Hisham Sati

L∞-Algebra Connections and Applications to String- and Chern-Simons n-Transport

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L ∞-algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization.

Twisted Differential String and Fivebrane Structures

In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of, differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted nonabelian cohomology, extending the differential generalized abelian cohomology as developed by Hopkins and Singer and shown by Freed to formalize the global description of anomaly cancellation problems in higher gauge theories arising in string theory. We demonstrate that the Green-Schwarz mechanism for the H3-field, as well as its magnetic dual version for the H7-field define cocycles in differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-, String(n)- and Fivebrane(n)- structures on target space, where the twist in each case is provided by the obstruction to lifting the classifying map of the gauge bundle through a higher connected cover of U(n) or O(n). We show that the twisted Bianchi identities in string theory can be captured by the (nonabelian) L∞-algebra valued differential form data provided by the differential refinements of these twisted cocycles.

Can D-branes wrap nonrepresentable cycles?

Sometimes a homology cycle of a nonsingular compactification manifold cannot be represented by a nonsingular submanifold. We want to know whether such nonrepresentable cycles can be wrapped by D-branes. A brane wrapping a representable cycle carries a K-theory charge if and only if its Freed-Witten anomaly vanishes. However some K-theory charges are only carried by branes that wrap nonrepresentable cycles. We provide two examples of Freed-Witten anomaly-free D6-branes wrapping nonrepresentable cycles in the presence of a trivial NS 3-form flux. The first occurs in type IIA string theory compactified on the Sp(2) group manifold and the second in IIA on a product of lens spaces. We find that the first D6-brane carries a K-theory charge while the second does not.

The E8 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 Hořava–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.

A higher twist in string theory

Abstract Considering the gauge field and its dual in heterotic string theory as a unified field, we show that the equations of motion at the rational level contain a twisted differential with a novel degree seven twist. This generalizes the usual degree three twist that lifts to twisted K-theory and raises the natural question of whether at the integral level the abelianized gauge fields belong to a generalized cohomology theory. Some remarks on possible such extension are given.

An approach to anomalies in M-theory via KSpin

Abstract The M-theory field strength and its dual, given by the integral lift of the left-hand side of the equation of motion, both satisfy certain cohomological properties. We study the combined fields and observe that the multiplicative structure on the product of the corresponding degree four and degree eight cohomology fits into that given by Spin K-theory. This explains some earlier results and leads naturally to the use of Spin characteristic classes. We reinterpret the one-loop term in terms of such classes and we show that it is a homotopy invariant. We argue that the various anomalies have natural interpretations within Spin K-theory. In the process, mod 3 reductions play a special role.

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 TOPOLOGICAL STRUCTURES RELATED TO M-BRANES II: TWISTED Wu AND Wuc STRUCTURES

Studying the topological aspects of M-branes in M-theory leads to various structures related to Wu classes. First we interpret Wu classes themselves as twisted classes and then define twisted notions of Wu structures. These generalize many known structures, including Pin- structures, twisted Spin structures in the sense of Distler–Freed–Moore, Wu-twisted differential cocycles appearing in the work of Belov–Moore, as well as ones introduced by the author, such as twisted Membrane and twisted Stringc structures. In addition, we introduce Wuc structures, which generalize Pinc structures, as well as their twisted versions. We show how these structures generalize and encode the usual structures defined via Stiefel–Whitney classes.

ANOMALIES OF E8 GAUGE THEORY ON STRING MANIFOLDS

In this paper we revisit the subject of anomaly cancelation in string theory and M-theory on manifolds with string structure and give three observations. First, that on string manifolds there is no E8 × E8 global anomaly in heterotic string theory. Second, that the description of the anomaly in the phase of the M-theory partition function of Diaconescu–Moore–Witten extends from the spin case to the string case. Third, that the cubic refinement law of Diaconescu–Freed–Moore for the phase of the M-theory partition function extends to string manifolds. The analysis relies on extending from invariants which depend on the spin structure to invariants which instead depend on the string structure. Along the way, the one-loop term is refined via the Witten genus.

Twisted Morava K-theory and E-theory

For a class H ∈ H n+2 (X; Z), we define twisted Morava K-theory K(n) ∗ (X;H )a t the prime 2, as well as an integral analogue. We explore properties of this twisted cohomology theory, studying a twisted Atiyah–Hirzebruch spectral sequence, a universal coefficient theorem (in the spirit of Khorami). We extend the construction to define twisted Morava E-theory, and provide applications to string theory and M-theory.