Chiral Integrable Boundary States in ABJM

• Chiral integrable boundary states are defined as special boundary states enforcing same-level parity pairing among Bethe roots, particularly within the ABJM spin chain context.
• The methodology employs untwisted transfer matrices and crossing relations to derive explicit overlap formulas and classify basis state families in the SU(4) alternating chain.
• These findings impact quantum quenches, Loschmidt echo analyses, and AdS/CFT boundary one-point functions by extending established integrable overlap paradigms.

Chiral integrable boundary states are boundary or initial states in integrable many-body systems whose nonzero overlaps with Bethe eigenstates require parity pairing within each Bethe-root species, rather than across different species related by a Dynkin-diagram involution. In the general overlap taxonomy of integrable boundary states, this is the “chiral” alternative to achiral pairing, and in the $SU(4)$ alternating spin chain of ABJM theory it is encoded by an untwisted parity condition on the transfer matrix. The first explicit chiral integrable boundary states in ABJM were identified in 2025 as specific two-site and four-site basis states of the planar two-loop scalar-sector chain, together with exact overlap formulae against on-shell Bethe states (Gombor et al., 2020, Liu et al., 4 Jul 2025).

1. Conceptual definition and chirality criterion

In lattice integrable models, an integrable boundary state is characterized by annihilation by all odd local conserved charges,

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$

A sufficient transfer-matrix formulation is

$\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$

with $\Pi$ the space-reflection operator. In the general framework of integrable overlaps, two pairing patterns arise. A chiral overlap pairs Bethe roots within the same species,

$\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$

while an achiral overlap pairs roots across species related by a diagram involution, $\mathbf u^{(a)}=-\mathbf u^{(a')}$. In the language of $K$-matrices, untwisted $K$-matrices correspond to chiral overlaps, and twisted $K$-matrices to achiral overlaps (Gombor et al., 2020).

The ABJM $SU(4)$ alternating chain refines this distinction because it carries two commuting transfer matrices, $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$0 and $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$1. The corresponding lattice conditions are

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$2

for the untwisted, chiral case, and

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$3

for the twisted, achiral case. Previously identified integrable boundary states in ABJM were exclusively achiral and enforced the pairing structure $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$4, $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$5. The chiral states found in the $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$6 alternating chain instead enforce $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$7, $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$8, $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$9, with no inter-level mixing (Liu et al., 4 Jul 2025).

2. ABJM spin-chain realization

The planar two-loop dilatation operator in the scalar sector of ABJM theory yields an $\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$0 alternating spin chain with odd sites in the fundamental $\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$1 and even sites in the anti-fundamental $\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$2. Its Hamiltonian is

$\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$3

Here $\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$4 is the permutation operator and $\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$5 the trace operator, acting in the alternating representation structure.

The chain is governed by four rational $\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$6-matrices,

$\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$7

$\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$8

which generate the monodromies

$\Pi\,\tau(u)\,\Pi\,|B\rangle=\tau(u)|B\rangle,$9

$\Pi$0

and transfer matrices

$\Pi$1

They satisfy

$\Pi$2

This alternating structure is decisive. The existence of both $\Pi$3 and $\Pi$4, together with the crossing parameter $\Pi$5, makes possible a specifically ABJM form of chiral integrability that differs from the better-known achiral constructions in the same theory (Liu et al., 4 Jul 2025).

3. Untwisted integrability and the operator criterion

The chiral integrability condition in the alternating chain is the untwisted relation

$\Pi$6

Using the crossing relations

$\Pi$7

and reversal of ordering under $\Pi$8, one obtains

$\Pi$9

hence

$\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$0

This immediately produces the chiral same-level parity constraints on Bethe roots.

To make the condition tractable on basis states, $\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$1 and $\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$2 are expanded as

$\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$3

$\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$4

The sufficient criterion used for classification is the termwise equality

$\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$5

The construction in the 2025 ABJM analysis is based on this parity/transfer-matrix method rather than on boundary $\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$6-matrices or reflection equations. The paper explicitly notes that it does not use the reflection equation $\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$7 to construct the states, and instead formulates integrability through the untwisted condition and the crossing relation between $\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$8 and $\mathbf u^{(a)}=\{u_1^{(a)},-u_1^{(a)},\dots\},$9 (Liu et al., 4 Jul 2025).

4. Explicit basis-state families in the $\mathbf u^{(a)}=-\mathbf u^{(a')}$0 alternating chain

The 2025 classification restricts attention to pure basis states of the alternating Hilbert space,

$\mathbf u^{(a)}=-\mathbf u^{(a')}$1

with $\mathbf u^{(a)}=-\mathbf u^{(a')}$2 and $\mathbf u^{(a)}=-\mathbf u^{(a')}$3. Solving the sufficient condition $\mathbf u^{(a)}=-\mathbf u^{(a')}$4, the analysis identifies exactly four classes of chiral integrable basis states (Liu et al., 4 Jul 2025).

Class	Pattern	Length condition
1	$\mathbf u^{(a)}=-\mathbf u^{(a')}$5, $\mathbf u^{(a)}=-\mathbf u^{(a')}$6	any $\mathbf u^{(a)}=-\mathbf u^{(a')}$7
2	$\mathbf u^{(a)}=-\mathbf u^{(a')}$8, $\mathbf u^{(a)}=-\mathbf u^{(a')}$9 distinct	even $K$0
3	$K$1, $K$2 distinct	even $K$3
4	$K$4, $K$5 distinct	even $K$6

The two-site class exists for any $K$7, whereas the three four-site patterns exist only for even $K$8. The two-site class corresponds to $K$9-BPS chiral primaries in ABJM.

The derivation is organized by the operator pairs $K$0. For $K$1 and similarly $K$2, parity already enforces uniform odd/even-site patterns. The decisive constraints arise from $K$3: for odd $K$4, consistency requires $K$5, hence $K$6; for even $K$7, the matching conditions require

$K$8

which yields the mutual distinctness conditions in the four-site classes. For $K$9 or $K$0, the relevant equalities were analytically simple only in special cases, and were numerically verified up to $K$1 for the candidate structures.

The claim that these four classes exhaust the basis-state sector is supported, rather than globally proved. For $K$2, numerical construction of Bethe eigenstates with unpaired roots produced basis states with nonzero coefficients that fail the sufficient condition; the numbers reported are $K$3 non-integrable basis states for $K$4, $K$5 for $K$6, and $K$7 for $K$8. None coincides with the four classes above. This supports the conclusion that, among basis states, only these patterns are chiral integrable (Liu et al., 4 Jul 2025).

5. Bethe ansatz, selection rules, and overlap formulae

The nested Bethe ansatz for the $K$9 alternating chain involves three sets of rapidities $SU(4)$0, $SU(4)$1, and $SU(4)$2, obeying

$SU(4)$3

$SU(4)$4

$SU(4)$5

with

$SU(4)$6

Two selection rules govern nonzero overlaps $SU(4)$7. The first is combinatorial: the basis state must occur with nonzero coefficient in the Bethe eigenstate. The second is the chiral pairing rule enforced by

$SU(4)$8

namely

$SU(4)$9

The explicit overlap analysis in the 2025 paper focuses on eight chiral boundary states “on the right side of the equator,” with nonzero overlaps in $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$00 or $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$01 subsectors. In the $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$02 subsector, overlaps reduce to XXX-chain Néel-state overlaps. For

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$03

the wavefunction component is

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$04

with

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$05

For on-shell parity-invariant roots, the overlap takes a Gaudin-determinant form. The corresponding norm satisfies

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$06

and for parity-invariant roots $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$07 factorizes into $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$08. The normalized overlap is

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$09

where

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$10

The $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$11 overlaps factorize. For example,

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$12

This is one of the central structural outcomes: chiral ABJM boundary states inherit the determinant-ratio pattern characteristic of integrable overlaps, but with same-level pairing at all three nested levels (Liu et al., 4 Jul 2025).

6. Reflection equations, fused constructions, and terminological boundaries

The 2025 basis-state construction is not formulated through reflection equations, but the broader conceptual relation between chirality and untwisted/twisted boundary data had already been established in the general theory of integrable boundary states. There, untwisted $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$13-matrices produce chiral overlaps, while twisted $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$14-matrices produce achiral overlaps, and the selection rule is read off from how the transfer matrix transforms under reflection (Gombor et al., 2020).

A direct reflection-equation realization for ABJM was developed later. The 2026 paper “Chiral Integrable Boundary States of ABJM Spin Chain from Reflection Equations” constructs $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$15-site chiral integrable matrix product states by fusion, based on reflection equations in the $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$16 alternating chain. For four-site chiral product states it proposes exact Gaudin-determinant overlap formulae with on-shell Bethe states and numerically investigates the chiral integrable subspaces. In particular, it reports a $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$17-dimensional chiral integrable subspace at $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$18 and a $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$19-dimensional one at $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$20, showing that the previously known basis states do not exhaust all chiral integrable states once more general matrix-product structures are admitted (Liu et al., 2 Feb 2026).

The term chiral boundary state is not uniform across subfields. In symmetric orbifold BCFT, for example, “chiral” refers to preservation of the extended chiral algebra $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$21 with trivial automorphism $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$22, and the relevant paper explicitly states that it does not claim integrability in the sense of boundary Yang–Baxter equations, factorized reflection, or reflection amplitudes (Belin et al., 2021). A common misconception is therefore to treat all “chiral boundary states” as equivalent objects. In the integrable-spin-chain literature relevant here, chirality refers instead to the root-pairing pattern in overlaps and to the untwisted parity condition on the transfer matrix.

7. Applications, analogies, and open problems

The immediate applications of chiral integrable boundary states in ABJM are to quantum quenches, Loschmidt echoes, and overlap-based observables in the $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$23 and $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$24 sectors. Because the overlaps are explicit and the selection rules are sharp, these states provide integrable initial data with chirality-sensitive mode content. A plausible implication is that relaxation and entanglement patterns can differ from those of achiral ABJM initial states, since chiral states couple to parity-invariant excitations within each nesting level rather than across levels (Liu et al., 4 Jul 2025).

The overlap technology also fits the broader AdS/CFT pattern in which boundary one-point functions reduce to normalized overlaps between integrable boundary states and Bethe eigenstates. In D3–D5 defect CFT, for instance, valence-bond or matrix-product boundary states lead to determinant formulae built from Baxter polynomials and ratios of Gaudin determinants, again with paired-rapidity selection rules. The ABJM chiral states therefore extend an established overlap paradigm into a regime where ABJM had previously exhibited only achiral boundary states (Kristjansen et al., 2020).

Several limitations remain explicit. The 2025 analysis is confined to the planar two-loop scalar sector and to basis states. The sufficient condition

$Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$25

is conjectured to be necessary within the basis-state class, but this is not proved. More general matrix product states satisfying $Q_{2k+1}|B\rangle=0,\qquad k\ge 1.$26 relations were left for future work, and extensions to fermionic sectors, higher loops, other gauge theories, and a complete classification of chiral integrable states remain open. The 2026 reflection-equation construction suggests that the chiral landscape is substantially larger than the original basis-state sector, while also making clear that a full classification problem is still unresolved (Liu et al., 4 Jul 2025).