Gauge Fixing and the Localization Deformation Complex

In a gauge theory, the localizing supercharge commonly squares to a field-dependent gauge transformation as well as rigid bosonic symmetries. A one-loop determinant cannot be read from the ungauge-fixed Hessian, whose gauge directions are null. Supersymmetry must be combined with BRST gauge fixing into one odd differential, and the resulting fluctuation complex—not a list of component operators—determines the determinant grading and zero modes.

Required background. $Q$-cohomology and path-integral deformation supplies the deformation argument. BRST cohomology and physical observables supplies ghosts and gauge-orbit cohomology.

Helpful background. Master equations and BV gauge fixing supplies the extension required for open or reducible gauge algebras.

The equivariant supersymmetry–BRST algebra

Write the supersymmetry acting on gauge-covariant fields as $Q_0$. At a generic field configuration its square has the form

$$Q_0^2=\mathcal H+\delta_{\rm gauge}(\Lambda),$$

where $\mathcal H$ is an even rigid transformation, often a combination of an isometry, an R rotation, and flavor rotations. The parameter $\Lambda$ may itself depend on the fields. Thus $Q_0$ is not a differential on the unreduced field space: its square still moves along a gauge orbit.

Introduce the Grassmann-odd gauge ghost $c$ and, in one common graded convention, set

$$\widehat QX=Q_0X+\delta_cX, \qquad \widehat Qc=-\Lambda-\frac12[c,c].$$

For every covariant physical field $X$,

$$\widehat Q^2X =\mathcal HX+ \delta_{\Lambda+\widehat Qc+\frac12[c,c]}X =\mathcal HX.$$

Closure on $c$, on $\Lambda$, and on any constant ghosts then fixes their remaining transformations. In concrete models a residual constant gauge transformation can be included in $\mathcal H$; this is why writing only the two displayed rules is not a substitute for checking the algebra field by field. Pestun carries out precisely this extension, including constant ghost variables, in Pestun 2012, §§4.1–4.4, especially Eqs. (4.10)–(4.13).

The result is equivariant nilpotence:

$$\widehat Q^2=\mathcal H.$$

It becomes strict nilpotence only on $\mathcal H$-invariant quantities. Changing the sign used for $\delta_c$ changes the displayed ghost signs but not the invariant statement: the gauge part of $Q_0^2$ must be absorbed, and the same $\mathcal H$ must close on every field.

For an irreducible gauge condition $G(A)=0$, add a nonminimal antighost–multiplier pair. After a convenient covariant redefinition it may be written

$$\widehat Q\bar c=B, \qquad \widehat QB=\mathcal H\bar c.$$

A standard gauge fermion is

$$\Psi_{\rm gf} =\int_M\operatorname{Tr}\!\left[ \bar c\left(G(A)+\frac\xi2B\right) \right].$$

The gauge-fixed deformation $\widehat Q(V_{\rm loc}+\Psi_{\rm gf})$ is invariant only if $\mathcal H(V_{\rm loc}+\Psi_{\rm gf})=0$ and no uncancelled boundary term is produced. These are hypotheses, not consequences of the notation $\widehat Q(\cdots)$.

The fluctuation complex

Let $\Phi_0$ be a chosen supersymmetric saddle in a specified bundle sector. Linearize the gauge action, the gauge-covariant localization equations, and any identities among those equations. The underlying deformation complex is

$$\Gamma(E_{-1}) \xrightarrow{\ D_{\rm gauge}\ } \Gamma(E_0) \xrightarrow{\ D_{\rm BPS}\ } \Gamma(E_1) \xrightarrow{\ D_{\rm id}\ } \Gamma(E_2),$$

with $D_{\rm BPS}D_{\rm gauge}=0$ and $D_{\rm id}D_{\rm BPS}=0$. Here:

• $E_{-1}$ contains infinitesimal gauge parameters, represented in the path integral by ghosts;
• $E_0$ contains bosonic fluctuations;
• $E_1$ contains the linearized BPS equations or their fermionic partners;
• $E_2$ records relations among equations. Higher reducibility extends the complex farther in either direction.

The gauge condition is related to this complex but is not normally the arrow $D_{\rm BPS}$: for example, a background-covariant slice adds $D_{\rm gauge}^{\dagger}x=0$. With chosen inner products, the gauge-fixed kinetic problem is governed by an operator such as

$$\mathscr D =D_{\rm gauge}^{\dagger}\oplus D_{\rm BPS}: \Gamma(E_0) \longrightarrow \Gamma(E_{-1})\oplus\Gamma(E_1),$$

together with the ghost operator and any identity complex. Mixing the gauge condition into $D_{\rm BPS}$ while still asserting $D_{\rm BPS}D_{\rm gauge}=0$ would be incorrect: a gauge slice is deliberately nonconstant along a gauge orbit.

The first three relevant cohomology groups are

$$H^{-1}=\ker D_{\rm gauge}, \qquad H^0=\frac{\ker D_{\rm BPS}}{\operatorname{im}D_{\rm gauge}}, \qquad H^1=\frac{\ker D_{\rm id}}{\operatorname{im}D_{\rm BPS}}.$$

$H^{-1}$ is the infinitesimal stabilizer of the saddle. $H^0$ is the physical tangent space to the localization locus. $H^1$ contains obstructions and, depending on the cohomological grading, unpaired fermionic directions. Calling every element of these spaces a “zero mode” hides three different operations: divide by a stabilizer, integrate a collective coordinate, or saturate a fermionic integral.

How the complex enters the determinant

Near an isolated saddle, first add the gauge-fixing pairs and resolve any identities so that the quadratic problem has a two-term Fredholm representative $D_{10}$. Then reorganize nonzero fluctuations into $\widehat Q$-doublets,

$$\widehat QX=\Psi, \qquad \widehat Q\Psi=\mathcal HX.$$

The precise $D_{10}$ depends on the cohomological variables and gauge fermion; it is often equivalent to $\mathscr D$ plus acyclic BRST pairs. Its bosonic and fermionic quadratic operators pair most modes. What survives is encoded by the equivariant virtual character

$$\operatorname{ind}_{\mathcal H}(D_{10}) =\operatorname{Tr}_{\ker D_{10}}e^{t\mathcal H} -\operatorname{Tr}_{\operatorname{coker}D_{10}}e^{t\mathcal H}.$$

Expanding this character into $\mathcal H$-weights gives the multiplicities that enter a regulated product. It does not by itself choose the real versus complex Gaussian normalization, a Pfaffian orientation, a spectral cut, or the measure on the kernel. Those data are supplied on the loci and determinants page.

Ellipticity, transverse ellipticity, and expansion data

For an elliptic complex, the symbol sequence is exact for every nonzero cotangent vector. On a compact manifold with an appropriate domain, its cohomology is finite dimensional and its index is stable under lower-order deformations.

If $\mathcal H$ generates a compact group action, the symbol may instead be exact only for covectors transverse to the group orbits. The complex is then transversely elliptic. Its equivariant index is a distribution, commonly represented by a formal Laurent series. Choosing how to expand factors such as $(1-t)^{-1}$ is part of the answer: opposite expansions can describe different polarizations of the same distribution. Five-dimensional contact localization supplies an explicit example Källén, Qiu, and Zabzine 2012, §§3–4.

Thus an assertion that “the index gives the determinant” is incomplete unless it also states the group action, fixed-point data, expansion prescription, regulator, and treatment of zero weights.

Reducible saddles and boundaries

At a reducible connection the stabilizer $G_{\Phi_0}$ is nontrivial. The Faddeev–Popov operator then has zero modes and the local quotient can be singular. One must work stratum by stratum, divide by the actual stabilizer rather than automatically by all of $G$, and retain any residual finite-dimensional integral. Relations among gauge transformations require ghosts-for-ghosts, or more generally a BV resolution.

On a manifold with boundary, an operator is specified by both its differential expression and its domain. Boundary conditions must make the variational problem well posed, be preserved by $\widehat Q$ and $\mathcal H$, and give a Fredholm complex—or be supplemented by boundary degrees of freedom that do. Two admissible boundary conditions can share the same bulk symbol yet have different kernels, cokernels, and determinants.

What must be established before taking a determinant

A complete derivation records:

• the full $Q_0^2$, including its field-dependent gauge parameter;
• every ghost, antighost, multiplier, auxiliary, and constant-ghost transformation;
• closure of $\widehat Q^2=\mathcal H$ and $\mathcal H$ invariance of the deformation and integration cycle;
• the bundle sector, saddle, stabilizer, and residual Weyl quotient;
• the complete symbol complex and whether it is elliptic or transversely elliptic;
• the domain and adjoint boundary conditions;
• $H^{-1}$, $H^0$, and $H^1$, with a separate prescription for each;
• the expansion, regulator, determinant-line orientation, and Gaussian normalization.

Leaving one item implicit may be harmless in a worked model with a standard convention, but it prevents the displayed determinant from being a definition in a new problem.

Exercises

1. Absorbing the gauge part of the square. Using the displayed graded identity, show that the chosen transformation of $c$ removes $\delta_{\rm gauge}(\Lambda)$ from $\widehat Q^2X$. Why is this not yet a proof of closure on every field?

Solution

Substitution gives $\Lambda+\widehat Qc+\tfrac12[c,c]=0$, so $\widehat Q^2X=\mathcal HX$ for a covariant physical field. Closure on $c$ additionally involves $\widehat Q\Lambda$; the nonminimal and constant-ghost fields have their own transformations. Each must be checked, and any residual constant gauge action must be included consistently in $\mathcal H$.

2. Three kinds of zero mode. Let $\epsilon\in H^{-1}$, $v\in H^0$, and $\psi\in H^1$. State the path-integral operation associated with each class.

Solution

$\epsilon$ generates a stabilizer transformation and is replaced by the residual gauge quotient or group integral. The bosonic class $v$ is a genuine tangent direction and becomes a collective coordinate with an induced measure. The fermionic class $\psi$ gives a Grassmann zero-mode integral; the contribution vanishes unless insertions or interaction terms saturate it. None belongs in the primed nonzero-mode determinant.

3. Why a transverse index needs a chamber. Expand $(1-t)^{-1}$ for $|t|<1$ and for $|t|>1$. What does the difference teach about a transversely elliptic character?

Solution

For $|t|<1$,

$$\frac1{1-t}=\sum_{n\geq0}t^n.$$

For $|t|>1$,

$$\frac1{1-t}=-\sum_{n\geq1}t^{-n}.$$

They are analytic representatives of the same rational expression in different domains but assign different formal weight multiplicities. A determinant extracted from such a character must therefore declare the expansion or polarization.

References

• Källén, Johan, Jian Qiu, and Maxim Zabzine. “The Perturbative Partition Function of Supersymmetric 5D Yang–Mills Theory with Matter on the Five-Sphere.” Journal of High Energy Physics 2012, no. 8 (2012): 157. doi:10.1007/JHEP08(2012)157. Open preprint.
• Pestun, Vasily. “Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops.” Communications in Mathematical Physics 313 (2012): 71–129. doi:10.1007/s00220-012-1485-0. Open preprint.

Next step

Use the completed complex to derive the supersymmetric loci, collective-coordinate measures, and regulated one-loop factors.