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The formula ⋆ mod ō ( ħ k ) of Kontsevich’s star-product with harmonic propagators was known in full at ħ k ⩾6 since 2018 for generic Poisson brackets, and since 2022 also at k = 7 for affine brackets. We discover that the mechanism of associativity for the star-product up to ō ( ħ 6 ) is different from the mechanism at order 7 for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod ō ( ħ 6 ), whereas at order ħ 7 and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.

The idea of a \\(\\)-system was introduced by Veselov in the study of rational solutions of the WDVV equations of associativity. These are algebraic/geometric conditions on the set of covectors that appear in rational solutions to the WDVV equations. Here, this idea is generalized to open WDVV equations, which are an additional set of PDEs originating from open Gromow-Witten Theory. We develop -- for rank-one extensions -- algebraic/geometric conditions on the covectors that supplement the \\(\\)-system to give rational solutions to the open WDVV equations. Examples, and the relation to superpotentials and to Dubrovin almost-duality, are given.

We introduce a non-associative model for the Hilbert scheme of points in arbitrary dimension. We define a smooth ambient space, which we call the non-associative Hilbert scheme, containing the classical nested Hilbert scheme \\(NHilb^d(A^n)\\) as the associativity, cut out by an explicit section of an associativity bundle. This construction yields canonical perfect obstruction theories and virtual fundamental classes on \\(NHilb^d(A^n)\\) for all \\((n, d)\\). Using virtual localization, we obtain closed formulas for these virtual classes as sums over admissible nested partitions. Over the punctual locus, we rewrite these as a single multivariable iterated residue formula governing all virtual integrals. Our construction works for all \\(n\\), produces positive-dimensional virtual classes when \\(n\\) is large compared to the number of points, and we expect that they extend the non-commutative matrix model and virtual class construction on Calabi-Yau threefolds.

We introduce para-associative algebroids as vector bundles whose sections form a ternary algebra with a generalised form of associativity. We show that a necessary and sufficient condition for local triviality is the existence of a differential connection, i.e., a connection that satisfies the Leibniz rule over the ternary product.

A bstract We show that associativity of the tree-level OPE in a celestial CFT imposes constraints on the coupling constants of the corresponding bulk theory. These constraints are the same as those derived in [ 9 ] from the Jacobi identity of the algebra of soft modes. The constrained theories are interesting as apparently well-defined celestial CFTs with a deformed w 1+ ∞ symmetry algebra. We explicitly work out the ramifications of these constraints on scattering amplitudes involving gluons, gravitons and scalars in these theories. We find that all four-point amplitudes constructible solely from holomorphic or anti-holomorphic three-point amplitudes vanish on the support of these constraints, which implies that all purely holomorphic or purely anti-holomorphic higher-point amplitudes vanish.

A bstract We study celestial chiral algebras appearing in celestial holography, using the light-cone gauge formulation of self-dual Yang-Mills theory and self-dual gravity, and explore also a deformation of the latter. The recently discussed w 1+ ∞ algebra in self-dual gravity arises from the soft expansion of an area-preserving diffeomorphism algebra, which plays the role of the kinematic algebra in the colour-kinematics duality and the double copy relation between the self-dual theories. The W 1+ ∞ deformation of w 1+ ∞ arises from a Moyal deformation of self-dual gravity. This theory is interpreted as a constrained chiral higher-spin gravity, where the field is a tower of higher-spin components fully constrained by the graviton component. In all these theories, the chiral structure of the operator-product expansion exhibits the colour-kinematics duality: the implicit ‘left algebra’ is the self-dual kinematic algebra, while the ‘right algebra’ provides the structure constants of the operator-product expansion, ensuring its associativity at tree level. In a scattering amplitudes version of the Ward conjecture, the left algebra ensures the classical integrability of this type of theories. In particular, it enforces the vanishing of the tree-level amplitudes via the double copy.

Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations-anyons-and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization-termed symmetry enriched topological order. When the global symmetry group G, which we take to be discrete, does not change topological superselection sectors-i.e. does not change one type of anyon into a different type of anyon-one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G, one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic 'twist' defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.

Brain computer interfaces (BCI) for the rehabilitation of motor impairments exploit sensorimotor rhythms (SMR) in the electroencephalogram (EEG). However, the neurophysiological processes underpinning the SMR often vary over time and across subjects. Inherent intra- and inter-subject variability causes covariate shift in data distributions that impede the transferability of model parameters amongst sessions/subjects. Transfer learning includes machine learning-based methods to compensate for inter-subject and inter-session (intra-subject) variability manifested in EEG-derived feature distributions as a covariate shift for BCI. Besides transfer learning approaches, recent studies have explored psychological and neurophysiological predictors as well as inter-subject associativity assessment, which may augment transfer learning in EEG-based BCI. Here, we highlight the importance of measuring inter-session/subject performance predictors for generalized BCI frameworks for both normal and motor-impaired people, reducing the necessity for tedious and annoying calibration sessions and BCI training.

Cobordism categories are known to be compact closed. They can therefore be used to define non-degenerate models of multiplicative linear logic by combining the Int construction with double glueing. In this work we detail such construction in the case of low-dimensional cobordisms, and exhibit a connexion between those models and the model of Interaction graphs introduced by Seiller. In particular, we exhibit how the so-called trefoil property is a consequence of the associativity of composition of higher structures, providing a first step toward establishing models as obtained from a double glueing construction. We discuss possible extensions to higher-dimensional cobordisms categories

The compensation equation is the law of conservation of a certain quantity with respect to a given class of admissible transformations. This paper discusses approaches to the construction of a compensating function for rational aggregation functions, including those with the associativity property.