Finite-Dimensional and Equivariant Localization

Finite-dimensional equivariant localization is a theorem about compact group actions on finite-dimensional manifolds. It replaces an integral of an equivariantly closed form by an integral over the fixed set, divided by the equivariant Euler class of the normal bundle. This theorem is the precise model for supersymmetric localization, but compactness, orientation, finite-dimensional measure theory, and invertibility of normal weights do not automatically survive in a quantum field theory.

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 acts like an equivariant differential.

The Cartan differential and fixed set

Let a torus $T$ act smoothly on a compact oriented manifold $M$. Choose a basis of generating vector fields $V_a$ and degree-two parameters $u^a$. On invariant polynomial-valued forms, define

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

Then

$$d_T^2=-u^a\mathcal L_{V_a}=0$$

on $T$-invariant forms. This is the finite-dimensional prototype of an odd symmetry whose square is a bosonic action.

Let $F=M^T$ be the fixed set, decomposed into connected components. If $\alpha(u)$ is equivariantly closed and the equivariant Euler classes of the normal bundles are inverted, the Atiyah–Bott–Berline–Vergne formula is

$$\int_M\alpha(u) =\sum_{F_i\subset M^T} \int_{F_i}\frac{\iota_i^*\alpha(u)}{e_T(N_{F_i})}.$$

For an isolated fixed point $p$, the denominator is the product of the nonzero tangent weights, with an orientation-dependent sign and the chosen $2\pi$ normalization. Nonisolated components remain integrals; they do not become isolated saddles by wishful thinking. The de Rham formulation and its relation to the moment map are developed in Atiyah and Bott 1984, §§3–7.

Why the integral localizes

Away from $F$, the vector field generated by a generic torus element is nonzero. An invariant metric produces a one-form

$$\lambda=\frac{g(V,\cdot)}{g(V,V)}$$

on $M\setminus F$. Equivariantly, $d_T\lambda$ has an invertible degree-zero part. This makes an equivariantly closed form exact after localizing the coefficient ring, so the contribution from $M\setminus F$ vanishes. A tubular neighborhood of each fixed component then reduces to the inverse Euler class of its normal bundle.

Every hypothesis is doing work:

• compactness removes a surface term at infinity;
• orientation fixes the sign of the Euler class;
• smoothness gives a normal bundle;
• nonzero normal weights make its equivariant Euler class invertible;
• equivariant closure supplies the exact deformation.

If the fixed set is singular, the action is noncompact, or zero normal weights remain, the displayed formula needs modification rather than a formal division by zero.

Worked example: rotation of the two-sphere

Take $S^2$ with coordinates $(\theta,\varphi)$, orientation

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

and the $U(1)$ action generated by $V=\partial_\varphi$. With moment map $\mu=\cos\theta$, one has $d\mu=\iota_V\omega$. Hence

$$\Omega_T=\omega+u\mu, \qquad d_T\Omega_T=0.$$

Direct integration of the degree-two part gives

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

The fixed points are the north and south poles, with moment-map values $+1$ and $-1$ and oriented tangent weights $+u$ and $-u$. In the convention where an isolated complex weight contributes $2\pi/w$,

$$I(u)=2\pi\left(\frac{e^u}{u}+\frac{e^{-u}}{-u}\right) =4\pi\frac{\sinh u}{u},$$

exactly matching the direct integral. The apparent poles cancel only after both fixed points are included.

What survives in a field theory

The analogy with a Euclidean path integral is

Finite-dimensional geometry	Formal field-theory counterpart
$M$	field space modulo gauge transformations
$d_T$	localizing supercharge plus gauge BRST differential
fixed set $M^T$	BPS configurations and topological sectors
normal bundle	gauge-fixed fluctuation complex
$e_T(N)$	regularized one-loop superdeterminant
compact orientation	integration cycle, convergence, and determinant phase

The right column contains serious analytic problems. Field space is generally noncompact and infinite dimensional; it may have singular gauge orbits and reducible connections; the “Euler class” is an infinite product; and the integration cycle becomes middle-dimensional only after complexification. Thus finite-dimensional localization supplies the architecture, not a general proof of a QFT formula. Schwarz and Zaboronsky isolate rigorous supermanifold conditions for odd-symmetry localization Schwarz and Zaboronsky 1997, pp. 463–476, while the transition to gauge theory remains model dependent.

Exercises

1. Small-$u$ check. Show that the fixed-point answer for $S^2$ has a finite $u\to0$ limit and interpret it.

Solution

Since $\sinh u/u=1+u^2/6+\cdots$, $I(0)=4\pi$. This is the ordinary symplectic area $\int_{S^2}\omega$. The cancellation of the separate $1/u$ poles is a completeness check on the fixed set.

2. Zero normal weight. Why can a zero weight not simply be omitted from $e_T(N_F)$?

Solution

A zero weight means the corresponding direction is fixed by the torus and therefore tangent to a larger fixed component, or that the fixed locus is singular. Omitting it misidentifies the normal bundle. One must enlarge or resolve the fixed component and integrate the remaining zero direction explicitly.

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.
• 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 by deriving the Ward identity for a $Q$-exact deformation of the path integral.