Dirac-Minimal Metrics in Spin Geometry

• Dirac-minimal metrics are defined on closed spin manifolds as those where the dimension of harmonic spinors exactly equals the topologically dictated index, eliminating any surplus.
• In twisted and conformal settings, these metrics arise through variational methods and analytic perturbation techniques, ensuring that the kernel of the Dirac operator attains its minimal size.
• The framework connects index theory, real KO-theory, and spectral geometry, providing insights into the structure and connectivity of the space of minimal metrics.

Dirac-minimal metrics arise in spin geometry as lower-bound-attaining metrics for the Dirac operator, and the term also appears in a variational sense on spin surfaces. In the index-theoretic setting, a Riemannian metric $g$ on a closed spin manifold is Dirac-minimal when the dimension of the space of harmonic spinors equals the absolute value of the analytic index of the Dirac operator. In the twisted setting along a map $f\colon M\to N$, one analogously calls a triple $(f,g,h)$ Dirac-minimal when the kernel of the twisted Dirac operator $D^f_{g,h}$ attains the lower bound coming from real $KO$-theory. In a separate variational literature on spin surfaces, a smooth conformal factor attaining the infimum of a normalized Dirac eigenvalue functional is called a Dirac-minimal or conformally critical metric. These usages share a common focus on extremality for Dirac operators, but they concern different functionals and different geometric mechanisms (Wittmann, 2018, Ammann et al., 2 Aug 2025, Martynyuk, 16 Apr 2026).

1. Index-theoretic notion on closed spin manifolds

Let $(M^n,g)$ be a closed Riemannian spin manifold, $\Sigma^g M\to M$ its spinor bundle, and

$D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$

the associated Dirac operator for a local $g$-orthonormal frame $(e_a)$. Since $f\colon M\to N$0 is elliptic and self-adjoint, its spectrum is real and discrete. When $f\colon M\to N$1 is even, one has the $f\colon M\to N$2-grading $f\colon M\to N$3 and the analytic index

$f\colon M\to N$4

Atiyah–Singer index theory implies that $f\colon M\to N$5 is independent of the metric; the data summarized by Ammann–Dahl states that it depends only on the spin-bordism class of $f\colon M\to N$6, in particular on the $f\colon M\to N$7-genus when $f\colon M\to N$8, or on a $f\colon M\to N$9-valued $(f,g,h)$0-invariant when $(f,g,h)$1 (Ammann et al., 2 Aug 2025).

The basic lower bound is

$(f,g,h)$2

A metric $(f,g,h)$3 is called Dirac-minimal if equality holds: $(f,g,h)$4 Thus Dirac-minimality means that the space of harmonic spinors has no excess dimension beyond what is forced topologically. In the notation of Ammann–Dahl, $(f,g,h)$5 denotes the space of all smooth Riemannian metrics with the $(f,g,h)$6-topology, and $(f,g,h)$7 denotes the open dense subset of Dirac-minimal metrics (Ammann et al., 2 Aug 2025).

This definition isolates the metric-dependent part of harmonic spinor theory. The index is topological, while the full kernel dimension can jump with the metric; Dirac-minimal metrics are precisely those for which no such jump occurs.

2. Twisted Dirac operators along maps and real $(f,g,h)$8-theory

Wittmann studies a twisted analogue in which $(f,g,h)$9 is a closed spin manifold, $D^f_{g,h}$0 is a closed manifold, $D^f_{g,h}$1 is smooth, and $D^f_{g,h}$2 and $D^f_{g,h}$3 are Riemannian metrics on $D^f_{g,h}$4 and $D^f_{g,h}$5, respectively. Writing $D^f_{g,h}$6 for the complex spinor bundle and $D^f_{g,h}$7 for the pull-back of the tangent bundle of $D^f_{g,h}$8, one forms the real tensor product bundle

$D^f_{g,h}$9

with tensor product connection

$KO$0

Clifford multiplication acts on the spinor factor, and the twisted Dirac operator is

$KO$1

given locally by

$KO$2

Because $KO$3 is a real, first-order elliptic Fredholm operator on a closed manifold, it has an analytic index

$KO$4

obtained from the symbol class in $KO$5 via the Thom isomorphism and the Atiyah–Singer push-forward (Wittmann, 2018).

The summarized standard facts are that this index is independent of the metrics $KO$6, depends only on $KO$7, and is homotopy-invariant in $KO$8. The kernel dimension satisfies the general lower bound

$KO$9

Wittmann therefore defines the triple $(M^n,g)$0, or simply the metric $(M^n,g)$1 with $(M^n,g)$2 fixed, to be Dirac-minimal if

$(M^n,g)$3

The surface case $(M^n,g)$4 is especially rigid because

$(M^n,g)$5

Under this identification the index is the mod-$(M^n,g)$6 reduction of the quaternionic dimension of the kernel: $(M^n,g)$7 Accordingly, Dirac-minimality on a spin surface means that $(M^n,g)$8 and that the index parity is realized (Wittmann, 2018).

3. Generic minimality for twisted operators on spin surfaces

The main genericity theorem in Wittmann’s work concerns closed $(M^n,g)$9-dimensional spin surfaces with fixed spin structure. If there exists one triple $\Sigma^g M\to M$0 for which $\Sigma^g M\to M$1 is minimal, then minimality is generic in each variable: for generic Riemannian metrics $\Sigma^g M\to M$2 on $\Sigma^g M\to M$3, the operator remains minimal; for generic Riemannian metrics $\Sigma^g M\to M$4 on $\Sigma^g M\to M$5, it remains minimal; and, if $\Sigma^g M\to M$6 is real analytic, then for a generic map $\Sigma^g M\to M$7 in the fixed homotopy class $\Sigma^g M\to M$8, the kernel remains minimal. Here “generic” means open in the $\Sigma^g M\to M$9-topology and dense in the $D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$0-topology (Wittmann, 2018).

The proof is described as an analytic perturbation argument of Maier–Anghel–Bär–Dahl type. The exceptional set where the kernel jumps above its minimal size is a finite union of analytic hypersurfaces, hence of codimension one. This identifies excess harmonic spinors as a nongeneric phenomenon once a single minimal configuration is known to exist.

Wittmann also gives explicit constructions. In the surface-to-circle case, if $D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$1 with its nontrivial line bundle, differences of non-equivalent spin structures on $D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$2 can be used to build a map $D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$3 such that

$D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$4

so that the twisted Dirac operator reduces to an untwisted Dirac operator for another spin structure. By choosing one spin structure of nonzero Hitchin $D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$5-index and one of zero index, one obtains $D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$6 (Wittmann, 2018).

A second family concerns maps from a spin surface to an odd-dimensional orientable target $D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$7. If $D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$8, then for a null-homotopic map $D_g=\sum_{a=1}^n e_a\!\cdot\!\nabla^g_{e_a}$9 and a suitable closed geodesic $g$0, there are metrics $g$1 on $g$2 and $g$3 on $g$4 for which

$g$5

The summary states that this yields Dirac-minimal examples in every homotopy class when the index is nonzero. Combined with the generic perturbation theorem, this shows that for most metrics, and for maps in the analytic-target setting, the kernel is as small as index theory permits (Wittmann, 2018).

4. Connectedness of the space of Dirac-minimal metrics in dimensions $g$6 and $g$7

Ammann–Dahl prove that if $g$8 is a closed connected spin manifold of dimension $g$9 or $(e_a)$0, then $(e_a)$1 is path-connected, hence connected (Ammann et al., 2 Aug 2025). This is a structural result about the topology of the space of lower-bound-attaining metrics rather than a pointwise existence statement.

The method is to stratify $(e_a)$2 by the number of extra harmonic spinors and to show that each higher stratum has codimension at least $(e_a)$3. The variation of the Dirac operator is analyzed after the Bourguignon–Gauduchon–Maier trivialization. The derivative of the transported operator is

$(e_a)$4

To control the infinitesimal behavior of small eigenspaces, the paper introduces an energy–momentum tensor $(e_a)$5 attached to harmonic spinors $(e_a)$6. The key rank estimate asserts that for nonzero $(e_a)$7 and $(e_a)$8, the linear map

$(e_a)$9

has real rank at least $f\colon M\to N$00, unless $f\colon M\to N$01 is conformal to a flat torus, or in dimension $f\colon M\to N$02 unless $f\colon M\to N$03. The Banach-manifold submersion theorem then implies that the nonminimal locus

$f\colon M\to N$04

is cut out by a submersion of codimension at least $f\colon M\to N$05 in each connected component. An abstract connectivity lemma shows that removing successive codimension-$f\colon M\to N$06 strata from a connected Banach manifold preserves connectedness (Ammann et al., 2 Aug 2025).

The same work records several consequences and examples. In dimension $f\colon M\to N$07, if $f\colon M\to N$08 has genus $f\colon M\to N$09, Hitchin’s inequality gives

$f\colon M\to N$10

On low-genus surfaces $f\colon M\to N$11, or whenever $f\colon M\to N$12 with $f\colon M\to N$13, every metric is automatically Dirac-minimal. For large genus, both Dirac-minimal and non-Dirac-minimal metrics can be exhibited via hyperelliptic methods, the Bär–Schmutz–Schaller dimension count, or minimal-surface spinorial Weierstrass representations of Ammann–Weiß–Witt (Ammann et al., 2 Aug 2025).

In dimension $f\colon M\to N$14, the index theorem gives

$f\colon M\to N$15

Aside from $f\colon M\to N$16, every closed spin $f\colon M\to N$17-manifold with nonzero $f\colon M\to N$18-genus supports only Dirac-minimal metrics, and even on $f\colon M\to N$19 the space of Dirac-minimal metrics is dense. The same summary notes that positive-scalar-curvature metrics make $f\colon M\to N$20 invertible, so when $f\colon M\to N$21, the psc-space lies inside $f\colon M\to N$22 (Ammann et al., 2 Aug 2025).

5. Critical metrics for normalized Dirac eigenvalues and harmonic maps

A separate line of work studies minimization of Dirac eigenvalues in a fixed conformal class. For a closed oriented Riemannian spin surface $f\colon M\to N$23, Karpukhin–Métras–Polterovich enumerate positive Dirac eigenvalues $f\colon M\to N$24 by quaternionic multiplicity and define

$f\colon M\to N$25

where $f\colon M\to N$26 is a conformal class and $f\colon M\to N$27 a spin structure. Their Euler–Lagrange theory states that if $f\colon M\to N$28 is a critical point for $f\colon M\to N$29, then there exist $f\colon M\to N$30-orthonormal eigenspinors $f\colon M\to N$31 such that

$f\colon M\to N$32

Conversely, such a collection, together with the multiplicity gap hypotheses, makes $f\colon M\to N$33 critical (Karpukhin et al., 2023).

Writing $f\colon M\to N$34 in the splitting $f\colon M\to N$35, one obtains a map

$f\colon M\to N$36

Using conformal covariance of the Dirac operator, the normalization forces $f\colon M\to N$37 to be conformal to the pull-back metric $f\colon M\to N$38, and one has

$f\colon M\to N$39

The map $f\colon M\to N$40 is harmonic and satisfies a quaternionic symmetry condition on its harmonic sequence; equivalently, it is twistor-horizontal. The paper therefore defines a quaternionic harmonic map as a harmonic map $f\colon M\to N$41 with this symmetry, and establishes a bijection between $f\colon M\to N$42-critical metrics in a conformal class and quaternionic harmonic maps (Karpukhin et al., 2023).

This correspondence recovers and extends classical spectral extremality statements. On $f\colon M\to N$43, one obtains a harmonic map $f\colon M\to N$44 of degree $f\colon M\to N$45, whence

$f\colon M\to N$46

Thus $f\colon M\to N$47, with equality only for the round metric, recovering Bär’s theorem by an energy comparison. On $f\colon M\to N$48, the same paper proves that for the trivial spin structure and for moduli with $f\colon M\to N$49 in the standard fundamental domain, the flat metric is the unique $f\colon M\to N$50-minimizer in its conformal class (Karpukhin et al., 2023).

These results concern criticality of normalized eigenvalues rather than minimality of the kernel. The shared appearance of harmonic spinors, conformal covariance, and extremal metrics suggests a close thematic relation, but the invariant being optimized is different.

6. Conformally critical metrics on spin surfaces and the broadened use of “Dirac-minimal”

Martynyuk formulates the conformal eigenvalue problem in a way that explicitly uses the terminology “Dirac-minimal.” For a closed spin manifold $f\colon M\to N$51, write

$f\colon M\to N$52

Conformal covariance implies that $f\colon M\to N$53 is a conformal invariant of $f\colon M\to N$54. One then defines

$f\colon M\to N$55

Introducing a generalized conformal factor $f\colon M\to N$56, the problem can be rewritten using the scale-invariant generalized eigenvalue

$f\colon M\to N$57

so that

$f\colon M\to N$58

A smooth $f\colon M\to N$59, hence a metric $f\colon M\to N$60, that attains this infimum is called a Dirac-minimal or conformally critical metric (Martynyuk, 16 Apr 2026).

The Euler–Lagrange system for a minimizer $f\colon M\to N$61 of the volume-renormalized functional $f\colon M\to N$62 has the form

$f\colon M\to N$63

for some spectral index $f\colon M\to N$64, coefficients $f\colon M\to N$65 with $f\colon M\to N$66, and $f\colon M\to N$67-orthonormal eigenspinors $f\colon M\to N$68. In dimension $f\colon M\to N$69, where $f\colon M\to N$70, this simplifies to

$f\colon M\to N$71

If a closed surface satisfies the strict Aubin-type inequality

$f\colon M\to N$72

then there exists a smooth $f\colon M\to N$73 and eigenspinors $f\colon M\to N$74 solving

$f\colon M\to N$75

so that $f\colon M\to N$76 achieves $f\colon M\to N$77. Its degeneration set $f\colon M\to N$78 is finite and satisfies

$f\colon M\to N$79

The summary states that this extends Ammann’s result for $f\colon M\to N$80 to all $f\colon M\to N$81 (Martynyuk, 16 Apr 2026).

On $f\colon M\to N$82, Martynyuk obtains the exact conformal spectrum

$f\colon M\to N$83

and consequently

$f\colon M\to N$84

for every metric $f\colon M\to N$85 on $f\colon M\to N$86. Equality forces $f\colon M\to N$87 disjoint “bubbles” in the conformal class, and for $f\colon M\to N$88 one recovers the round sphere. The same paper records strict monotonicity in dimension $f\colon M\to N$89,

$f\colon M\to N$90

and interprets higher-genus minimizing sequences via a bubble-tree phenomenon analogous to the Yamabe problem (Martynyuk, 16 Apr 2026).

Taken together, these works show that “Dirac-minimal metrics” names two mathematically precise notions. In the index-theoretic literature, the term refers to metrics for which the kernel of the Dirac operator is exactly as small as the index permits. In Martynyuk’s variational formulation, it refers to conformal metrics realizing the infimum of a normalized eigenvalue functional. The first notion is governed by index theory, transversality, and the topology of metric spaces; the second by conformal covariance, nonlinear spinorial Euler–Lagrange equations, and blow-up analysis.