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We prove that the categories of weight modules over the simple $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator algebras at fractional admissible levels and central charges are rigid (and hence the categories of weight modules are braided ribbon categories) and that the decomposition formulae of fusion products of simple projective modules conjectured by Thomas Creutzig, David Ridout and collaborators hold (including when the decomposition involves summands that are indecomposable yet not simple). In addition to solving this old open problem, we develop new techniques for the construction of intertwining operators by means of integrating screening currents over certain cycles, which are expected to be of independent interest, due to their applicability to many other algebras. In the example of $\mathfrak{sl}(2)$ these new techniques allow us to give explicit formulae for a logarithmic intertwining operator from a pair of simple projective modules to the projective cover of the tensor unit, namely, the vertex operator algebra as a module over itself., Comment: 62 pages, 3 figures; updated version: corrected some minor typos and added references

Dust particle impacts on the Wind spacecraft were detected with its plasma wave instrument Wind/WAVES. Frequency analysis on this dust impact time series revealed spectral peaks indicative of a solar rotation signature. We investigated whether this solar rotation signature is embedded in the interplanetary or interstellar dust (ISD) and whether it is caused by co-rotating interaction regions (CIRs), by the sector structure of the interplanetary magnetic field (IMF), or by external effects. We performed frequency analysis on subsets of the data to investigate the origin of these spectral peaks, comparing segments of Wind's orbit when the spacecraft moved against or with the ISD inflow direction and comparing the time periods of the ISD focusing and defocusing phases of the solar magnetic cycle. A superposed epoch analysis of the number of dust impacts during CIRs was used to investigate the systematic effect of CIRs. Case studies of time periods with frequent or infrequent occurrences of CIRs were compared to synthetic data of dust impacts affected by CIRs. We performed similar case studies for time periods with a stable or chaotic IMF sector structure. The superposed epoch analysis was repeated for a time series of the spacecraft floating potential. Spectral peaks were found at the solar rotation period of ~27d and its harmonics at 13.5d and 9d. This solar rotation signature may affect both interplanetary and ISD. The appearance of this signature correlates with the occurrence of CIRs but not with the stability of the IMF sector structure. The CIRs cause, on average, a reduction in the number of dust impact detections. Periodic changes of the spacecraft's floating potential were found to partially counteract this reduction by enhancing the instrument's sensitivity to dust impacts; these changes of the floating potential are thus unlikely to be the cause of the solar rotation signature.

Let $L(\lambda)$ be a highest weight Harish-Chandra module with highest weight $\lambda$. When the associated variety of $L(\lambda)$ is not maximal, that is, not equal to the nilradical of the corresponding parabolic subalgebra, we prove that the unitarity of $L(\lambda)$ can be determined by a simple condition on the value of $z = (\lambda + \rho, \beta^{\vee})$, where $\rho$ is half the sum of positive roots and $\beta$ is the highest root. In the proof, certain distinguished antichains of positive noncompact roots play a key role. By using these antichains, we are also able to provide a uniform formula for the Gelfand--Kirillov dimension of all highest weight Harish-Chandra modules, generalizing our previous result for the case of unitary highest weight Harish-Chandra modules., Comment: 20 pages

We introduce sets $\mathcal{R}(\sigma)$ consisting of semistandard tableaux of shape $\sigma$, subject to a certain restriction on the initial column. These $\mathcal{R}(\sigma)$ are generalizations of the rational, symplectic, or orthogonal tableaux (which furnish a weight basis for the irreducible finite-dimensional representations $U_\sigma$ of the classical groups $H = \operatorname{GL}_k$, $\operatorname{Sp}_{2k}$, or $\operatorname{O}_k$, respectively). As our main result, we show that the cardinality of $\mathcal{R}(\sigma)$ gives the Bernstein degree (i.e., multiplicity) of the modules of covariants (of type $U_\sigma$) of each classical group $H$. Via Howe duality, these modules can also be viewed as $(\mathfrak{g},K)$-modules of unitary highest weight representations of a real reductive group $G_{\mathbb{R}}$. Notably, our result using $\mathcal{R}(\sigma)$ is valid regardless of the rank of $H$, whereas the previous result of Nishiyama-Ochiai-Taniguchi (expressing the Bernstein degree in terms of $\dim U_\sigma$) holds only when $k$ is at most the real rank of $G_{\mathbb{R}}$. In effect, as $k$ increases beyond this range, $|\mathcal{R}(\sigma)|$ interpolates between $\dim U_\sigma$ and the dimension of a certain "limiting" $K$-module., Comment: 16 pages; certain parts of the exposition overlap with our paper arXiv:2312.16749, which contains further details

We carried out very deep VLT/SPHERE imaging polarimetry of the nearby system Eps Eri based on 38.5 hours of integration time with a 600 - 900 nm broadband filter to search for polarized scattered light from a planet or from circumstellar dust using AO, coronagraphy, high precision differential polarimetry, and angular differential imaging. We have improved several data reduction and post-processing techniques and also developed new ones to further increase the sensitivity of SPHERE/ZIMPOL. The data provide unprecedented contrast limits, but no significant detection of a point source or an extended signal from circumstellar dust. For each observing epoch, we obtained a point source contrast for the polarized intensity between $2\cdot 10^{-8}$ and $4\cdot 10^{-8}$ at the expected separation of the planet Eps Eri b of 1'' near quadrature phase. The polarimetric contrast limits are about six to 50 times better than the intensity limits because polarimetric imaging is much more efficient in speckle suppression. Combining the entire 14-month data set to the search for a planet moving on a Keplerian orbit with the K-Stacker software further improves the contrast limits by a factor of about two, to about $8 \cdot 10^{-9}$ at 1''. This would allow the detection of a planet with a radius of about 2.5 Jupiter radii. The surface brightness contrast limits achieved for the polarized intensity from an extended scattering region are about 15 mag arcsec$^{-2}$ at 1'', or up to 3 mag arcsec$^{-2}$ deeper than previous limits. For Eps Eri, these limits exclude the presence of a narrow dust ring and they constrain the dust properties. This study shows that the polarimetric contrast limits for reflecting planets with SPHERE/ZIMPOL can be improved to a level $<10^{-8}$ simply by collecting more data over many nights and using the K-Stacker software., Comment: 26 pages, 20 figures, 5 tables. Accepted for publication in A&A

Let $n\in\mathbb{N}, n\geq 2$. We prove that the strongly singular differential operator \[\left(\Delta^2 +c|x|^{-4}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}, \quad c \in \mathbb{R}, \] is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if and only if \[c\geq \begin{cases}3(n+2)(6-n)&\mbox{for $2\leq n\leq 5$};\\[5pt] {\displaystyle -\frac{n(n+4)(n-4)(n-8)}{16}}&\mbox{for $n\geq 6$}.\end{cases}\] Via separation of variables, our proof reduces to studying the essential self-adjointness on the space $C_0^{\infty}((0,\infty))$ of fourth-order Euler-type differential operators of the form \[ \frac{d^4}{dr^4}+c_1\left(\frac{1}{r^2}\frac{d^2}{dr^2}+\frac{d^2}{dr^2}\frac{1}{r^2}\right)+\frac{c_2}{r^4},\quad r\in(0,\infty),\quad(c_1,c_2)\in \mathbb{R}^2,\] in $L^2((0,\infty);dr)$. Our methods generalize to differential operators related to higher-order powers of the Laplacian, however, there are some nontrivial subtleties that arise. For example, the natural expectation that for $m,n\in\mathbb{N}$, $n \geq 2$, there exist $c_{m,n}\in\mathbb{R}$ such that $\left(\Delta^m+c|x|^{-2m}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}$ is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if and only if $c \geq c_{m,n}$, turns out to be false. Indeed, for $n=20$, we prove that the differential operator \[ \left((-\Delta)^5+c|x|^{-10}\right)\big|_{C_0^{\infty}(\mathbb{R}^{20} \backslash \{0\})}, \quad c \in \mathbb{R},\] is essentially self-adjoint in $L^2\big( \mathbb{R}^{20}; d^{20} x\big)$ if and only if $c\in [0,\beta]\cup [\gamma,\infty)$, where $\beta\approx 1.0436\times 10^{10}$, and $\gamma\approx 1.8324\times 10^{10}$ are the two real roots of the quartic equation \begin{align*}&3125z^4-83914629120000z^3+429438995162964368031744 z^2\\&\quad+1045471534388841527438982355353600z\\&\quad +629847004905001626921946285352115240960000=0.\end{align*}, Comment: 24 pages

We prove a simple formula that calculates the associated variety of a highest weight Harish-Chandra module directly from its highest weight. We also give a formula for the Gelfand--Kirillov dimension of highest weight Harish-Chandra module which is uniform across Cartan types and is valid for arbitrary infinitesimal character., Comment: 34 pages

We reconsider an old problem, namely the dimension of the $G$-invariant subspace in $V^{\otimes p} \otimes V^{*\otimes q}$, where $G$ is one of the classical groups ${\rm GL}(V)$, ${\rm SL}(V)$, ${\rm O}(V)$, ${\rm SO}(V)$, or ${\rm Sp}(V)$. Spanning sets for the invariant subspace have long been well known, but linear bases are more delicate. Beginning in the broader setting of polynomial invariants, we write down multigraded linear bases which we realize as certain arc diagrams (which may include hyperedges); then the bases for tensor invariants are obtained by restricting our attention to the diagrams that are 1-regular. We survey the many equivalent ways -- some old, some new -- to enumerate these diagrams., Comment: 16 pages + appendix

For a complex reductive group $H$ with finite-dimensional representations $W$ and $U$, the module of covariants for $W$ of type $U$ is the space of all $H$-equivariant polynomial functions $W \longrightarrow U$. In this paper, we take $H$ to be one of the classical groups $\operatorname{GL}(V)$, $\operatorname{Sp}(V)$, or $\operatorname{O}(V)$ arising in Howe's dual pair setting, where $W$ is a direct sum of copies of $V$ and $V^*$. Our main result is a uniform combinatorial model for Stanley decompositions of the modules of covariants, using visualizations that we call jellyfish. Our decompositions allow us to interpret the Hilbert series as a positive combination of rational expressions which have concrete combinatorial interpretations in terms of lattice paths; significantly, this interpretation does not depend on the Cohen-Macaulay property. As a corollary, we recover a major result of Nishiyama-Ochiai-Taniguchi (2001) regarding the Bernstein degree of unitary highest weight $(\mathfrak{g},K)$-modules. We also extend our methods to compute the Hilbert series of the invariant rings for the groups $\operatorname{SL}(V)$ and $\operatorname{SO}(V)$, as well as the Wallach representations of type ADE., Comment: 46 pages; this article draws heavily from, and largely supersedes, the second half of our previous article arXiv:2310.04581. The main result in the present article generalizes and uniformizes the many examples in that previous article. Version 3 corrects typos and improves the exposition

We define the notion of strongly interlocked for indecomposable generalized modules for a vertex operator algebra, and show that the notion of graded pseudo-trace is well defined for modules which satisfy this property. We prove that the graded pseudo-trace is a symmetric linear operator that satisfies the logarithmic derivative property. As an application, we prove that all the indecomposable reducible generalized modules for the rank one Heisenberg (one free boson) vertex operator algebras are strongly interlocked, independent of the choice of conformal vector and thus have well-defined graded pseudo-traces. We also completely characterize which indecomposable reducible generalized modules for the universal Virasoro vertex operator algebras induced from the level zero Zhu algebra are strongly interlocked. In particular, we prove that the universal Virasoro vertex operator algebra with central charge c has modules induced from the level zero Zhu algebra with conformal weight h that are strongly interlocked if and only if either (c,h) is outside the extended Kac table, or the central charge is either c = 1 or 25, the conformal weight satisfies a certain property, and the level zero Zhu algebra module being induced is determined by a Jordan block of size less than a certain specified parameter. We give several examples of graded pseudo-traces for Heisenberg and Virasoro strongly interlocked modules., Comment: The title of the paper was changed to better reflect the central role that the notion of strongly interlocked module plays in the theory of graded pseudo-traces. A proof that graded pseudo-traces for any strongly interlocked modules are symmetric, linear, and satisfy the logarithmic derivative property was added. We rewrote and reorganized parts of the exposition and fixed minor typos

We consider the generalized eigenvalue problem for the classical Euler differential equation and demonstrate its intimate connection with Meijer's $G$-functions. In the course of deriving the solution of the generalized Euler eigenvalue equation we review some of the basics of generalized hypergeometric functions and Meijer's $G$-functions and some of its special cases where the underlying Mellin-type integrand exhibits higher-order poles., Comment: 21 pages

We consider essential self-adjointness on the space $C_0^{\infty}((0,\infty))$ of even order, strongly singular, homogeneous differential operators associated with differential expressions of the type \[ \tau_{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x > 0, \; n \in \mathbb{N}, \; c \in \mathbb{R}, \] in $L^2((0,\infty);dx)$. While the special case $n=1$ is classical and it is well-known that $\tau_2(c)\big|_{C_0^\infty((0,\infty))}$ is essentially self-adjoint if and only if $c \geq 3/4$, the case $n \in \mathbb{N}$, $n \geq 2$, is far from obvious. In particular, it is not at all clear from the outset that \[ \text{ there exists } c_n \in \mathbb{R}, \, n \in \mathbb{N}, \text{ such that } \tau_{2n}(c)\big|_{C_0^\infty((0,\infty))} \, \text{ is essentially self-adjoint if and only if } c \geq c_n. \tag{*}\label{0.1} \] As one of the principal results of this paper we indeed establish the existence of $c_n$, satisfying $c_n \geq (4n-1)!!\big/2^{2n}$, such that property \eqref{0.1} holds. In sharp contrast to the analogous lower semiboundedness question, \[ \text{ for which values of } c \, \text{\it is } \tau_{2n}(c)\big|_{C_0^{\infty}((0,\infty))} \, \text{ bounded from below?}, \] which permits the sharp (and explicit) answer $c \geq [(2n -1)!!]^{2}\big/2^{2n}$, $n \in \mathbb{N}$, the answer for \eqref{0.1} is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, \[ c_1 = 3/4, \quad c_2= 45, \quad c_3 = 2240 \big(214+7 \sqrt{1009}\,\big)\big/27, \] and remark that $c_n$ is the root of a polynomial of degree $n-1$. We demonstrate that for $n=6,7$, $c_n$ are algebraic numbers not expressible as radicals over $\mathbb{Q}$ (and conjecture this is in fact true for general $n \geq 6$)., Comment: 36 pages

Epigraphy increasingly turns to modern artificial intelligence (AI) technologies such as machine learning (ML) for extracting insights from ancient inscriptions. However, scarce labeled data for training ML algorithms severely limits current techniques, especially for ancient scripts like Old Aramaic. Our research pioneers an innovative methodology for generating synthetic training data tailored to Old Aramaic letters. Our pipeline synthesizes photo-realistic Aramaic letter datasets, incorporating textural features, lighting, damage, and augmentations to mimic real-world inscription diversity. Despite minimal real examples, we engineer a dataset of 250,000 training and 25,000 validation images covering the 22 letter classes in the Aramaic alphabet. This comprehensive corpus provides a robust volume of data for training a residual neural network (ResNet) to classify highly degraded Aramaic letters. The ResNet model demonstrates high accuracy in classifying real images from the 8th century BCE Hadad statue inscription. Additional experiments validate performance on varying materials and styles, proving effective generalization. Our results validate the model's capabilities in handling diverse real-world scenarios, proving the viability of our synthetic data approach and avoiding the dependence on scarce training data that has constrained epigraphic analysis. Our innovative framework elevates interpretation accuracy on damaged inscriptions, thus enhancing knowledge extraction from these historical resources., Comment: 41 pages, 19 images

The first half of this paper is largely expository, wherein we present a systematic combinatorial approach to the theory of polynomial (semi)invariants and multilinear invariants of several vectors and covectors, for the classical groups. This culminates in a graphical description of graded linear bases. By applying well-known results of lattice path combinatorics to Weyl's fundamental theorems of classical invariant theory, we write down Stanley decompositions and Hilbert-Poincare series in terms of families of non-intersecting lattice paths, enumerated with respect to certain corners. In the second half of the paper, we revisit the (semi)invariants in the first half as a special case of a much broader phenomenon. On one hand, polynomial invariants of a group $H$ can be generalized to modules of covariants, i.e., $H$-equivariant polynomial functions between $H$-modules. On the other hand, from the perspective of Roger Howe's theory of dual pairs, these modules of covariants can be viewed as infinite-dimensional simple $(\mathfrak{g}, K)$-modules. This suggests an expanded program in which our goal is to apply combinatorial techniques involving lattice paths in order to write down Hilbert series for arbitrary unitarizable highest-weight $(\mathfrak{g},K)$-modules. As a preview of future work in this program, we present examples showing how modules of covariants -- even those which are not Cohen-Macaulay, and therefore which we would not expect to be combinatorially nice -- can be decomposed in terms of lattice paths. We also extend these methods beyond the classical groups., Comment: 59 pages

We discuss the synergies between heliospheric and dust science, the open science questions, the technological endeavors and programmatic aspects that are important to maintain or develop in the decade to come. In particular, we illustrate how we can use interstellar dust in the solar system as a tracer for the (dynamic) heliosphere properties, and emphasize the fairly unexplored, but potentially important science question of the role of cosmic dust in heliospheric and astrospheric physics. We show that an Interstellar Probe mission with a dedicated dust suite would bring unprecedented advances to interstellar dust research, and can also contribute-through measuring dust - to heliospheric science. This can, in particular, be done well if we work in synergy with other missions inside the solar system, thereby using multiple vantage points in space to measure the dust as it `rolls' into the heliosphere. Such synergies between missions inside the solar system and far out are crucial for disentangling the spatially and temporally varying dust flow. Finally, we highlight the relevant instrumentation and its suitability for contributing to finding answers to the research questions., Comment: 18 pages, 7 Figures, 5 Tables. Originally submitted as white paper for the National Academies Decadal Survey for Solar and Space Physics 2024-2033

End-to-end event-based computation has the potential to push the envelope in latency and energy efficiency for edge AI applications. Unfortunately, event-based sensors (e.g., DVS cameras) and neuromorphic spike-based processors (e.g., Loihi) have been designed in a decoupled fashion, thereby missing major streamlining opportunities. This paper presents ColibriES, the first-ever neuromorphic hardware embedded system platform with dedicated event-sensor interfaces and full processing pipelines. ColibriES includes event and frame interfaces and data processing, aiming at efficient and long-life embedded systems in edge scenarios. ColibriES is based on the Kraken system-on-chip and contains a heterogeneous parallel ultra-low power (PULP) processor, frame-based and event-based camera interfaces, and two hardware accelerators for the computation of both event-based spiking neural networks and frame-based ternary convolutional neural networks. This paper explores and accurately evaluates the performance of event data processing on the example of gesture recognition on ColibriES, as the first step of full-system evaluation. In our experiments, we demonstrate a chip energy consumption of 7.7 \si{\milli\joule} and latency of 164.5 \si{\milli\second} of each inference with the DVS Gesture event data set as an example for closed-loop data processing, showcasing the potential of ColibriES for battery-powered applications such as wearable devices and UAVs that require low-latency closed-loop control.

An almost self-conjugate (ASC) partition has a Young diagram in which each arm along the diagonal is exactly one box longer than its corresponding leg. Classically, the ASC partitions and their conjugates appear in two of Littlewood's symmetric function identities. These identities can be viewed as Euler characteristics of BGG complexes of the trivial representation, for classical Hermitian symmetric pairs. In this paper, we consider partitions in which the arm-leg difference is an arbitrary constant $m$. Regarding these partitions and their conjugates as highest weights, we prove an identity yielding an infinite family of dimension equalities between $\mathfrak{gl}_n$- and $\mathfrak{gl}_{n+m}$-modules. We then interpret this combinatorial result in the context of blocks in parabolic category $\mathcal{O}$: using Enright-Shelton reduction, we find six infinite families of congruent blocks whose corresponding posets of highest weights consist of the partitions in question. These posets, in turn, lead to generalizations of the Littlewood identities and their corresponding BGG complexes. Our results in this paper shed light on the surprising combinatorics underlying the work of Enright and Willenbring (2004)., Comment: 33 pages, 5 figures, 4 tables

Using the Zhu algebra for a certain category of $\mathbb{C}$-graded vertex algebras $V$, we prove that if $V$ is finitely $\Omega$-generated and satisfies suitable grading conditions, then $V$ is rational, i.e. has semi-simple representation theory, with one dimensional level zero Zhu algebra. Here $\Omega$ denotes the vectors in $V$ that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element $\omega_\mu$ parameterized by $\mu \in \mathbb{C}$, and prove that for certain non-integer values of $\mu$, these vertex algebras, which are non-integer graded, are rational, with one dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate $\mathbb{C}$-graded Weyl vertex algebras of arbitrary ranks., Comment: Final version. Typos corrected and bibliography updated. We thank the referee for their comments and suggestions. To appear in Journal of Mathematical Physics

Most geophysical inverse problems are nonlinear and rely upon numerical forward solvers involving discretization and simplified representations of the underlying physics. As a result, forward modeling errors are inevitable. In practice, such model errors tend to be either completely ignored, which leads to biased and over-confident inversion results, or only partly taken into account using restrictive Gaussian assumptions. Here, we rely on deep generative neural networks to learn problem-specific low-dimensional probabilistic representations of the discrepancy between high-fidelity and low-fidelity forward solvers. These representations are then used to probabilistically invert for the model error jointly with the target geophysical property field, using the computationally-cheap, low-fidelity forward solver. To this end, we combine a Markov-chain-Monte-Carlo (MCMC) inversion algorithm with a trained convolutional neural network of the spatial generative adversarial network (SGAN) type, whereby at each MCMC step, the simulated low-fidelity forward response is corrected using a proposed model-error realization. Considering the crosshole ground-penetrating radar traveltime tomography inverse problem, we train SGAN networks on traveltime discrepancy images between: (1) curved-ray (high fidelity) and straight-ray (low fidelity) forward solvers; and (2) finite-difference-time-domain (high fidelity) and straight-ray (low fidelity) forward solvers. We demonstrate that the SGAN is able to learn the spatial statistics of the model error and that suitable representations of both the subsurface model and model error can be recovered by MCMC. In comparison with inversion results obtained when model errors are either ignored or approximated by a Gaussian distribution, we find that our method has lower posterior parameter bias and better explains the observed traveltime data.[...], Comment: Original publication: Geophys. J. Int., 23 September 2021

We present the multiple stellar systems observed within the SpHere INfrared survey for Exoplanet (SHINE). SHINE searched for substellar companions to young stars using high contrast imaging. Although stars with known stellar companions within SPHERE field of view (<5.5 arcsec) were removed from the original target list, we detected additional stellar companions to 78 of the 463 SHINE targets observed so far. 27% of the systems have three or more components. Given the heterogeneity of the sample in terms of observing conditions and strategy, tailored routines were used for data reduction and analysis, some of which were specifically designed for these data sets. We then combined SPHERE data with literature and archival ones, TESS light curves and Gaia parallaxes and proper motions, to characterise these systems as completely as possible. Combining all data, we were able to constrain the orbits of 25 systems. We carefully assessed the completeness of our sample for the separation range 50-500 mas (period range a few years - a few tens of years), taking into account the initial selection biases and recovering part of the systems excluded from the original list due to their multiplicity. This allowed us to compare the binary frequency for our sample with previous studies and highlight some interesting trends in the mass ratio and period distribution. We also found that, for the few objects for which such estimate was possible, the values of the masses derived from dynamical arguments were in good agreement with the model predictions. Stellar and orbital spins appear fairly well aligned for the 12 stars having enough data, which favour a disk fragmentation origin. Our results highlight the importance of combining different techniques when tackling complex problems such as the formation of binaries and show how large samples can be useful for more than one purpose., Comment: 59 pages, 11 tables, 18 figures

We present high-precision photometry and polarimetry for the protoplanetary disk around HD142527, with a focus on determining the light scattering parameters of the dust. We re-reduced polarimetric differential imaging data of HD142527 in the VBB (735 nm) and H-band (1625 nm) from the ZIMPOL and IRDIS subinstruments of SPHERE/VLT. With polarimetry and photometry based on reference star differential imaging, we were able to measure the linearly polarized intensity and the total intensity of the light scattered by the circumstellar disk with high precision. We used simple Monte Carlo simulations of multiple light scattering by the disk surface to derive constraints for three scattering parameters of the dust: the maximum polarization of $P_{\rm max}$, the asymmetry parameter $g$, and the single-scattering albedo $\omega$. We measure a reflected total intensity of $51.4\pm1.5$ mJy and $206\pm12$ mJy and a polarized intensity of $11.3\pm0.3$ mJy and $55.1\pm3.3$ mJy in the VBB and H-band, respectively. We also find in the visual range a degree of polarization that varies between $28\%$ on the far side of the disk and $17\%$ on the near side. The disk shows a red color for the scattered light intensity and the polarized intensity, which are about twice as high in the near-infrared when compared to the visual. We determine with model calculations the scattering properties of the dust particles and find evidence for strong forward scattering ($g\approx 0.5-0.75$), relatively low single-scattering albedo ($\omega \approx 0.2-0.5$), and high maximum polarization ($P_{\rm max} \approx 0.5-0.75$) at the surface on the far side of the disk for both observed wavelengths. The optical parameters indicate the presence of large aggregate dust particles, which are necessary to explain the high maximum polarization, the strong forward-scattering nature of the dust, and the observed red disk color., Comment: 20 pages, 14 figures

Polarisation is a decisive method to study the inner region of active galactic nuclei (AGNs) since it is not affected by contrast issues similarly to how classical imaging is. When coupled to high angular resolution (HAR), polarisation can help to disentangle the location of the various polarising mechanisms and then give an insight on the physics taking place on the core of AGNs. We obtained a new data set of HAR polarimetric images of the archetypal Seyfert 2 nucleus of NGC 1068 observed with SPHERE/VLT and we aim in this paper at presenting the polarisation maps and at spatially separating the location of the polarising mechanisms, thus deriving constraints on the organisation of the dust material in the inner region of this AGN. We then compared these measurements to radiative transfer simulations of scattering and dichroic absorption processes, using the Monte-Carlo code MontAGN. We establish a detailed table of the relative importance of the polarising mechanism as a function of the aperture and of the wavelength. We are able to separate the dominant polarising mechanisms in the three regions of the ionisation cone, the extended envelop of the torus and the very central bright source of the AGN. Thus, we estimate the contribution of the different polarisation mechanisms to the observed polarisation flux in these regions. Dichroic absorption is estimated to be responsible for about 99 % of the polarised flux coming from the photo-centre. However, this contribution would be restricted to this location only, double scattering process being the most important contributor to polarisation in the equatorial plane of the AGN and single scattering being dominant in the polar outflow bi-cone.