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The heterotic $SU(3)$ system, also known as the Hull--Strominger system, arises from compactifications of heterotic string theory to six dimensions. This paper investigates the local structure of the moduli space of solutions to this system on a compact 6-manifold $X$, using a vector bundle $Q=(T^{1,0}X)^* \oplus {End}(E) \oplus T^{1,0}X$, where $E\to X$ is the classical gauge bundle arising in the system. We establish that the moduli space has an expected dimension of zero. We achieve this by studying the deformation complex associated to a differential operator $\bar{D}$, which emulates a holomorphic structure on $Q$, and demonstrating an isomorphism between the two cohomology groups which govern the infinitesimal deformations and obstructions in the deformation theory for the system. We also provide a Dolbeault-type theorem linking these cohomology groups to \v{C}ech cohomology, a result which might be of independent interest, as well as potentially valuable for future research., Comment: v2: 34 pages, expanded introduction, example added, references updated

This paper initiates a classification programme of flows of $\mathrm{SU}(2)$-structures on $4$-manifolds which have short-time existence and uniqueness. Our approach adapts a representation-theoretic method originally due to Bryant in the context of $\mathrm{G}_2$ geometry. We show how this strategy can also be used to deduce the number of geometric flows of a given $H$-structure; we illustrate this in the $\mathrm{G}_2$, $\mathrm{Spin}(7)$ and $\mathrm{SU}(3)$ cases. Our investigation also leads us to derive explicit expressions for the Ricci and self-dual Weyl curvature in terms of the intrinsic torsion of the underlying $\mathrm{SU}(2)$-structure. We compute the first variation formulae of all the quadratic functionals in the torsion; these provide natural building blocks for $\mathrm{SU}(2)$ gradient flows. In particular, our results demonstrate that both the negative gradient flow of the Dirichlet energy of the intrinsic torsion and the Ricci harmonic flow are parabolic after a modified DeTurck's trick.

We explore three versions of the Laplacian coflow of $G_2$-structures on circle fibrations over Calabi--Yau 3-folds, interpreting their dimensional reductions to the K\"ahler geometry of the base. Precisely, we reduce Ans\"atze for the Laplacian coflow, modified or not by de Turck's trick, both on trivial products $CY^3\times S^1$ and on contact Calabi--Yau 7-manifolds, obtaining in each case a natural modification of the K\"ahler--Ricci flow.

We introduce the coupled instanton equations for a metric, a spinor, a three-form, and a connection on a bundle, over a spin manifold. Special solutions in dimensions $6$ and $7$ arise, respectively, from the Hull--Strominger and the heterotic $\operatorname{G}_2$ system. The equations are motivated by recent developments in theoretical physics and can be recast using generalized geometry; we investigate how coupled instantons relate to generalized Ricci-flat metrics and also to Killing spinors on a Courant algebroid. We present two open questions regarding how these different geometric conditions are intertwined, for which a positive answer is expected from recent developments in the physics literature by De la Ossa, Larfors and Svanes, and in the mathematics literature on Calabi--Yau manifolds, in recent work by the second-named author with Gonz\'alez Molina. We give a complete solution to the first of these problems, providing a new method for the construction of instantons in arbitrary dimensions. For $\operatorname{G}_2$-structures with torsion coupled to $\operatorname{G}_2$-instantons, in dimension $7$, we also establish results around the second problem. The last part of the present work carefully studies the approximate solutions to the heterotic $\operatorname{G}_2$-system constructed by the third and fourth authors on contact Calabi--Yau $7$-manifolds, for which we prove the existence of approximate coupled $\operatorname{G}_2$-instantons and generalized Ricci-flat metrics., Comment: 45 pages, new Theorem 2.32 with complete solution to Problem 1, in arbitrary dimensions. Improved version of Theorem 4.9. New Remark 4.10 about the Spin(7) case. Submitted for consideration in the International Journal of Mathematics special issue "At the interface of complex geometry and string theory"

This work seeks to advance the understanding of the smooth structure of the moduli space of self-dual contact instantons (SDCI) on Sasakian 7-manifolds M. A neighborhood of a smooth point of M is locally modeled on the first cohomological group of an elliptic complex (1.4). There is a cohomological obstruction to the smoothness for the moduli space, in terms of a second basic cohomological group, in this paper we study conditions under which this obstruction disappears, by computing a Weitzenb\"ock formula and using a Bochner-type method to obtain a vanishing theorem. Given an SDCI on a Sasakian bundle E, we find sufficient conditions for the vanishing of the obstruction in the positivity of a couple of operators R and F depending on the curvatures of the connection and the Riemann curvature of the Sasakian metric g. In particular, we find that if M is transversely Ricci positive and F positive, the moduli space of SDCI must be smooth. However, in general, the operator F is not positive definite and we describe bundles over the Stiefel manifold for which it is the case. Finally, we show that when the energy of the curvature is less than the first non-zero eigenvalue of RicT the obstruction vanishes.

The pursuit of artificial consciousness requires conceptual clarity to navigate its theoretical and empirical challenges. This paper introduces a composite, multilevel, and multidimensional model of consciousness as a heuristic framework to guide research in this field. Consciousness is treated as a complex phenomenon, with distinct constituents and dimensions that can be operationalized for study and for evaluating their replication. We argue that this model provides a balanced approach to artificial consciousness research by avoiding binary thinking (e.g., conscious vs. non-conscious) and offering a structured basis for testable hypotheses. To illustrate its utility, we focus on "awareness" as a case study, demonstrating how specific dimensions of consciousness can be pragmatically analyzed and targeted for potential artificial instantiation. By breaking down the conceptual intricacies of consciousness and aligning them with practical research goals, this paper lays the groundwork for a robust strategy to advance the scientific and technical understanding of artificial consciousness.

We propose a machine learning approach to study topological quantities related to the Sasakian and $G_2$-geometries of contact Calabi-Yau $7$-manifolds. Specifically, we compute datasets for certain Sasakian Hodge numbers and for the Crowley-N\"ordstrom invariant of the natural $G_2$-structure of the $7$-dimensional link of a weighted projective Calabi-Yau $3$-fold hypersurface singularity, for 7549 of the 7555 possible $\mathbb{P}^4(\textbf{w})$ projective spaces. These topological quantities are then machine learnt with high performance scores, where learning the Sasakian Hodge numbers from the $\mathbb{P}^4(\textbf{w})$ weights alone, using both neural networks and a symbolic regressor which achieve $R^2$ scores of 0.969 and 0.993 respectively. Additionally, properties of the respective Gr\"obner bases are well-learnt, leading to a vast improvement in computation speeds which may be of independent interest. The data generation and analysis further induced novel conjectures to be raised., Comment: 10 pages; 5 figures; 1 table

Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that the inequality \[ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1 \] has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| \leq C|t|$ and $|F_t(X,Y)| \leq |t|^{2 -\varepsilon}$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided., Comment: 27 pages

We formulate the gradient Dirichlet flow of $Sp(2)Sp(1)$-structures on $8$-manifolds, as the first systematic study of a geometric quaternion-K\"ahler (QK) flow. Its critical condition of \emph{harmonicity} is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the $G_2$ case. We also draw several comparisons with $Spin(7)$-structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via $\epsilon$-regularity. We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold $SU(3)$. We also study compactness and the formation of singularities., Comment: 32 pages

We develop an abstract theory of flows of geometric $H$-structures, i.e., flows of tensor fields defining $H$-reductions of the frame bundle, for a closed and connected subgroup $H\subset SO(n)$, on any connected and oriented $n$-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of $H$-structures, by way of the natural infinitesimal action of $\mathrm{GL}(n,\mathbb{R})$ on tensors combined with various bundle decompositions induced by $H$-structures. We compute evolution equations for the intrinsic torsion under general flows of $H$-structures and, as applications, we obtain general Bianchi-type identities for $H$-structures, and, for closed manifolds, a general first variation formula for the $L^2$-Dirichlet energy functional $\mathcal{E}$ on the space of $H$-structures. We then specialise the theory to the negative gradient flow of $\mathcal{E}$ over isometric $H$-structures, i.e., their harmonic flow. The core result is an almost monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulae by Chen-Struwe for the harmonic map heat flow. This yields an $\varepsilon$-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, relative to the $L^\infty$-norm of initial torsion, in the spirit of Chen-Ding. Moreover, below a certain energy level, the absence of a torsion-free isometric $H$-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat $n$-tori, so long as $[\mathbb{S}^n,SO(n)/H]$ contains more than one element and the universal cover of $SO(n)/H$ is a sphere; e.g. when $n=7$ and $H=\rm G_2$, or $n=8$ and $H=\rm Spin(7)$., Comment: v4: 61 pages, no figures. Final version, to appear in Journal f\"ur die reine und angewandte Mathematik. Completely revised article. Main corrections and clarifications: Lemma 2.4, proofs of Theorems 2.5, 2.7 and 2.10; Example 2.17 and Remark 2.18; proof of Lemma 2.23; Proposition 2.24; and proof of Theorem 2.29