M-theory ($R_{11}=g_s\sqrt{{\alpha }^{\prime }}$)

An eleven-dimensional nonperturbative framework suggested by superstring dualities, with eleven-dimensional supergravity as its low-energy limit. Type IIA at coupling $g_s$ has an interpretation on a circle of radius $R_{11}=g_s\sqrt{{\alpha }^{\prime }}$, so an additional dimension becomes visible at strong coupling. Perturbative string descriptions represent particular limits of this framework.

M-theory on a K3 surface and the heterotic string on a three-torus have matching seven-dimensional moduli and charge lattices. The K3 intersection lattice supplies twenty-two vector charges; an M5-brane wrapped on K3 becomes the heterotic fundamental string. The strong-coupling interpretation and low-energy matching are derived in section 4.7 of String Theory Dynamics in Various Dimensions.

When appropriate negative-square two-cycles of a K3 surface collapse, wrapped M2-branes become massless. Their intersection pattern is an ADE root diagram, and the massless charged states enlarge the abelian gauge algebra to the corresponding simply laced Lie algebra.

// Target: supersymmetry.bigb

A wrapped M5-brane fixes the heterotic string tension proportional to $V/{\ell }_p^6$. Reduction of the gravitational coupling gives $G_7\propto {\ell }_p^9/V$. Therefore $G_7/{\ell }_{s,H}^5\propto (V/{\ell }_p^4{)}^{3/2}$, yielding the displayed dimensionless coupling relation. This derivation separates the K3 volume modulus from its shape Narain moduli space.

M-theory on a two-torus and type IIB superstring theory on a circle describe the same nine-dimensional physics. Torus shape supplies the axion-dilaton; torus area supplies the circle-radius modulus after matching metrics. An M2-brane on a one-cycle gives a (p,q) string; an M5-brane on the whole torus gives a D3-brane. Matching brane charges and tensions establishes the dictionary in The Power of M Theory.

Compactifying M-theory on a circle gives type IIA superstring theory. The circle radius sets its coupling. Multiplying M2-brane and M5-brane tensions by the circle circumference gives the fundamental string and D4-brane tensions, while unwrapped objects give D2-branes and NS5-branes. Momentum and a Kaluza-Klein monopole supply D0-brane and D6-brane charges. The circle dictionary and worldvolume reductions are developed in Four Lectures on M-theory.

D0-branes are circle momentum states, D2-branes are unwrapped M2-branes, D4-branes are circle-wrapped M5-branes, and D6-branes arise from Kaluza-Klein monopoles with the M-circle as fibre. Wrapped M2-branes give fundamental strings, and unwrapped M5-branes give NS5-branes. The D8-brane is a source for Romans mass and is outside this ordinary massless-circle dictionary. Tension matching uses $R_{11}=g_s{\ell }_s$ and ${\ell }_p^3=g_s{\ell }_s^3$.

An M-brane is a brane in M-theory. Its elementary membrane and five-brane carry electric and magnetic charges for the three-form potential of eleven-dimensional supergravity. Compactification and wrapping relate them to strings and D-branes.

The M5-brane has five spatial dimensions and carries the magnetic charge dual to the M2-brane's electric three-form charge. Wrapping it on a compact cycle lowers its effective spatial dimension and multiplies its brane tension by the cycle volume.

The M2-brane has two spatial dimensions and couples electrically to the supergravity three-form. Its supersymmetric worldvolume description is the supermembrane.

The transverse harmonic function $H$ solves the field equations away from membrane sources. Positive centres contribute $Q_a/|y-y_a{|}^6$ in eight dimensions. Its Killing spinors are $H^{-1/6}{ϵ}_0$ with one oriented membrane projector, giving sixteen supersymmetries. The three-form's sign and membrane orientation must be chosen together.

For $H\sim R^6/r^6$, changing radial coordinate to $z=R^3/(2r^2)$ gives an AdS metric of radius $R/2$ times a round sphere of radius $R$. The Freund-Rubin compactification flux has magnitude $6/R$ in the AdS volume form. This is the regular positive-charge membrane throat, rather than the limit of an arbitrary singular harmonic multipole.

// Target: supersymmetry.bigb

For $F={\mathrm{vol}}_{012}\wedge d{H}^{-1}$ and $F=dA$, choose $A=(1-H^{-1}){\mathrm{vol}}_{012}$. A parallel probe then has $q=-1$ in the supermembrane action, so its induced-volume term $-T_2{H}^{-1}$ and potential term $-T_2(1-H^{-1})$ add to a position-independent constant. This proves the cancellation without a weak-field approximation. An oppositely charged probe has no such cancellation.

The supersymmetric M2-brane worldvolume theory includes embedding bosons and fermions and has kappa symmetry. In eleven dimensions its fermionic embedding coordinate is a thirty-two-component Majorana spinor.

The first term uses the induced worldvolume metric from the supervielbein; the second is the Wess-Zumino brane coupling to the super-three-form. Its fermionic dependence is retained before bosonic truncation. The relative charge magnitude equals the brane tension, while $q=\pm 1$ sets the orientation and matching kappa symmetry projector sign. In an on-shell eleven-dimensional supergravity superspace, this action has kappa symmetry.