Baryon Vertices and Wrapped Branes

The main idea

A single heavy external quark in the fundamental representation is represented holographically by a fundamental string ending on the AdS boundary. A baryon is different. In an $SU(N)$ gauge theory, a color-singlet baryonic source is not made from one fundamental index; it is made by antisymmetrizing $N$ fundamental indices:

$$B \sim \epsilon_{i_1\cdots i_N} q^{i_1} q^{i_2}\cdots q^{i_N}.$$

The bulk dual must therefore contain an object on which $N$ fundamental strings can end. In the canonical $\mathrm{AdS}_5\times S^5$ duality, that object is a D5-brane wrapped on the entire $S^5$ and localized as a particle in $\mathrm{AdS}_5$. This wrapped D5-brane is called the baryon vertex.

The most important point is not its shape but its charge. The background carries $N$ units of Ramond-Ramond five-form flux through $S^5$:

$$\int_{S^5} F_5 \propto N.$$

Because of a Wess-Zumino coupling on the D5-brane worldvolume, this flux induces $N$ units of electric charge for the D5 worldvolume gauge field. A fundamental string endpoint carries one unit of the opposite charge. A wrapped D5-brane in $\mathrm{AdS}_5\times S^5$ therefore requires $N$ string endpoints. This is the bulk version of the statement that $N$ fundamentals can be contracted with $\epsilon_{i_1\cdots i_N}$ to form an $SU(N)$ singlet.

The baryon vertex in $\mathrm{AdS}_5\times S^5$. The D5-brane wraps the internal $S^5$ and is pointlike in $\mathrm{AdS}_5$. The five-form flux induces $N$ units of worldvolume electric charge, so Gauss law requires $N$ fundamental strings to end on the wrapped brane.

This page explains the baryon vertex in three increasingly general languages: the $SU(N)$ color-singlet structure in the boundary theory, the wrapped-brane Gauss law in the bulk, and the broader role of wrapped branes as heavy nonlocal or determinant-like observables in holography.

The boundary baryon as an antisymmetric color singlet

For a theory with dynamical fundamental fields $q^i$, an ordinary baryonic operator has the schematic form

$$B(x) = \epsilon_{i_1\cdots i_N} q^{i_1}(x)q^{i_2}(x)\cdots q^{i_N}(x),$$

with spin, flavor, and derivative structure suppressed. Under $q^i\to U^i{}_j q^j$ with $U\in SU(N)$,

$$\epsilon_{i_1\cdots i_N} U^{i_1}{}_{j_1}\cdots U^{i_N}{}_{j_N} = (\det U)\epsilon_{j_1\cdots j_N} = \epsilon_{j_1\cdots j_N},$$

so the operator is gauge invariant. The number $N$ is not a dynamical accident. It is the rank of the gauge group and the number of fundamental indices needed to make a singlet.

In the canonical $\mathcal N=4$ $SU(N)$ theory, however, the elementary dynamical fields are in the adjoint representation. There are no dynamical quarks $q^i$ in the fundamental representation unless one adds flavor branes or external probes. Therefore the simplest baryonic object in the original $\mathrm{AdS}_5/\mathrm{CFT}_4$ example is better thought of as a baryonic Wilson-line configuration: $N$ heavy external fundamental sources, with their color indices contracted by an epsilon tensor.

A useful schematic expression is

$$W_B(C_1,\ldots,C_N) = \frac{1}{N!} \epsilon_{i_1\cdots i_N} \epsilon^{j_1\cdots j_N} \prod_{a=1}^N U(C_a)^{i_a}{}_{j_a},$$

where $U(C_a)$ is a fundamental Wilson line along the curve $C_a$. In $\mathcal N=4$ SYM, the supersymmetric line operator usually uses the Maldacena-Wilson connection

$$U(C) = P\exp\int_C ds \left( i A_\mu \dot x^\mu + |\dot x|\, n^I\Phi_I \right),$$

where $n^I$ specifies a direction on $S^5$. The color structure is the point: $N$ fundamental lines can join into a singlet, just as $N$ strings can join on the D5 baryon vertex.

There is a small but important subtlety. If all curves are literally identical and the gauge group is exactly $SU(N)$, then the determinant of a single fundamental Wilson line is trivial. The physically interesting baryonic Wilson configurations have separated external quark worldlines, or a junction structure at initial and final times, so that the epsilon tensor supplies a nontrivial color-singlet boundary condition.

The wrapped D5-brane baryon vertex

The canonical background is

$$\mathrm{AdS}_5\times S^5,$$

with $N$ units of self-dual five-form flux. The baryon vertex is a D5-brane with worldvolume

$$\mathbb R_t \times S^5.$$

It wraps the whole internal sphere and sits at some radial position and spatial location in $\mathrm{AdS}_5$. From the $\mathrm{AdS}_5$ viewpoint, it is a heavy charged particle. The fundamental strings stretching from the boundary to the D5-brane represent the $N$ heavy external quarks.

The D5-brane action is of Dirac-Born-Infeld plus Wess-Zumino form,

$$S_{D5} = -T_5\int d^6\xi\, e^{-\Phi} \sqrt{-\det(P[G]+2\pi\alpha' F)} + \mu_5\int P\!\left[\sum_q C_q\right] e^{2\pi\alpha' F+B}.$$

For the baryon vertex, the decisive term is the coupling between the worldvolume gauge field $A$ and the background five-form flux:

$$S_{\mathrm{WZ}} \supset \mu_5(2\pi\alpha') \int_{\mathbb R_t\times S^5} A\wedge F_5,$$

up to an orientation-dependent sign. Flux quantization gives

$$\mu_5(2\pi\alpha')\int_{S^5}F_5=N,$$

so the wrapped D5 action contains the effective coupling

$$S_{\mathrm{WZ}} \supset N\int dt\, A_t.$$

This is a tadpole for the worldvolume electric field. Since the D5-brane wraps a compact $S^5$, Gauss law does not allow an uncancelled net electric charge. A fundamental string endpoint on a D-brane is electrically charged under the brane’s worldvolume $U(1)$ gauge field. Therefore $N$ fundamental strings must end on the D5-brane to cancel the charge induced by the background flux.

This is the cleanest derivation of the baryon vertex. The bulk statement

$$\text{D5 on } S^5 + N \text{ string endpoints}$$

is the holographic image of the boundary statement

$$\epsilon_{i_1\cdots i_N} \times N \text{ fundamental color indices}.$$

Why the number of strings is exactly $N$

The integer $N$ appears three times, in three different languages:

Language	Meaning of $N$
Gauge theory	rank of $SU(N)$ and number of indices in $\epsilon_{i_1\cdots i_N}$
String background	number of D3-branes and five-form flux units through $S^5$
D5 worldvolume	induced electric charge that must be cancelled by string endpoints

The equality of these three meanings is one of the satisfying early checks of the duality. The wrapped D5-brane does not merely look like a baryon vertex; its worldvolume Gauss law enforces the same combinatorics as the boundary color tensor.

One sometimes hears the phrase “a baryon is made of $N$ strings.” That is close but slightly too loose. The more precise statement is:

A baryonic Wilson configuration in $SU(N)$ holography is represented by $N$ fundamental strings whose endpoints on a wrapped D5-brane cancel the electric tadpole induced by $N$ units of RR five-form flux.

This phrasing keeps both sides honest. The boundary baryon is a gauge-invariant color contraction. The bulk baryon vertex is a wrapped brane with a quantized worldvolume charge constraint.

Energetics and large-$N$ scaling

The wrapped D5-brane is heavy. This is exactly what one expects: a baryon in a large-$N$ gauge theory has mass of order $N$ in the usual large-$N$ counting.

To see the scaling, use Poincaré coordinates for $\mathrm{AdS}_5$,

$$ds^2 = \frac{L^2}{z^2} \left(-dt^2+d\vec x^2+dz^2\right) +L^2 d\Omega_5^2.$$

A static D5-brane wrapped on $S^5$ at radial position $z=z_v$ has a redshifted DBI energy of order

$$E_{D5}(z_v) \sim T_5 e^{-\Phi} L^5 \mathrm{Vol}(S^5)\frac{L}{z_v}.$$

Using

$$T_5=\frac{1}{(2\pi)^5 g_s\alpha'^3}, \qquad \mathrm{Vol}(S^5)=\pi^3, \qquad L^4=4\pi g_sN\alpha'^2, \qquad \lambda=4\pi g_sN,$$

one obtains, in these conventions,

$$E_{D5}(z_v) = \frac{N\sqrt\lambda}{8\pi}\frac{1}{z_v},$$

before including the detailed electric flux and string profile. The exact coefficient is less universal than the lesson:

$$E_{D5}\sim \frac{N\sqrt\lambda}{z_v}.$$

The $N$ strings also contribute an energy of order $N\sqrt\lambda$. The complete baryonic Wilson configuration is found by minimizing the total action of the wrapped D5-brane plus the attached strings, with force balance at the vertex. For quarks separated by a characteristic boundary distance $R$ in a conformal theory, dimensional analysis implies

$$E_B(R) \propto -\frac{N\sqrt\lambda}{R},$$

with a numerical coefficient depending on the geometry of the $N$ quark positions and on the precise saddle. The factor $N$ is robust; it is the bulk reflection of large-$N$ baryon physics.

This differs from the quark-antiquark potential computed from one U-shaped string,

$$V_{q\bar q}(R) \propto -\frac{\sqrt\lambda}{R}.$$

A mesonic source is order one in the number of fundamental strings. A baryonic source is order $N$.

The force-balance picture

A useful classical picture is this:

1. $N$ string endpoints are fixed on the boundary at the positions of external quarks.
2. The strings descend into AdS and meet at a D5-brane wrapped on $S^5$.
3. The D5-brane can move in the AdS radial direction.
4. The stable saddle, when it exists, satisfies force balance between the string tensions and the effective weight of the wrapped D5-brane.

The force balance is not merely Newtonian mechanics drawn in curved space. It follows from varying the combined action

$$S_{\mathrm{total}} = \sum_{a=1}^N S_{F1}^{(a)}+S_{D5},$$

subject to the worldvolume Gauss-law constraint. At the string endpoints on the D5-brane, the boundary variation of the Nambu-Goto action must be balanced by the variation of the D5 action and its electric flux.

For many purposes one does not need the full solution. The qualitative lessons are already powerful:

• baryons are heavy objects with energy of order $N$;
• the baryon vertex is forced by flux quantization;
• baryonic observables are sensitive to wrapped branes, not only to supergravity fields;
• in finite-temperature or confining geometries, the position of the vertex and the fate of the attached strings diagnose screening, confinement, and baryon stability.

Baryon vertices versus baryonic operators

There are two related but distinct uses of the word “baryon” in holography.

External baryonic Wilson configurations

The D5 baryon vertex in $\mathrm{AdS}_5\times S^5$ represents $N$ external fundamental charges. This is the object discussed above. It is best viewed as a baryonic Wilson-line configuration, especially in pure $\mathcal N=4$ SYM where there are no dynamical fundamental quarks.

Determinant-like baryonic operators

In more general AdS$_5$/CFT$_4$ examples, especially quiver gauge theories dual to $\mathrm{AdS}_5\times X_5$, the internal space $X_5$ can have nontrivial cycles. D3-branes wrapped on three-cycles $\Sigma_3\subset X_5$ can represent determinant-like baryonic operators. Their dimensions scale like $N$ because the wrapped D3-brane mass is proportional to $1/g_s$.

For unit-radius internal metrics, the standard estimate is

$$\Delta_{\Sigma_3} = T_3 L^4 \mathrm{Vol}(\Sigma_3) = \frac{\pi N}{2} \frac{\mathrm{Vol}(\Sigma_3)}{\mathrm{Vol}(X_5)}.$$

The precise operator depends on the quiver theory and on the homology class of $\Sigma_3$. The important lesson is that wrapped branes are the natural bulk duals of heavy operators whose dimensions scale as $N$.

In $S^5$ itself, $H_3(S^5)=0$, so there are no topologically stable D3-branes wrapped on nontrivial three-cycles. Determinant and subdeterminant operators built from adjoint scalars in $\mathcal N=4$ SYM are instead related to giant gravitons and other D3-brane configurations. They are important, but they are not the same object as the D5 baryon vertex.

Wrapped branes as a general dictionary entry

The pages so far have mostly discussed two kinds of bulk objects:

• supergravity fields, dual to single-trace local operators;
• fundamental strings, dual to Wilson-line observables.

Wrapped branes add a third major class. They are extended objects in ten or eleven dimensions which may become particles, strings, domain walls, or defects in the noncompact AdS spacetime after wrapping internal cycles.

Bulk object	Boundary interpretation	Typical scaling
fundamental string ending on the boundary	Wilson line in the fundamental representation	exponent $\sim \sqrt\lambda$
D5 on $S^5$ with $N$ strings	baryonic Wilson vertex for $N$ external quarks	energy $\sim N\sqrt\lambda$
D3 on $\Sigma_3\subset X_5$	determinant-like baryonic operator	dimension $\sim N$
wrapped brane extending along an AdS subspace	defect, interface, or extended operator	depends on brane and embedding
brane with worldvolume electric flux	Wilson loops in higher representations or bound-state probes	often $\sim N$ at fixed $k/N$

This table is deliberately schematic. The detailed dictionary depends on the internal geometry, fluxes, supersymmetry, and boundary conditions. But the organizing principle is stable: internal cycles and background fluxes turn branes into charged objects in AdS, and those charges become global, topological, or representation-theoretic data in the CFT.

Center charge and $N$-ality

The baryon vertex is also a geometric avatar of $N$-ality. A single fundamental Wilson line transforms nontrivially under the center of $SU(N)$, while a product of $N$ fundamentals is center-neutral:

$$\left(e^{2\pi i/N}\right)^N=1.$$

The D5 vertex implements the same rule dynamically. A single F-string cannot simply end in empty AdS; it must end on the boundary, on a suitable brane, on a horizon, or on another object carrying the right charge. A wrapped D5 in $\mathrm{AdS}_5\times S^5$ can absorb exactly $N$ units of fundamental-string endpoint charge because the background five-form flux induces exactly $N$ units of worldvolume electric charge.

This does not mean that every configuration with fewer than $N$ strings is impossible in every background. Horizons can absorb strings, flavor branes can carry endpoint charges, and D-branes with worldvolume flux can describe higher-representation Wilson loops. The narrower statement is that the standalone baryon vertex in the vacuum $\mathrm{AdS}_5\times S^5$ background is the object that makes $N$ external fundamental charges into a singlet.

Baryons in confining and finite-temperature geometries

In a conformal theory, no intrinsic mass scale exists, so the baryonic potential for external quarks separated by $R$ must scale like $1/R$. In confining holographic backgrounds, the story changes. Strings can lie along an IR wall or cap, producing linear potentials. A baryon vertex may sit near the IR region, with $N$ strings stretching to boundary quarks. The resulting baryon energy can contain a term proportional to the total length of confining strings, closer to the qualitative picture of a baryon in a confining gauge theory.

At finite temperature, black-brane horizons introduce screening. Strings can fall into the horizon, and sufficiently large baryonic configurations may cease to exist as connected saddles. The vertex may be pulled toward the horizon, and the preferred configuration can transition from a connected baryon to screened quark-like pieces. This is the baryonic analogue of the connected-to-disconnected transition for the quark-antiquark Wilson loop.

These applications are useful, but one should keep the hierarchy of claims clear:

• the existence of the D5 vertex and the requirement of $N$ strings is a topological/flux statement;
• the detailed baryon potential is a dynamical saddle calculation;
• phenomenological baryons in QCD-like bottom-up models are model-dependent.

Common mistakes

Mistake 1: Treating the baryon vertex as a dynamical quark

The wrapped D5-brane is not one of the quarks. It is the junction that contracts $N$ fundamental color indices. The quarks are represented by the $N$ string endpoints on the boundary.

Mistake 2: Forgetting the Wess-Zumino term

The DBI volume term tells you that a wrapped D5-brane is heavy. It does not explain why $N$ strings must end on it. The quantized number of string endpoints comes from the Wess-Zumino coupling to $F_5$.

Mistake 3: Calling every wrapped brane a baryon vertex

A D5 on $S^5$ with $N$ attached F-strings is a baryon vertex. A D3 wrapped on a nontrivial three-cycle may be dual to a determinant-like baryonic operator. A D-brane with electric flux may describe a higher-representation Wilson loop. These are related members of the wrapped-brane dictionary, but they are not interchangeable.

Mistake 4: Importing too much QCD intuition into $\mathcal N=4$ SYM

The canonical baryon vertex lives in a conformal, supersymmetric, adjoint-matter theory with external fundamental probes. It captures large-$N$ color-singlet structure beautifully, but it is not automatically a realistic proton.

Mistake 5: Ignoring gauge group subtleties

The epsilon tensor is an invariant tensor of $SU(N)$. In a $U(N)$ theory, a product of $N$ fundamentals is not neutral under the overall $U(1)$ unless additional ingredients are included or the decoupled center-of-mass $U(1)$ is treated appropriately. In the AdS/CFT limit, the interacting theory is usually discussed as $SU(N)$ or $PSU(N)$ data with subtleties in line operators and one-form symmetries.

Exercises

Exercise 1: The epsilon tensor and gauge invariance

Let $q^i$ transform in the fundamental representation of $SU(N)$. Show that

$$B=\epsilon_{i_1\cdots i_N}q^{i_1}\cdots q^{i_N}$$

is gauge invariant.

Solution

Under $q^i\to U^i{}_j q^j$,

$$B\to \epsilon_{i_1\cdots i_N} U^{i_1}{}_{j_1}\cdots U^{i_N}{}_{j_N} q^{j_1}\cdots q^{j_N}.$$

The determinant identity gives

$$\epsilon_{i_1\cdots i_N} U^{i_1}{}_{j_1}\cdots U^{i_N}{}_{j_N} = (\det U)\epsilon_{j_1\cdots j_N}.$$

For $U\in SU(N)$, $\det U=1$, hence

$$B\to \epsilon_{j_1\cdots j_N}q^{j_1}\cdots q^{j_N}=B.$$

This is the boundary reason that a baryon contains exactly $N$ fundamentals.

Exercise 2: The D5 worldvolume charge

Use

$$\mu_5=\frac{1}{(2\pi)^5\alpha'^3}, \qquad \int_{S^5}F_5=(2\pi)^4\alpha'^2N,$$

and the Wess-Zumino term

$$S_{\mathrm{WZ}} \supset \mu_5(2\pi\alpha') \int A\wedge F_5$$

to show that the wrapped D5-brane carries $N$ units of electric charge under its worldvolume gauge field.

Solution

Integrating the Wess-Zumino coupling over the internal $S^5$ gives

$$S_{\mathrm{WZ}} \supset \mu_5(2\pi\alpha') \left(\int_{S^5}F_5\right) \int dt\, A_t.$$

Substituting the normalizations,

$$\mu_5(2\pi\alpha')\int_{S^5}F_5 = \frac{1}{(2\pi)^5\alpha'^3} (2\pi\alpha') (2\pi)^4\alpha'^2N =N.$$

Thus

$$S_{\mathrm{WZ}} \supset N\int dt\, A_t.$$

The D5 worldvolume has an induced electric charge $N$. Since a fundamental string endpoint carries one unit of worldvolume electric charge, Gauss law requires $N$ string endpoints with the opposite orientation.

Exercise 3: Scaling of the wrapped D5 energy

In the Poincaré metric

$$ds^2 = \frac{L^2}{z^2}(-dt^2+d\vec x^2+dz^2)+L^2d\Omega_5^2,$$

estimate the DBI energy of a D5-brane wrapped on $S^5$ at $z=z_v$. Use

$$T_5=\frac{1}{(2\pi)^5g_s\alpha'^3}, \qquad \mathrm{Vol}(S^5)=\pi^3, \qquad L^4=4\pi g_sN\alpha'^2, \qquad \lambda=4\pi g_sN.$$

Solution

The wrapped D5-brane is pointlike in AdS and wraps the internal $S^5$. The redshift from the time direction contributes a factor $L/z_v$, while the internal volume contributes $L^5\mathrm{Vol}(S^5)$. Thus

$$E_{D5} = T_5 L^5\mathrm{Vol}(S^5)\frac{L}{z_v} = \frac{1}{(2\pi)^5g_s\alpha'^3} \pi^3L^6\frac{1}{z_v}.$$

Therefore

$$E_{D5} = \frac{L^6}{32\pi^2 g_s\alpha'^3}\frac{1}{z_v}.$$

Now

$$\frac{L^6}{g_s\alpha'^3} = \frac{L^4}{\alpha'^2}\frac{L^2}{\alpha'}\frac{1}{g_s} = \lambda\sqrt\lambda\frac{1}{g_s}.$$

Since $g_s=\lambda/(4\pi N)$,

$$\frac{L^6}{g_s\alpha'^3}=4\pi N\sqrt\lambda.$$

Hence

$$E_{D5} = \frac{N\sqrt\lambda}{8\pi}\frac{1}{z_v}.$$

The exact coefficient can be modified by worldvolume electric flux and by the full force-balanced embedding, but the scaling

$$E_{D5}\sim \frac{N\sqrt\lambda}{z_v}$$

is the key result.

Exercise 4: Dimension of a wrapped D3-brane baryonic operator

Consider type IIB string theory on $\mathrm{AdS}_5\times X_5$, where $X_5$ is an Einstein space with unit-radius volume $\mathrm{Vol}(X_5)$. A D3-brane wraps a three-cycle $\Sigma_3\subset X_5$ and is pointlike in $\mathrm{AdS}_5$. Assuming

$$L^4=\frac{4\pi^4 g_sN\alpha'^2}{\mathrm{Vol}(X_5)}, \qquad T_3=\frac{1}{(2\pi)^3g_s\alpha'^2},$$

show that its conformal dimension is

$$\Delta_{\Sigma_3} = \frac{\pi N}{2} \frac{\mathrm{Vol}(\Sigma_3)}{\mathrm{Vol}(X_5)}.$$

Solution

A particle of mass $m$ in global AdS has leading large-mass dimension

$$\Delta\simeq mL.$$

For a D3-brane wrapped on $\Sigma_3$,

$$m=T_3 L^3\mathrm{Vol}(\Sigma_3),$$

where the cycle volume is measured in the unit-radius metric on $X_5$. Thus

$$\Delta=mL=T_3L^4\mathrm{Vol}(\Sigma_3).$$

Substituting the given expressions,

$$\Delta = \frac{1}{(2\pi)^3g_s\alpha'^2} \frac{4\pi^4 g_sN\alpha'^2}{\mathrm{Vol}(X_5)} \mathrm{Vol}(\Sigma_3).$$

Simplifying,

$$\Delta = \frac{4\pi^4}{8\pi^3} N \frac{\mathrm{Vol}(\Sigma_3)}{\mathrm{Vol}(X_5)} = \frac{\pi N}{2} \frac{\mathrm{Vol}(\Sigma_3)}{\mathrm{Vol}(X_5)}.$$

The dimension scales as $N$, as expected for determinant-like baryonic operators.

Exercise 5: Why fewer than $N$ strings are not a standalone baryon

Explain why a wrapped D5-brane in vacuum $\mathrm{AdS}_5\times S^5$ cannot be the endpoint of only $k<N$ fundamental strings if no other charged object is present.

Solution

The Wess-Zumino coupling induces $N$ units of worldvolume electric charge on the D5-brane. A fundamental string endpoint supplies one unit of the opposite charge. If only $k$ strings end on the D5-brane, the net worldvolume charge is proportional to $N-k$, which violates Gauss law on the compact wrapped $S^5$ unless some other object supplies the missing charge.

On the boundary side, $k<N$ isolated fundamental indices cannot be contracted with $\epsilon_{i_1\cdots i_N}$ to form a pure $SU(N)$ baryon. More elaborate configurations can exist if additional ingredients are included, such as horizons, flavor branes, anti-strings, or branes carrying worldvolume flux. But the standalone baryon vertex in the vacuum canonical background requires $N$ strings.

Further reading

• Edward Witten, “Baryons And Branes In Anti de Sitter Space”.
• David Gross and Hirosi Ooguri, “Aspects of Large $N$ Gauge Theory Dynamics as Seen by String Theory”.
• A. Brandhuber, N. Itzhaki, J. Sonnenschein, and S. Yankielowicz, “Baryons from Supergravity”.
• Curtis Callan, Alberto Guijosa, and Konstantin Savvidy, “Baryons and String Creation from the Fivebrane Worldvolume Action”.
• Ofer Aharony, Steven Gubser, Juan Maldacena, Hirosi Ooguri, and Yaron Oz, “Large $N$ Field Theories, String Theory and Gravity”.

Takeaway

The baryon vertex is one of the sharpest places where the gauge-theory color algebra becomes literal geometry. The invariant tensor $\epsilon_{i_1\cdots i_N}$ becomes a wrapped D5-brane; the $N$ fundamental color indices become $N$ fundamental strings; and the statement that the configuration is a singlet becomes a worldvolume Gauss law enforced by $N$ units of five-form flux.