The CP(N-1) Model

The $\mathrm{CP}^{N-1}$ model turns a constrained complex vector into a projective field by identifying its local phase. That redundancy produces a composite $U(1)$ connection, quantized flux on suitable closed Euclidean spacetimes, and a theta angle. At large $N$ the connection and the constraint multiplier become useful independent integration variables, but their saddle parameters must still be translated into gauge-invariant observables. Claims about the phase at $\theta=\pi$ depend on $N$, global structure, boundary conditions, and the controlled regime.

Required background. Sigma-model target geometry and control supplies the quotient construction, while gauge fields, redundancy, and observables distinguishes a redundant phase from a physical symmetry. Helpful background. Theta terms and vacuum sectors fixes the general periodicity question, and large-N normalizations fixes the order of limits.

Projective field and emergent connection

Let

$$z(x)\in\mathbb C^N,\qquad z^\dagger z=1,$$

with the local equivalence

$$z(x)\sim e^{i\alpha(x)}z(x).$$

The physical field is the rank-one projector

$$P=zz^\dagger,\qquad P^2=P,\qquad \operatorname{tr}P=1,$$

which is invariant under the redundant phase. Define

$$A_\mu=-i\,z^\dagger\partial_\mu z, \qquad D_\mu z=(\partial_\mu-iA_\mu)z .$$

Then $A_\mu\mapsto A_\mu+\partial_\mu\alpha$ and $D_\mu z\mapsto e^{i\alpha}D_\mu z$. The two-derivative Euclidean action is

$$S_E =\frac{1}{g_0^2} \int\mathrm d^2x\, (D_\mu z)^\dagger D_\mu z +i\theta Q .$$

The overall factor is a convention; here it is chosen so that the large-$N$ gap equation below has the same integral normalization as the chapter’s $O(N)$ example. The faithful continuous global symmetry acting on local gauge-invariant fields is $PSU(N)=SU(N)/\mathbb Z_N$, not an extra copy of the redundant $U(1)$.

Varying the action with respect to an independent algebraic $A_\mu$ gives

$$A_\mu=-i\,z^\dagger\partial_\mu z,$$

so the gauge-like presentation is exactly equivalent to the projective kinetic term at this classical level. Quantum mechanically it is often more useful to integrate over $A_\mu$ and impose the constraint with a multiplier. A Maxwell term is then generated by matter fluctuations even though none was present in the original two-derivative action.

Flux and topological charge

The curvature of the composite connection is

$$F_{\mu\nu} =\partial_\mu A_\nu-\partial_\nu A_\mu =-i\left[ (\partial_\mu z)^\dagger\partial_\nu z -(\partial_\nu z)^\dagger\partial_\mu z \right].$$

On an oriented closed Euclidean two-manifold, when the projective field defines a smooth line bundle and the gauge transformations have the standard $U(1)$ periods,

$$Q =\frac{1}{2\pi}\int F \in\mathbb Z.$$

Therefore $e^{-i\theta Q}$ in the Euclidean path-integral weight is $2\pi$-periodic in $\theta$. On a manifold with boundary, the bulk integral need not be an integer by itself; boundary conditions, edge degrees of freedom, or a boundary counterterm must be specified. Twisted bundles can also fractionalize the charge carried by individual semiclassical events while preserving the periodicity of the full partition function.

For $N=2$, $\mathrm{CP}^1\simeq S^2$, and the charge agrees with the degree of the corresponding $O(3)$ sigma-model map after matching orientations and action normalizations. This special equivalence should not be extrapolated to arbitrary $N$.

Large-N auxiliary saddle

Set $t_0=Ng_0^2$ and rescale $w=z/g_0$, so $w^\dagger w=N/t_0$. A useful regulated action is

$$S_E[w,A,\lambda] =\int\mathrm d^2x\, \left[ \lvert D_\mu w\rvert^2 +\lambda\left( w^\dagger w-\frac{N}{t_0} \right) \right] +i\theta Q .$$

Choosing the multiplier contour through a real positive saddle and integrating the $N$ complex components gives, at $\theta=0$,

$$\frac{S_{\mathrm{eff}}[A,\lambda]}{N} =\operatorname{Tr}\ln(-D^2+\lambda) -\frac{1}{t_0}\int\mathrm d^2x\,\lambda .$$

The translation-invariant saddle $A_\mu=0$, $\lambda=m^2>0$ obeys

$$\frac{1}{t_0} =\int_{\lvert p\rvert<\Lambda} \frac{\mathrm d^2p}{(2\pi)^2} \frac{1}{p^2+m^2} =\frac{1}{4\pi} \ln\!\left(1+\frac{\Lambda^2}{m^2}\right).$$

Thus $m\sim\Lambda e^{-2\pi/t_0}$ at weak bare coupling. This is a controlled leading large-$N$ saddle and a dimensional-transmutation scale Witten 1979, §§ 2–3.

It is not automatically the pole mass of a physical $z$ particle. The field $z$ is gauge-charged under the redundant $U(1)$ and cannot create an isolated gauge-invariant asymptotic state. The saddle controls the inverse range in the charged propagator after gauge fixing and the threshold entering the induced effective action. Physical masses must be extracted from gauge-invariant operators such as

$$\mathcal O^A=z^\dagger T^A z$$

or from finite-volume energy levels. At leading large $N$, integrating out $w$ also generates a Maxwell term for $A_\mu$; its long-distance dynamics and the constraint field reorganize the gauge-invariant spectrum.

Dual variables and their conditions

The figure collects three transformations used in this chapter. None is an unconditional identification of elementary particles: every arrow carries a constraint, measure, anomaly, or boundary condition, and the round trip is checked on observables.

Dual and auxiliary variables expose otherwise hidden dynamics, but they do not erase the definition of the theory. In $\mathrm{CP}^{N-1}$ the $U(1)$ quotient and bundle data accompany the connection; in Gross–Neveu the Hubbard–Stratonovich field is fixed by its equation of motion and determinant; in the Schwinger model the current normalization, anomaly, flux sector, and external-probe definition accompany bosonization. The diagram is schematic rather than a spectrum plot.

The model’s position relative to the other strong-coupling examples is summarized by the laboratory map and its regime table.

Theta claims require a regime

Three statements are often conflated:

• Exact kinematics: integer $Q$ under the stated global conditions implies $2\pi$ periodicity of the partition function.
• Large N: the vacuum energy is organized into branches whose envelope restores $2\pi$ periodicity; a branch crossing can occur at $\theta=\pi$.
• Semiclassics: on a specified weakly coupled compactification, dilute integer or fractional events can generate a calculable theta-dependent potential.

Agreement of periodicity and anomalies is a cross-check, but the functional form of the vacuum energy and the infrared phase are dynamical. In particular, the familiar gapless $\theta=\pi$ behavior of $\mathrm{CP}^1\simeq O(3)$ is not a universal result for every $\mathrm{CP}^{N-1}$ model. The detailed separation is developed in Theta Terms and Topological Effects in Sigma Models.

Common pitfalls

Treating A as a new microscopic photon. In the minimal projective action, $A_\mu$ begins as an algebraic composite or auxiliary variable. Its kinetic term and range are induced and depend on the fields that have been integrated out.

Calling z a physical particle. The local phase is redundant, so an isolated $z$ operator is not gauge invariant. A physical mass statement must name a gauge-invariant correlator or a state in a properly dressed Hilbert space.

Assuming integer charge without global conditions. The flux integral is integer on a suitable closed manifold with the standard line-bundle quantization. Boundaries, twists, and quotient backgrounds change the intermediate description and must be included explicitly.

Exercises

1. Show that eliminating the algebraic connection from $\lvert D_\mu z\rvert^2$ gives

$$(D_\mu z)^\dagger D_\mu z =\partial_\mu z^\dagger\partial_\mu z -\lvert z^\dagger\partial_\mu z\rvert^2.$$

Solution

Expanding with $z^\dagger z=1$ gives

$$\lvert D_\mu z\rvert^2 =\partial_\mu z^\dagger\partial_\mu z +iA_\mu\left( z^\dagger\partial_\mu z -\partial_\mu z^\dagger z \right) +A_\mu^2.$$

Since $\partial_\mu z^\dagger z=-z^\dagger\partial_\mu z$, stationarity gives $A_\mu=-i z^\dagger\partial_\mu z$. Substitution yields the stated projector onto directions orthogonal to the redundant phase.

2. Cover $S^2$ with north and south patches and suppose $A_N-A_S=\mathrm d\alpha$ on the equator. Show that $Q$ equals the winding of $e^{i\alpha}$.

Solution

Stokes’ theorem on the two patches gives

$$\int_{S^2}F =\oint_{\mathrm{equator}}(A_N-A_S) =\oint_{\mathrm{equator}}\mathrm d\alpha .$$

Single-valuedness of the transition function $e^{i\alpha}$ implies $\oint\mathrm d\alpha=2\pi k$ with $k\in\mathbb Z$. Hence $Q=k$. The argument also shows exactly which bundle and closed-manifold assumptions produce integrality.

Continue

Use Gap Equations, Dimensional Transmutation, and Physical Mass to distinguish the saddle scale from a gauge-invariant pole, then Theta Terms and Topological Effects in Sigma Models to compare large-$N$, semiclassical, and anomaly information.

References

• Gaiotto, Davide, Anton Kapustin, Zohar Komargodski, and Nathan Seiberg. “Theta, Time Reversal, and Temperature.” Journal of High Energy Physics 2017, no. 5 (2017): 091. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, §§ 6.3–6.4. DOI.
• Witten, Edward. “Instantons, the Quark Model, and the 1/N Expansion.” Nuclear Physics B 149 (1979): 285–320. DOI.