In this article, we provide a detailed characterization of the optical properties of the metamaterial associated with Schwarzschild space in the presence of a cosmological constant, using the analogy between spacetime and anisotropic metamaterials. The scalar construction is a trajectory-specific inverse-designed GRIN model, engineered per ray, so that its agreement with the geodesics verifies the numerical implementation rather than establishing an independent spacetime–medium correspondence; the tensorial Plebański medium, by contrast, is fixed by the metric alone. We explicitly show the refractive index associated with the metamaterial and find that the cosmological constant provides interesting optical properties to the metamaterial. For Λ > 0 we show that the cosmological term displaces the critical impact parameter to bc=33M/1−9M2Λ while leaving the photon sphere at r=3M, so that rays which are marginally supercritical in the asymptotically flat case are captured. We further prove that the Rindler–Ishak correction factor coincides with the local anisotropy ratio nϕ/nr=f of the Plebański medium, and that this identifies the effect with the splitting between the ray and wave-normal directions that any anisotropic medium produces: the Rindler–Ishak angle is the geometric mean of the two, tanψinv=tanψraytanθk, and both angles are separately measurable in the analogue system. Fixing the emission condition in the measured rather than the coordinate sense, we find that a positive cosmological term suppresses the deflection recorded between a static emitter and receiver at finite radius, by 80% at the value used here, so that what it enhances is the capture window and not the bending. Conversely, for Λ < 0 the absence of a global Euclidean embedding translates into an imaginary scalar index acting as an algebraic censor, while the exact tensorial mapping yields a matched uniaxial zero-index medium (λϕ=λz→0, λr=1, Z=1) whose radial constitutive component remains that of vacuum. The two causal boundaries unify into a single optical statement: de Sitter has infinite optical thickness at finite radius, Anti-de Sitter finite optical thickness at infinite radius.
We construct and classify exact complex plane-wave solutions of the vacuum SU(2) Yang–Mills equations using a simple ansatz for the gauge potentials. By reducing the nonlinear Yang–Mills equations to algebraic constraint conditions, we obtain a complete classification. For nonzero coupling and nonvanishing transverse and color-space parameters, the solutions can be reduced to a single fixed color generator by an SU(2) gauge transformation if and only if β→×ℓ→=0. Therefore, the genuinely non-Abelian solutions are precisely those with β→×ℓ→≠0, occurring in subbranches of Cases (iii) and (vi). Their field strengths are of plane-wave type, satisfy the standard dispersion relation ω=ck, and contain a nonlinear second harmonic proportional to (β→×ℓ→)·S→. Although their quadratic Lorentz invariants vanish, the gauge-invariant Hermitian energy density can remain nonzero and exhibit a bounded modulation. We also show that a common amplitude rescaling generates an exact zero-action critical family connecting the vacuum to genuinely non-Abelian waves, with the phase-averaged Hermitian energy density growing nonlinearly with the amplitude. We further extend the solutions to arbitrary smooth wave profiles and construct a complex external current that generates a non-Abelian pulse from vacuum. Using this pulse as a background, we obtain an explicit relation between a probe gluon’s incoming and outgoing color amplitudes. These results provide a complete and transparent classification of Abelian-type and genuinely non-Abelian propagating waves in SU(2) Yang–Mills theory.
We discuss a Wakimoto-like free field realisation of b(2, 1; alpha)(1), whose sl(2, R) subalgebra has level k = 1, that requires no gauging, i.e., realises the current algebra exactly. We then compute the partition function of the theory and show that, by combining this with the ghost contribution, the full, on-shell projected string partition function reproduces precisely the single-particle partition function of the Sym(N) (S'(2)(0)) theory, i.e. the symmetric orbifold theory of 8 free fermions, 1 compact free boson, and 1 non-compact free boson. We also discuss other aspects such as DDF operators and BRST and physical state conditions.
