Search

Let Z ⊂ A k be an affine scheme over C and J Z its jet scheme. It is well-known that C [ J Z ] , the coordinate ring of J Z , has the structure of a commutative vertex algebra. This paper develops the orbifold theory for C [ J Z ] . A finite-order linear automorphism g of Z acts by vertex algebra automorphisms on C [ J Z ] . We show that C [ J g Z ] , where J g Z is the scheme of g-twisted jets has the structure of a g-twisted C [ J Z ] module. We consider spaces of orbifold coinvariants valued in the modules C [ J g Z ] on orbicurves [ Y / G ] , with Y a smooth projective curve and G a finite group, and show that these are isomorphic to C [ Z G ] .

We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the representation possess a compatible grading, and conditions on the support of the module. Quotients by these ideals lead to combinatorial Hopf algebras which can be interpreted as Hall algebras of certain sub-categories of modules. In the case of the free commutative monoid on n generators, we obtain a co-commutative Hopf algebra structure on n-dimensional skew shapes, whose underlying associative product amounts to a “stacking” operation on the skew shapes. The primitive elements of this Hopf algebra correspond to connected skew shapes, and form a graded Lie algebra by anti-symmetrizing the associative product. We interpret this Hopf algebra as the Hall algebra of a certain category of coherent torsion sheaves on A / F 1 n supported at the origin, where F 1 denotes the field of one element. This Hopf algebra may be viewed as an n-dimensional generalization of the Hopf algebra of symmetric functions, which corresponds to the case n = 1 .

Given a graded monoid A with 1, one can construct a projective monoid scheme MProj ( A ) analogous to Proj ( R ) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves on MProj ( A ) , and we prove several basic results regarding these. We show that: 1. every quasicoherent sheaf F on MProj ( A ) can be constructed from a graded A-set in analogy with the construction of quasicoherent sheaves on Proj ( R ) from graded R-modules 2. if F is coherent on MProj ( A ) , then F ( n ) is globally generated for large enough n, and consequently, that F is a quotient of a finite direct sum of invertible sheaves 3. if F is coherent on MProj ( A ) , then Γ ( MProj ( A ) , F ) is finitely generated over A 0 (and hence a finite set if A 0 = { 0 , 1 } ). The last part of the paper is devoted to classifying coherent sheaves on P 1 in terms of certain directed graphs and gluing data. The classification of these over F 1 is shown to be much richer and combinatorially interesting than in the case of ordinary P 1 , and several new phenomena emerge.

In this text, we generalize Cech cohomology to sheaves F with values in blue B-modules where B is a blueprint with −1. If X is an object of the underlying site, then the cohomology sets H l ( X , F ) turn out to be blue B-modules. For a locally free O X -module F on a monoidal scheme X, we prove that H l ( X , F ) + = H l ( X + , F + ) where X + is the scheme associated with X and F + is the locally free O X + -module associated with F . In an appendix, we show that the naive generalization of cohomology as a right derived functor is infinite-dimensional for the projective line over F 1 .

We study commutative ring structures on the integral span of rooted trees and $n$-dimensional skew shapes. The multiplication in these rings arises from the smash product operation on monoid representations in pointed sets. We interpret these as Grothendieck rings of indecomposable monoid representations over $\fun$ - the "field" of one element. We also study the base-change homomorphism from $\mt$-modules to $k[t]$-modules for a field $k$ containing all roots of unity, and interpret the result in terms of Jordan decompositions of adjacency matrices of certain graphs., arXiv admin note: text overlap with arXiv:1706.03900

Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields., 25 pages

Let \(\mathrm{A }\) be a finitely generated semigroup with 0. An \(\mathrm{A }\)-module over \(\mathbb F _1\) (also called an \(\mathrm{A }\)-set), is a pointed set \((M,*)\) together with an action of \(\mathrm{A }\). We define and study the Hall algebra \(\mathbb H _{\mathrm{A }}\) of the category \(\mathcal C _{\mathrm{A }}\) of finite \(\mathrm{A }\)-modules. \(\mathbb H _{\mathrm{A }}\) is shown to be the universal enveloping algebra of a Lie algebra \(\mathfrak n _{\mathrm{A }}\), called the Hall Lie algebra of \(\mathcal C _{\mathrm{A }}\). In the case of \(\langle t \rangle \)—the free monoid on one generator \(\langle t \rangle \), the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent \(\langle t \rangle \)-modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when \(\mathrm{A }\) is a quotient of \(\langle t \rangle \) by a congruence, and the monoid \(G \cup \{ 0\}\) for a finite group \(G\).

We define and study the category C o h n ( P 1 ) of normal coherent sheaves on the monoid scheme P 1 (equivalently, the M 0 -scheme P 1 / F 1 in the sense of Connes–Consani–Marcolli, Connes (2009) [2] ). This category resembles in most ways a finitary abelian category, but is not additive. As an application, we define and study the Hall algebra of C o h n ( P 1 ) . We show that it is isomorphic as a Hopf algebra to the enveloping algebra of the product of a non-standard Borel in the loop algebra L g l 2 and an abelian Lie algebra on infinitely many generators. This should be viewed as a ( q = 1 ) version of Kapranov’s result relating (a certain subalgebra of) the Ringel–Hall algebra of P 1 over F q to a non-standard quantum Borel inside the quantum loop algebra U ν ( s l 2 ) , where ν 2 = q .

We consider natural representations of the Connes-Kreimer Lie algebras on rooted trees/Feynman graphs arising from Hecke correspondences in the categories LRF ,LFG constructed by K. Kremnizer and the author. We thus obtain the insertion/elimination representations constructed by Connes-Kreimer as well as an isomorphic pair we term top-insertion/top-elimination. We also construct graded finite-dimensional sub/quotient representations of these arising from ”truncated” correspondences.

We construct symmetric monoidal categories \({\mathcal{LRF}, \mathcal{LFG}}\) of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of \({\mathcal{LRF}, \mathcal{LFG}}\), \({{\bf H}_\mathcal{LRF}, {\bf H}_\mathcal{LFG}}\) are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman diagrams. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras.

We examine the structure of the insertion-elimination Lie algebra on rooted trees introduced in \cite{CK}. It possesses a triangular structure $\g = \n_+ \oplus \mathbb{C}.d \oplus \n_-$, like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a "lowest weight" $\lambda \in \mathbb{C}$. We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.

We present a superfield formulation of the chiral de Rham complex (CDR) of Malikov-Schechtman-Vaintrob in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N=1 structure on CDR (action of the N=1 super--Virasoro, or Neveu--Schwarz, algebra). If the metric is K"ahler, and the manifold Ricci-flat, this is augmented to an N=2 structure. Finally, if the manifold is hyperk"ahler, we obtain an N=4 structure. The superconformal structures are constructed directly from the Levi-Civita connection. These structures provide an analog for CDR of the extended supersymmetries of nonlinear sigma-models., Comment: References added

Suppose that a finite group G G acts on a smooth complex variety X X . Then this action lifts to the Chiral de Rham complex Ω X ch \Omega ^{\operatorname {ch}}_{X} of X X and to its cohomology by automorphisms of the vertex algebra structure. We define twisted sectors for Ω X ch \Omega ^{\operatorname {ch}}_{X} (and their cohomologies) as sheaves of twisted vertex algebra modules supported on the components of the fixed-point sets X g , g ∈ G X^{g}, g \in G . Each twisted sector sheaf carries a BRST differential and is quasi-isomorphic to the de Rham complex of X g X^{g} . Putting the twisted sectors together with the vacuum sector and taking G G -invariants, we recover the additive and graded structures of Chen-Ruan orbifold cohomology. Finally, we show that the orbifold elliptic genus is the partition function of the direct sum of the cohomologies of the twisted sectors.

Starting with a vertex algebra V, a finite group G of automorphisms of V, and a suitable collection of twisted V-modules, we construct (twisted) D-modules on the stack of pointed G-covers, introduced by Jarvis, Kaufmann, and Kimura. The fibers of these sheaves are spaces of orbifold conformal blocks defined in joint work with Edward Frenkel. The key ingredient is a G-equivariant version of the Virasoro uniformization theorem.

Given a compact complex algebraic variety with an effective action of a finite group G, and a class � 2 H 2 (G,U(1)), we introduce an orbifold elliptic genus with discrete torsion �, denoted Ell �(X,G,q,y). We give an interpretation of this genus in terms of the chiral de Rham complex attached to the orbifold (X/G). If X is Calabi-Yau and G preserves the volume form, Ell �(X,G,q,y) is a weak Jacobi form. We also obtain a formula for the generating function of the elliptic genera of symmetric products with discrete torsion.

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let $V$ be a vertex algebra, $H$ a finite group of automorphisms of $V$, and $C$ an algebraic curve such that $H \subset \on{Aut}(C)$. We show that a suitable collection of twisted $V$--modules gives rise to a section of a certain sheaf on the quotient $X=C/H$. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac-Moody vertex algebras. We also give a chiral algebra interpretation of twisted modules.

We construct Wakimoto modules for twisted a!ne Lie algebras , and interpret this construction in terms of vertex algebras and their twisted modules. Using the Wakimoto construction, we prove the Kac-Kazhdan conjecture on the characters of irreducible modules with generic critical highest weights in the twisted case. We provide explicit formulas for the twisted fields in the case of A (2) .

We present V light curves of P Cygni from JD 2446200-9300 (1985-1993). Inspection of the light curves, and Fourier and autocorrelation analysis, show that the star varies on time scales of about 40 to several hundred days. No one time scale stands out, though 40, 120 and 160 days are noticeable in the autocorrelation diagram of the combined 1985-1993 data.