= Solution
The three identifications are
$$
\boxed{\text{M-theory on }S^1\longleftrightarrow\text{type IIA},\qquad
\text{M-theory on }T^2\longleftrightarrow\text{type IIB on }S^1,\qquad
\text{M-theory on K3}\longleftrightarrow\text{heterotic on }T^3.}
$$
They match the same lower-dimensional theory, including its <moduli of a string compactification>, charged states and extended <branes>. Different weakly coupled descriptions occupy different limits of its <moduli space>.
For the <M-theory circle duality>, write $\ell_s=\sqrt{\alpha'}$ and let $\ell_p$ be the eleven-dimensional <Planck length>. The standard length dictionary is
$$
\boxed{R_{11}=g_A\ell_s,\qquad\ell_p^3=g_A\ell_s^3.}
$$
The circle radius is the <type IIA superstring theory> coupling modulus. The eleven-dimensional metric supplies the ten-dimensional metric, <dilaton> and <Ramond–Ramond potential> $C_1$ through <Kaluza-Klein theory>; the three-form supplies $C_3$ and the <Neveu–Schwarz two-form> $B_2$ by $A_3=C_3+B_2\wedge dy$. Thus the field content also agrees with <type IIA supergravity>.
The <M2-brane> and <M5-brane> tensions are $T_2=(2\pi)^{-2}\ell_p^{-3}$ and $T_5=(2\pi)^{-5}\ell_p^{-6}$. Wrapping one spatial direction multiplies the effective <brane tension> by $2\pi R_{11}$. Consequently:
* A wrapped <M2-brane> gives the <fundamental string>, since $2\pi R_{11}T_2=1/(2\pi\alpha')$; an unwrapped <M2-brane> gives the <D2-brane>.
* A wrapped <M5-brane> gives the <D4-brane>, with tension $(2\pi)^{-4}g_A^{-1}\ell_s^{-5}$; an unwrapped <M5-brane> gives the <NS5-brane>, with tension $(2\pi)^{-5}g_A^{-2}\ell_s^{-6}$.
* Momentum $n/R_{11}$ gives $n$ units of <D0-brane> charge. The <Kaluza-Klein monopole> with the M-circle as its fibre gives the <D6-brane>.
These tension and charge relations explain why increasing $g_A$ reveals an eleventh dimension. The simple circle dictionary describes massless <type IIA supergravity>; a <D8-brane> involves its massive extension and <Romans mass>.
For the <M-theory torus duality>, the <complex structure> of the torus is the IIB <axion-dilaton>
$$
\boxed{\tau=C_0+i/g_B.}
$$
Its area supplies the remaining radius modulus. In a common nine-dimensional mass normalization, an <M2-brane> wrapped on the entire torus has mass $T_2\mathcal A$, identified with $1/R_B$; hence $R_B$ is inversely related to the torus area in that normalization. Radii quoted in different string or Einstein metrics also include their <Weyl rescaling>. A rectangular torus with cycle radii $R_{10},R_{11}$ has $g_B=R_{11}/R_{10}$. Equivalently, first reduce to IIA and use <T-duality>:
$$
R_B=\frac{\alpha'}{R_A},\qquad g_B=\frac{g_A\ell_s}{R_A},
$$
where $R_A,R_B$ in this formula use the corresponding string metrics. Shrinking the torus area at fixed $\tau$ decompactifies the IIB circle. The torus <mapping class group> $SL(2,\mathbb Z)$ becomes IIB <S-duality>.
An <M2-brane> wrapped on a primitive $(p,q)$ one-cycle gives a <(p,q) string>, whose tension is the membrane tension times the cycle length before conversion to the IIB metric. Momentum on the torus matches winding of these strings on the IIB circle. An <M5-brane> wrapped on the whole torus gives an unwrapped <D3-brane>; an unwrapped <M2-brane> gives a <D3-brane> wrapped on the IIB circle. An <M5-brane> wrapped on one torus cycle gives a <(p,q) five-brane> wrapped on the IIB circle. The unwrapped <M5-brane> is the magnetic dual of the IIB circle's <Kaluza-Klein mode>, represented by its <Kaluza-Klein monopole> five-brane. These identifications match effective spatial dimensions as well as electric and magnetic charges.
For the <M-theory K3 duality>, the common noncompact spacetime has seven dimensions and sixteen <supersymmetry generators>. A <Ricci flat> <K3 surface> has a positive three-plane in its second real <cohomology>, equipped with the <K3 intersection lattice> of signature $(3,19)$. The unit-volume <moduli space of Ricci-flat K3 metrics> has local form
$$
\frac{SO(3,19)}{SO(3)\times SO(19)},\qquad \dim=57,
$$
with a discrete lattice identification and one additional positive volume modulus. This is the heterotic <Narain moduli space> on $T^3$, together with the seven-dimensional coupling: its torus metric has six parameters, its antisymmetric tensor three, and its sixteen gauge <Wilson lines> forty-eight, giving $6+3+48=57$ before the <dilaton>.
The coupling–volume relation can be checked dimensionally rather than guessed. A wrapped <M5-brane> gives a heterotic <fundamental string> with tension $T_5V$, so its squared <string length> is proportional to $\ell_p^6/V$. The seven-dimensional gravitational coupling is proportional to $\ell_p^9/V$. Their dimensionless ratio gives
$$
\boxed{g_{H,7}^2\propto\frac{\ell_p^9/V}{(\ell_p^6/V)^{5/2}}
=\left(\frac V{\ell_p^4}\right)^{3/2}.}
$$
Thus the large-volume M-description corresponds to strong heterotic coupling. Overall constants depend on the normalization of the lower-dimensional coupling.
There are twenty-two <gauge fields> from reducing $A_3$ on the twenty-two harmonic two-forms of K3, matching the heterotic sixteen gauge vectors plus three metric and three two-form vectors. The remaining seven-dimensional three-form is dual to the heterotic two-form. A wrapped <M2-brane> gives an electrically charged particle in the shared charge lattice, matching heterotic momentum, winding and gauge charges; an <M5-brane> on a two-cycle gives its magnetic three-brane. An <M5-brane> wrapping all of K3 gives the heterotic string, while an unwrapped <M2-brane> corresponds to a heterotic <NS5-brane> wrapped on $T^3$. At suitable collapsing two-cycles, charged membrane states become massless and yield <ADE gauge enhancement>. The geometry accounts for both generic abelian charges and nonabelian enhancement. There is no extra K3 two-form modulus in the M-description: its fundamental potential is a three-form, and $b_3(\mathrm{K3})=0$.
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