Exotic Pair in Topology & Quantum Systems

• Exotic pair is a term describing two objects that share a coarse equivalence (e.g., homeomorphism or isotopy) yet differ in refined structures (e.g., diffeomorphism or smooth isotopy).
• In smooth 4-manifold topology, exotic pairs arise via cork twists and stabilization methods, revealing deeper insights into low-dimensional geometric structures.
• In hadronic spectroscopy and quantum many-body systems, exotic pairs highlight nonstandard pairing mechanisms, including Majorana pairings and spin-exotic molecules, with significant implications for particle interactions and superconductivity.

Searching arXiv for recent and relevant uses of “exotic pair” across disciplines. {"^^^^^^^^3query3 pair\"^^^^^^^^3 OR ti:\3^^^^^^^^"exotic pair\"","max_results":^^^^3all:\3query3^^^^,"sort_by":"submittedDate","sort_order":"descending"} I’ll also search for exact provided papers to ensure accurate citation coverage. {"^^^^^^^^3query3 OR id:(&&&^^^^3all:\3^^^^&&&) OR id:(&&&^^^^3 OR ti:\3^^^^&&&) OR id:(Tsvelik, 2020) OR id:(Chen et al., 2021) OR id:(Feiguin et al., 2010) OR id:(Jiang et al., 2015) OR id:(Ueda et al., 4 Apr 2025) OR id:(Kasprowski et al., 14 Aug 2025) OR id:(Dai et al., 4 Jun 2026) OR id:(&&&^^^^3all:\3query3^^^^&&&) OR id:(&&&^^^^3all:\3all:\3^^^^&&&) OR id:(&&&^^^^3all:\3 OR ti:\3^^^^&&&)","max_results":^^^^3 OR ti:\3query3^^^^,"sort_by":"relevance","sort_order":"descending"} “Exotic pair” is a field-dependent technical term rather than a single universally fixed concept. In contemporary arXiv usage, it most commonly denotes either a pair of objects that agree in a coarser category but differ in a finer one—most notably smooth PRESERVED_PLACEHOLDER_^^^^3query3^^^^-manifolds that are homeomorphic but not diffeomorphic, or embedded surfaces that are topologically isotopic but not smoothly isotopic—or a nonstandard pairing structure built from unconventional degrees of freedom such as Majoranas, doublons and holons, high-spin atoms, or hadronic molecules with spin-exotic quantum numbers (&&&^^^^3query3^^^^&&&, &&&^^^^3all:\3^^^^&&&, &&&^^^^3 OR ti:\3^^^^&&&, Tsvelik, 2020, Feiguin et al., 2010, Jiang et al., 2015, Ueda et al., 4 Apr 2025).

^^^^3all:\3^^^^. Principal meanings of the term

Across the cited literature, the term is used in several technically distinct senses.

Domain	Object called an “exotic pair”	Criterion
Smooth PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^-manifolds	Two manifolds PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^	Homeomorphic but not diffeomorphic (&&&^^^^3query3^^^^&&&)
Stable $4$-manifold theory	Two closed smooth $4$-manifolds $M,M'$	Stably homeomorphic but not stably diffeomorphic under connected sum with $S^2\times S^2$ (Kasprowski et al., 14 Aug 2025)
Embedded surfaces or disks	Two properly embedded surfaces $\Sigma,\Sigma'\subset X$	Topologically isotopic relative to $\partial X$, but not smoothly isotopic (&&&^^^^3all:\3^^^^&&&)
Hadronic molecules	A vector molecule and its $J^{PC}=1^{-+}$ partner	Same charmed-meson constituents, opposite PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^-parity, with one state spin-exotic (&&&^^^^3 OR ti:\3^^^^&&&)
Many-body pairing problems	Nonstandard bound or condensed pairs	Pairing of Majoranas, Cooper pairs, quintet pairs, or doublon–holon pairs in unconventional symmetry channels (Tsvelik, 2020, Feiguin et al., 2010, Jiang et al., 2015, Ueda et al., 4 Apr 2025)
BSM model building	A vector-like pair of color-triplet scalars	A gauge-invariant bilinear mass term violates baryon number by PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^ (&&&^^^^3 OR ti:\3^^^^8&&&)
BRS cohomology	A ghost-number PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^ invariant plus a ghost-number PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^ anomaly candidate	A paired cohomological structure dependent on pseudofields and a constant spinor (&&&^^^^3 OR ti:\3^^^^9&&&)

This distribution suggests that the phrase usually marks a controlled departure from a standard equivalence class, symmetry assignment, or pairing channel.

^^^^3 OR ti:\3^^^^. Exotic pairs in smooth PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^4-manifold topology

In PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^5-manifold topology, two smooth manifolds PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^6 and PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^7 are exotic if they are homeomorphic as topological manifolds but not diffeomorphic as smooth manifolds (&&&^^^^3query3^^^^&&&). Takahashi studies a compact exotic pair with boundary,

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^8

obtained by a cork twist along the Akbulut cork, and proves

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9

where PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3query3^^^^ is the relative trisection genus (&&&^^^^3query3^^^^&&&). The paper emphasizes that the new content is the explicit construction of genus-PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^ relative trisections together with a proof of minimality; before that work, the smallest known trisection genus of any exotic pair satisfying “same trisection genus” was PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^ (&&&^^^^3query3^^^^&&&). In that setting, the lower bound

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^

for hyperbolic boundary PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4-manifolds is the key obstruction to lower genus (&&&^^^^3query3^^^^&&&).

A related stable notion is developed for closed nonorientable PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5-manifolds with spin universal cover. A pair PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6 is stably exotic if it is stably homeomorphic but not stably diffeomorphic, where stabilization means connected sum with copies of PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7 (Kasprowski et al., 14 Aug 2025). The same work records two structural restrictions: orientable stable exotica do not exist by a result of Gompf, and stably exotic pairs can occur only when PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8 is nonorientable and the universal cover is spin (Kasprowski et al., 14 Aug 2025). Under the hypothesis PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9, the paper gives a complete description in terms of $4$^^^^3query3^^^^, with the first obstruction

$4$^^^^3all:\3^^^^

and a second obstruction expressed through a universal class $4$^^^^3 OR ti:\3^^^^ (Kasprowski et al., 14 Aug 2025).

The relative–absolute distinction is sharpened in work on absolutely exotic compact $4$3-manifolds with boundary. There, “absolute” means that the exotic structure is not relative to a particular parameterization of the boundary (&&&^^^^3all:\3all:\3^^^^&&&). Starting from any compact smooth $4$4-manifold $4$5 with boundary that admits a relatively exotic structure, Akbulut and Ruberman produce a pair of codimension-zero submanifolds homotopy equivalent to $4$6 that are absolutely exotic copies of each other (&&&^^^^3all:\3all:\3^^^^&&&). Applied to corks, this yields absolutely exotic contractible $4$7-manifolds (&&&^^^^3all:\3all:\3^^^^&&&).

The cut-and-paste mechanism behind many such pairs is encoded by corks and plugs. One paper states that every exotic pair in $4$8-dimension is obtained each other by twisting a cork or plug, and then introduces order-$4$9 and infinite-order versions (&&&^^^^3all:\3 OR ti:\3^^^^&&&). Its main result is the existence of a plug $4$^^^^3query3^^^^ with infinite order, and twisting $4$^^^^3all:\3^^^^ gives compact exotic manifolds with boundary from enlargements of $4$^^^^3 OR ti:\3^^^^ (&&&^^^^3all:\3 OR ti:\3^^^^&&&). In this usage, an exotic pair is the output of a controlled boundary twist.

3. Exotic pairs of embedded surfaces and slice disks

For embedded surfaces, the relevant equivalence relation is isotopy rather than diffeomorphism of ambient manifolds. In a smooth $4$3-manifold $4$4 with boundary, properly embedded surfaces $4$5 with

$4$6

form an exotic pair if there exists a topological isotopy of $4$7 relative to $4$8 taking $4$9 to $M,M'$^^^^3query3^^^^, but no smooth isotopy relative to $M,M'$^^^^3all:\3^^^^ does so (&&&^^^^3all:\3^^^^&&&).

Lin constructs such a pair in the punctured $M,M'$^^^^3 OR ti:\3^^^^ surface $M,M'$3. The surfaces $M,M'$4 and $M,M'$5 are topologically isotopic relative to the boundary but not smoothly so; their complements are diffeomorphic; and they remain exotic after one external stabilization in

$M,M'$6

(&&&^^^^3all:\3^^^^&&&). The obstruction is phrased in terms of the $M,M'$7-equivariant family Bauer–Furuta invariant, together with vanishing theorems for diffeomorphisms on $M,M'$8 and $M,M'$9 (&&&^^^^3all:\3^^^^&&&).

A different stabilization problem is addressed for surfaces with boundary in $S^2\times S^2$^^^^3query3^^^^. Miller shows that there are exotic disks in the four-ball with arbitrarily large stabilization distance, giving the first examples of exotic behavior in the four-ball for which “one is not enough” (&&&^^^^3all:\3query3^^^^&&&). In that paper, the stabilization distance measures how many internal stabilizations are required before two disks become smoothly isotopic, and the lower bounds come from Floer-theoretic techniques together with the behavior of satellite operations (&&&^^^^3all:\3query3^^^^&&&).

More recently, singular instanton Floer homology with the Chern–Simons filtration has been used to produce exotic pairs of slice disks (Dai et al., 4 Jun 2026). The same work constructs a strongly invertible $S^2\times S^2$^^^^3all:\3^^^^-slice knot for which any symmetric pair of $S^2\times S^2$^^^^3 OR ti:\3^^^^-disks are exotic, and remain exotic after stabilizing by $S^2\times S^2$3 or $S^2\times S^2$4, or by standard $S^2\times S^2$5 or $S^2\times S^2$6, for any $S^2\times S^2$7 (Dai et al., 4 Jun 2026). Here the obstruction is transported to the branched double cover and then analyzed using instanton $S^2\times S^2$8-invariants and involutive Heegaard Floer theory (Dai et al., 4 Jun 2026).

4. Exotic pairs in hadronic spectroscopy

In hadron physics, “exotic pair” refers not to topological inequivalence but to a pair of near-threshold hadronic molecules built from the same open-charm mesons and differing by charge conjugation, one of which has spin-exotic quantum numbers (&&&^^^^3 OR ti:\3^^^^&&&). The basic constituents are the charmed meson pairs

$S^2\times S^2$9

viewed as S-wave $\Sigma,\Sigma'\subset X$^^^^3query3^^^^ molecules (&&&^^^^3 OR ti:\3^^^^&&&).

The central example is a $\Sigma,\Sigma'\subset X$^^^^3all:\3^^^^ molecule with $\Sigma,\Sigma'\subset X$^^^^3 OR ti:\3^^^^, predicted as the exotic partner of the vector state $\Sigma,\Sigma'\subset X$3 interpreted as a $\Sigma,\Sigma'\subset X$4 molecule (&&&^^^^3 OR ti:\3^^^^&&&). The same constituent pair can form both $\Sigma,\Sigma'\subset X$5 and $\Sigma,\Sigma'\subset X$6 states because the relative sign between $\Sigma,\Sigma'\subset X$7 and $\Sigma,\Sigma'\subset X$8 flips the $\Sigma,\Sigma'\subset X$9-parity while keeping $\partial X$^^^^3query3^^^^ and $\partial X$^^^^3all:\3^^^^ (&&&^^^^3 OR ti:\3^^^^&&&). Since $\partial X$^^^^3 OR ti:\3^^^^ cannot be realized in a simple $\partial X$3 meson, that partner is spin-exotic (&&&^^^^3 OR ti:\3^^^^&&&).

The paper also gives concrete phenomenology. Assuming a common binding energy $\partial X$4 MeV, the masses of the $\partial X$5 and $\partial X$6 partners are estimated as

$\partial X$7

(&&&^^^^3 OR ti:\3^^^^&&&). The radiative transition

$\partial X$8

is predicted to have

$\partial X$9

and the cross section for

$J^{PC}=1^{-+}$^^^^3query3^^^^

is estimated to peak around $J^{PC}=1^{-+}$^^^^3all:\3^^^^ GeV with magnitude $J^{PC}=1^{-+}$^^^^3 OR ti:\3^^^^ pb (&&&^^^^3 OR ti:\3^^^^&&&). The angular distribution

$J^{PC}=1^{-+}$3

is proposed as a diagnostic of the molecular $J^{PC}=1^{-+}$4 assignment (&&&^^^^3 OR ti:\3^^^^&&&). In this literature, the “pair” is exotic because one member is a conventional vector while the other lies outside the $J^{PC}=1^{-+}$5 quantum-number pattern.

5. Exotic pairing states in quantum many-body systems

A separate usage appears in condensed-matter and cold-atom physics, where “exotic” qualifies the pairing channel rather than a pair of inequivalent objects.

In arrays of Majorana–Cooper pair boxes, bond-directed interactions generated through metallic nanowires can simulate the hexagonal Kitaev model, a Kitaev Kondo lattice, and various spin models with three-spin interactions (Tsvelik, 2020). In that architecture, exotic pair phenomena occur at several levels: Majorana pairing within and between MCBs, overscreened Kondo pairing between conduction electrons and Majorana-based spins, and higher-order multi-Majorana “three-body pairing” processes (Tsvelik, 2020). In the Kitaev realization, the emergent spin liquid is described by itinerant $J^{PC}=1^{-+}$6 Majoranas moving in a static $J^{PC}=1^{-+}$7 gauge background built from bond operators $J^{PC}=1^{-+}$8, so that the “idle” Majoranas become the propagating fermions of the spin liquid (Tsvelik, 2020).

In driven-dissipative bosonic arrays, pair injection and collective pair dissipation stabilize an exotic state with bosons condensed along the modes of a closed manifold in Fourier space (Chen et al., 2021). The relevant Bose surface is

$J^{PC}=1^{-+}$9

and in the regime PRESERVED_PLACEHOLDER_^^^^3all:\3query3query3^^^^ with PRESERVED_PLACEHOLDER_^^^^3all:\3query3all:\3^^^^, the asymptotic solution has occupation only on that closed manifold, with

PRESERVED_PLACEHOLDER_^^^^3all:\3query3 OR ti:\3^^^^

and balanced mode amplitudes satisfying PRESERVED_PLACEHOLDER_^^^^3all:\3query33^^^^ and PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^4 (Chen et al., 2021). When PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^5, constants of motion

PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^6

force persistent oscillations, giving self-oscillatory condensates analogous to superfluid time crystals (Chen et al., 2021).

On a two-leg ladder with spin-dependent hopping, DMRG evidence was presented for a Cooper-pair Bose-metal, a fully paired state with a gap for fermion excitations in which Cooper pairs remain uncondensed (Feiguin et al., 2010). In that phase, the pair momentum distribution has singularities at finite momenta inherited from the mismatched noninteracting Fermi points, rather than a dominant PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^7 peak as in the conventional superfluid (Feiguin et al., 2010). The same model also exhibits a phase of paired Cooper pairs with d-wave symmetry, identified by a negative binding energy for two Cooper pairs and by a smooth single-pair momentum distribution together with singularities in the pair-density structure factor (Feiguin et al., 2010).

For a ^^^^3all:\3^^^^D spin-PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^8 Fermi gas at an PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^9-symmetric integrable point, tuning the singlet and quintet channels preserves spin singlet and quintet Cooper pairs in two sets of PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3query3^^^^ spin subspaces (Jiang et al., 2015). The model supports FFLO-like pair correlations, exact Bethe-ansatz thermodynamics, and trapped-gas shell structures in which spin singlet and quintet pairs form multiple shells under the local density approximation (Jiang et al., 2015). Here the exotic aspect is that stable s-wave pairing can occur not only in the spin-singlet channel but also in spin-quintet channels organized by the PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3all:\3^^^^ structure (Jiang et al., 2015).

A more recent ladder example is the photodoped Mott insulator, where DMRG reveals a doublon–holon pairing state characterized by quasi-long-ranged doublon–holon correlations (Ueda et al., 4 Apr 2025). The phase exhibits doublon–holon pairing correlations with opposite signs along the rung and chain directions, reminiscent of d-wave pairing in chemically doped ladder systems, and appears between the spin-singlet phase and the charge-density-wave/PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3 OR ti:\3^^^^-pairing phase (Ueda et al., 4 Apr 2025). In this case the paired objects are a doublon and a holon rather than two electrons.

6. Algebraic and beyond-the-Standard-Model usages

In BRS cohomology of the Wess–Zumino model, “exotic pair” has an algebraic meaning. Dixon uses the spectral sequence method with pseudofields and a constant spinor to identify, at dimension zero, a new set of invariants together with a closely related new set of possible supersymmetry anomalies, and calls this an “exotic pair” (&&&^^^^3 OR ti:\3^^^^9&&&). The invariants of the exotic pairs are all dependent on the pseudofields, which means that the field parts of these invariants are not supersymmetric, though the invariants are in the cohomology space of supersymmetry (&&&^^^^3 OR ti:\3^^^^9&&&). At dimension one, an additional ghost charge PRESERVED_PLACEHOLDER_^^^^3all:\3all:\33^^^^ term appears; this is called a “change”, and the resulting three-term structure is an “exotic triplet” (&&&^^^^3 OR ti:\3^^^^9&&&). The pair and triplet are constrained by simple equations arising from the spectral sequence, such as PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^4, PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^5, and their higher-dimensional analogues (&&&^^^^3 OR ti:\3^^^^9&&&).

In a BSM context, “exotic vector-like pair” denotes two color-triplet scalars,

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^6

with baryon numbers

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^7

together with a Majorana fermion PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^8 and a scalar PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^9 generating PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3query3^^^^ (&&&^^^^3 OR ti:\3^^^^8&&&). The pair is called exotic because the bilinear mass term

PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3all:\3^^^^

violates baryon number as PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3 OR ti:\3^^^^ (&&&^^^^3 OR ti:\3^^^^8&&&). Through the Yukawas

PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\33^^^^

this scalar pair mediates an effective PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^4 operator with

PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^5

yielding neutron–antineutron oscillations, and it also drives post-sphaleron baryogenesis through PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^6 or PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^7 decays (&&&^^^^3 OR ti:\3^^^^8&&&). The same construction allows a light PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^8 in the PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^9–PRESERVED_PLACEHOLDER_^^^^3all:\33query3^^^^ GeV range, in which case PRESERVED_PLACEHOLDER_^^^^3all:\33all:\3^^^^ can be a WIMP-like dark matter candidate and PRESERVED_PLACEHOLDER_^^^^3all:\33 OR ti:\3^^^^ oscillation can proceed indirectly through PRESERVED_PLACEHOLDER_^^^^3all:\333^^^^ and PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^4 oscillations (&&&^^^^3 OR ti:\3^^^^8&&&).

Taken together, these usages indicate that “exotic pair” functions as a cross-disciplinary marker for paired structures that evade the standard classification of the ambient theory. In topology, the evasion is categorical—homeomorphism without diffeomorphism, or topological isotopy without smooth isotopy. In spectroscopy and many-body physics, it is dynamical or representation-theoretic—pairing channels, quantum numbers, or condensates that are inaccessible in the conventional minimal setting. In cohomology and model building, it identifies paired algebraic or field-theoretic structures whose defining feature is a nonstandard coupling or obstruction.