Multifold Exceptional Structures

• Multifold exceptional structures are systems defined by exact multiplicity through exact coverings, symmetry-enforced degeneracies, or exceptional algebraic and geometric organization.
• They occur across disciplines such as coding theory, topological semimetals, non-Hermitian physics, and supersymmetric field theories, revealing novel symmetry and multiplicity patterns.
• Their hierarchical organization and codimension control enable precise deformation responses and protection mechanisms, underpinning robust physical and mathematical phenomena.

Multifold exceptional structures are higher-multiplicity objects whose defining property is either exact $u$-fold covering, symmetry-enforced $n$-fold degeneracy, or exceptional algebraic and geometric organization. In current arXiv usage, the phrase spans several distinct technical settings: multifold $1$-perfect codes in Hamming graphs, multifold fermions in topological semimetals, multifold exceptional points in non-Hermitian band theory, exceptional moduli spaces of $4$d $\mathcal N=3$ theories, exceptional complex structures in $E_{6(6)}\times \mathbb R^+$ generalised geometry, and the finite-dimensional extensions collected under Exceptional Periodicity (Krotov, 2022, Robredo et al., 2024, Yoshida et al., 2024, Kaidi et al., 2022, Tennyson et al., 2021, Truini et al., 2017). Taken together, these literatures indicate a common structural pattern: multiplicity is not incidental, but is fixed by arithmetic conditions, crystalline or anti-unitary symmetry, generalized similarity, or exceptional moduli-space data.

1. Terminological scope and structural motifs

The adjective “multifold” refers to exact multiplicity. In coding theory, a $p$-fold $1$-perfect code is a subset $C$ such that every radius-$1$ ball contains exactly $n$0 codewords, equivalently

$n$1

In crystalline band theory, multifold fermions are symmetry-enforced $n$2-, $n$3-, $n$4-, or $n$5-fold crossings of Bloch bands at high-symmetry $n$6-points. In non-Hermitian physics, an EP$n$7 is an $n$8-fold spectral degeneracy at which both eigenvalues and eigenvectors coalesce. In supersymmetric field theory, exceptional structure refers to quotient moduli spaces $n$9 labelled by exceptional crystallographic complex reflection groups, or to exceptional-generalised reductions such as $1$0 and $1$1 structures (Krotov, 2022, Robredo et al., 2024, Yoshida et al., 2024, Kaidi et al., 2022, Tennyson et al., 2021).

A recurrent mathematical feature is codimension control. Generic EP$1$2s require codimension $1$3, while symmetry-protected EP$1$4s require codimension $1$5. Multifold $1$6-perfect codes in $1$7 require the arithmetic condition $1$8 together with sphere-packing divisibility. Exceptional $1$9 moduli spaces take the orbifold form $4$0, where $4$1 is constrained by invariant theory. This suggests that “multifold exceptional structure” is best understood as a family resemblance across disciplines rather than as a single formal category.

2. Symmetry-enforced multifold band structures in solids

In topological semimetals, multifold band crossings are high-symmetry degeneracies where more than two bands meet at a point in momentum space. In spin-orbit-coupled crystals, they are the irreducible corepresentations of the little group at high-symmetry $4$2-points, and the allowed cases in the review literature are $4$3-, $4$4-, $4$5-, and $4$6-fold. These crossings are symmetry-protected, especially by non-symmorphic operations such as screw axes and glide planes, and their exhaustive classification is carried out over the $4$7 magnetic space groups. The same literature distinguishes Sohnke magnetic space groups, which lack mirrors, inversion, and rotoinversions, as the setting in which topologically charged multifold fermions can occur (Robredo et al., 2024).

The low-energy description often takes the generalized Weyl form

$4$8

with $4$9 in a higher-dimensional representation. For a $\mathcal N=3$0-fold fermion, the spin-$\mathcal N=3$1 case yields band Chern numbers $\mathcal N=3$2. For a $\mathcal N=3$3-fold spin-$\mathcal N=3$4 fermion, the band Chern numbers are $\mathcal N=3$5, while a more exotic $\mathcal N=3$6-fold phase proposed for BaAsPt carries $\mathcal N=3$7. Sixfold nodes are built from two $\mathcal N=3$8-folds; in chiral type-II magnetic space groups they realize a double spin-$\mathcal N=3$9 fermion whose lower bands carry $E_{6(6)}\times \mathbb R^+$0 and $E_{6(6)}\times \mathbb R^+$1, giving total charge $E_{6(6)}\times \mathbb R^+$2 at half filling. Eightfold nodes are the highest possible in that review and are usually topologically neutral. A common misconception is that these quasiparticles should fit the Poincaré classification of elementary particles; the review explicitly states the opposite, because the relevant symmetry is crystalline rather than relativistic (Robredo et al., 2024).

Large Chern numbers control the surface phenomenology. A node with $E_{6(6)}\times \mathbb R^+$3 requires four surface channels, and the reviewed chiral materials exhibit “double Fermi arcs” and “long Fermi arcs” that can span a large fraction of the surface Brillouin zone. The same band topology produces strong gyrotropic magnetic and circular photogalvanic responses. In the optical-response analysis, multifold fermions can have zero Berry curvature yet finite gyrotropic magnetic effect, and the circular photogalvanic effect is quantized and frequency-independent when the active optical transition surfaces are closed. The natural unit is

$E_{6(6)}\times \mathbb R^+$4

and the quantized plateaus are set by band Chern numbers rather than by two-band Weyl physics alone (Flicker et al., 2018).

The field response of multifold nodes is highly tunable. In CoSi, biaxial strain in the $E_{6(6)}\times \mathbb R^+$5-plane breaks the threefold rotation while preserving the twofold screw symmetries and time reversal, lifting the original fourfold node at $E_{6(6)}\times \mathbb R^+$6 and sixfold node at $E_{6(6)}\times \mathbb R^+$7. Under $E_{6(6)}\times \mathbb R^+$8 strain, the fourfold degeneracy at $E_{6(6)}\times \mathbb R^+$9 is replaced by four Weyl points of charge $p$0, while four charge-$p$1 Weyl points appear near $p$2. On the $p$3 surface, the long Fermi arcs survive but reconnect between the split Weyl nodes; on the $p$4 surface, the unstrained closed loop turns into small open arcs near the surface-zone center. The effective $p$5 $p$6 model reproduces this by adding a perturbation that commutes with the screws but not with the broken threefold rotation (Bose et al., 2021).

Symmetry-adapted effective models sharpen this picture in the presence of spin-orbit coupling. For the threefold fermion at the $p$7 point of space group $p$8 (No. 199), the symmetry-complete linearized model is

$p$9

with a Zeeman term

$1$0

The threefold node at $1$1 has monopole charge $1$2. Under $1$3 it splits into four magnetic monopoles, three with $1$4 and one with $1$5; under $1$6 it splits into two nodes with $1$7. The associated tight-binding model captures the global Brillouin-zone motion and pair annihilation of these monopoles, which the local $1$8 theory cannot describe (Satow et al., 4 Aug 2025).

Multifold fermions also enter superconductivity. Several noncentrosymmetric superconductors, including Li$1$9Pd$C$0B, Li$C$1Pt$C$2B, Mo$C$3Al$C$4C, PdBiSe, Y$C$5C$C$6, and La$C$7C$C$8, are identified as multifold fermion metals whose normal-state topology had been largely ignored in superconductivity studies. In Li$C$9Pd$1$0B, the $1$1-centered pockets carry total Chern number $1$2, the remaining pockets carry $1$3, and the $1$4 surface exhibits $1$5 Fermi arcs. With extended $1$6-wave $1$7 pairing, the DIII invariant

$1$8

can take the value $1$9, implying four Majorana cones on the $n$00 surface. Li$n$01Pt$n$02B, by contrast, is described as a likely nodal topological superconductor with dispersionless Majorana surface modes (Gao et al., 2020).

3. Multifold exceptional points and non-Hermitian topology

A multifold exceptional point EP$n$03 is an $n$04-fold non-Hermitian degeneracy where the characteristic polynomial has an $n$05-fold root and the Hamiltonian becomes defective. A systematic classification uses resultants of the characteristic polynomial and its derivatives: $n$06 For generic systems, the resultant vector has $n$07 real components, so EP$n$08s have codimension $n$09. For symmetry-protected cases such as $n$10-symmetric systems, the resultants become real and the codimension drops to $n$11. The corresponding topological invariant is the resultant winding number, defined by the degree of the normalized resultant map from a sphere surrounding the EP to a target sphere of the same dimension (Yoshida et al., 2024).

Local anti-unitary symmetries make this codimension reduction operational. For $n$12, pseudo-Hermiticity, $n$13, and pseudo-chirality, the relevant resultant conditions are real, so a $n$14-fold EP is stable in $n$15 dimensions. The resulting real “resultant vector” yields an integer invariant: for EP3s it is a winding number on $n$16, and more generally it is the degree of a map into $n$17. The frictional shallow-water model provides an explicit $n$18-symmetric realization: the reduced $n$19 Hamiltonian exhibits EP3s at

$n$20

with topological charges $n$21. The same model shows EP3 merging and transitions into and out of a forbidden-propagation regime (Delplace et al., 2021).

The later mathematical development broadens this picture from symmetry protection to local similarity relations and connects the resultant construction to the Hermitian tenfold way, vector bundles, and Mayer–Vietoris arguments. The Hermitian “resultant Hamiltonian”

$n$22

places generic EP$n$23s in class AIII, while pseudo-Hermitian even-order EPs fall into class A and are associated with Chern numbers rather than winding numbers. The same framework proves Abelian doubling theorems and predicts $n$24-protected bulk Fermi arcs induced by non-local symmetries in odd-$n$25 cases (Stålhammar et al., 2024).

An important correction to a common oversimplification is that EP$n$26s need not appear as isolated objects. Under generalized similarities—pseudo-Hermiticity, pseudo anti-Hermiticity, and self skew-similarity—an EP$n$27 is typically embedded in a hierarchy of lower-order exceptional manifolds. In a pseudo-Hermitian $n$28-band model in $n$29D, the EP4 condition is $n$30, but the EP4 points sit inside EP2 surfaces defined by the vanishing of the discriminant and EP3 arcs defined by $n$31 and $n$32. In the self-skew-similar $n$33-band case, EP4s and EP2 surfaces occur but EP3s are forbidden. In a $n$34-band model with multiple similarities, EP6s coexist with EP2 surfaces, EP3 arcs, and EP4 arcs, while EP5s are absent (Montag et al., 4 Aug 2025).

Hopf exceptional points add a second layer of topology beyond resultant winding. For HEP3s in five dimensions and symmetry-protected HEP5s, the normalized resultant map is classified by $n$35, which the paper relates to the Witten anomaly. This yields the statement that HEP3s and symmetry-protected HEP5s act as their own “antiparticles.” For symmetry-protected HEP4s, the relevant invariant is the Hopf invariant $n$36, with the map $n$37. The same program predicts finite-group topologies such as $n$38, $n$39, and $n$40, explicitly beyond the Bernard–LeClair periodic table (Yoshida et al., 17 Apr 2025).

A further extension moves from momentum space to combined frequency-momentum space. For nonlinear eigenvalue problems

$n$41

the frequency-momentum winding number is defined from the real vector of $n$42 and its $n$43-derivatives. This produces a unified proof of the doubling theorem for EP$n$44s in nonlinear systems, with or without $n$45 and $n$46 constraints. In the linear limit, the same construction yields a $n$47 classification for $n$48-symmetric EP2s, refining the previously reported $n$49 picture (Yoshida, 1 Apr 2026).

4. Exact-multiplicity covering structures in coding theory

In graph-theoretic coding theory, a multifold $n$50-perfect code is a set $n$51 of vertices such that every vertex lies within distance at most $n$52 from exactly $n$53 elements of $n$54. In a regular graph, such a code is a special case of a completely regular code, and in a $n$55-ary Hamming graph $n$56 the code and its complement form an equitable partition. The quotient matrix is

$n$57

and double counting gives the sphere-packing relation

$n$58

The paper proves a complete parameter characterization when $n$59 is a prime power. For unrestricted codes, a $n$60-fold $n$61-perfect code in $n$62 exists if and only if

$n$63

For additive $n$64-linear codes, the characterization is

$n$65

together with an equivalent extra condition $n$66. The additive theory is translated into multiset geometry through multispreads, defined by

$n$67

The exact equivalence between additive radius-$n$68 completely regular codes and $n$69-spreads is the central bridge between coding and combinatorial design. The same work proves that the multispreads required by the parameter classification always exist, so every admissible additive parameter set is realizable, and every admissible parameter set in the unrestricted case is realizable by taking appropriate unions of cosets. A notable caveat is that for non-prime $n$70, not every admissible parameter set is obtainable from linear codes or unions of cosets of linear multifold perfect codes; the paper highlights a $n$71-fold $n$72-perfect code in $n$73 that cannot be obtained in that way but can be constructed additively. The framework also situates multifold perfect codes within list decoding, multifold ball packings, multiple coverings, and the theory of intriguing sets (Krotov, 2022).

5. Exceptional moduli, quotient geometries, and categorified structures

For genuine $n$74d $n$75 SCFTs, a general expectation is that the vacuum moduli space takes the global orbifold form

$n$76

with $n$77 a crystallographic complex reflection group. Exceptional CCRGs play here the role that exceptional Weyl groups play in Lie theory. The paper shows that non-geometric quotients of type-$n$78 $n$79d $n$80 theories realize nearly all exceptional moduli spaces beyond Weyl groups, specifically

$n$81

with the missing exceptions

$n$82

The construction starts from M-theory on $n$83 with a diagonal quotient

$n$84

and is extended by outer-automorphism twists, which replace $n$85 by Hecke groups $n$86. The identification of the quotient group is made by matching degrees of invariant polynomials; for example, $n$87 has degrees $n$88, and $n$89 has degrees $n$90. The same work uses Hasse diagrams of singular strata and the Shapere–Tachikawa relation to test consistency. A recurrent misunderstanding is that exceptional CCRG labels are merely formal algebraic options; the paper argues the opposite by giving explicit string/M-theory realizations (Kaidi et al., 2022).

Exceptional-generalised geometry provides a different meaning of “exceptional structure.” In $n$91 generalised geometry, an exceptional complex structure is an integrable $n$92 structure encoded by a maximally isotropic involutive subbundle $n$93, where in the M-theory case

$n$94

The defining conditions include

$n$95

Its refinement to an $n$96 structure introduces a triplet $n$97 satisfying

$n$98

together with the moment-map condition

$n$99

All ECSs are classified by type and class: only types $1$00 and $1$01, and classes $1$02 and $1$03, occur. For class-zero structures, the hypermultiplet moduli reproduce the fluxless Calabi–Yau result,

$1$04

even in the presence of flux (Tennyson et al., 2021).

A categorified moduli-space picture extends the familiar bundle story from invertible symmetries to noninvertible ones. For Calabi–Yau SCFTs, the prototype is the Bagger–Witten line bundle, whose square is the Hodge line bundle,

$1$05

The proposal for $1$06 and $1$07 SCFT moduli spaces is a stack of fusion categories acting as a noninvertible analogue of the Hodge/Bagger–Witten bundle. The relevant sectors are the tricritical Ising category for $1$08 and the Ising category for $1$09, with fusion rules such as

$1$10

in the Ising case and

$1$11

in the tricritical Ising case. The closed-string sector is described by a fusion-ring bundle, while the D-brane sector requires a stack of module categories, because associators and $1$12-symbols are essential. This is conjectural, but it is explicitly motivated by the observation that anomalous invertible symmetries can still induce global structures over moduli space, suggesting a parallel role for noninvertible symmetries (Perez-Lona et al., 24 Jun 2025).

6. Exceptional periodicity and cross-domain synthesis

Exceptional Periodicity proposes a finite-dimensional infinite extension of the classical exceptional chain

$1$13

to

$1$14

indexed by

$1$15

At $1$16, the classical exceptional objects are recovered; for higher $1$17, one obtains finite-dimensional “extended-root” analogues that are no longer Lie algebras in the usual sense, because the Jacobi identity is only partially preserved. The extended roots of $1$18 are

$1$19

with an even number of plus signs. The “Magic Star” projection organizes these roots around a central $1$20 and three Jordan pairs. One tip of the Magic Star gives a cubic Vinberg $1$21-algebra, which generalizes $1$22 and admits trace, sharp, and cubic norm operations analogous to the exceptional Jordan algebra. The associated algebra

$1$23

is defined by a generalized Cartan-style bracket with an asymmetry function resembling the cocycle of lattice vertex algebras. The proposal is therefore not an affine or hyperbolic extension, but a finite-dimensional periodic family of exceptional-like objects (Truini et al., 2017).

Taken together, these literatures suggest three recurring principles. First, multiplicity is exact: every vertex is covered exactly $1$24 times, every node is exactly $1$25-fold, every quotient is by a specific exceptional group, and every generalized structure is cut out by precise isotropy or involutivity conditions. Second, protection mechanisms are explicit: arithmetic divisibility in Hamming graphs, crystalline and non-symmorphic symmetry in solids, anti-unitary symmetry or generalized similarity in non-Hermitian systems, and invariant-theoretic or generalized-geometric constraints in moduli problems. Third, multifold exceptional structures are usually hierarchical rather than isolated. Multispreads encode additive codes; EP$1$26s sit on manifolds of EP$1$27s; multifold fermions can split into Weyl nodes under strain or field; exceptional SCFT moduli spaces contain singular strata supporting lower-rank theories; and Exceptional Periodicity organizes classical exceptional objects into an infinite but finite-dimensional sequence. This suggests that the unifying content of the subject lies less in a shared ontology than in a shared architecture of exact multiplicity, constrained deformation, and exceptional organization.