In particular, we demonstrate that two new relations are found by comparing the fifth order terms of each series formula.
Abstract: This thesis presents a novel path to the explicit construction of models of `higher quantum groups' by studying positively-shifted Poisson structures on a generic semi-free commutative differential graded algebra and investigating exactly how they induce infinitesimal deformations of the symmetric monoidal structure on the dg-category of semi-free dg-modules over such a commutative differential graded algebra.
First we develop a graphical calculus to characterise the shifted Poisson structures on finitely generated semi-free commutative differential graded algebras in terms of families of homotopy-coherent data adjoined to the corresponding finite-dimensional higher Lie algebra. Such a characterisation allows us to show that we recover the notion of higher Lie quasibialgebra due to Bai, Sheng and Zhu. When applied to an ordinary Lie algebra, we recover Safronov's result that the 1 and 2-shifted Poisson structure in this case are given by quasi-Lie bialgebra structures and invariant symmetric 2-tensors, respectively. We generalise these results to the case of a Lie 2-algebra and obtain 1, 2, 3 and 4-shifted Poisson structures which we interpret as semiclassical data of higher quantum groups. In particular, if we assume a 2-term differential graded Lie algebra and trivialise part of the data of a 2-shifted Poisson structure then we recover a special case of the notion of a symmetric quasi-invariant tensor in a differential crossed module due to Cirio and Faria Martins.
It is then shown that every 2-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a concrete infinitesimal 2-braiding on the homotopy 2-category of the symmetric monoidal dg-category of finitely generated semi-free dg-modules over A. This provides a concrete realisation, to first order in the deformation parameter, of the abstract deformation quantisation results in derived algebraic geometry due to Calaque, Pantev, Toën, Vaquié and Vezzosi. Of particular interest is the case when A is the Chevalley-Eilenberg algebra of a higher Lie algebra, where the braided monoidal deformations developed in this thesis may be interpreted as candidates for representation categories of higher quantum groups.
Upon choosing the usual ansatz of the braiding and associator as, respectively, the exponential and Drinfeld's Knizhnik-Zamolodchikov series of the infinitesimal 2-braiding, it is demonstrated that the hexagon axioms are obstructed at second order by modifications. These modifications are shown to satisfy the requisite axioms of a braided monoidal 2-category (i.e., the modifications are shown to be `infinitesimal hexagonators') provided that the infinitesimal 2-braiding is totally y-equivariant and coherent in Cirio and Faria Martins' sense. We show that those infinitesimal 2-braidings induced by 2-shifted Poisson structures are indeed totally y-equivariant and we conjecture that coherency also holds by relating the condition to the third-weight component of the Maurer-Cartan equation that a 2-shifted Poisson structure definitively satisfies. We then demonstrate that the pentagonator is nontrivial by determining its third-order term.
Finally, we study the problem of sylleptic deformations and show that a 3-shifted Poisson structure induces ``infinitesimal syllepses" which can be trivially integrated to all orders. We also show that a ``coboundary" 2-shifted Poisson structure induces ``infinitesimal coboundary syllepses" and these admit of a Cartier integration alongside that of infinitesimal 2-braidings and we carry this out to second and third order.
C. Kemp, R. Laugwitz and A. Schenkel, Shifted Poisson structures on higher Chevalley-Eilenberg algebras, Lett. Math. Phys. 116, no.19 (2026) [arXiv:2412.12804 [math.QA]].
Abstract: This paper develops a graphical calculus to determine the n-shifted Poisson structures on finitely generated semi-free commutative differential graded algebras. When applied to the Chevalley-Eilenberg algebra of an ordinary Lie algebra, we recover Safronov's result that the (n=1)- and (n=2)-shifted Poisson structures in this case are given by quasi-Lie bialgebra structures and, respectively, invariant symmetric tensors. We generalize these results to the Chevalley-Eilenberg algebra of a Lie 2-algebra and obtain 1, 2, 3 and 4-shifted Poisson structures in this case, which we interpret as semi-classical data of 'higher quantum groups'.
C. Kemp, R. Laugwitz and A. Schenkel, Infinitesimal 2-braidings from 2-shifted Poisson structures, J. Geom. Phys. 212, 105456 (2025) [arXiv:2408.00391 [math.QA]].
Abstract: It is shown that every 2-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra A defines a very explicit infinitesimal 2-braiding on the homotopy 2-category of the symmetric monoidal dg-category of finitely generated semi-free A-dg-modules. This provides a concrete realization, to first order in the deformation parameter ℏ, of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, Toën, Vaquié and Vezzosi. Of particular interest is the case when A is the Chevalley-Eilenberg algebra of a Lie N-algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of ‘higher quantum groups’.
C. J. D. Kemp, N. R. Cooper and F. N. Ünal, Nested-sphere description of the N-level Chern number and the generalized Bloch hypersphere, Phys. Rev. Research 4, 023120 (2022) [arXiv:2110.06934 [cond-mat.quant-gas]].
Abstract: The geometric interpretation of (pseudo)spin 1/2 systems on the Bloch sphere has been appreciated across different areas ranging from condensed matter to quantum information and high energy physics. Although similar notions for larger Hilbert spaces are established in mathematics, they have been so far less explored beyond the two-level case for practical usage in condensed matter settings, or have involved restrictions to sub manifolds within the full Hilbert space. We here employ a coherence vector description to theoretically characterize a general N-level system on the higher dimensional generalized Bloch (hyper)sphere by respecting the structure of the underlying SU(N) algebra and construct physically intuitive geometric pictures for topological concepts. Focusing on two spatial dimensions, we reveal a geometric interpretation for the Chern number in larger Hilbert spaces in terms of a nested structure comprising N-1 two-spheres. We demonstrate that for the N-level case, there is an exterior two-sphere that provides a useful characterization of the system, notably by playing a primary role in determining the Chern number. The external sphere can be directly measured in ultracold atoms via well-established band mapping techniques, thereby imparting knowledge of the topological nature of state. We also investigate how the time evolution of the coherence vector defined on the generalized Bloch hypersphere can be utilized to extract the full state vector in experiments, allowing us to develop a tomography scheme involving quenches for three-level systems. Our geometric description opens up a new avenue for the interpretation of the topological classification and the dynamical illustration of multi-level systems, which in turn is anticipated to help in the design of new experimental probes.
M. N. Crowe, C. J. D. Kemp and E. R. Johnson, The decay of Hill’s vortex in a rotating flow, J. Fluid Mech. 919 (2021).
Abstract: Hill's vortex is a classical solution of the incompressible Euler equations which consists of an axisymmetric spherical region of constant vorticity matched to an irrotational external flow. This solution has been shown to be a member of a one-parameter family of steady vortex rings and as such is commonly used as a simple analytic model for a vortex ring. Here, we model the decay of a Hill's vortex in a weakly rotating flow due to the radiation of inertial waves. We derive analytic results for the modification of the vortex structure by rotational effects and the generated wave field using an asymptotic approach where the rotation rate, or inverse Rossby number, is taken to be small. Using this model, we predict the decay of the vortex speed and radius by combining the flux of vortex energy to the wave field with the conservation of peak vorticity. We test our results against numerical simulations of the full axisymmetric Navier–Stokes equations.
Preprints (all in review for publication)
C. Kemp, Cartier integration of infinitesimal 2-braidings via 2-holonomy of the CMKZ 2-connection, II: The pentagonator, arXiv:2603.22694 [math.QA].
Abstract: This is a continuation of the previous paper (arXiv:2508.01944) in this series. We recontextualise Cirio and Martins’ work to motivate our fundamental conjecture that the DrinfeldKohno (Lie) 2-algebra has trivial cohomology. It is then shown that this conjecture implies the following: given a coherent totally symmetric infinitesimal 2-braiding t, every modification endomorphic on the zero transformation vanishes if it is made up of the four-term relationators and whiskerings by t. The power of such an implication is that, in our context, one need only construct the data of a braided monoidal 2-category and it will automatically satisfy the axioms. We thus conclude by constructing the pentagonator via Cirio and Martins’ Knizhnik-Zamolodchikov 2-connection over the configuration space of 4 distinguishable particles on the complex line, Y4. In particular, we make use of Bordemann, Rivezzi and Weigel’s pentagon in Y4.
Abstract: Given a homotopy Lie algebra (i.e. an L∞-algebra) g, we show concretely how the Lada-Markl g-modules (i.e. representations) assemble into a symmetric monoidal dg-category. Considering the homotopy 2-category of that dg-category, we construct infinitesimal 2-braidings from 2-shifted Poisson structures then show that such infinitesimal 2-braidings are coherent in Cirio and Faria Martins' sense. We then explicitly determine the differential of the Chevalley-Eilenberg algebra associated with a finite-dimensional homotopy Lie algebra and construct the symmetric monoidal dg-equivalence between the category of representations and the category of semi-free dg-modules over the Chevalley-Eilenberg algebra.
C. J. D. Kemp, Cartier integration of infinitesimal 2-braidings via 2-holonomy of the CMKZ 2-connection, I: Hexagonators and the Breen polytope, arXiv:2508.01944 [math.QA].
Abstract: This paper follows on from Syllepses from 3-shifted Poisson structures and second-order integration of infinitesimal 2-braidings. Given a symmetric strict infinitesimal 2-braiding on a symmetric strict monoidal cochain 2-category, we construct a candidate hexagonator series as a 2-holonomy with respect to the Cirio-Martins-Knizhnik-Zamolodchikov (CMKZ) fake flat 2-connection over the configuration space of 3 distinguishable particles on the complex line. The second-order term of the hexagonator series is computed and found to agree with the 'infinitesimal hexagonator' from the aforementioned work. Finally, we assume a coherent totally symmetric strict infinitesimal 2-braiding and prove that the Breen polytope axiom is satisfied by translating it into a 2-loop in the configuration space of 3 distinguishable particles on the complex line.
C. J. D. Kemp, Syllepses from 3-shifted Poisson structures and second-order integration of infinitesimal 2-braidings, arXiv:2505.01949 [math.QA].
Abstract: This paper follows on from Infinitesimal 2-braidings from 2-shifted Poisson structures. It is demonstrated that the hexagonators appearing at second order satisfy the braided monoidal 2-category axioms provided that the infinitesimal 2-braiding is "totally symmetric" and "coherent" (in Cirio and Faria Martins' sense). We show that those infinitesimal 2-braidings induced by 2-shifted Poisson structures are indeed totally symmetric and we relate coherency to the third-weight component of the Maurer-Cartan equation that a 2-shifted Poisson structure must satisfy. Furthermore, we show that 3-shifted Poisson structures and 'coboundary' 2-shifted Poisson structures induce syllepses.