Torsion Classes Research Articles

Axioms for higher twisted torsion invariants of smooth bundles

This paper attempts to investigate the space of various characteristic classes for smooth manifold bundles with local system on the total space inducing a finite holonomy covering. These classes are known as twisted higher torsion classes. We will give a system of axioms that we require these cohomology classes to satisfy. Higher Franz Reidemeister torsion and twisted versions of the higher Miller-Morita-Mumford classes will satisfy these axioms. We will show that the space oftwisted torsion invariants is two dimensional or one dimensional depending on the torsion degree and spanned by these two classes. The proof will greatly depend on results on the equivariant Hatcher constructions developed in a separate paper.

Modularity lifting beyond the Taylor–Wiles method

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor–Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions—one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on the automorphic side, applies to automorphic forms on the group $$\mathrm {GL}(n)$$ over a general number field; it is contingent on a conjecture which, in particular, predicts the existence of Galois representations associated to torsion classes in the cohomology of the associated locally symmetric space. We show that if this conjecture holds, then our main theorem implies the following: if E is an elliptic curve over an arbitrary number field, then E is potentially automorphic and satisfies the Sato–Tate conjecture. In addition, we also prove some unconditional results. For example, in the setting of $$\mathrm {GL}(2)$$ over $$\mathbf {Q}$$ , we identify certain minimal global deformation rings with the Hecke algebras acting on spaces of p-adic Katz modular forms of weight 1. Such algebras may well contain p-torsion. Moreover, we also completely solve the problem (for p odd) of determining the multiplicity of an irreducible modular representation $$\overline{\rho }$$ in the Jacobian $$J_1(N)$$ , where N is the minimal level such that $$\overline{\rho }$$ arises in weight two.

$\boldsymbol{\tau}$ -Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors

The class of support $\tau$-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $\tau$-tilting finite algebras, i.e. finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $\tau$-rigid modules. We show that $A$ is $\tau$-tilting finite if and only if very torsion class in $\mod A$ is functorially finite. We observe that cones generated by $g$-vectors of indecomposable direct summands of each support $\tau$-tilting module form a simplicial complex $\Delta(A)$. We show that if $A$ is $\tau$-tilting finite, then $\Delta(A)$ is homeomorphic to an $(n-1)$-dimensional sphere, and moreover the partial order on support $\tau$-tilting modules can be recovered from the geometry of $\Delta(A)$. Finally we give a bijection between indecomposable $\tau$-rigid $A$-modules and bricks of $A$ satisfying a certain finiteness condition, which is automatic for $\tau$-tilting finite algebras.

Type II string theory on Calabi-Yau manifolds with torsion and non-Abelian discrete gauge symmetries

We provide the first explicit example of Type IIB string theory compactification on a globally defined Calabi-Yau threefold with torsion which results in a four-dimensional effective theory with a non-Abelian discrete gauge symmetry. Our example is based on a particular Calabi-Yau manifold, the quotient of a product of three elliptic curves by a fixed point free action of {mathbb{Z}}_2times {mathbb{Z}}_2 . Its cohomology contains torsion classes in various degrees. The main technical novelty is in determining the multiplicative structure of the (torsion part of) the cohomology ring, and in particular showing that the cup product of second cohomology torsion elements goes non-trivially to the fourth cohomology. This specifies a non-Abelian, Heisenberg-type discrete symmetry group of the cfour-dimensional theory.

A canonical torsion theory for pro-p Iwahori–Hecke modules

Let F be a locally compact nonarchimedean field with residue characteristic p and G the group of F-rational points of a connected split reductive group over F. We define a torsion pair in the category Mod(H) of modules over the pro-p-Iwahori Hecke k-algebra H of G, where k is an arbitrary field. We prove that, under a certain hypothesis, the torsionfree class embeds fully faithfully into the category ModI(G) of smooth k-representations of G generated by their pro-p-Iwahori fixed vectors.If the characteristic of k is different from p then this hypothesis is always satisfied and the torsionfree class is the whole category Mod(H).If k contains the residue field of F then we study the case G=SL2(F). We show that our hypothesis is satisfied, and we describe explicitly the torsionfree and the torsion classes. If F≠Qp and p≠2, then an H-module is in the torsion class if and only if it is a union of supersingular finite length submodules; it lies in the torsionfree class if and only if it does not contain any nonzero supersingular finite length module. If F=Qp, the torsionfree class is the whole category Mod(H), and we give a new proof of the fact that Mod(H) is equivalent to ModI(G). These results are based on the computation of the H-module structure of certain natural cohomology spaces for the pro-p-Iwahori subgroup I of G.

LATTICE STRUCTURE OF TORSION CLASSES FOR HEREDITARY ARTIN ALGEBRAS

Let $\unicode[STIX]{x1D6EC}$ be a connected hereditary artin algebra. We show that the set of functorially finite torsion classes of $\unicode[STIX]{x1D6EC}$-modules is a lattice if and only if $\unicode[STIX]{x1D6EC}$ is either representation-finite (thus a Dynkin algebra) or $\unicode[STIX]{x1D6EC}$ has only two simple modules. For the case of $\unicode[STIX]{x1D6EC}$ being the path algebra of a quiver, this result has recently been established by Iyama–Reiten–Thomas–Todorov and our proof follows closely some of their considerations.

Geometric models of twisted differential K-theory I

This is the first in a series of papers constructing geometric models of twisted differential K-theory. In this paper we construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion class. By differential twists we will mean smooth U(1)-gerbes with connection, and we use twisted vector bundles with connection as cocycles. The model we construct satisfies the axioms of Kahle and Valentino, including functoriality, naturality of twists, and the hexagon diagram. This paper confirms a long-standing hypothetical idea that twisted vector bundles with connection define twisted differential K-theory.

Torsion classes, wide subcategories and localisations

For a finite dimensional algebra $A$, we establish correspondences between torsion classes and wide subcategories in $mod(A)$. In case $A$ is representation finite, we obtain an explicit bijection between these two classes of subcategories. Moreover, we translate our results to the language of ring epimorphisms and universal localisations. It turns out that universal localisations over representation finite algebras are classified by torsion classes and support $\tau$-tilting modules.

Non-vanishing superpotentials in heterotic string theory and discrete torsion

We study the non-perturbative superpotential in E8 × E8 heterotic string theory on a non-simply connected Calabi-Yau manifold X, as well as on its simply connected covering space tilde{X} . The superpotential is induced by the string wrapping holomorphic, isolated, genus 0 curves. According to the residue theorem of Beasley and Witten, the non-perturbative superpotential must vanish in a large class of heterotic vacua because the contributions from curves in the same homology class cancel each other. We point out, however, that in certain cases the curves treated in the residue theorem as lying in the same homology class, can actually have different area with respect to the physical Kahler form and can be in different homology classes. In these cases, the residue theorem is not directly applicable and the structure of the superpotential is more subtle. We show, in a specific example, that the superpotential is non-zero both on tilde{X} and on X. On the non-simply connected manifold X, we explicitly compute the leading contribution to the superpotential from all holomorphic, isolated, genus 0 curves with minimal area. The reason for the non-vanishing of the superpotental on X is that the second homology class contains a finite part called discrete torsion. As a result, the curves with the same area are distributed among different torsion classes and, hence, do not cancel each other.

Potential Automorphy and the Leopoldt conjecture

We study in this paper Hida's $p$-adic Hecke algebra for ${\rm GL}_n$ over a CM field $F$. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his conjecture in the case $F={\Bbb Q}$ implies the classical Leopoldt conjecture for a number field $K$ of degree $n$ over ${\Bbb Q}$, if one assumes further the existence of automorphic induction of characters for the extension $K/{\Bbb Q}$. We study Hida's conjecture using the automorphy lifting techniques adapted to the ${\rm GL}_n$ setting by Calegari-Geraghty. We prove an automorphy lifting result in this setting, conditional on existence and local-global compatibility of Galois representations arising from torsion classes in the cohomology of the corresponding symmetric manifolds. Under the same conditions we show that one can deduce the classical (abelian) Leopoldt conjectures for a totally real number field $K$ and a prime $p$ using Hida's non-abelian Leopoldt conjecture for $p$-adic Hecke algebra for ${\rm GL}_n$ over CM fields without needing to assume automorphic induction of characters for the extension $K/{\Bbb Q}$. For this methods of potential automorphy results are used.

On the exactness of products in the localization of (Ab.4⁎) Grothendieck categories

Let C be an (Ab.4⁎) Grothendieck category, that is, products are exact in C. Given a hereditary torsion class T⊆C, we study the exactness of products in the Gabriel localization C/T of C. We show that, under suitable assumptions on C, the k+1-th derived functor of the product vanishes, provided the Gabriel dimension of C/T is smaller than k. As a consequence, we deduce that, under suitable hypotheses, the derived category D(C/T) is left-complete.

On aspects of holographic thermal QCD at finite coupling

In the context of string theoretic dual of thermal QCD-like theories at finite gauge/string coupling of [1] (as part of the ‘MQGP’ limit of [2]), we obtain the QCD deconfinement temperature compatible with lattice results for the right number of light flavors Nf=3, and the correct mass scale of the light (first generation) quarks. The type IIB background of [1] is also shown to be thermodynamically stable. Further, we show that the temperature dependence of DC electrical conductivity mimics a one-dimensional Luttinger liquid, and the requirement of the Einstein relation (ratio of electrical conductivity and charge susceptibility equal to the diffusion constant) to be satisfied requires a specific dependence of the Ouyang embedding parameter on the horizon radius. These results arise due to the non-Kählerity and non-conformality of the type IIB background. On the geometrical side we quantify the former (non-Kählerity) by evaluating the SU(3)/G2-structure torsion classes of the local type IIA mirror/M-theory uplift. Analogous to what was shown for the type IIB background in [5], we first show that the type IIA delocalized SYZ mirror (after fine tuning) can also be approximately supersymmetric. We then work out the G2-structure torsion classes of the local M-theory uplift of the mirror type IIA metric – in the large-N limit at finite coupling, G2 structure approaches G2 holonomy.

G-stable support τ-tilting modules

Motivated by τ-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra Λ with action by a finite group G; we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over Λ; G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective Λ-modules, and G-stable functorially finite torsion classes in the category of finitely generated left Λ-modules. In the case when Λ is the endomorphism of a G-stable cluster-tilting object T over a Hom-finite 2-Calabi-Yau triangulated category C with a G-action, these are also in bijection with G-stable cluster-tilting objects in C. Moreover, we investigate the relationship between stable support τ-tilitng modules over Λ and the skew group algebra ΛG

S-folds and 4d N $$ \mathcal{N} $$ = 3 superconformal field theories

S-folds are generalizations of orientifolds in type IIB string theory, such that the geometric identifications are accompanied by non-trivial S-duality transformations. They were recently used by Garcia-Etxebarria and Regalado to provide the first construction of four dimensional N=3 superconformal theories. In this note, we classify the different variants of these N=3 preserving S-folds, distinguished by an analog of discrete torsion, using both a direct analysis of the different torsion classes and the compactification of the S-folds to three dimensional M-theory backgrounds. Upon adding D3-branes, these variants lead to different classes of N=3 superconformal field theories. We also analyze the holographic duals of these theories, and in particular clarify the role of discrete gauge and global symmetries in holography.

On the image of complex conjugation in certain Galois representations

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$.

Algebraically closed and existentially closed Abelian lattice-ordered groups

If \({\mathcal{G}}\) is an Abelian lattice-ordered (l-) group, then \({\mathcal{G}}\) is algebraically (existentially) closed just in case every finite system of l-group equations (equations and inequations), involving elements of \({\mathcal{G}}\), that is solvable in some Abelian l-group extending \({\mathcal{G}}\) is solvable already in \({\mathcal{G}}\). This paper establishes two systems of axioms for algebraically (existentially) closed Abelian l-groups, one more convenient for modeltheoretic applications and the other, discovered by Weispfenning, more convenient for algebraic applications. Among the model-theoretic applications are quantifierelimination results for various kinds of existential formulas, a new proof of the amalgamation property for Abelian l-groups, Nullstellensatze in Abelian l-groups, and the display of continuum-many elementary-equivalence classes of existentially closed Archimedean l-groups. The algebraic applications include demonstrations that the class of algebraically closed Abelian l-groups is a torsion class closed under arbitrary products, that the class of l-ideals of existentially closed Abelian l-groups is a radical class closed under binary products, and that various classes of existentially closed Abelian l-groups are closed under bounded Boolean products.

The Gysin sequence for quantum lens spaces

We define quantum lens spaces as `direct sums of line bundles' and exhibit them as `total spaces' of certain principal bundles over quantum projective spaces. For each of these quantum lens spaces we construct an analogue of the classical Gysin sequence in K-theory. We use the sequence to compute the K-theory of the quantum lens spaces, in particular to give explicit geometric representatives of their K-theory classes. These representatives are interpreted as `line bundles' over quantum lens spaces and generically define `torsion classes'. We work out explicit examples of these classes.

Deformations of $G_2$-structures with torsion

We consider non-infinitesimal deformations of G2-structures on 7-dimensional manifolds and derive an exact expression for the torsion of the deformed G2-structure. We then specialize to a case when the deformation is defined by a vector v and we explicitly derive the expressions for the different torsion components of the new G2-structure in terms of the old torsion components and derivatives of v. In particular this gives a set of differential equations for the vector v which have to be satisfied for a transition between G2-structures with particular torsions. For some specific torsion classes we find that these equations have no solutions.

C-sortable words as green mutation sequences

Let $Q$ be an acyclic quiver and $\mathbf{s}$ be a sequence with elements in the vertex set $Q_0$. We describe an induced sequence of simple (backward) tilting in the bounded derived category $\mathcal{D}(Q)$, starting from the standard heart $\mathcal{H}_Q=\operatorname{mod}\mathbf{k}Q$ and ending at another heart $\mathcal{H}_\mathbf{s}$ in $\mathcal{D}(Q)$. Then we show that $\mathbf{s}$ is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to $\mathcal{H}_Q[-1]$; it is maximal if and only if $\mathcal{H}_\mathbf{s}=\mathcal{H}_Q[-1]$. This provides a categorical way to understand green mutations. Further, fix a Coxeter element $c$ in the Coxeter group $W_Q$ of $Q$, which is admissible with respect to the orientation of $Q$. We prove that the sequence $\widetilde{\mathbf{w}}$ induced by a $c$-sortable word $\mathbf{w}$ is a green mutation sequence. As a consequence, we obtain a bijection between $c$-sortable words and finite torsion classes in $\mathcal{H}_Q$. As byproducts, the interpretations of inversions, descents and cover reflections of a $c$-sortable word $\mathbf{w}$ are given in terms of the combinatorics of green mutations.

Torsion Classes and t-Structures in Higher Homological Algebra

Higher homological algebra was introduced by Iyama. It is also known as $n$-homological algebra where $n \geq 2$ is a fixed integer, and it deals with $n$-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with $n+2$ objects. This was recently formalised by Jasso in the theory of $n$-abelian categories. There is also a derived version of $n$-homological algebra, formalised by Geiss, Keller, and Oppermann in the theory of $( n+2 )$-angulated categories (the reason for the shift from $n$ to $n+2$ is that angulated categories have triangulated categories as the base case). We introduce torsion classes and t-structures into the theory of $n$-abelian and $( n+2 )$-angulated categories, and prove several results to motivate the definitions. Most of the results concern the $n$-abelian and $( n+2 )$-angulated categories ${\mathcal M}( \Lambda )$ and ${\mathcal C}( \Lambda )$ associated to an $n$-representation finite algebra $\Lambda$, as defined by Iyama and Oppermann. We characterise torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in ${\mathcal M}( \Lambda )$ and intermediate t-structures in ${\mathcal C}( \Lambda )$ which is a category one can reasonably view as the $n$-derived category of ${\mathcal M}( \Lambda )$. We hint at the link to $n$-homological tilting theory.