Narrow Results

This paper is a review of current developments in the study of moduli spaces of G₂ manifolds. G₂ manifolds are seven-dimensional manifolds with the exceptional holonomy group G₂. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi–Yau manifolds in seven dimensions. They play an important role in physics as natural candidates for supersymmetric vacuum solutions of M-theory compactifications. Despite the physical motivation, many of the results are of purely mathematical interest. Here we cover the basics of G₂ manifolds, local deformation theory of G₂ structures and the local geometry of the moduli spaces of G₂ structures.

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 sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spin^c case and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the Gromov–Lawson–Rosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

We characterize the integral cohomology and the rational homotopy type of the maximal Borel-equivariantization of the combined Hopf/twistor fibration, and find that subtle relations satisfied by the cohomology generators are just those that govern Hořava–Witten’s proposal for the extension of the Green–Schwarz mechanism from heterotic string theory to heterotic M-theory. We discuss how this squares with the Hypothesis H that the elusive mathematical foundation of M-theory is based on charge quantization in tangentially twisted unstable Cohomotopy theory.

In the quest for mathematical foundations of M-theory, the Hypothesis H that fluxes are quantized in Cohomotopy theory, implies, on flat but possibly singular spacetimes, that M-brane charges locally organize into equivariant homotopy groups of spheres. Here, we show how this leads to a correspondence between phenomena conjectured in M-theory and fundamental mathematical concepts/results in stable homotopy, generalized cohomology and Cobordism theory $Mf$ :

• —stems of homotopy groups correspond to charges of probe $p$-branes near black $b$-branes;
• —stabilization within a stem is the boundary-bulk transition;
• —the Adams d-invariant measures $G_4$-flux;
• —trivialization of the d-invariant corresponds to $H_3$-flux;
• —refined Toda brackets measure $H_3$-flux;
• —the refined Adams e-invariant sees the $H_3$-charge lattice;
• —vanishing Adams e-invariant implies consistent global $C_3$-fields;
• —Conner–Floyd’s e-invariant is the $H_3$-flux seen in the Green–Schwarz mechanism;
• —the Hopf invariant is the M2-brane Page charge (${\tilde{G}}_7$-flux);
• —the Pontrjagin–Thom theorem associates the polarized brane worldvolumes sourcing all these charges.

In particular, spontaneous K3-reductions with 24 branes are singled out from first principles :

• —Cobordism in the third stable stem witnesses spontaneous KK-compactification on K3-surfaces;
• —the order of the third stable stem implies the 24 NS5/D7-branes in M/F-theory on K3.

Finally, complex-oriented cohomology emerges from Hypothesis H, connecting it to all previous proposals for brane charge quantization in the chromatic tower: K-theory, elliptic cohomology, etc. :

• —quaternionic orientations correspond to unit $H_3$-fluxes near M2-branes;
• —complex orientations lift these unit $H_3$-fluxes to heterotic M-theory with heterotic line bundles.

In fact, we find quaternionic/complex Ravenel-orientations bounded in dimension; and we find the bound to be 10, as befits spacetime dimension $10+1$.

A novel Chern–Simons E₈ gauge theory of gravity in D = 15 based on an octicE₈ invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extension (to incorporate spacetime fermions) of this Chern–Simons E₈ gauge theory. We review the construction showing why the ordinary 11D Chern–Simons gravity theory (based on the Anti de Sitter group) can be embedded into a Clifford-algebra valued gauge theory and that an E₈ Yang–Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E₈ gauge bundle formulation was instrumental in understanding the topological part of the 11-dim M-theory partition function. The nature of this 11-dim E₈ gauge theory remains unknown. We hope that the Chern–Simons E₈ gauge theory of gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional reduction.

The massless supermultiplet of 11-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F₄ into those of its maximal compact subgroup Spin(9). In an earlier paper, a dynamical Kaluza–Klein origin of this observation is proposed with internal space the Cayley plane, 𝕆P², and topological aspects are explored. In this paper we consider the geometric aspects and characterize the corresponding forms which contribute to the action as well as cohomology classes, including torsion, which contribute to the partition function. This involves constructions with bilinear forms. The compatibility with various string theories are discussed, including reduction to loop bundles in ten dimensions.

We discuss a class of three-dimensional Chern–Simons (CS) quiver gauge models obtained from M-theory compactifications on singular complex four-dimensional hyper-Kähler (HK) manifolds, which are realized explicitly as a cotangent bundle over two-Fano toric varieties V². The corresponding CS gauge models are encoded in quivers similar to toric diagrams of V². Using toric geometry, it is shown that the constraints on CS levels can be related to toric equations determining V².

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 String^c structures. In addition, we introduce Wu^c structures, which generalize Pin^c structures, as well as their twisted versions. We show how these structures generalize and encode the usual structures defined via Stiefel–Whitney classes.

We show that the supermembrane theory compactified on a torus is invariant under T-duality. There are two different topological sectors of the compactified supermembrane (M2) classified according to a vanishing or nonvanishing second cohomology class. We find the explicit T-duality transformation that acts locally on the supermembrane theory and we show that it is an exact symmetry of the theory. We give a global interpretation of the T-duality in terms of bundles. It has a natural description in terms of the cohomology of the base manifold and the homology of the target torus. We show that in the limit when the torus degenerate into a circle and the M2 mass operator restricts to the string-like configurations, the usual closed string T-duality transformation between the type IIA and type IIB mass operators is recovered. Moreover, if we just restrict M2 mass operator to string-like configurations but we perform a generalized T-duality we find the SL(2,Z) nonperturbative multiplet of IIA.

M2-branes couple to a 3-form potential, which suggests that their description involves a non-abelian 2-gerbe or, equivalently, a principal 3-bundle. We show that current M2-brane models fit this expectation: they can be reformulated as higher gauge theories on such categorified bundles. We thus add to the still very sparse list of physically interesting higher gauge theories.

Recently, a new gauging procedure called Sculpting mechanism was proposed to obtain the M-theory origin of type II gauged Supergravity theories in 9D. We study this procedure in detail and give a better understanding of the different deformations and changes in fiber bundles that are able to generate new relevant physical gauge symmetries in the theory. We discuss the geometry involved in the standard approach (Noether-like) and in the new Sculpting-like one and comment on possible new applications.

It is described how the Extended Relativity Theory in $C$-spaces (Clifford spaces) allows a unified formulation of point particles, strings, membranes and $p$-branes, moving in ordinary target spacetime backgrounds, within the description of a single polyparticle moving in $C$-spaces. The degrees of freedom of the latter are provided by Clifford polyvector-valued coordinates (antisymmetric tensorial coordinates). A correspondence between the $p$-brane ($p$-loop) “Schrödinger-like” equations of Ansoldi–Aurilia–Spallucci and the polyparticle wave equation in $C$-spaces is found via the polyparticle/$p$-brane correspondence. This correspondence might provide another unexplored avenue to quantize $p$-branes (a notoriously difficult and unsolved problem) from the more straightforward quantization of the polyparticle in $C$-spaces, even in the presence of external interactions. We conclude with comments about the compositeness nature of the polyvector-valued coordinate operators in terms of ordinary $p$-brane coordinates via the evaluation of $n$-ary commutators.

Using M-theory compactification, we develop a three-factor separation for the scalar submanifold of $N=2$ seven-dimensional supergravity associated with 2-cycles of the K3 surface. Concretely, we give an interplay between the three-scalar submanifold factors and the extremal black holes obtained from M2-branes wrapping such 2-cycles. Then, we show that the corresponding black hole charges are linked to one, two and four qubit systems.

Exploiting an M-brane system whose structure and symmetries are inspired by those of graphene (what we call a graphene-brane), we propose here a similitude between two layers of graphene joined by a nanotube and wormholes scenarios in the brane world. By using the symmetries and mathematical properties of the M-brane system, we show here how to possibly increase its conductivity, to the point of making it as a superconductor. The questions of whether and under which condition this might point to the corresponding real graphene structures becoming superconducting are briefly outlined.

In this paper, we investigate the deflection angle and the trajectories of the light rays near black holes in M-theory scenarios. Using the Gauss–Bonnet theorem, we first discuss the deflection angle of the light rays near four- and seven-dimensional AdS black holes derived from the M-theory compactifications on the real spheres on $S^7$ and $S^4$, respectively. Then, we examine brane number effect and the rotating parameter on such light behaviors. We finish this work by providing a study of the light ray trajectories via the equations of motion associated with $M2$ and $M5$-branes.

A brief introduction of the Extended Relativity Theory in Clifford Spaces ($C$-space) paves the way to the explicit construction of the generalized relativistic transformations of the Clifford multivector-valued coordinates in $C$-spaces. The most general transformations furnish a full mixing of the grades of the multivector-valued coordinates. The transformations of the multivector-valued momenta follow leading to an invariant generalized mass ${ℳ}$ in $C$-spaces which differs from $m$. No longer the proper mass appearing in the relativistic dispersion relation $E^2-p\cdot p=m^2$ remains invariant under the generalized relativistic transformations. It is argued how this finding might shed some light into the cosmological constant problem, dark energy, and dark matter. We finalize with some concluding remarks about extending these transformations to phase spaces and about Born reciprocal relativity. An appendix is included with the most general (anti) commutators of the Clifford algebra multivector generators.

In this work, we reconsider the study of black holes and black strings in the compactification of M-theory on a Calabi–Yau three-fold, considered as a complete intersection of hypersurfaces in a product of weighted projective spaces given by ${𝕎\mathbb{P}}^4(\omega ,1,1,1,1)\times {\mathbb{P}}^1$. Using the $N=2$ supergravity formalism in five dimensions, we examine the BPS and non-BPSsolutions by wrapping M-branes on appropriate cycles in such a Calabi–Yau geometry. For the black hole case, we compute certain thermodynamical quantities. In particular, we calculate the entropy taking a maximal value corresponding to the ordinary projective space ${\mathbb{P}}^4$ with $\omega =1$. Using extended black hole entropies, we evaluate the temperature involving a minimal value for ${\mathbb{P}}^4$. Then, we approach the stability of the non-BPS black holes via the recombination factor. In the allowed electric charge regions, we show that such states are unstable. For the black string solutions, we calculate the tension taking a minimal value corresponding to ${\mathbb{P}}^4$. Computing the recombination factor, we show that the associated non-BPS black string states are stable in the allowed magnetic charge regions of the moduli space.