Related papers: Duality and cohomology in M-theory with boundary

T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H-flux, as we motivate using E_8 and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, Lens…

We show that the homological properties of a 5-manifold M with fundamental group G are encapsulated in a G-invariant stable form on the dual of the third syzygy of Z. In this notation one may express an even stronger version of Poincare…

R. M. Brown's theorem on mixed Dirichlet and Neumann boundary conditions is extended in two ways for the special case of polyhedral domains. A (1) more general partition of the boundary into Dirichlet and Neumann sets is used on (2)…

While M- and F-theory compactifications describe a much larger class of vacua than perturbative string compactifications, they typically need singularities to generate non-abelian gauge fields and charged matter. The physical explanation…

The study of the partition function in M-theory involves the use of index theory on a twelve-dimensional bounding manifold. In eleven dimensions, viewed as a boundary, this is given by secondary index invariants such as the…

We discuss the geometry of string and M-theory gauge fields in Deligne cohomology. In particular, we show how requiring string structure (or loop space Spin-C structure) on the five-brane leads to topological conditions on the flux in the…

Above a Laurent polynomial f one makes grow a vector space of vanishing cycles (after the work of Sabbah, singularity setting), a graded Milnor ring (after the work of Kouchnirenko) and an orbifold cohomology ring (after the work of…

We introduce a new Hermitian metric on the cohomology ring of compact K\"ahlerian manifolds with a pair $(v,w)$ satisfying certain Hodge-Riemann relations. An Hermitian metric on the exterior algebra of the cotangent bundle is also defined…

In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a…

We set up a cochain complex $C^\bullet_{\mathrm{mix}}(\mathcal{M})$ whose cohomology controls deformations of the mixed associator of a module category $\mathcal{M}$ over a $\Bbbk$-linear monoidal category $\mathcal{C}$. We show that…

Let $M$ be a complete hyperbolic $n$-manifold, $n\geq 2$. Via integration over geodesic simplices, any closed bounded differential 2-form on $M$ defines a bounded cohomology class in $H^2_b(M)$. It was proved by Barge and Ghys (for $n=2$)…

By analyzing super-torsion and brane super-cocycles, we derive a new duality in M-theory, which takes the form of a higher version of T-duality in string theory. This involves a new topology change mechanism abelianizing the 3-sphere…

We introduce a cohomological invariant arising from a class in nonabelian cohomology. This invariant generalizes the Dixmier-Douady class and encodes the obstruction to a C*-algebra bundle being the fixed-point algebra of a gauge action. As…

On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into…

We consider discontinuous operations of a group $G$ on a contractible $n$-dimensional manifold $X$. Let $E$ be a finite dimensional representation of $G$ over a field $k$ of characteristics 0. Let $\mathcal{E}$ be the sheaf on the quotient…

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially…

By fibering the duality between the $E_{8}\times E_{8}$ heterotic string on $T^{3}$ and M-theory on K3, we study heterotic duals of M-theory compactified on $G_{2}$ orbifolds of the form $T^{7}/\mathbb{Z}_{2}^{3}$. While the heterotic…

Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization $\tilde{\mathcal{M}}_{Dol}^G$ of the moduli space $\mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree…

We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta-invariants upon variation of the Spin structure. The main…

We give a precise formulation of the M-theory 3-form potential C in a fashion applicable to topologically nontrivial situations. In our model the 3-form is related to the Chern-Simons form of an E8 gauge field. This leads to a precise…