Finite-Dimensional and Equivariant Localization

The Atiyah–Bott–Berline–Vergne formula is a theorem about a compact group acting on a finite-dimensional manifold. It replaces the integral of an equivariantly closed form by integrals over the fixed components, weighted by inverse equivariant Euler classes. This is the clean mathematical model for supersymmetric localization. It is not, by itself, a theorem about an infinite-dimensional quantum-field-theory path integral.

Required background. Differential forms, integration, and Stokes’ theorem supplies integration of closed forms. Fredholm and Dirac index theorems supplies the later infinite-dimensional replacement for a normal-bundle Euler class. Lie groups, Lie algebras, and adjoint actions supplies torus actions and weights.

Helpful background. Topological and holomorphic twists motivates why an odd symmetry can act like an equivariant differential.

Equivariant forms in the Cartan model

Let a compact torus $T$ with Lie algebra $\mathfrak t$ act smoothly on a manifold $M$. Choose a basis $e_a$ of $\mathfrak t$, let $V_a$ be the corresponding generating vector fields, and let $u^a$ be the dual polynomial variables. The Cartan complex is

$$\Omega_T(M) =\bigl(S(\mathfrak t^*)\otimes\Omega(M)\bigr)^T, \qquad \deg u^a=2.$$

The total degree is twice the polynomial degree plus the ordinary form degree. In particular,

$$\deg\!\left((u^1)^{k_1}\cdots(u^r)^{k_r}\omega_p\right) =2\sum_a k_a+p.$$

With the sign convention used on this page,

$$d_T=d-u^a\iota_{V_a}.$$

Cartan’s identity gives

$$d_T^2=-u^a\mathcal L_{V_a}.$$

Therefore $d_T^2=0$ on the $T$-invariant complex. An element $\alpha(u)$ satisfying $d_T\alpha=0$ is equivariantly closed. When $M$ is oriented and either compact or the form has compact support, equivariant integration means taking the ordinary top-form component and integrating it over $M$:

$$\int_M:\Omega_T(M)\longrightarrow S(\mathfrak t^*).$$

It lowers total degree by $\dim M$. Rational functions of the $u^a$ enter only after localizing the coefficient ring.

For a Hamiltonian circle action, this convention is easy to test. If $V$ preserves a symplectic form $\omega$ and

$$d\mu=\iota_V\omega,$$

then

$$\Omega_T=\omega+u\mu, \qquad d_T\Omega_T=u\bigl(d\mu-\iota_V\omega\bigr)=0.$$

Authors who use $d_T=d+u\iota_V$ use the opposite sign for the moment-map term. Mixing those two conventions reverses fixed-point exponentials.

The fixed-component theorem

Atiyah–Bott–Berline–Vergne formula. Let $T$ act smoothly on a compact oriented manifold $M$ without boundary. Let

$$F=M^T=\coprod_i F_i$$

be its fixed set, and let $\iota_i:F_i\hookrightarrow M$ be the inclusions. Each $F_i$ is a smooth closed submanifold. Orient its normal bundle $N_i$ consistently with the orientations of $M$ and $F_i$. For every equivariantly closed $\alpha$, after inverting the nonzero torus weights,

$$\int_M\alpha =\sum_i\int_{F_i} \frac{\iota_i^*\alpha}{e_T(N_i)}.$$

The equality lives in the localized equivariant coefficient ring. The exponential of an equivariant two-form is interpreted coefficient by coefficient, or in the corresponding completed ring. The topological localization theorem and integration formula are given in Atiyah and Bott 1984, §3, especially Eq. (3.8); the differential-form version is developed in their §4. The independent fixed-zero formulation appears in Berline and Vergne 1983, Théorème 1.6.

At an isolated fixed point $p$ of a $2n$-dimensional manifold, the tangent representation splits into oriented real two-planes. Let their weights be $w_{p,1},\ldots,w_{p,n}\in\mathfrak t^*$. In the convention

$$e_T(T_pM)=\prod_{j=1}^n\frac{w_{p,j}}{2\pi},$$

the contribution is

$$\frac{\iota_p^*\alpha}{e_T(T_pM)} =(2\pi)^n \frac{\iota_p^*\alpha}{\prod_j w_{p,j}}.$$

The factors of $2\pi$ are conventional; the numerator and Euler class must use the same Chern–Weil normalization. A positive-dimensional fixed component remains an integral over that component. Localization does not turn it into an isolated point.

Why only the fixed set remains

The proof has two distinct ingredients: exactness away from the fixed set and the Thom isomorphism near it.

Choose a generic $\xi\in\mathfrak t$ whose vector field $V_\xi$ vanishes precisely on $F$. Average a metric over $T$, and on $M\setminus F$ define

$$\lambda=\frac{g(V_\xi,\,\cdot)}{\lVert V_\xi\rVert^2}.$$

Then $\iota_{V_\xi}\lambda=1$. For the evaluated Cartan differential $d_\xi=d-\iota_{V_\xi}$,

$$d_\xi\lambda=d\lambda-1.$$

Its ordinary-form-degree-zero part is $-1$, so it has a finite geometric-series inverse. Because $\lambda$ is invariant,

$$d_\xi\!\left(\frac{\lambda}{d_\xi\lambda}\right)=1 \qquad\text{on }M\setminus F.$$

Consequently every $d_\xi$-closed form is $d_\xi$ exact away from $F$. Remove small tubular neighborhoods of the $F_i$ and apply Stokes’ theorem. The new boundary terms are converted by the equivariant Thom isomorphism into

$$\iota_i^*\iota_{i*}(\beta)=e_T(N_i)\,\beta,$$

which explains the inverse Euler class. Compactness and the absence of an original boundary ensure that no additional surface term survives.

This proof also explains why every hypothesis matters. The fixed set of a smooth compact-torus action is already a smooth submanifold, and its normal representation has no trivial $T$-weight. If a denominator vanishes after substituting a special $\xi$, then $\xi$ is nongeneric: its one-parameter subgroup fixes more directions, so one must use the larger fixed set rather than delete the zero factor.

Worked example: rotation of the two-sphere

Give $S^2$ the orientation and area form

$$\omega=\sin\theta\,d\theta\wedge d\varphi, \qquad \int_{S^2}\omega=4\pi.$$

Let $U(1)$ rotate $\varphi$, so $V=\partial_\varphi$. For $\mu=\cos\theta$,

$$\iota_V\omega=-\sin\theta\,d\theta=d\mu.$$

Hence $\Omega_T=\omega+u\mu$ is equivariantly closed. Since $\omega^2=0$ on $S^2$, the ordinary two-form part of $e^{\Omega_T}$ is $e^{u\mu}\omega$, and direct integration gives

$$\begin{aligned} I(u) &=\int_{S^2}e^{\Omega_T}\\ &=\int_0^{2\pi}d\varphi\int_0^\pi e^{u\cos\theta}\sin\theta\,d\theta\\ &=4\pi\frac{\sinh u}{u}. \end{aligned}$$

The fixed points are the north and south poles. Their moment-map values are $+1$ and $-1$. With the orientation induced by $\omega$, their tangent weights are $+u$ and $-u$, so

$$\begin{aligned} I(u) &=\frac{e^u}{u/(2\pi)} +\frac{e^{-u}}{-u/(2\pi)}\\ &=4\pi\frac{\sinh u}{u}. \end{aligned}$$

This is the Duistermaat–Heckman specialization of the fixed-point formula Atiyah and Bott 1984, §§6–7. Each fixed-point term has a pole at $u=0$, but the complete sum does not. The cancellation is a useful check that both the fixed set and the orientations have been handled consistently.

Mathematical failure modes

The displayed theorem must be modified in several common situations:

• if $M$ is noncompact, a contribution from infinity may survive;
• if $M$ has a boundary, Stokes’ theorem produces an explicit boundary term;
• if $M$ is not orientable, the pushforward requires the appropriate orientation local system;
• if the space is singular or an orbifold, one needs a localization theorem for that category, including stabilizer factors;
• if one specializes the equivariant parameters onto a weight hyperplane, the fixed locus of the chosen subgroup enlarges;
• if $\alpha$ is not equivariantly closed, there is no cohomology class to localize.

These are changes to the theorem, not minor numerical corrections.

The exact boundary of the QFT analogy

The structural dictionary is useful:

Finite-dimensional theorem	Formal field-theory counterpart
compact manifold $M$	integration cycle in field space
Cartan differential $d_T$	supersymmetry combined with gauge BRST
fixed set $M^T$	BPS configurations in every topological sector
normal bundle $N_F$	gauge-fixed transverse fluctuation complex
equivariant Euler class	regularized one-loop superdeterminant
tangent orientation	measure orientation and determinant phase
compactness without boundary	convergence and vanishing field-space flux

Only the left column is covered by the theorem above. In QFT, field space is infinite dimensional and generally noncompact; the quotient by gauge transformations can be singular; reducible saddles have stabilizers; the Euler class becomes an infinite product; and the contour may be a middle-dimensional cycle in a complexified space. Gauge fixing, zero modes, phases, regulators, anomalies, and contour ends are extra data.

There is a rigorous finite-dimensional supermanifold analogue. Schwarz and Zaboronsky assume a compact supermanifold, a volume-preserving odd vector field, and a compact bosonic square; their localization theorem then concentrates a $Q$-invariant integral near the zero set Schwarz and Zaboronsky 1997, §3, Theorem 1. Those hypotheses are informative precisely because an ordinary continuum QFT path integral has not automatically satisfied them. Finite-dimensional localization supplies the architecture for the next page, not a universal proof of it.

Common pitfalls

Dividing by a zero weight. A zero normal weight signals a nongeneric equivariant parameter or an incorrectly chosen fixed component. Recompute the fixed set before writing an Euler denominator.

Dropping a positive-dimensional fixed component. The numerator and inverse Euler class must still be integrated over that component, including its zero modes.

Calling the QFT column a theorem. Every entry in that column requires a model-specific definition and analytic justification.

Exercises

1. Nilpotence on invariants. Verify $d_T^2=-u^a\mathcal L_{V_a}$.

Solution

Using $d^2=0$, $\iota_{V_a}\iota_{V_b}+\iota_{V_b}\iota_{V_a}=0$, and Cartan’s identity $d\iota_{V_a}+\iota_{V_a}d=\mathcal L_{V_a}$,

$$d_T^2 =-u^a\bigl(d\iota_{V_a}+\iota_{V_a}d\bigr) +u^au^b\iota_{V_a}\iota_{V_b} =-u^a\mathcal L_{V_a}.$$

The last term vanishes because $u^au^b$ is symmetric while the contractions anticommute. Hence $d_T^2=0$ on $T$-invariant forms.

2. The ordinary-area limit. Show that the fixed-point answer for $S^2$ has a finite $u\to0$ limit.

Solution

Expanding $\sinh u=u+u^3/6+\cdots$ gives

$$I(u)=4\pi\left(1+\frac{u^2}{6}+\cdots\right),$$

so $I(0)=4\pi=\int_{S^2}\omega$. The separate $1/u$ poles cancel only after adding both fixed points.

3. A special equivariant parameter. Suppose a $T^2$ normal representation has weights $w_1(u_1,u_2)=u_1$ and $w_2(u_1,u_2)=u_1-u_2$. What goes wrong at $u_1=u_2$?

Solution

The second evaluated weight vanishes, so the chosen one-parameter subgroup acts trivially in that normal direction. Its fixed set is therefore larger than the fixed set used in the isolated-point formula. One must localize with respect to the larger fixed component, or keep generic $(u_1,u_2)$ until the complete fixed-point sum is formed and only then study a regular limit. Simply omitting $w_2$ changes the normal bundle and is incorrect.

References

• Atiyah, Michael F., and Raoul Bott. “The Moment Map and Equivariant Cohomology.” Topology 23 (1984): 1–28. doi:10.1016/0040-9383(84)90021-1.
• Berline, Nicole, and Michèle Vergne. “Zéros d’un champ de vecteurs et classes caractéristiques équivariantes.” Duke Mathematical Journal 50 (1983): 539–549. doi:10.1215/S0012-7094-83-05024-X. Open PDF.
• Schwarz, Albert, and Oleg Zaboronsky. “Supersymmetry and Localization.” Communications in Mathematical Physics 183 (1997): 463–476. doi:10.1007/BF02506415. Open preprint.

Next step

The field-theory argument begins with the finite-parameter Ward identity for a $Q$-exact path-integral deformation.