Twisted Cohomology Research Articles

One-loop double copy relation from twisted (co)homology

We propose a geometric relation between closed and open string amplitudes at one-loop. After imposing a homological splitting on the world-sheet torus, twisted intersection theory is used to establish a one-loop double copy relation. The latter expresses a closed string amplitude by a pair of open string amplitudes and twisted intersection numbers. These inner products on the vector space of twisted differential forms are related to the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral.

Intersection numbers, polynomial division and relative cohomology

We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential n-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology groups and the polynomial division technique, recently proposed in the literature. We show that delta-forms capture the leading behaviour of the intersection numbers in presence of evanescent analytic regulators, whose use is, therefore, bypassed. This simplified algorithm is applied to derive the complete decomposition of two-loop planar and non-planar Feynman integrals in terms of a master integral basis. More generally, it can be applied to derive relations among twisted period integrals, relevant for physics and mathematical studies.

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

AbstractWe compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the rational homology of the unit tangent bundles of the surfaces. These coefficients define a covariant functor over the classical category associated to mapping class groups, rather than a contravariant one, and are thus out of the scope of the traditional framework to study twisted cohomological stability. A remarkable property is that the computed stable twisted cohomology is not free as a module over the stable cohomology algebra with constant rational coefficients. For comparison, we also compute the stable cohomology group with coefficients in the first rational cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.

On unstable twisted rational cohomology groups of the automorphism groups of free groups

In this paper, we consider low-dimensional twisted rational cohomology groups of the automorphism groups of free groups. In particular, we study behaviors of Kawazumi’s cocycles by observing their restrictions to the IA-automorphism groups. We show the nontrivialities of certain unstable twisted cocycles constructed from Kawazumi’s cocycles, and compute the second and the third cohomology groups of the automorphism group of the free group of rank three with coefficients in the exterior powers of the abelianization of the free group.

A double copy from twisted (co)homology at genus one

We study the twisted (co)homology of a family of genus-one integrals — the so called Riemann-Wirtinger integrals. These integrals are closely related to one-loop string amplitudes in chiral splitting where one leaves the loop-momentum, modulus and all but one puncture un-integrated. While not actual one-loop string integrals, they share many properties and are simple enough that the associated twisted (co)homologies have been completely characterized [1]. Using intersection numbers — an inner product on the vector space of allowed differential forms — we derive the Gauss-Manin connection for two bases of the twisted cohomology providing an independent check of [2]. We also use the intersection index — an inner product on the vector space of allowed contours — to derive a double-copy formula for the closed-string analogues of Riemann-Wirtinger integrals (one-dimensional integrals over the torus). Similar to the celebrated KLT formula between open- and closed-string tree-level amplitudes, these intersection indices form a genus-one KLT-like kernel defining bilinears in meromorphic Riemann-Wirtinger integrals that are equal to their complex counterparts.

Erratum: Twisted cohomology of the complement of theta divisors in an abelian surface

Erratum: Twisted cohomology of the complement of theta divisors in an abelian surface

Cosmology meets cohomology

The cosmological polytope and bootstrap programs have revealed interesting connections between positive geometries, modern on-shell methods and bootstrap principles studied in the amplitudes community with the wavefunction of the Universe in toy models of FRW cosmologies. To compute these FRW correlators, one often faces integrals that are too difficult to evaluate by direct integration. Borrowing from the Feynman integral community, the method of (canonical) differential equations provides an efficient alternative for evaluating these integrals. Moreover, we further develop our geometric understanding of these integrals by describing the associated relative twisted cohomology. Leveraging recent progress in our understanding of relative twisted cohomology in the Feynman integral community, we give an algorithm to predict the basis size and simplify the computation of the differential equations satisfied by FRW correlators.

Twisted Frölicher-type inequality on foliations

We study the twisted Bott-Chern and Aeppli cohomologies on transversely Hermitian foliations. And we study the Frölicher type inequality of these twisted cohomologies. As an application, we prove the Hodge isomorphism on transverse Kähler foliations.

Bulk-to-boundary anyon fusion from microscopic models

Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2 + 1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted Z2 as well as Z3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.

Superconformal quantum mechanics and growth of sheaf cohomology

We give a geometric interpretation for superconformal quantum mechanics defined on a hyper-Kähler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes on the resolved space and their index degeneracies can also be related to the Euler characteristic computed in equivariant sheaf cohomology. In the special case of the Hilbert scheme of K points on ℂ2, we obtain a rigorous estimate for the exponential growth of the index degeneracies of BPS states as K → ∞. This growth serves as a toy model for our recently proposed duality between a seven dimensional black hole and superconformal quantum mechanics.

Real time lattice correlation functions from differential equations

We report on an exact calculation of lattice correlation functions on a finite four-dimensional lattice with either Euclidean or Minkowskian signature. The lattice correlation functions are calculated by the method of differential equations. This method can be used for Euclidean and Minkowskian signature alike. The lattice correlation functions have a power series expansion in 1/ sqrt{lambda } , where λ is the coupling. We show that this series is convergent for all non-zero values of λ. At small coupling we quantify the accuracy of perturbative approximations. At the technical level we systematically investigate the interplay between twisted cohomology and the symmetries of the twist function.

Anyonic defect branes and conformal blocks in twisted equivariant differential (TED) K-theory

We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension[Formula: see text]2 defect branes, such as of [Formula: see text]-branes in IIB/F-theory on [Formula: see text]-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special [Formula: see text]-monodromy charges not seen for other branes, at least partially reflected in conformal blocks of the [Formula: see text]-WZW model over their transverse punctured complex curve. Indeed, it has been argued that all “exotic” branes of string theory are defect branes carrying such U-duality monodromy charges — but none of these had previously been identified in the expected brane charge quantization law given by K-theory. Here we observe that it is the subtle (and previously somewhat neglected) twisting of equivariant K-theory by flat complex line bundles appearing inside orbi-singularities (“inner local systems”) that makes the secondary Chern character on a punctured plane inside an [Formula: see text]-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman and Varchenko showed realizes [Formula: see text]-conformal blocks, here in degree 1 — in fact it gives the direct sum of these over all admissible fractional levels [Formula: see text]. The remaining higher-degree [Formula: see text]-conformal blocks appear similarly if we assume our previously discussed “Hypothesis H” about brane charge quantization in M-theory. Since conformal blocks — and hence these twisted equivariant secondary Chern characters — solve the Knizhnik–Zamolodchikov equation and thus constitute representations of the braid group of motions of defect branes inside their transverse space, this provides a concrete first-principles realization of anyon statistics of — and hence of topological quantum computation on — defect branes in string/M-theory.

Intersection numbers of twisted homology and cohomology groups associated to the Riemann–Wirtinger integral

The Riemann–Wirtinger integral is an analogue of the hypergeometric integral, which is defined on a one-dimensional complex torus. We study the intersection forms on the twisted homology and cohomology groups associated with the Riemann–Wirtinger integral. We derive explicit formulas of some intersection numbers, and apply them to studies of the monodromy representation, connection problems, and contiguity relations.

LAURICELLA HYPERGEOMETRIC FUNCTIONS, UNIPOTENT FUNDAMENTAL GROUPS OF THE PUNCTURED RIEMANN SPHERE, AND THEIR MOTIVIC COACTIONS

AbstractThe goal of this paper is to raise the possibility that there exists a meaningful theory of ‘motives’ associated with certain hypergeometric integrals, viewed as functions of their parameters. It goes beyond the classical theory of motives, but should be compatible with it. Such a theory would explain a recent and surprising conjecture arising in the context of scattering amplitudes for a motivic Galois group action on Gauss’s ${}_2F_1$ hypergeometric function, which we prove in this paper by direct means. More generally, we consider Lauricella hypergeometric functions and show, on the one hand, how the coefficients in their Taylor expansions can be promoted, via the theory of motivic fundamental groups, to motivic multiple polylogarithms. The latter are periods of ordinary motives and admit an action of the usual motivic Galois group, which we call the local action. On the other hand, we define lifts of the full Lauricella functions as matrix coefficients in a Tannakian category of twisted cohomology, which inherit an action of the corresponding Tannaka group. We call this the global action. We prove that these two actions, local and global, are compatible with each other, even though they are defined in completely different ways. The main technical tool is to prove that metabelian quotients of generalised Drinfeld associators on the punctured Riemann sphere are hypergeometric functions. We also study single-valued versions of these hypergeometric functions, which may be of independent interest.

Macaulay matrix for Feynman integrals: linear relations and intersection numbers

We elaborate on the connection between Gel’fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations for Feynman Integrals. We propose a novel, more efficient algorithm to compute Macaulay matrices, which are used to derive Pfaffian systems of differential equations. The Pfaffian matrices are then employed to obtain linear relations for mathcal{A} -hypergeometric (Euler) integrals and Feynman integrals, through recurrence relations and through projections by intersection numbers.

Twisted translation flows and effective weak mixing

We introduce a twisted cohomology cocycle over the Teichmüller flow and prove a “spectral gap” for its Lyapunov spectrum with respect to the Masur–Veech measures. We then derive Hölder estimates on spectral measures and bounds on the speed of weak mixing for almost all translation flows in every stratum of Abelian differentials on Riemann surfaces, as well as bounds on the deviation of ergodic averages for product translation flows on the product of a translation surface with a circle.

Twisted cohomology pairings of knots II; to some Blanchfield pairings

In this paper, we study some comparison between a bilinear cohomology pairing in local coefficients and the Blanchfield pairing of a knot. We show that the former pairing is an [Formula: see text]-equivalent invariant, and give a criterion to a relation between the two pairings. We also observe that the pairings of some knots are equivalent, and that the pairings of other knots are not equivalent.

Twisted basic Dolbeault cohomology on transverse Kähler foliations

In this paper, we study the twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem on a transverse Kähler foliation. And we give some properties for \(\Delta _\kappa \)-harmonic forms and prove the Kodaira–Serre-type duality and Dolbeault isomorphism for the twisted basic Dolbeault cohomology.

Cohomology jump loci and absolute sets for singular varieties

We extend the notion of absolute subsets of Betti moduli spaces of smooth algebraic varieties to the case of normal varieties. As a consequence we prove that twisted cohomology jump loci in rank one over a normal variety are a finite union of translated subtori. We show that the same holds for jump loci twisted by a unitary local system in the case where the underlying variety X is projective with H1(X,Q) pure of weight one. Lastly, we study the interaction of these loci with Hodge theoretic data naturally associated to the representation variety of fundamental groups of smooth projective varieties.

A very short note on the (rational) graded Hori map

The graded Hori map has been recently introduced by Han-Mathai in the context of T-duality as a -graded transform whose homogeneous components are the Hori-Fourier transforms in twisted cohomology associated with integral multiples of a basic pair of T-dual closed 3-forms. We show how in the rational homotopy theory approximation of T-duality, such a map is naturally realized as a pull-iso-push transform, where the isomorphism part corresponds to the canonical equivalence between the left and the right gerbes associated with a T-duality configuration.