A recent numerical evaluation of the spherical universal log coefficient in the Maxwell free–energy and its decomposition into bulk and edge contributions via a bounded hyperbolic geometry mode calculation is shown to be equivalent to an existing compact formulation for p-forms, at p=1, on a conically deformed sphere. The edge mode seems to be a ghost (p-1)–form. Some numbers are given.

An oldish question is resurrected concerning the significance of the ambiguous `b-type' terms encountered in calculations of the vacuum, Casimir energy on the Einstein Universe for conformally coupled scalar fields. Some remarks in the literature are hopefully clarified and the relevance of much earlier evaluations is pointed out. A consistency principle is suggested.

The contribution, E, of hyperbolic elements to the scalar Casimir energy on a compact quotient of the upper half hyperbolic plane is computed for a propagation operator conformal in three dimensions. Due to the proliferation of prime closed geodesics, the series form for the Casimir energy has an IR divergence. The expression for E is given as a sum of polylogarithms which allows the divergence to be isolated and rendered finite by an ad hoc Ramanujan renormalisation. The remaining part of E is computed using the specific lower length spectrum of the (2,3,7) triangle and a universal asymptotic form for larger lengths. The tentative value of E found is such as to make the total conformal Casimir energy on the triangle probably negative.

Some exact high temperature expansions are derived using a temperature inversion symmetry of the internal energy for conformal scalars and spinors on the Einstein Universe.

Making use of an 1847 result of Newman, a (known) closed formula for a log-sine integral is rapidly obtained in terms of Riemann Zeta and Clausen functions.

A symmetrising shift employed by Frenkel and Hartnoll in the approximate computation of the elements of matrix spherical harmonics is further explored and shown to be related to the permutation symmetry of 3-j symbols yielding an extension of the Edmonds classical limit. Some graphs are displayed.

The details are expounded of an old treatment of the limit of the su(N) structure constants as N tends to infinity. A recently derived parity property of the series expansion is shown to be the same as the known mirror symmetry of 6-j symbols.

The contribution of elliptic fixed points to the scalar Casimir energy on compact quotients of the upper half hyperbolic plane is computed for a propagation operator conformal in three dimensions. The expression involves derivatives of two-dimensional Barnes zeta-functions which are reduced to Hurwitz zeta-functions for numerical purposes. The values are all positive for any elliptic order.

The Dirac APS eta invariant on a Berger sphere of dimension $2n-1$ is discovered, numerically, to coincide, up to spin factors, with the Dirac conformal anomaly on a round sphere of even dimension, $n$. The analytical expression, given in terms of a generalised Bernoulli polynomial, is shown to equal a known conjecture for the eta invariant. Weingart's generating function is also obtained with no extra work.

Two theorems involving curl eigenfields on the 3--sphere are obtained using angular momentum theory. Spinor hyperspherical harmonics are shown to form an explicit, convenient basis. In particular, a spin--one vector calculus is reviewed. An easy proof of the vanishing of `odd' eigenfields is given and related to the sign change of fermionic spinors under 2$\pi$ rotations. The theorem that curl eigenfields with constant norm have to be proportional to a fundamental eigenfield (Hopf field) is also rapidly obtained. Attention is drawn to the relevance of early work of Schr\"odinger on Maxwell theory in an expanding universe.

A discrete Funk--Hecke formula is set up using the analogy between ordinary and operator spherical harmonics. It is the fuzzy sphere analogue of the conventional theory. An example is related, in the classical limit, to the Rayleigh partial wave expansion.

It is shown that the functional determinant ($\sim$ effective action) for a scalar field propagating on the mixed signature product of unit spheres, S$^q\times$S$^p$, according to the GJMS operator, depends, if $d$ is odd, only on $d=p+q$ and on whether $p$ is even or odd. In the first case the effective action equals twice the standard quantity on S$^d$ and vanishes in the second.

For a free–field flat monodromy defect, a formula for the finite part of the correlator is obtained as a double power series in (1 − x ) and (1 − x ) where x and x are lightcone coordinates. It takes the particular form of a series in (1 − x ) with coefficients finite sums of hypergeometric functions of 1 − x and is identified with a bulk block expansion. An expression for the coefficient of the (1 − x ) n (1 − x ) m term is thereby found as an explicit function of the flux and dimension. Some typical examples are presented. A transformation allows the bulk block expansion to be written as an Appell F 3 function which has simplifying consequences.

Some elementary algebraic points regarding the Green function for a localised flux tube are developed. A calculation of the effective action density is included.

The effect of a spherical monodromy defect on the entanglement entropy and central charge C_T of a free conformal scalar field propagating on an odd-dimensional sphere is investigated. As on even spheres the central charge becomes negative for a range of the flux parameter, δ, with possible implications for reflection positivity. The work is mostly numerical but closed forms are found for the Z_2 monodromy (δ=1/2) yielding an explicit d→∞ limit, C_T→ -6/π

For a scalar CFT with a monodromy defect, a `subtracted Green function' is derived in terms of an Appell $F_1$ function. A conjectured relation of Gimenez-Grau and Liendo is thereby proved and extended and shown to hold for generalised free fields. A possible means of determining the bulk block expansion is outlined. Finite coincidence limits are expressed as combinations of Beta functions.

The central charge CT is computed for scalar and Dirac fields propagating according to GJMS-type kinetic operators acting on odd ddimensional spheres in the presence of a spherical monodromy. The relation of CT to the derivatives of the free energy on the conically deformed sphere via the Perlmutter factor leads to a numerical quadrature. The variation of CT with the monodromy flux, δ, displays sign changes, exactly as in even dimensions. Closed forms for CT are derived when δ equals 0 or 1/2 with the derivative order either even or odd and shown to agree with existing, even d expressions. The infinite d limits are also derived in these special cases.

An earlier contour expression for the Green function of a free complex scalar field in the presence of a conical singularity with localised magnetic flux is shown to yield expressions for the field correlator and defect block expansions that have been more recently found in connection with monodromy defects in conformal field theory. The Green function appears as the Picard integral representation of the Appell $F_1$ function. This is shown to transform into a confluent Horn function, corresponding to a different defect block expansion. Other transformations are discussed.

Explicit polynomial forms for Rényi and entanglement entropies are given on even –dimensional spheres which possess a codimension–2 U(1) monodromy defect. Free scalar and Dirac fields are treated and higher-derivative propagation operators employed. The central charge, CT , is also calculated. A comparison with existing results indicates that some adjustment might be needed in the identifications made in the underlying theory used there.

A Casimir--type analysis of the effect of dividing the two--sphere by several lines of latitude is done for conformally invariant Dirichlet and Neumann scalars and for spinors. An effective action combination is shown to have minima for symmetrical arrangements of the circles. For a domain of two caps and a slice, the Dirichlet slice expands to fill the whole sphere while the Neumann one adjusts to an angular separation of 57.92 degrees. The fermion expression is written in terms of the Weber class function, $\gf$, and a connection is noted with an earlier calculation of twisted scalar effective actions on the tetrahedron.