Twisted Cohomology Research Articles

On the push-forwards for motivic cohomology theories with invertible stable Hopf element

We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces.

The Wess-Zumino-Witten term of the M5-brane and differential cohomotopy

We combine rational homotopy theory and higher Lie theory to describe the Wess-Zumino-Witten (WZW) term in the M5-brane sigma model. We observe that this term admits a natural interpretation as a twisted 7-cocycle on super-Minkowski spacetime with coefficients in the rational 4-sphere. This exhibits the WZW term as an element in twisted cohomology, with the twist given by the cocycle of the M2-brane. We consider integration of this rational situation to differential cohomology and differential cohomotopy.

Twisted Chiral de Rham Complex, Generalized Geometry, and T-duality

The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold $Z$, and contains the ordinary de Rham complex at weight zero. Given a closed 3-form $H$ on $Z$, we construct the twisted chiral de Rham differential $D_H$, which coincides with the ordinary twisted differential in weight zero. We show that its cohomology vanishes in positive weight and coincides with the ordinary twisted cohomology in weight zero. As a consequence, we propose that in a background flux, Ramond-Ramond fields can be interpreted as $D_H$-closed elements of the chiral de Rham complex. Given a T-dual pair of principal circle bundles $Z, \widehat{Z}$ with fluxes $H, \widehat{H}$, we establish a degree-shifting linear isomorphism between a central quotient of the $i \mathbb{R}[t]$-invariant chiral de Rham complexes of $Z$ and $\widehat{Z}$. At weight zero, it restricts to the usual isomorphism of $S^1$-invariant differential forms, and induces the usual isomorphism in twisted cohomology. This is interpreted as T-duality in type II string theory from a loop space perspective. A key ingredient in defining this isomorphism is the language of Courant algebroids, which clarifies the notion of functoriality of the chiral de Rham complex.

Some problems of hypergeometric integrals associated with hypersphere arrangement

The $n$ dimensional hypergeometric integrals associated with a hypersphere arrangement $S$ are formulated by the pairing of $n$ dimensional twisted cohomology $H_{\nabla}^{n} (X, \Omega^{\cdot} (*S))$ and its dual. Under the condition of general position there are stated some results and conjectures which concern a representation of the standard form by a special basis of the twisted cohomology, the variational formula of the corresponding integral in terms of special invariant 1-forms using Calyley-Menger minor determinants, a connection relation of the unique twisted $n$-cycle identified with the unbounded chamber to a special basis of twisted $n$-cycles identified with bounded chambers. General conjectures are presented under a much weaker assumption.

Spherical T-Duality

We introduce spherical T-duality, which relates pairs of the form $(P,H)$ consisting of a principal $SU(2)$-bundle $P\rightarrow M$ and a 7-cocycle $H$ on $P$. Intuitively spherical T-duality exchanges $H$ with the second Chern class $c_2(P)$. Unless $dim(M)\leq 4$, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. Nonetheless, we prove that all spherical T-dualities induce a degree-shifting isomorphism on the 7-twisted cohomologies of the bundles and, when $dim(M)\leq 7$, also their integral twisted cohomologies and, when $dim(M)\leq 4$, even their 7-twisted K-theories. While spherical T-duality does not appear to relate equivalent string theories, it does provide an identification between conserved charges in certain distinct IIB supergravity and string compactifications.

Topological T-duality for torus bundles with monodromy

We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficient conditions for existence of a T-dual in the case of affine torus bundles. This is general enough to include all principal torus bundles as well as torus bundles with arbitrary monodromy representations. We show that isomorphisms in twisted cohomology, twisted K-theory and of Courant algebroids persist in this general setting. We also give an example where twisted K-theory groups can be computed by iterating T-duality.

Computing Bases of Twisted Cohomology Groups for Sparse Polynomials

We present an algorithm to construct bases of twisted cohomology groups for the complement of a generic sparse polynomial over a generic local system of rank 1. The algorithm is based on the theory given in a recent preprint [1].

Exotic Twisted Equivariant Cohomology of Loop Spaces, Twisted Bismut–Chern Character and T-Duality

We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Chern character form, a loop space refinement of the twisted Chern character form, which represent classes in the completed periodic exotic twisted $S^1$-equivariant cohomology of $LZ$. We establish a localisation theorem for the completed periodic exotic twisted $S^1$-equivariant cohomology for loop spaces and apply it to establish T-duality in a background flux in type II String Theory from a loop space perspective.

The E 8 Moduli 3-Stack of the C-Field in M-Theory

The higher gauge field in 11-dimensional supergravity -- the C-field -- is constrained by quantum effects to be a cocycle in some twisted version of differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization of the full smooth moduli 3-stack of configurations of the C-field, the gravitational field/background, and the (auxiliary) E8-field. We show that the truncation of this moduli 3-stack to a bare 1-groupoid of field configurations reproduces the differential integral Wu structures that Hopkins-Singer had shown to formalize Witten's argument on the nature of the C-field. We give a similarly simple and natural characterization of the moduli 2-stack of boundary C-field configurations and show that it is equivalent to the moduli 2-stack of anomaly free heterotic supergravity field configurations. Finally we show how to naturally encode the Horava-Witten boundary condition on the level of moduli 3-stacks, and refine it from a condition on 3-forms to a condition on the corresponding full differential cocycles.

On quantum groups and Lie bialgebras related to sl(n)

Given an arbitrary field of characteristic 0, we study Lie bialgebra structures on sl(n, ), based on the description of the corresponding classical double. For any Lie bialgebra structure δ, the classical double D(sl(n, ),δ) is isomorphic to sl(n, ) ⊗A, where A is either [ε], with ε2 = 0, or ⊗ or a quadratic field extension of . In the first case, the classification leads to quasi-Frobenius Lie subalgebras of sl(n, ). In the second and third cases, a Belavin-Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n, ), up to gauge equivalence. The Belavin-Drinfeld untwisted and twisted cohomology sets associated to an r-matrix are computed.

Variation of Hodge structure for generalized complex manifolds

A generalized complex manifold which satisfies the ∂∂¯-lemma admits a Hodge decomposition in twisted cohomology. Using a Courant algebroid theoretic approach we study the behavior of the Hodge decomposition in smooth and holomorphic families of generalized complex manifolds. In particular we define period maps, prove a Griffiths transversality theorem and show that for holomorphic families the period maps are holomorphic. Further results on the Hodge decomposition for various special cases including the generalized Kähler case are obtained.

On a spectral sequence for twisted cohomologies

Let (Ω*(M), d) be the de Rham cochain complex for a smooth compact closed manifolds M of dimension n. For an odd-degree closed form H, there is a twisted de Rham cochain complex (Ω*(M), d + H ∧) and its associated twisted de Rham cohomology H*(M,H). The authors show that there exists a spectral sequence {E , d r } derived from the filtration $F_p (\Omega ^ * (M)) = \mathop \oplus \limits_{i \geqslant p} \Omega ^i (M)$ of Ω*M, which converges to the twisted de Rham cohomology H*(M,H). It is also shown that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.

Principal $$\infty $$ ∞ -bundles: general theory

The theory of principal bundles makes sense in any infinity-topos, such as that of topological, of smooth, or of otherwise geometric infinity-groupoids/infinity-stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure infinity-group G these G-principal infinity-bundles reproduce the theories of ordinary principal bundles, of bundle gerbes/principal 2-bundles and of bundle 2-gerbes and generalize these to their further higher and equivariant analogs. The induced associated infinity-bundles subsume the notions of gerbes and higher gerbes in the literature. We discuss here this general theory of principal infinity-bundles, intimately related to the axioms of Giraud, Toen-Vezzosi, Rezk and Lurie that characterize infinity-toposes. We show a natural equivalence between principal infinity-bundles and intrinsic nonabelian cocycles, implying the classification of principal infinity-bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber infinity-bundles associated to principal infinity-bundles subsumes a theory of infinity-gerbes and of twisted infinity-bundles, with twists deriving from local coefficient infinity-bundles, which we define, relate to extensions of principal infinity-bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice infinity-topos. In a companion article [NSSb] we discuss explicit presentations of this theory in categories of simplicial (pre)sheaves by hyper-Cech cohomology and by simplicial weakly-principal bundles; and in [NSSc] we discuss various examples and applications of the theory.

The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podleś sphere

We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of ‘modular’ spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podleś sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class which is in dimension 2.

A note on Reidemeister torsion and pleated surfaces

This paper uses the notion of ℂ-symplectic chain complex and proves an explicit formula for the Reidemeister torsion of an arbitrary ℂ-symplectic chain complex in terms of intersection forms of the homologies. In applications, the formula is applied to closed manifolds and also compact manifolds with boundary by using the homologies with coefficients in complex numbers field. Moreover, an explicit formula for the Reidemeister torsion of representations from the fundamental group of a closed oriented hyperbolic surface to PSL2(ℂ) is presented in terms of the cup product of twisted cohomologies, which is related with Weil–Petersson form and thus the Thurston symplectic form. The formula is also applied to pleated surfaces.

Non-commutative Hodge structures: Towards matching categorical and geometric examples

The subject of the present work is the de Rham part of non-commutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas for the corresponding connection on the periodic cyclic homology viewed as a bundle over the punctured formal disk. Our main result says that for the category of matrix factorizations of a polynomial the formulas reproduce, up to a certain shift, a well-known connection on the associated twisted de Rham cohomology which plays a central role in the geometric approach to the Hodge theory of isolated singularities.

Topological T-duality for general circle bundles

We extend topological T-duality to the case of general circle bundles. In this setting we prove existence and uniqueness of T-duals. We then show that T-dual spaces have isomorphic twisted cohomology, twisted $K$-theory and Courant algebroids. A novel feature is that we must consider two kinds of twists in de Rham cohomology and $K$-theory, namely by degree 3 integral classes and a less familiar kind of twist using real line bundles. We give some examples of T-dual non-oriented circle bundles and calculate their twisted $K$-theory.

An ∞-categorical approach to R -line bundles, R -module Thom spectra, and twisted R -homology

We develop a generalization of the theory of Thom spectra using the language of ∞-categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parameterized spectra, and our definition is motivated by the geometric definition of Thom spectra of May–Sigurdsson. For an A∞-ring spectrum R, we associate a Thom spectrum to a map of ∞-categories from the ∞-groupoid of a space X to the ∞-category of free rank one R-modules, which we show is a model for BGL1R; we show that BGL1R classifies homotopy sheaves of rank one R-modules, which we call R-line bundles. We use our R-module Thom spectrum to define the twisted R-homology and cohomology of R-line bundles over a space classified by a map X→BGL1R, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory.

METABELIAN SL(N, C) REPRESENTATIONS OF KNOT GROUPS, III: DEFORMATIONS

Given a knot K and an irreducible metabelian SL(n,C) representation we establish an equality for the dimension of the first twisted cohomology. In the case of equality, we prove that the representation must have finite image and that it is conjugate to an SU(n) representation. In this case we show it determines a smooth point x in the SL(n,C) character variety, and we use a deformation argument to establish the existence of a smooth (n-1)-dimensional family of characters of irreducible SL(n,C) representations near x. Combining this with our previous existence results, we deduce the existence of large families of irreducible SU(n) and SL(n,C) non-metabelian representation for knots K in homology 3-spheres S with nontrivial Alexander polynomial. We then relate the condition on twisted cohomology to a more accessible condition on untwisted cohomology of a certain metabelian branched cover of S branched along K.

Twisted Cyclic Cohomology and Modular Fredholm Modules

Connes and Cuntz showed in [Comm. Math. Phys. 114 (1988), 515-526] that suitable cyclic cocycles can be represented as Chern characters of finitely summable semifinite Fredholm modules. We show an analogous result in twisted cyclic cohomology using Chern characters of modular Fredholm modules. We present examples of modular Fredholm modules arising from Podle\'s spheres and from $SU_q(2)$.