Moduli-Space Metrics and Quantum Corrections

The equations defining a moduli space, its complex or hyperkähler structure, and its metric are different layers of information. Quantum effects can leave the set of vacua unchanged while modifying the two-derivative kinetic term, or can instead deform the vacuum equations themselves. A reliable claim of protection must name the layer, the dimension and supersymmetry, the branch, and the assumptions excluding mixing with additional light fields.

Required background. Kähler and hyperkähler quotients gives the classical geometry, and Kähler sigma models relates its metric to kinetic terms. Helpful background. The 1PI effective action distinguishes a quantum effective metric from the microscopic one.

Reading a metric from a local effective action

Let $m^i$ be massless scalar coordinates on a smooth patch of the vacuum space. First choose a Wilsonian matching scale $\mu_{\mathrm W}$ below every field that has been integrated out and above the external momenta of interest. When $|p|\ll M_{\mathrm{heavy}}(m)$, the local two-derivative action begins

$$S_{\mathrm W}(\mu_{\mathrm W}) =\int d^dx\left[ g^{\mathrm W}_{i\bar j}(m,\bar m;\mu_{\mathrm W}) \,\partial_\mu m^i\partial^\mu\bar m^{\bar j} +O(\partial^4) \right].$$

This Wilsonian metric is a local two-derivative coupling, defined only up to field redefinitions and matching-scheme choices. In an $\mathcal N=1$ description it is locally

$$g^{\mathrm W}_{i\bar j} =\partial_i\partial_{\bar j}K_{\mathrm W}.$$

Rigid four-dimensional $\mathcal N=1$ supersymmetry enforces this Kähler structure Zumino 1979, pp. 203–206, but it does not protect the real function $K_{\mathrm W}$ or its metric from quantum corrections.

The full 1PI action requires a separate qualification. Loops of the massless fields retained in $S_{\mathrm W}$ can generate nonanalytic momentum dependence such as $p^2\log(-p^2/\mu_{\mathrm W}^2)$, so the 1PI action need not admit a local derivative expansion at $p=0$. One may speak of a 1PI metric only for its local analytic two-derivative part, at a stated nonexceptional momentum or with an infrared regulator, and only when the remainder is parametrically controlled. Throughout this page, an unqualified “effective metric” means the Wilsonian two-derivative metric; any 1PI claim will be labeled explicitly. The finite-cutoff locality and massless infrared obstruction are made explicit in Bilal 2007, introduction, p. 2, and §§ 2.1–2.2, pp. 3–6.

The functional and infrared distinction is developed systematically in Nonrenormalization Theorems: Wilsonian, 1PI, and Infrared Scope, including why taking $\mu_{\mathrm W}\to0$ does not preserve locality in a massless theory.

Holomorphy of a superpotential does not protect the real function $K_{\mathrm W}$. Loops of fields whose masses depend on $m$ can therefore change the metric even when $W_{\mathrm W}$ is exactly known. Near a locus where a supposedly integrated-out mass $M(m)$ vanishes, terms such as $\log(|M(m)|/\mu_{\mathrm W})$ become singular and the reduced field description fails. Standard supersymmetric effective-action examples are reviewed in Weinberg 2000, chs. 27–29.

Three questions should be kept separate:

• Does the quantum theory still have a vacuum branch?
• Which complex, symplectic, or hyperkähler structures survive?
• What is the exact metric on the smooth part of that branch?

The first can be constrained by holomorphy and indices. The second follows from the supersymmetry algebra acting on massless multiplets. The third is generally the most dynamical.

When protection claims apply

Read each row independently; the table is not an implication chain. The first five rows concern intrinsic local two-derivative branch metrics, understood Wilsonianly. The remaining rows state different objects explicitly: massless 1PI locality, local Wilsonian superpotentials, and a protected BPS index. Every conclusion travels with the assumptions in its own row. On a narrow screen, scroll horizontally to compare the five fields without shrinking the text; in print, all five fields are kept within the page width.

When protection claims apply: claim domains and limits.

Assumptions	Conclusion	Exactness status	Failure modes	Where the argument lives
4d N=1 local Wilsonian Kähler metric Rigid N=1; smooth chiral patch; local Wilsonian action at finite matching scale; perturbative heavy thresholds integrated out.	Supersymmetry enforces Kähler structure, not the classical metric. Heavy thresholds correct the local Kähler potential and metric.	Corrected in general. There is no general metric nonrenormalization theorem; nonperturbative D-term corrections are model dependent.	Restore a field when its mass reaches the matching window. Finite local terms depend on scheme and coordinates. Do not identify the result with a massless zero-momentum 1PI metric.	Threshold calculation Zumino 1979, pp. 203–206; Grisaru–Roček–von Unge 1996, § 3, Eq. (3.5)
4d N=2 Coulomb-branch metric Rigid renormalizable N=2 theory; smooth Coulomb patch with only Abelian vector multiplets light; local special coordinates.	The metric is rigid special Kähler. In the standard renormalizable setting the prepotential is perturbatively one-loop exact and also receives instanton corrections.	Structure exact; classical metric not exact. Exact global data require periods and electric–magnetic patching.	At a singular fiber restore the light monopole or dyon. A single electric prepotential is not global. Extra supersymmetry or different matter content can cancel terms.	Coulomb-branch metrics Seiberg–Witten 1994, §§ 2–3 and 5.1
4d N=2 Higgs-branch metric Rigid theory with eight unbroken supercharges; smooth pure Higgs locus; intrinsic neutral-hypermultiplet sigma model; a weakly coupled path that stays on that locus.	The intrinsic two-derivative Higgs-branch metric equals its classical hyperkähler metric under these assumptions.	Conditionally protected on the smooth pure branch, not a theorem about the full moduli space.	Coulomb or mixed-branch intersections add massless vector multiplets. Gravity, gauging, compactification, or fewer supercharges changes the argument.	Pure Higgs-branch argument Argyres–Plesser–Seiberg 1996, § 3
3d N=4 Higgs-branch metric Rigid theory with eight unbroken supercharges; smooth surviving Higgs patch; field content, real masses, and FI triplets specified.	The intrinsic metric of the surviving Higgs branch is not quantum corrected. Real masses can lift branches; FI triplets can deform or resolve the quotient.	Conditionally protected after the deformation parameters and surviving branch are fixed.	A lifted branch has no metric to protect. Changing FI data changes the classical target. Branch intersections, compactification data, gravity, or reduced supersymmetry require a new analysis.	Three-dimensional branch metrics Intriligator–Seiberg 1996, §§ 1–3.1
3d N=4 Coulomb-branch metric Rigid theory; vector scalars and dual photons form a smooth Coulomb patch; gauge group, matter, masses, and monopole zero modes specified.	Matter loops and allowed BPS-monopole instantons can correct the Coulomb-branch metric.	Quantum corrected and model dependent; supersymmetry preserves hyperkähler structure, not the classical metric.	Abelian and non-Abelian theories differ; the origin may require new coordinates or light fields. Monopole operators are branch coordinates or observables, not themselves the correction mechanism.	Three-dimensional branch metrics Intriligator–Seiberg 1996, §§ 1–3.1; Seiberg–Witten 1996, §§ 2.3–2.4
Massless 1PI locality All modes, including massless ones, are integrated to zero momentum; one asks whether a local two-derivative coefficient exists.	Locality is conditional. A gap, infrared regulator, or stated nonexceptional subtraction point can support a local analytic part; the full massless 1PI action need not be local at zero momentum.	Not an unconditional theorem; locality is an additional infrared hypothesis.	Branch cuts, inverse derivatives, and terms such as p² log(−p²) obstruct a zero-momentum derivative expansion. Taking the Wilsonian cutoff to zero can destroy the premise.	Wilsonian and 1PI domains Bilal 2007, §§ 2.1–2.2; Seiberg 1993, §§ 2.1, 3, 4.1
Perturbative Wilsonian superpotential 4d rigid N=1; local Wilsonian action at finite cutoff; holomorphic variables; supersymmetric regulator; perturbation theory on a nonsingular patch.	Loops do not generate or renormalize the local Wilsonian superpotential in holomorphic variables.	All-order perturbative statement within the named Wilsonian and locality assumptions.	Wavefunction and D-term corrections remain. Canonical couplings can run. Nonperturbative F-terms, higher-derivative F-terms, and massless 1PI infrared terms lie outside the theorem.	Perturbative local theorem Seiberg 1993, §§ 2.1, 3, 4.1
Nonperturbative Wilsonian superpotential Specified 4d N=1 SU(Nc) SQCD with 0 < Nf < Nc, no tree superpotential, det M nonzero, fixed holomorphic scale and branch, plus controlled generation and decoupling input.	The ADS Wilsonian superpotential is exact for this theory and mesonic patch when its scale convention and dynamical normalization are carried with it.	Conditionally exact for the stated field content, global form, patch, and evidence chain; no generic nonperturbative-superpotential theorem follows.	Symmetry alone does not prove generation. Extra zero modes, different gauge global form or matter, additional invariants, singular coordinates, or a changed scale convention alter the conclusion.	ADS hypotheses and derivation Seiberg 1993, §§ 2.1, 3, 4.1; Intriligator–Leigh–Seiberg 1994, §§ 3.1–3.2
Primitive wall crossing 4d N=2; generic codimension-one wall; primitive charges γ1 and γ2; fixed DSZ pairing and chamber orientation; constituent indices unchanged; support property satisfied.	The protected-index jump is the primitive product formula determined by the DSZ pairing and the two constituent indices.	Conditional exact corollary for the named primitive two-charge crossing, not a formula for the unprotected spectrum.	Nonprimitive charges, simultaneous alignments, continuum effects, constituent jumps, or changed sign and wall-orientation conventions require the full wall-crossing structure. The formula does not prove a state exists.	Primitive wall-crossing derivation Gaiotto–Moore–Neitzke 2010, § 2.2

“Protected” never means that a singular coordinate chart is trustworthy, that global identifications are automatic, that higher-derivative terms vanish, or that the full massless 1PI functional is local at zero momentum. Follow the link in the last column before exporting a row to a new theory.

Why the 4d N=2 Higgs metric is special

On a smooth pure Higgs branch of a rigid four-dimensional $\mathcal N=2$ theory, the massless coordinates sit in neutral hypermultiplets, whereas gauge couplings and Coulomb-branch moduli sit in vector multiplets. Eight unbroken supercharges forbid a general dependence of the intrinsic hypermultiplet two-derivative metric on background vector-multiplet scalars. If one can move to a weakly coupled region without leaving that smooth pure branch, the same intrinsic metric can be computed there and equals the classical hyperkähler quotient metric Argyres, Plesser, and Seiberg 1996, § 3, pp. 21–22.

The argument has a domain. At an intersection with a Coulomb or mixed branch, extra vector multiplets are massless and a sigma model containing only Higgs coordinates is incomplete Argyres, Plesser, and Seiberg 1996, introduction, p. 3, and § 3, p. 21. Coupling to supergravity changes the target geometry from hyperkähler to quaternionic Kähler. Gauging a flavor symmetry or compactifying can introduce new scales and new light sectors. Each modification changes an assumption used above.

Coulomb-branch metrics and special coordinates

For a rank-$r$ four-dimensional $\mathcal N=2$ Coulomb branch, choose local special coordinates $a^I$. The two-derivative abelian action is controlled locally by a holomorphic prepotential $\mathcal F(a)$,

$$\tau_{IJ}(a)=\frac{\partial^2\mathcal F}{\partial a^I\partial a^J}, \qquad ds^2=\operatorname{Im}\tau_{IJ}\,da^I d\bar a^J.$$

Holomorphy strongly constrains $\mathcal F$ but does not force it to equal its classical value. In the standard renormalizable rigid theories the prepotential is perturbatively one-loop exact, while instantons generate nonperturbative terms Seiberg and Witten 1994, § 2, pp. 8–9, Eqs. (2.14)–(2.16), and § 5.1, pp. 21–22. Its second derivatives give the special-Kähler metric Seiberg and Witten 1994, § 3, pp. 9–10, Eqs. (3.1)–(3.2). Moreover, $(a^I,a_{D,I})$ undergo electric–magnetic monodromy, so no single prepotential need cover the entire branch.

This chapter uses special geometry only as a metric diagnostic. The dimension-specific construction—including the symplectic section, positivity conditions, patch changes, and global limitations—is developed in Prepotentials and Special Kähler Geometry.

For pure $SU(2)$ theory at $|a|\gg\Lambda$, dimensional analysis and the one-loop beta function give schematically

$$\mathcal F(a) =\frac{i}{2\pi}a^2\log\frac{a^2}{\Lambda^2} +a^2\sum_{k\ge1}c_k\left(\frac{\Lambda}{a}\right)^{4k},$$

up to convention-dependent quadratic terms. The logarithm and instanton series change the metric. Near a singular point, a monopole or dyon becomes massless; including that multiplet restores a regular local effective description even though the metric written only in the original electric coordinate appears singular. This exact $SU(2)$ Coulomb-branch mechanism is the Seiberg–Witten solution Seiberg and Witten 1994, §§ 2–5.1, pp. 8–22, arXiv:hep-th/9407087. Its curve, periods, singular fibers, and strong-coupling patches are developed in The Pure SU(2) Seiberg–Witten Solution.

Three-dimensional N=4 Higgs and Coulomb metrics

In a rigid three-dimensional $\mathcal N=4$ theory, the Higgs- and Coulomb-branch coordinates transform under opposite factors of the $SU(2)\times SU(2)$ R-symmetry. On a smooth surviving Higgs branch, eight unbroken supercharges protect the intrinsic metric from quantum corrections. This statement holds after the deformation data are fixed: real masses can lift Higgs directions without deforming the metric that survives, while an FI triplet changes the quotient level and can deform or resolve the target Intriligator and Seiberg 1996, §§ 1–2, pp. 1–5.

The Coulomb branch instead contains vector-multiplet scalars and dual photons. Charged-matter loops can change its asymptotic metric, as the explicit Abelian examples show Intriligator and Seiberg 1996, § 3.1, pp. 7–8, Eqs. (3.1)–(3.4). In non-Abelian theories, BPS-monopole instantons can also correct the metric when their fermion zero modes permit the two-derivative interaction Seiberg and Witten 1996, §§ 2.3–2.4, pp. 8–12, and p. 25. Monopole operators may furnish coordinates or observables on the quantum branch; their existence is not itself the correction mechanism.

A threshold calculation as a diagnostic

Take one canonically normalized heavy chiral multiplet $H$ and one light modulus $X$, with

$$W=\frac12M(X)H^2, \qquad M(X)=m+\lambda X.$$

In a convenient mass-independent subtraction convention, integrating out $H$ gives the local one-loop matching term

$$\Delta K_{\mathrm W} =-\frac{1}{32\pi^2}|M(X)|^2 \left[ \log\frac{|M(X)|^2}{\mu_{\mathrm W}^2}-1 \right].$$

Grisaru, Roček, and von Unge compute the constant-background 1PI effective Kähler potential Grisaru, Roček, and von Unge 1996, § 3, p. 4, Eq. (3.5). Here the loop contains only the heavy field with $|M|\gg\mu_{\mathrm W}$, so its analytic local part agrees with the heavy-field Wilsonian matching term up to a finite local subtraction and corrections of order $\mu_{\mathrm W}^2/|M|^2$. This identification is not a statement that a massless zero-momentum 1PI metric is local.

Let $u=|M|^2$. Since $\partial_Xu=\lambda\overline M$ and $\partial_X\partial_{\bar X}u=|\lambda|^2$, differentiating rather than merely asserting a correction gives

$$\Delta g^{\mathrm W}_{X\bar X} =\partial_X\partial_{\bar X}\Delta K_{\mathrm W} =-\frac{|\lambda|^2}{32\pi^2} \left[ \log\frac{|M(X)|^2}{\mu_{\mathrm W}^2}+1 \right].$$

For $N_H$ identical heavy multiplets, multiply this result by $N_H$. Changing the finite local subtraction shifts the constant $1$, but not the coefficient of the logarithm. This is a local Wilsonian matching correction, not the complete massless 1PI functional. It is reliable only where $|p|\ll\mu_{\mathrm W}\ll|M(X)|$ and perturbation theory is controlled. Its logarithmic behavior as $M\to0$ is not proof that the full theory is singular; it announces that $H$ must be restored to the light-field description.

For threshold singularities of this kind, first identify which mass used in matching is approaching zero. Reintroduce that degree of freedom before interpreting the geometry.

How to state a protection claim

A useful statement has the form:

In a rigid theory with specified dimension and supersymmetry, on the named branch away from mixed-branch intersections, the stated two-derivative structure is protected from the listed quantum effects; this does not fix global identifications, higher-derivative terms, or physics at loci with additional massless fields.

Then check five independent inputs: the unbroken supercharges, the multiplets containing couplings and moduli, the absence of branch mixing, the Wilsonian matching window, and whether an asserted 1PI coefficient is insulated from massless infrared nonlocalities. Holomorphy can constrain F-terms; it does not by itself establish a metric theorem.

Common pitfalls

Using the classical quotient metric everywhere. The quotient metric is a classical starting point. In an $\mathcal N=1$ theory it is generally renormalized, and near a singular stratum its coordinate-only EFT may be invalid.

Calling holomorphic data a metric. A prepotential or chiral-ring relation constrains the metric only after the appropriate supersymmetry structure is supplied. Even then, monodromy can require multiple special-coordinate patches.

Overextending Higgs-branch protection. The standard 4d $\mathcal N=2$ result assumes a rigid theory and a pure Higgs-branch region. Mixed branches, compactification, gauging, and gravity must be treated separately.

Exercises

Let $K=X^\dagger X-(c/\Lambda^2)(X^\dagger X)^2$ with real $c$.

1. Compute $g_{X\bar X}$.
2. Determine the region in which this truncated metric is positive.
3. Explain why violation of positivity does not prove that the UV theory is inconsistent.

Solution

Writing $r=X\bar X$ gives $g_{X\bar X}=\partial_X\partial_{\bar X}(r-cr^2/\Lambda^2)=1-4cr/\Lambda^2$. For $c>0$, the truncated metric is positive when $|X|^2<\Lambda^2/(4c)$; for $c<0$ it is positive throughout this truncation. If the bound is approached, higher powers of $r/\Lambda^2$ are not parametrically smaller, so the derivative and field expansion has left its controlled domain. One must use the full effective action or the UV completion before drawing a consistency conclusion.

Wilsonian versus 1PI threshold. Suppose a field $H$ has $X$-dependent mass $M(X)$ and is integrated out, while another retained field remains exactly massless. The matching calculation produces a local $\Delta K_{\mathrm W}(X,\bar X)$, whereas massless loops also produce a 1PI form factor proportional to $p^2\log(-p^2/\mu_{\mathrm W}^2)$.

1. State the scale inequalities needed for the local Wilsonian expansion.
2. Which of the two terms defines a local metric coefficient at zero momentum?
3. What must be done as $M(X)\to0$?

Solution

Choose external momenta and a matching scale so that $|p|\ll\mu_{\mathrm W}\ll|M(X)|$ on the patch being described. The local matching term $\Delta K_{\mathrm W}$ changes the Wilsonian metric after two field derivatives. The logarithmic 1PI form factor is nonanalytic at $p=0$; it belongs to the full momentum-dependent vertex and cannot be renamed a local zero-momentum metric without an infrared regulator or a stated nonexceptional subtraction point. When $M(X)$ approaches the matching window, the heavy-field expansion fails. Retain $H$ explicitly and rematch in a description whose light spectrum is complete.

References

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