M2, M5, and Higher-Dimensional Brane Examples

The near-horizon limits of M2- and M5-branes produce two classical Freund–Rubin backgrounds of eleven-dimensional supergravity: AdS$_4\times S^7$ and AdS$_7\times S^4$. Flux quantization fixes both curvature radii in Planck units. Reducing the Einstein term then gives $L_4^2/G_4\propto N^{3/2}$ for M2-branes and $L_7^5/G_7\propto N^3$ for M5-branes. The first scaling has a same-observable check from the localized ABJM sphere partition function; the second appears in protected six-dimensional anomalies as well as large-$N$ gravity. These agreements are powerful but differently scoped: ABJM has a conventional three-dimensional Lagrangian, whereas the interacting six-dimensional $(2,0)$ theory is well established through mutually consistent non-Lagrangian data, not through an ordinary non-Abelian two-form action.

Required background. Near-horizon brane dictionaries supplies the brane-to-boundary map. Quantized flux and harmonic spectra supplies the relation between integer charge, curvature, and compact towers.

Helpful background. Higher-dimensional origins and duality frames explains circle reduction of the $(2,0)$ theory. Dimensions and reality conditions fixes the supercharge count and real forms. Higher-dimensional fixed points and anomalies develops the intrinsic tensor-branch and anomaly evidence.

For a quick route, first fix the two radius conventions, follow the Newton-constant derivation, compare the M2 and M5 evidence, and then execute the adversarial Lagrangian test.

Two Freund–Rubin throats

Use unit-radius metrics inside $ds^2_{\mathrm{AdS}}$ and $ds^2_S$. Let $L_4$ and $L_7$ denote the physical AdS radii, while $R_7$ and $R_4$ denote the physical radii of $S^7$ and $S^4$. The subscripts name the factor, not a power. In the eleven-dimensional Planck-length convention

$$2\kappa_{11}^2=(2\pi)^8\ell_p^9, \qquad G_{11}=\frac{\kappa_{11}^2}{8\pi} :=16\pi^7\ell_p^9.$$

For $N$ M2-branes on the standard $\mathbb C^4/\mathbb Z_k$ orbifold, with no discrete-torsion shift, the near-horizon metric is

$$ds_{11}^2 :=L_4^2 ds_{\mathrm{AdS}_4}^2 +R_7^2 ds_{S^7/\mathbb Z_k}^2, \qquad R_7=2L_4.$$

The electric flux integer is

$$N :=\frac{1}{(2\pi\ell_p)^6} \int_{S^7/\mathbb Z_k}\! *F_4,$$

where the orientation is chosen so that $N>0$. On this Freund–Rubin background the possible Chern–Simons term in the Page charge has no pullback contribution to the integration cycle. At the two-derivative supergravity level, before finite-$N$ charge shifts, flux quantization gives

$$R_7^6=32\pi^2Nk\,\ell_p^6, \qquad L_4=\frac{\ell_p}{2}(32\pi^2Nk)^{1/6}.$$

Higher-curvature terms at the orbifold fixed point and nonzero discrete torsion shift the effective M2 charge beyond this leading map Bergman and Hirano 2009, § 3, p. 7, eq. (3.24), PDF.

At $k=1$ this is the round-sphere M2 throat. Its superconformal algebra is $OSp(8|4)$, with bosonic algebra $\mathfrak{so}(3,2)\oplus\mathfrak{so}(8)$. For generic $k>2$, the orbifold preserves the $OSp(6|4)$ algebra of three-dimensional $\mathcal N=6$ ABJM theory; at $k=1,2$ the supersymmetry enhances to $\mathcal N=8$ Aharony et al. 2008, § 2.3, pp. 8–9, text after eq. (2.14), PDF. The quotient geometry and its M-theory/type-IIA parameter map are developed in Aharony et al. 2008, § 4.1, pp. 21–24, eqs. (4.1)–(4.7), PDF.

For $N$ coincident M5-branes, the magnetic four-form flux threads $S^4$:

$$ds_{11}^2 :=L_7^2 ds_{\mathrm{AdS}_7}^2 +R_4^2 ds_{S^4}^2, \qquad L_7=2R_4,$$ $$N :=\frac{1}{(2\pi\ell_p)^3} \int_{S^4}\!F_4, \qquad R_4^3=\pi N\ell_p^3, \qquad L_7=2\ell_p(\pi N)^{1/3}.$$

The superconformal algebra is $OSp(8^*|4)$, whose bosonic algebra is $\mathfrak{so}(6,2)\oplus\mathfrak{usp}(4)$, with $USp(4)\simeq Spin(5)$ the R-symmetry. The notation $OSp(6,2|4)$ found in some older literature gestures at the same bosonic factors but does not name the standard real form as precisely. Older papers may also call one chirality choice $(0,2)$; this site uses the modern $(2,0)$ label. Maldacena’s original construction gives both decoupling limits, harmonic functions, radius ratios, and $G_{11}$ normalization Maldacena 1998, §§ 3.1–3.2, pp. 9–10, eqs. (3.1)–(3.3), PDF.

The factor of two in each product is indispensable. Calling every radius simply $L$ hides powers of two in both the flux–radius relation and the compact volume, changing the numerical coefficients.

Dimensional reduction counts effective degrees of freedom

Because these backgrounds are unwarped direct products, the lower-dimensional Einstein term is obtained by integrating over the physical compact factor. The unit-sphere volumes are

$$\operatorname{Vol}(S^7_{\mathrm{unit}})=\frac{\pi^4}{3}, \qquad \operatorname{Vol}(S^4_{\mathrm{unit}})=\frac{8\pi^2}{3}.$$

For the M2 quotient, $\operatorname{Vol}(S^7/\mathbb Z_k)$ is smaller by $k$. Therefore

$$\begin{aligned} G_4 &=\frac{G_{11}}{\operatorname{Vol}(S^7_{R_7}/\mathbb Z_k)} =\frac{48\pi^3k\,\ell_p^9}{R_7^7},\\ G_7 &=\frac{G_{11}}{\operatorname{Vol}(S^4_{R_4})} =\frac{6\pi^5\ell_p^9}{R_4^4}. \end{aligned}$$

Substitute the flux–radius relations only after this reduction. The dimensionless gravitational normalizations become

$$\boxed{ \frac{L_4^2}{G_4} =\frac{2\sqrt2}{3}\,\sqrt{k}\,N^{3/2} }, \qquad \boxed{ \frac{L_7^5}{G_7} =\frac{16}{3\pi^2}N^3 }.$$

This calculation is reproducible from the four displayed inputs: $G_{11}$, the unit-sphere volumes, the two radius ratios, and flux quantization. It also passes an immediate dimensional check: $[G_4]=\text{length}^2$ and $[G_7]=\text{length}^5$.

The powers of $N$ are not literal counts of gauge-variant fields. They describe how normalized observables controlled by the bulk Einstein term—sphere free energies, stress-tensor correlators, Weyl anomalies, or thermal coefficients—grow in a specified limit. Different observables can have different subleading terms.

M2 boundary data: ABJM tests the coefficient

The round $k=1$ throat motivates a dual three-dimensional $\mathcal N=8$ M2-brane SCFT. The more explicit family is ABJM theory: $U(N)_k\times U(N)_{-k}$ Chern–Simons matter theory, conjecturally dual to M-theory on AdS$_4\times S^7/\mathbb Z_k$. Here $k$ changes the global geometry, supersymmetry, spectrum, and calculational frame; it cannot be omitted from “the M2 theory.”

The renormalized Euclidean AdS$_4$ Einstein action predicts for a round boundary three-sphere

$$F_{S^3}^{\mathrm{grav}} :=\frac{\pi L_4^2}{2G_4} =\frac{\pi\sqrt{2k}}{3}N^{3/2}.$$

On the boundary, define $F_{S^3}:=-\log\lvert Z_{S^3}\rvert$. Supersymmetric localization reduces the ABJM path integral to a finite-dimensional matrix integral Kapustin, Willett, and Yaakov 2010, § 4, pp. 20–22, eq. (4.4), PDF. Its Fermi-gas analysis at fixed positive integer $k$ gives

$$F_{S^3}^{\mathrm{ABJM}} =\frac{\pi\sqrt{2k}}{3}N^{3/2} +O(N^{1/2}), \qquad N\to\infty\ \text{at fixed }k,$$

including the same coefficient, not just the same power Mariño and Putrov 2012, §§ 4.2 and 5.3, pp. 19–20 and 27–31, eqs. (4.21)–(4.27), (5.37), and (5.65)–(5.67), PDF. Mariño and Putrov define $F=\log Z$, so their displayed leading term is negative; the convention here, $F_{S^3}=-\log\lvert Z_{S^3}\rvert$, reverses that sign. This is a strong same-observable normalization check: the boundary result comes from a localized QFT integral, while the bulk result comes from flux quantization and the gravitational on-shell action. Localization and the leading asymptotic coefficient do not make generic non-BPS observables exact.

The dual frame depends on how $N$ and $k$ scale. The Hopf-fiber radius in Planck units is

$$\frac{R_7}{k\ell_p} =(32\pi^2)^{1/6} \left(\frac{N}{k^5}\right)^{1/6}.$$

Thus eleven-dimensional supergravity on the quotient requires $N\gg k^5$ as well as weak local curvature. When $k$ is large, type-IIA supergravity on AdS$_4\times\mathbb{CP}^3$ instead has the parametric window

$$1\ll\lambda:=\frac Nk, \qquad \frac{N^{1/2}}{k^{5/2}}\ll1,$$

equivalently $k\ll N\ll k^5$ up to order-one factors Aharony et al. 2008, § 4.1, pp. 22–24, eqs. (4.2)–(4.7), PDF. Large $N$ without a scaling prescription for $k$ does not select a bulk frame.

M5 boundary data: anomalies without an ordinary action

The AdS$_7\times S^4$ throat is associated with the interacting six-dimensional type-$A_{N-1}$ $(2,0)$ theory. The stack also has one decoupled free center-of-mass tensor multiplet, represented holographically by a boundary doubleton rather than an ordinary propagating bulk mode Günaydin and Takemae 2000, abstract and § 3.2, PDF. Keeping or subtracting this free sector changes exact finite-$N$ formulas.

Let $T$ be the tangent bundle to the six-dimensional worldvolume and $\mathcal N_\perp$ its rank-five normal, or R-symmetry, bundle. In an integral Pontryagin-class convention, one free tensor multiplet has

$$I_8^{\mathrm{tens}} :=\frac1{48} \left[ p_2(\mathcal N_\perp)-p_2(T) +\frac14\bigl(p_1(T)-p_1(\mathcal N_\perp)\bigr)^2 \right].$$

Anomaly inflow for the full stack gives

$$I_8[N\ \mathrm{M5}] =N I_8^{\mathrm{tens}} +\frac{N^3-N}{24}p_2(\mathcal N_\perp).$$

Subtracting the center-of-mass tensor leaves the interacting theory:

$$\boxed{ I_8[A_{N-1}] =(N-1)I_8^{\mathrm{tens}} +\frac{N^3-N}{24}p_2(\mathcal N_\perp) }.$$

The cubic term arises because the eleven-dimensional Chern–Simons inflow is cubic in the M5 charge; the linear pieces retain finite-$N$ information Harvey, Minasian, and Moore 1998, § 2.1, pp. 1–3, eqs. (2.2)–(2.6), PDF. Tensor-branch anomaly matching reproduces the ADE formula by a largely field-theoretic route Ohmori et al. 2014, §§ 1.1 and 2.3, pp. 3–4 and 9–10, PDF. The $N=1$ check is especially transparent: after subtracting the one free tensor, the formula vanishes, as it must for the trivial interacting $A_0$ sector.

Logical status. The anomaly is an exact protected datum, and its $N^3$ term agrees with the scaling of $L_7^5/G_7$. It is not a computation of a generic finite-$N$ correlator. The anomaly-inflow route shares M-theory brane input with the holographic construction; tensor-branch matching supplies a distinct consistency route, but protection still limits what the agreement tests.

What large N controls—and what it does not

There is no independent string coupling in the displayed eleven-dimensional frame. Curvature and quantum effects are controlled by Planck-length ratios and the effective lower-dimensional Newton coupling:

$$\frac{L_4}{\ell_p} =\frac12(32\pi^2Nk)^{1/6}, \qquad \frac{G_4}{L_4^2} =\frac{3}{2\sqrt2\,\sqrt{k}\,N^{3/2}},$$ $$\frac{L_7}{\ell_p} =2(\pi N)^{1/3}, \qquad \frac{G_7}{L_7^5} =\frac{3\pi^2}{16N^3}.$$

For fixed $k$, large $N$ makes both throats weakly curved and suppresses bulk loops. For the M2 quotient it must also keep the orbifold circle larger than $\ell_p$ if the eleven-dimensional description is used. The first omitted higher-derivative or loop order is observable-dependent; writing a small parameter does not prove that every coefficient at that order is nonzero or bound the remainder.

Large $N$ does not remove the compact sphere. Because $R_7=2L_4$ and $L_7=2R_4$, sphere harmonics have

$$m_{\mathrm{KK}}L_{\mathrm{AdS}}=O(1).$$

There is no parametric gap between the full Kaluza–Klein tower and the AdS scale. A four- or seven-dimensional gauged-supergravity sector can still be useful when it is a consistent truncation, but not because every omitted compact mode is heavy.

M2 and M5 dictionaries, controls, and claim ceilings

Datum	M2 / AdS4 × S7/ℤk	M5 / AdS7 × S4	Status or caveat
Flux and radii	R76 = 32π2Nkℓp6; R7 = 2L4	R43 = πNℓp3; L7 = 2R4	Classical eleven-dimensional solution plus quantized charge; discrete torsion would modify the M2 global data.
Boundary realization	ABJM gives a conventional 3D Chern–Simons-matter Lagrangian for the orbifold family.	The interacting type-AN−1 (2,0) SCFT is specified through symmetry, charge lattice, tensor branch, anomalies, compactifications, and brane constructions.	“Non-Lagrangian” is not “undefined”; it identifies the missing ordinary six-dimensional action, not an absence of theory data.
Einstein normalization	L42/G4 = (2√2/3)√k N3/2	L75/G7 = 16N3/(3π2)	Exact within the two-derivative product reduction; it controls leading large-N observables, not every finite-N term.
Protected or localized check	Localized S3 free energy reproduces π√(2k)N3/2/3 at fixed k.	The exact anomaly polynomial contains (N3−N)p2(𝒩⊥)/24.	Strong normalization and consistency evidence; neither result licenses all unprotected dynamics.
Supergravity control	Large Nk controls local curvature; the 11D quotient frame additionally needs N ≫ k5.	Large N controls curvature and loops.	Both compact towers remain at the AdS scale; consistent truncation is a separate property.
Generic finite-N observable	The Lagrangian defines it in principle, although strong-coupling computation may be difficult.	No ordinary local non-Abelian tensor action supplies a comparable direct calculation.	Large-N bulk predictions are conditional and asymptotic; protected data have a higher evidential ceiling than generic correlators.

Adversarial test: the missing six-dimensional Lagrangian

Consider the following tempting argument:

1. postulate a local non-Abelian two-form field $B_{\mu\nu}$ with self-dual field strength;
2. write a conventional Lorentz-covariant six-dimensional action with manifest $(2,0)$ supersymmetry;
3. quantize that action and compute a generic finite-$N$ correlator;
4. compare the correlator with an AdS$_7$ loop expansion.

The argument fails at step 2. No accepted action with all of those properties is known for the interacting theory. Self-duality already complicates an ordinary covariant action for one Abelian tensor, and the non-Abelian interacting generalization is not obtained by replacing derivatives with a familiar matrix-valued covariant derivative. Treating the desired action as if it existed would hide the decisive hypothesis inside notation.

Circle compactification gives a powerful but narrower statement. At energies well below the inverse circle radius, the theory reduces to five-dimensional maximally supersymmetric Yang–Mills. Instanton particles carry the charge expected of Kaluza–Klein momentum, suggesting that nonperturbative five-dimensional sectors remember the sixth dimension. Whether this relation alone supplies a complete definition at all energies is an additional proposal, not a conventional six-dimensional Lagrangian theorem Douglas 2011, §§ 1–2.2, pp. 1–9, PDF.

The strongest claim that survives is therefore substantial: the type-$A_{N-1}$ $(2,0)$ SCFT is a well-established interacting theory in standard physics usage, constrained by its superconformal algebra, self-dual-string lattice, tensor branch, exact anomalies, compactifications, and controlled M-theory realizations. What does not survive is a generic finite-$N$ correlator derived from an invented six-dimensional action. Such a correlator must instead be labeled as a protected result, a compactification result, a large-$N$ holographic prediction, or an open calculation according to the route that actually supports it.

Common pitfalls

Using one radius for both factors. The sphere and AdS radii differ by two, in opposite directions for the M2 and M5 products. State which radius appears before reducing the Einstein term.

Reading $N^{3/2}$ or $N^3$ as a literal field count. These powers describe the leading growth of specified normalized observables. They do not provide a basis-independent count of local microscopic fields.

Assuming large $N$ removes Kaluza–Klein modes. It suppresses curvature and quantum-gravity corrections, but the compact radii remain of order the AdS radius. A small-field truncation needs a consistency argument.

Calling every ABJM limit “M-theory.” At fixed $k$ and sufficiently large $N$, the eleven-dimensional frame is appropriate. A correlated large-$k$ limit instead produces a type-IIA window, while weak ’t Hooft coupling is a boundary perturbative regime.

Mistaking the absence of a string coupling for exactness. Eleven-dimensional M-theory has no independent $g_s$, but it does have higher-derivative and loop corrections controlled by Planck-length and Newton-coupling ratios. Large $N$ suppresses those corrections without erasing them.

Equating non-Lagrangian with nonexistent. The $(2,0)$ theory has extensive exact and structural data. The missing object is a conventional manifestly six-dimensional non-Abelian tensor action, not all meaningful definitions or observables.

Exercises

1. Reproduce both Newton normalizations

Starting from $G_{11}=16\pi^7\ell_p^9$, the two unit-sphere volumes, and the flux–radius relations, derive $L_4^2/G_4$ and $L_7^5/G_7$. Keep the M2 quotient factor $1/k$ in the compact volume.

Solution

For M2,

$$G_4 =\frac{16\pi^7\ell_p^9} {\pi^4R_7^7/(3k)} =\frac{48\pi^3k\ell_p^9}{R_7^7}.$$

Using $L_4=R_7/2$ gives

$$\frac{L_4^2}{G_4} =\frac{R_7^9}{192\pi^3k\ell_p^9}.$$

Since $R_7^6=32\pi^2Nk\ell_p^6$,

$$R_7^9 =(32\pi^2Nk)^{3/2}\ell_p^9 =128\sqrt2\,\pi^3(Nk)^{3/2}\ell_p^9,$$

and hence

$$\frac{L_4^2}{G_4} =\frac{2\sqrt2}{3}\sqrt{k}\,N^{3/2}.$$

For M5,

$$G_7 =\frac{16\pi^7\ell_p^9} {(8\pi^2/3)R_4^4} =\frac{6\pi^5\ell_p^9}{R_4^4}.$$

With $L_7=2R_4$ and $R_4^3=\pi N\ell_p^3$,

$$\frac{L_7^5}{G_7} =\frac{16R_4^9}{3\pi^5\ell_p^9} =\frac{16N^3}{3\pi^2}.$$

2. Diagnose the M2 duality frame

Show that the Hopf circle is large in Planck units precisely when $N/k^5$ is large. For a correlated limit $k\sim N^\alpha$ with $\alpha\geq0$, classify the eleven-dimensional, type-IIA supergravity, and weak-boundary-coupling regions. Then recover the leading fixed-$k$ ABJM sphere free energy from the gravitational normalization.

Solution

The quotient circle has radius $R_7/k$, so

$$\frac{R_7}{k\ell_p} =\frac{(32\pi^2Nk)^{1/6}}{k} =(32\pi^2)^{1/6} \left(\frac{N}{k^5}\right)^{1/6}.$$

It is parametrically larger than $\ell_p$ when $N\gg k^5$. If $k\sim N^\alpha$, this is the eleven-dimensional region $0\leq\alpha<1/5$. For $1/5<\alpha<1$, one has $k\ll N\ll k^5$, which is the type-IIA supergravity window. The endpoint $\alpha=1/5$ is a crossover where the quotient circle is only order $\ell_p$; at $\alpha=1$, $\lambda=N/k$ is order one and type-IIA curvature is not parametrically small. For $\alpha>1$, $\lambda\to0$, so the boundary theory is weakly coupled rather than described by classical type-IIA supergravity.

Fixed $k$ corresponds to $\alpha=0$. In that regime,

$$F_{S^3}^{\mathrm{grav}} =\frac{\pi}{2}\frac{L_4^2}{G_4} =\frac{\pi\sqrt{2k}}{3}N^{3/2},$$

which is the leading fixed-$k$ localization result. The calculation does not determine its $O(N^{1/2})$ and smaller corrections.

3. Remove the center-of-mass tensor

Starting from the anomaly of the full $N$-M5 stack, subtract the free center-of-mass tensor. Evaluate the result at $N=1$ and explain why the check says nothing about a generic unprotected correlator.

Solution

Subtracting one $I_8^{\mathrm{tens}}$ gives

$$I_8[A_{N-1}] =(N-1)I_8^{\mathrm{tens}} +\frac{N^3-N}{24}p_2(\mathcal N_\perp).$$

At $N=1$, both coefficients vanish, leaving no interacting $A_0$ sector. This checks the finite-$N$ bookkeeping and the free-sector subtraction. An anomaly is protected and records a symmetry obstruction; it does not determine the full operator spectrum or generic position-dependent correlators.

What the comparison establishes

The two throats share one reproducible chain:

$$\text{quantized brane charge} \longrightarrow \text{sphere and AdS radii} \longrightarrow G_{d+1} \longrightarrow \text{leading large-}N\text{ observable}.$$

For M2-branes, ABJM localization checks the full leading coefficient in a specified fixed-$k$ limit. For M5-branes, the exact anomaly polynomial establishes protected $N^3-N$ data while two-derivative gravity captures the leading cubic normalization. Neither comparison creates a Kaluza–Klein gap, removes approximation conditions, or supplies a conventional six-dimensional action.

Continue to D1-D5 systems and AdS$_3$ top-down data for a duality with a tractable two-dimensional CFT locus and a nontrivial moduli-space interpolation. For the methods used here, Volume X develops three-dimensional sphere localization and six-dimensional tensor-branch anomalies, while Volume XIV states the domain and failure conditions of supergravity as an EFT. Stringy and quantum corrections organizes the corrections omitted by the classical backgrounds. Chapter 5 then compares objects, target sectors, evidence, and falsifiers for nonperturbative definitions.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Aharony, O., Bergman, O., Jafferis, D. L., and Maldacena, J. (2008), “$\mathcal N=6$ Superconformal Chern–Simons-Matter Theories, M2-Branes and Their Gravity Duals,” Journal of High Energy Physics 2008(10), 091. DOI. Open PDF.
• Bergman, O., and Hirano, S. (2009), “Anomalous Radius Shift in AdS$_4$/CFT$_3$,” Journal of High Energy Physics 2009(07), 016. DOI. Open PDF.
• Douglas, M. R. (2011), “On D=5 Super Yang–Mills Theory and (2,0) Theory,” Journal of High Energy Physics 2011(2), 011. DOI. Open PDF.
• Günaydin, M., and Takemae, S. (2000), “Unitary Supermultiplets of $OSp(8^*|4)$ and the AdS$_7$/CFT$_6$ Duality,” Nuclear Physics B 578(1–2), 405–448; erratum 697(1–2), 399–402 (2004). DOI. Open PDF.
• Harvey, J. A., Minasian, R., and Moore, G. (1998), “Non-Abelian Tensor-Multiplet Anomalies,” Journal of High Energy Physics 1998(9), 004. DOI. Open PDF.
• Kapustin, A., Willett, B., and Yaakov, I. (2010), “Exact Results for Wilson Loops in Superconformal Chern–Simons Theories with Matter,” Journal of High Energy Physics 2010(3), 089. DOI. Open PDF.
• Maldacena, J. M. (1998), “The Large $N$ Limit of Superconformal Field Theories and Supergravity,” Advances in Theoretical and Mathematical Physics 2, 231–252. DOI. Open PDF.
• Mariño, M., and Putrov, P. (2012), “ABJM Theory as a Fermi Gas,” Journal of Statistical Mechanics: Theory and Experiment 2012, P03001. DOI. Open PDF.
• Ohmori, K., Shimizu, H., Tachikawa, Y., and Yonekura, K. (2014), “Anomaly Polynomial of General 6D SCFTs,” Progress of Theoretical and Experimental Physics 2014, 103B07. DOI. Open PDF.