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 $10+1$.

We discuss the asymptotic properties of quantum states density for fundamental p-branes which can yield a microscopic interpretation of the thermodynamic quantities in M-theory. The matching of the BPS part of spectrum for superstring and supermembrane gives the possibility of getting membrane's results via string calculations. In the weak coupling limit of M-theory, the critical behavior coincides with the first-order phase transition in the standard string theory at temperature less than the Hagedorn's temperature T_H. The critical temperature at large coupling constant is computed by considering M-theory on manifold with topology . Alternatively we argue that any finite temperature can be introduced in the framework of membrane thermodynamics.

Free field equations, with various spins, for space–time algebras with second-rank tensor (instead of the usual vector) momentum are constructed. Similar algebras are appearing in superstring/M theories. Special attention is paid to gauge invariance properties, in particular the spin-two equations with gauge invariance are constructed for dimensions 2+2 and 2+4, and the connection with Einstein equation and diffeomorphism invariance is established.

We consider M-theory on AdS₄ × N^0,1,0 where N^0,1,0 = (SU(3) × (2))/(SU(2) × U(1)). We review a Penrose limit of AdS₄ × N^0,1,0 that provides the pp-wave geometry of AdS₄ × S⁷. There exists a subsector of three-dimensional dual gauge theory, by taking both the conformal dimension and R-charge large with the finiteness of their difference, which has enhanced maximal supersymmetry. We identify operators in the gauge theory with supergravity KK excitations in the pp-wave geometry and describe how the gauge theory operators originating from both short vector multiplet and long gravitino multiplet fall into supermultiplets.

Little groups for preon branes (i.e. configurations of branes with maximal (n-1)/n fraction of survived supersymmetry) for dimensions d=2,3,…,11 are calculated for all massless, and partially for massive orbits. For massless orbits little groups are semidirect product of d-2 translational group T_d-2 on a subgroup of (SO(d-2) × R-invariance) group. E.g. at d=9 the subgroup is exceptional G₂ group. It is also argued, that 11D Majorana spinor invariants, which distinguish orbits, are actually invariant under d=2+10 Lorentz group. Possible applications of these results include construction of field theories in generalized spacetimes with brane charges coordinates, different problems of group's representations decompositions, spin-statistics issues.

In ten dimensions, SO(8) triality encourages confusion between fermions ans bosons, but in eleven dimensions, SO(9) has no such features. Yet this is the domain of M — theory whose infrared limit is the divergent supergravity field theory. Curtright ascribes this divergence to an incomplete cancellation among the Dynkin indices between the supergravity representations. We speculate that an infinite number of the recently discovered Euler triplets, describing massless particles of higher spin, can cancel the divergences of eleven dimensional supergravity, perhaps providing a clue to M — theory.

We consider d=10, N=1 supersymmetry algebra with maximal number of tensor charges Z and introduce a class of orbits of Z, invariant w.r.t. the T₈ subgroup of massless particles' little group T₈⋉SO(8). For that class of orbits we classify all possible orbits and little groups, which appear to be semidirect products T₈⋉SO(k₁)×⋯×SO(k_n), with k₁+⋯+k_n=8, where compact factor is embedded into SO(8) by triality map. We define actions of little groups on supercharge Q and construct corresponding supermultiplets. In some particular cases we show the existence of supermultiplets with both Fermi and Bose sectors consisting of the same representations of tensorial Poincaré. In addition, complete classification of supermultiplets (not restricted to T₈-invariant orbits) with rank-2 matrix of supersymmetry charges anticommutator, is given.

After a brief review of string and M-theory we point out some deficiencies. Partly to rectify them, we present several arguments for "F-theory", enlarging spacetime to (2, 10) signature, following the original suggestion of C. Vafa. We introduce a suggestive supersymmetric 27-plet of particles, associated to the exceptional symmetric hermitian space E₆/Spin^c(10). Several possible future directions, including using projective rather than metric geometry, are mentioned. We should emphasize that F-theory is yet just a very provisional attempt, lacking clear dynamical principles.

We consider a few topics in E₁₁ approach to superstrings/M-theory: even subgroups (Z₂ orbifolds) of E_n, n = 11, 10, 9 and their connection to Kac–Moody algebras, particularly to EE₁₁ subgroup of E₁₁; possible form of supersymmetry relation in E₁₁; decomposition of first fundamental representation l₁ w.r.t. the SO(10, 10) and its square-root at first few levels; particle orbit of l₁ ⋉ E₁₁. Possible relevance of coadjoint orbits method is noticed, based on a self-duality form of equations of motion in E₁₁.

By compactifying the Bagger–Lambert–Gustavsson model on ℝ^1,1×S¹, we obtain a new two-dimensional massless field theory with dynamical fields valued in the Lie three-algebra coupled with an SO(1, 1) scalar and vector field which are valued in the set of the endomorphisms of the Lie three-algebra. In the limit g_BLG→∞ the theory reduces to a supersymmetric Lie three-valued generalization of the Green–Schwarz superstring in the light-cone gauge.

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.

We find a class of Hermitian generalized Jordan triple systems (HGJTSs) and Hermitian (ϵ, δ)-Freudenthal–Kantor triple systems (HFKTSs). We apply one of the most simple HGJTSs which we find to a field theory and obtain a typical u(N) Chern–Simons gauge theory with a fundamental matter.

Recently, Padmanabhan has discussed that the expansion of the cosmic space is due to the difference between the number of degrees of freedom on the boundary surface and the number of degrees of freedom in a bulk region. Now, a natural question arises that how these degrees of freedom emerged from nothing? We try to address this issue in a new theory which is more complete than M-theory and reduces to it with some limitations. In M-theory, there is no stable object like stable M3-branes that our universe is formed on it and for this reason cannot help us to explain cosmological events. In this research, we propose a new theory, named G-theory which could be the mother of M-theory and superstring theory. In G-theory, at the beginning, two types of G0-branes, one with positive energy and one with negative energy are produced from nothing in 14 dimensions. Then, these branes are compactified on three circles via two different ways (symmetrically and anti-symmetrically), and two bosonic and fermionic parts of action for M0-branes are produced. By joining M0-branes, supersymmetric Mp-branes are created which contain the equal number of degrees of freedom for fermions and bosons. Our universe is constructed on one of Mp-branes and other Mp-brane and extra energy play the role of bulk. By dissolving extra energy which is produced by compacting actions of Gp-branes, into our universe, the number of degrees of freedom on it and also its scale factor increase and universe expands. We test G-theory with observations and find that the magnitude of the slow-roll parameters and the tensor-to-scalar ratio in this model are very much smaller than one which are in agreement with predictions of experimental data. Finally, we consider the origin of the extended theories of gravity in G-theory and show that these theories could be anomaly free.

In string theory with ten dimensions, all Dp-branes are constructed from D0-branes whose action has two-dimensional brackets of Lie 2-algebra. Also, in M-theory, with 11 dimensions, all Mp-branes are built from M0-branes whose action contains three-dimensional brackets of Lie 3-algebra. In these theories, the reason for difference between bosons and fermions is unclear and especially in M-theory there is not any stable object like stable M3-branes on which our universe would be formed on it and for this reason it cannot help us to explain cosmological events. For this reason, we construct G-theory with M dimensions whose branes are formed from G0-branes with N-dimensional brackets. In this theory, we assume that at the beginning there is nothing. Then, two energies, which differ in their signs only, emerge and produce 2M degrees of freedom. Each two degrees of freedom create a new dimension and then M dimensions emerge. M-N of these degrees of freedom are removed by symmetrically compacting half of M-N dimensions to produce Lie-N-algebra. In fact, each dimension produces a degree of freedom. Consequently, by compacting M-N dimensions from M dimensions, N dimensions and N degrees of freedom is emerged. These N degrees of freedoms produce Lie-N-algebra. During this compactification, some dimensions take extra i and are different from other dimensions, which are known as time coordinates. By this compactification, two types of branes, Gp and anti-Gp-branes, are produced and rank of tensor fields which live on them changes from zero to dimension of brane. The number of time coordinates, which are produced by negative energy in anti-Gp-branes, is more sensible to number of times in Gp-branes. These branes are compactified anti-symmetrically and then fermionic superpartners of bosonic fields emerge and supersymmetry is born. Some of gauge fields play the role of graviton and gravitino and produce the supergravity. The question may arise that what is the physical reason which shows that this theory is true. We shown that G-theory can be reduced to other theories like nonlinear gravity theories in four dimensions. Also, this theory, can explain the physical properties of fermions and bosons. On the other hand, this theory explains the origin of supersymmetry. For this reason, we can prove that this theory is true. By reducing the dimension of algebra to three and dimension of world to 11 and dimension of brane to four, G-theory is reduced to F(R)-gravity.

It is shown that besides the standard real algebraic framework for M-theory a consistent octonionic realization can be introduced. The octonionic M-superalgebra and superconformal M-algebra are derived. The first one involves 52 real bosonic generators and presents a novel and surprising feature, its octonionic M5 (super-5-brane) sector coincides with the M1 and M2 sectors. The octonionic superconformal M-algebra is given by OS_p(1,8∣O) and admits 239 bosonic and 64 fermionic generators.

It is shown that the twisted sector spectrum, as well as the associated Chern–Simons interactions, can be determined on M-theory orbifold fixed planes that do not admit gravitational anomalies. This is demonstrated for the seven-planes arising within the context of an explicit R⁶ × S¹/Z₂ × T⁴/Z₂ orbifold, although the results are completely general. Local anomaly cancellation in this context is shown to require fractional anomaly data that can only arise from a twisted sector on the seven-planes, thus determining the twisted spectrum up to a small ambiguity. These results open the door to the construction of arbitrary M-theory orbifolds, including those containing fixed four-planes which are of phenomenological interest.

We analyze the structure of heterotic M-theory on K3 orbifolds by presenting a comprehensive sequence of M-theoretic models constructed on the basis of local anomaly cancellation. This is facilitated by extending the technology developed in our previous papers to allow one to determine "twisted" sector states in nonprime orbifolds. These methods should naturally generalize to four-dimensional models, which are of potential phenomenological interest.