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 gravity duals of SO/USp superconformal quiver gauge theories on M5-branes which are localized on top of ℝ⁵/ℤ₂ and wrapping on a Riemann surface of genus g. We concentrate on Riemann surfaces with no punctures and show that the gravity solutions are classified by the genus g of the Riemann surface and the torsion part of the four-form flux.

We find that, apart from the instanton contributions, the all genus partition function of the ABJM matrix model sums up to the Airy function. We present the result, discuss its implication and also summarize some further progress.

We show that the ABJM theory, which is an superconformal U(N) × U(N) Chern-Simons matter theory, can be studied for arbitrary N at arbitrary coupling constant by applying a simple Monte Carlo method to the matrix model derived by using the localization method. Here we calculate the free energy, and show that some results obtained by the Fermi gas approach can be clearly understood from the constant map contribution obtained by the genus expansion.