Related papers: Prismatic crystals and $q$-Higgs fields
In this note, we introduce and study the Cartier--Witt stack $\mathrm{WCart}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in…
We compare several different notions of filtered derived commutative ring, discussing HKR-filtered Hochschild homology, Hodge-filtered de Rham cohomology, and the lesser-known Hodge-filtered infinitesimal cohomology. Our main result is that…
For any prism $(A, d)$, we construct an analogue of Fontaine's map $W_r(A/d) \to A/d\phi(d)\cdots\phi^{r-1}(d)$. Subsequently, we define a canonical map from de Rham-Witt forms to prismatic cohomology in the perfect case and prove that it…
We prove that rigid cohomology can be computed as the cohomology of a site analogous to the crystalline site. Berthelot designed rigid cohomology as a common generalization of crystalline and Monsky-Washnitzer cohomology. Unfortunately,…
We explain how the action of the Heisenberg algebra on the space of q-deformed wedges yields the Heisenberg crystal structure on charged multipartitions, by using the boson-fermion correspondence and looking at the action of the Schur…
A framework is developed to describe the Zariski topologies on the prime and primitive spectra of a quantum algebra $A$ in terms of the (known) topologies on strata of these spaces and maps between the collections of closed sets of…
We obtain a decomposition for the Hochschild cochain complex of a split algebra and we study some properties of the cohomology of each term of this decomposition. Then, we consider the case of trivial extensions, specially of Frobenius…
A mathematical framework is developed to describe tilted perovskites using a tensor description of octahedral deformations. The continuity of octahedral tilts through the crystal is described using an operator which relates the deformations…
We define the big crystalline site for a log scheme and prove the basic properties. In particular, we show the boundedness, base change, and perfectness theorems for the crystalline higher direct image of quasi-coherent crystals between…
Let $\g$ be an affine Kac-Moody Lie algebra. Let $\U^+$ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to $\g$. We construct a basis of $\U^+$ which is related to the Kashiwara-Lusztig global crystal basis…
In local relative $p$-adic Hodge theory, we show that the Galois cohomology of a finite height crystalline representation (up to a twist) is essentially computed via the (Fontaine--Messing) syntomic complex with coefficients in the…
We propose a generic framework in which the Higgs and the unparticle are both composite. The underlying theories are four dimensional, asymptotically free, nonsupersymmetric gauge theories with fermionic matter. We sketch a possible uni-…
We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining…
We show that there exists a unique crystal base of a parabolic Verma module over a quantum orthosymplectic superalgebra, which is induced from a $q$-analogue of a polynomial representation of a general linear Lie superalgebra.
A general concept for the derivation of symmetry-based pseudo spin Hamiltonians is described. It systematically bridges the gap between the atomistic basis and various pseudo spin models presented in literature. It thus allows the…
In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded…
In this paper the problem of the theory of a quasicrystal structures - the determination of coordinates of each atom of quasicrystal in analytical form - is solved. Within the framework of the proposed model a periodic crystal can be…
We characterize the relative prismatic cohomology of Bhatt and Scholze by a universal property by endowing it with the additional structure of a ``derived $\delta$-ring". This involves introducing an analogue of prismatic envelopes in the…
We show a comparison theorem between log prismatic cohomology and log crystalline cohomology for a $p$-adic formal scheme with semistable reduction. Combined with the prismatic-\'etale comparison theorem recently proved by Tian, this…
The ordering dynamics of the Higgs field is studied, using techniques inspired by the study of phase ordering in condensed matter physics, as a first step to understanding the evolution of cosmic structure through the formation of…