Narrow Results

This is a review of exceptional field theory: a generalisation of Kaluza–Klein theory that unifies the metric and $p$-form gauge field degrees of freedom of supergravity into a generalised or extended geometry, whose additional coordinates may be viewed as conjugate to brane winding modes. This unifies the maximal supergravities, treating their previously hidden exceptional Lie symmetries as a fundamental geometric symmetry. Duality orbits of solutions simplify into single objects, that in many cases have simple geometric interpretations, for instance as wave or monopole-type solutions. It also provides a route to explore exotic or nongeometric aspects of M-theory, such as exotic branes, $U$-folds, and more novel sorts of non-Riemannian spaces.

We study $d$-dimensional black holes surrounded by dark energy (DE), embedded in $D$-dimensional M-theory/superstring inspired models having ${\mathrm{\text{AdS}}}_d\times {𝕊}^{d+k}$ space–time where $D=2d+k$. We focus first on the thermodynamical Hawking–Page phase transitions, whose microscopical origin is linked to $N$ coincident $(d-2)$-branes supposed to live in such inspired models. Interpreting the cosmological constant as the number of colors, we compute various thermodynamical quantities in terms of the brane number, the entropy and the DE contribution. Calculating the ordinary chemical potential conjugated to the number of colors, we show that a generic black hole is more stable for a larger number of branes in lower dimensions. In the presence of DE, however, we find that the DE state parameter ${\omega }_q$ takes particular values, for $(D,d,k)$ models, providing nontrivial phase transitions. Then, we examine some optical properties. Concretely, we investigate shadow behaviors of quintessential black holes in terms of $(d-2)$-brane physics. In terms of certain ratios, we find similar behaviors for critical quantities and shadow radius.

We study the shadows of four-dimensional black holes in M-theory inspired models. We first inspect the influence of M2-branes on such optical aspects for nonrotating solutions. In particular, we show that the M2-brane number can control the circular shadow size. This geometrical behavior is distorted for rotating solutions exhibiting cardioid shapes in certain moduli space regions. Implementing a rotation parameter, we analyze the geometrical shadow deformations. Among others, we recover the circular behaviors for a large M2-brane number. Investigating the energy emission rate at high energies, we find, in a well-defined approximation, that the associated peak decreases with the M2-brane number. Moreover, we investigate a possible connection with observations (from Event Horizon Telescope or future devices) from a particular M-theory compactification by deriving certain constraints on the M$2$-brane number in the light of the $M87^{\star }$ observational parameters.

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 $10+1$.

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.

We report on the exact computation of the S³ partition function of U(N)_k × U(N)_-k ABJM theory for k = 1, N = 1, …, 19. The result is a polynomial in π^-1 with rational coefficients. As an application of our results, we numerically determine the coefficient of the membrane 1-instanton correction to the partition function.

In this paper, we will first derive the Wheeler–DeWitt equation for the generalized geometry which occurs in M-theory. Then we will observe that M2-branes act as probes for this generalized geometry, and as M2-branes have an extended structure, their extended structure will limits the resolution to which this generalized geometry can be defined. We will demonstrate that this will deform the Wheeler–DeWitt equation for the generalized geometry. We analyze such a deformed Wheeler–DeWitt equation in the minisuperspace approximation, and observe that this deformation can be used as a solution to the problem of time. This is because this deformation gives rise to time crystals in our universe due to the spontaneous breaking of time reparametrization invariance.

These are notes for four lectures on higher structures in M-theory as presented at workshops at the Erwin Schrödinger Institute and Tohoku University. The first lecture gives an overview of systems of multiple M5-branes and introduces the relevant mathematical structures underlying a local description of higher gauge theory. In the second lecture, we develop the corresponding global picture. A construction of non-abelian superconformal gauge theories in six dimensions using twistor spaces is discussed in the third lecture. The last lecture deals with the problem of higher quantization and its relation to loop space. An appendix summarizes the relation between 3-Lie algebras and Lie 2-algebras.

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 present analytic solutions for membrane metric function based on transverse k-center instanton geometries. The membrane metric functions depend on more than two transverse coordinates and the solutions provide realizations of fully localized type IIA D2/D6 and NS5/D6 brane intersections. All solutions have partial preserved supersymmetries.

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.

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 review our recent proposal for a background-independent formulation of a holographic theory of quantum gravity. The present paper incorporates the necessary background material on geometry of canonical quantum theory, holography and space–time thermodynamics, Matrix theory, as well as our specific proposal for a dynamical theory of geometric quantum mechanics, as applied to Matrix theory. At the heart of this review is a new analysis of the conceptual problem of time and the closely related and phenomenologically relevant problem of vacuum energy in quantum gravity. We also present a discussion of some observational implications of this new viewpoint on the problem of vacuum energy.

The first lecture gives a colloquium-level overview of string theory and M-theory. The second lecture surveys various attempts to construct a viable model of particle physics. A recently proposed approach, based on F-theory, is emphasized.

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.

In this paper, we review inflationary cosmology in M-theory with quantum corrections. In old days the inflation was proposed as a resolution to the cosmological problems, and nowadays models of the inflation are severely restricted by the observations. Among them, the predictions of the Starobinsky model, which contains scalar curvature squared term, is consistent with the observations. The higher curvature terms will come from quantum effect of the gravity, and it is natural to ask its origin in superstring theory or M-theory. We investigate inflationary solution in the M-theory with higher curvature terms. We show that higher curvature terms induce an exponentially expanding solution in the early universe, and the inflation naturally ends when the corrections are suppressed. We also discuss that the ambiguity of the higher curvature terms do not affect the inflationary scenario in the M-theory.

The bosonic large-$N$ master field of the IIB matrix model can, in principle, give rise to an emergent classical spacetime. The task is then to calculate this master field as a solution of the bosonic master-field equation. We consider a simplified version of the algebraic bosonic master-field equation and take dimensionality $D=2$ and matrix size $N=6$. For an explicit realization of the pseudorandom constants entering this simplified algebraic equation, we establish the existence of a solution and find, after diagonalization of one of the two obtained matrices, a band-diagonal structure of the other matrix.

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.