Uma Iyer
We use the definition of differential operators on noncommutative rings given by V.Lunts and A.Rosenberg to find the differential operators on Azumaya algebras and the Heisenberg algebras.
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We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo $p.$
For a smooth toric variety X over a field of positive characteristic, a T-equivariant étale cover Y \rightarrow T^*X^{(1)} trivializing the sheaf of crystalline differential operators on X is constructed. This trivialization is used to show that the sheaf of differential operators is a trivial Azumaya algebra along the fibers of the moment map. This result is then extended to certain Azumaya algebras on hypertoric varieties, whose global sections are central reductions of the hypertoric enveloping algebra in positive characteristic. A criteria for a derived Beilinson-Bernstein localization the…
Let $A$ be a ring equipped with a derivation $δ$. We study differential Azumaya $A$ algebras, that is, Azumaya $A$ algebras equipped with a derivation that extends $δ$. We calculate the differential automorphism group of the trivial differential algebra, $M_{n}(A)$ with coordinatewise differentiation. We introduce the $δ$-flat Grothendieck topology to show that any differential Azumaya $A$ algebra is locally isomorphic to a trivial one and then construct, as in the non-differential setting, the embedding of the differential Brauer group into $H^{2}(A_{δ-pl},G_{m,δ}) $. We conclude by showing t…
In this Part II of D(11), we introduce new objects: super-$C^k$-schemes and Azumaya super-$C^k$-manifolds with a fundamental module (or, synonymously, matrix super-$C^k$-manifolds with a fundamental module), and extend the study in D(11.1) ([L-Y3], arXiv:1406.0929 [math.DG]) to define the notion of `differentiable maps from an Azumaya/matrix supermanifold with a fundamental module to a real manifold or supermanifold'. This allows us to introduce the notion of `fermionic D-branes' in two different styles, one parallels Ramond-Neveu-Schwarz fermionic string and the other Green-Schwarz fermionic …
We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous maps, $k$-times differentiable maps, and smooth maps from an Azumaya/matrix manifold with a fundamental module to a (commutative) real manifold $Y$ is developed. Such maps are meant to describe D-branes or matrix branes in string theory when these branes are light and soft with only small enough or even zero brane-tension. When $Y$ is a symplec…
A note on Azumaya algebras and one-forms
本文证明若光滑簇X存在非闭的整体一次微分形式，则其Frobenius扭转X'上由晶体微分算子诱导的非分裂Azumaya代数在X'的某个一次覆盖上仍不分裂，回答了Sasha Petrov的问题。
In this continuation of [L-Y1], [L-L-S-Y], [L-Y2], and [L-Y3] (arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], arXiv:0901.0342 [math.AG], arXiv:0907.0268 [math.AG]), we study D-branes in a target-space with a fixed $B$-field background $(Y,α_B)$ along the line of the Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the current work. We focus first on the gauge-field-twist effect of $B$-field to the Chan-Paton module on D-branes. Basic properties of the moduli space of D-branes, as morphisms from Azumaya schemes with a twisted fundamental module to $(Y,α_B)$, ar…
Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of rank r, and Hmo(R) the category of noncommutative motives with R-coefficients. Assume that 1/r belongs to R. Under this assumption, we prove that the noncommutative motives with R-coefficients of X and A are isomorphic. As an application, we show that all the R-linear additive invariants of X and A are exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homolo…
We consider q-deformations of multiplicative hypertoric varieties, for q a non-zero element of an algebraically closed field of characteristic 0. We construct an algebra Dq of q-difference operators as a Heisenberg double in a braided monoidal category. We then focus on the case where q is specialized to a root of unity. In this setting, we use Dq to construct an Azumaya algebra on an l-twist of the multiplicative hypertoric variety, before showing that this algebra splits over the fibers of both the moment and resolution maps. Finally, we sketch a derived localization theorem for these Azumay…
We define the $p^m$-curvature map on the sheaf of differential operators of level $m$ on a scheme of positive characteristic $p$ as dual to some divided power map on infinitesimal neighborhhods. This leads to the notion of $p^m$-curvature on differential modules of level $m$. We use this construction to recover Kaneda's description of a semi-linear Azumaya splitting of the sheaf of differential operators of level $m$. Then, using a lifting modulo $p^2$ of Frobenius, we are able to define a Frobenius map on differential operators of level $m$ as dual to some divided Frobenius on infinitesimal n…
For a smooth variety $Y$ over a perfect field of positive characteristic, the sheaf $D_Y$ of crystalline differential operators on $Y$ (also called the sheaf of $PD$-differential operators) is known to be an Azumaya algebra over $T^*_{Y'},$ the cotangent space of the Frobenius twist $Y'$ of $Y.$ Thus to a sheaf of modules $M$ over $D_Y$ one can assign a closed subvariety of $T^*_{Y'},$ called the $p$-support, namely the support of $M$ seen as a sheaf on $T^*_{Y'}.$ We study here the family of $p$-supports assigned to the reductions modulo primes $p$ of a holonomic $\mathcal{D}$-module. We prov…
For a smooth variety $X$ over an algebraically closed field of characteristic $p$, to a differential 1-form $α$ on the Frobenius twist $X^{(1)}$ one can associate an Azumaya algebra $\mathcal D_{X,α}$, defined as a certain central reduction of the algebra $\mathcal D_X$ of "crystalline differential operators" on $X$. For a resolution of singularities $π:X\to Y$ of an affine variety $Y$, we study for which $α$ does the class $[\mathcal D_{X,α}]$ in the Brauer group $\mathrm{Br}(X^{(1)})$ descend to $Y^{(1)}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations …
We construct explicitly noncommutative deformations of categories of holomorphic line bundles over higher dimensional tori. Our basic tools are Heisenberg modules over noncommutative tori and complex/holomorphic structures on them introduced by A. Schwarz. We obtain differential graded (DG) categories as full subcategories of curved DG categories of Heisenberg modules over the complex noncommutative tori. Also, we present the explicit composition formula of morphisms, which in fact depends on the noncommutativity.
Let X a proper smooth curve over the field of complex numbers. Localization of the Heisenberg algebra gives the algebra of global sections of the ring of differential operators on the Jacobian J of X. It seems natural to ask for same kind of relations over a characteristic p > 0 field replacing the Heisenberg algebra by its entire form. We propose a conjecture about the constuction of an arrow between the two objects when X is an elliptic curve.
Using the curved bc-beta-gamma system (a tensor product of a Heisenberg and a Clifford vertex algebra) we introduce quantum analogy of Lichnerowicz differential. As follows we suggest new machinery for finding the Lichnerowicz-Poisson cohomology groups for any Poisson manifold. Moreover, the defined provides new invariant. Keywords: Poisson manifold, Lichnerowicz differential, Chiral de Rham complex, cohomologies, vertex algebras, deformation theory, Nambu-Poisson bracket, n-Lie algebras, Gromov-Witten theory.
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