Non-Invertible Time-Reversal Symmetry

• Non-Invertible Time-Reversal Symmetry is defined as a conserved anti-linear operator whose fusion with its adjoint yields a nontrivial condensation defect rather than the identity.
• It emerges in gauge theories at rational θ-angles, defect categories, and open quantum systems, thereby constraining infrared dynamics and influencing selection rules.
• This symmetry challenges traditional notions by substituting group inversion with projection or partial isometry mechanisms, offering fresh insights into anomaly inflow and enforced gaplessness.

Searching arXiv for recent and foundational papers on non-invertible time-reversal symmetry. First I’ll retrieve the core paper on non-invertible time-reversal symmetry and nearby related work. Using arXiv search now. Non-invertible time-reversal symmetry is a generalization of ordinary time-reversal in which the symmetry operation reverses time or implements an anti-linear transformation, yet fails to admit an inverse in the usual group-theoretic sense. In the recent literature, the term refers to several related but distinct structures. In gauge theory, a non-invertible time-reversal symmetry is a conserved anti-linear topological operator whose fusion with its adjoint yields a nontrivial condensation defect rather than the identity, and which exists at rational values of the $\theta$-angle such as $\theta=\pi p/N$ (Choi et al., 2022). In defect and categorical settings, anti-unitary symmetries may satisfy algebras such as $\mathsf T^2=C$ or $\mathsf T^2=(-1)^F C$, so that fusion with the orientation-reversed defect produces a sum of sectors rather than a single inverse (Cordova et al., 2023). In open quantum systems, a distinct notion of non-invertibility appears when a time-reversal-symmetric Lindblad-type evolution acquires a factor $\mathrm{sgn}(t)$: the resulting dynamical map is non-invertible through $t=0$, even though the equation remains fully time-reversal invariant (Guff et al., 2023). These developments place non-invertible time-reversal at the intersection of higher-form symmetry, topological defects, anomaly inflow, generalized Wigner structures, and nonequilibrium quantum dynamics.

1. Definition and basic algebra

In a theory with a $2\pi$-periodic $\theta$-angle, ordinary time-reversal is usually represented only at $\theta=0$ or $\theta=\pi$ by an anti-linear, invertible operator $\theta=\pi p/N$0 satisfying $\theta=\pi p/N$1 and an algebra such as $\theta=\pi p/N$2 or $\theta=\pi p/N$3 (Choi et al., 2022). The non-invertible case departs from this by replacing the anti-unitary operator with an anti-linear topological operator that still commutes with the Hamiltonian and flips the sign of the time coordinate, but is not invertible. At rational $\theta=\pi p/N$4, one may construct an anti-linear topological operator $\theta=\pi p/N$5 that is conserved and time-reversing, yet obeys

$\theta=\pi p/N$6

where $\theta=\pi p/N$7 is a condensation defect (Choi et al., 2022). This failure of $\theta=\pi p/N$8 to equal the identity is the defining hallmark of non-invertibility in the gauge-theory setting.

A related formulation appears in defect categories. In $\theta=\pi p/N$9-dimensional topological orders equipped with charge conjugation $\mathsf T^2=C$0, one can adjoin an antiunitary defect $\mathsf T^2=C$1 with algebra

$\mathsf T^2=C$2

so that

$\mathsf T^2=C$3

rather than a single identity sector (Cordova et al., 2023). Here non-invertibility is encoded directly in fusion.

A further formulation arises from a generalized Wigner perspective. If the requirement of bijectivity is dropped while preserving transition probabilities, then a non-invertible symmetry may be realized as

$\mathsf T^2=C$4

where $\mathsf T^2=C$5 is unitary or antiunitary and $\mathsf T^2=C$6 is a positive semi-definite, non-invertible Hermitian operator, with $\mathsf T^2=C$7 acting as the identity on the physical subspace (Ortiz et al., 29 Sep 2025). In operator-theoretic terms, such a $\mathsf T^2=C$8 is a partial isometry satisfying

$\mathsf T^2=C$9

with $\mathsf T^2=(-1)^F C$0 and $\mathsf T^2=(-1)^F C$1 orthogonal projections (Ortiz et al., 29 Sep 2025). Specializing to time-reversal gives a non-invertible antiunitary transformation of the form $\mathsf T^2=(-1)^F C$2 with $\mathsf T^2=(-1)^F C$3, $\mathsf T^2=(-1)^F C$4, and $\mathsf T^2=(-1)^F C$5 (Ortiz et al., 29 Sep 2025). This formulation is conceptually distinct from topological-defect constructions, but it gives a quantum-mechanical template for non-invertible anti-linear symmetry.

2. Gauge-theory construction at rational $\mathsf T^2=(-1)^F C$6

The canonical gauge-theoretic construction appears in free Maxwell theory and massive QED at rational $\mathsf T^2=(-1)^F C$7-angle (Choi et al., 2022). For free Maxwell theory,

$\mathsf T^2=(-1)^F C$8

The naive time reversal $\mathsf T^2=(-1)^F C$9 acts anti-linearly by

$\mathrm{sgn}(t)$0

Because this flips $\mathrm{sgn}(t)$1, it fails to preserve the theory at generic rational $\mathrm{sgn}(t)$2. The remedy is a codimension-one topological interface $\mathrm{sgn}(t)$3 that shifts $\mathrm{sgn}(t)$4, realized by a $\mathrm{sgn}(t)$5 fractional quantum Hall state on the interface $\mathrm{sgn}(t)$6:

$\mathrm{sgn}(t)$7

with

$\mathrm{sgn}(t)$8

where $\mathrm{sgn}(t)$9 is a dynamical $t=0$0 one-form on $t=0$1 (Choi et al., 2022).

The resulting time-reversal defect is the composition

$t=0$2

This operator is topological and anti-linear, flips $t=0$3, and inserts the FQH layer. Its non-invertibility is expressed by the fusion relation

$t=0$4

with $t=0$5 the condensation defect obtained by summing over magnetic one-form symmetry lines on $t=0$6 (Choi et al., 2022).

The same work extends the construction beyond abelian Maxwell theory. In massive QED at $t=0$7, the composition $t=0$8 remains unbroken and flows in the infrared to the Maxwell non-invertible $t=0$9 (Choi et al., 2022). In non-Abelian gauge theory, including $2\pi$0 $2\pi$1 super Yang-Mills along $2\pi$2, the non-invertible time-reversal takes the form

$2\pi$3

with fusion

$2\pi$4

the $2\pi$5 condensation defect (Choi et al., 2022). At special points $2\pi$6 or $2\pi$7, this operator factorizes into known non-invertible duality or triality defects composed with the invertible $2\pi$8 at $2\pi$9 (Choi et al., 2022).

This framework modifies the standard statement that time-reversal exists only at $\theta$0 or $\theta$1. More precisely, that statement applies to invertible anti-unitary time-reversal. At rational $\theta$2, there is instead a conserved anti-linear topological operator without an inverse (Choi et al., 2022).

3. Ward identities, selection rules, and conserved defects

Because the non-invertible time-reversal defect extends along a constant-time slice, it commutes with the Hamiltonian:

$\theta$3

For any local operator $\theta$4,

$\theta$5

These relations imply $\theta$6, so the defect is conserved, and they yield Ward identities in correlation functions (Choi et al., 2022). In particular,

$\theta$7

from which certain time-odd correlators vanish or are related to time-reversed correlators up to insertions of the condensation defect $\theta$8 (Choi et al., 2022).

The same logic persists in massive QED, where correlators of charged fields satisfy analogous time-reversal Ward identities up to $\theta$9 (Choi et al., 2022). These identities show that non-invertible time-reversal does not simply mimic ordinary $\theta=0$0 time reversal with a missing inverse; rather, the missing inverse is replaced by a defect sector that modifies operator selection rules.

Defect-worldvolume constructions furnish an analogous structure. In $\theta=0$1-dimensional self-dual theories with $\theta=0$2 one-form symmetry, the duality defect $\theta=0$3 carries a hidden anti-unitary time-reversal $\theta=0$4 on its worldvolume (Apte et al., 2022). Geometrically, this is constructed as $\theta=0$5, where a $\theta=0$6-rotation around an axis in the defect swaps $\theta=0$7, and composition with gauging $\theta=0$8 reverses defect orientation while preserving the boundary condition (Apte et al., 2022). On defect anyons,

$\theta=0$9

and sequential application yields

$\theta=\pi$0

where $\theta=\pi$1 is charge conjugation (Apte et al., 2022). This is a non-invertible anti-unitary action internal to the defect worldvolume TQFT.

A different but related phenomenon occurs in the $\theta=\pi$2-dimensional lattice Kramers–Wannier setting. There the exact non-invertible duality operator $\theta=\pi$3 obeys

$\theta=\pi$4

with $\theta=\pi$5 the on-site $\theta=\pi$6 spin-flip and $\theta=\pi$7 the lattice translation (Seiberg et al., 2024). In the presence of the defect $\theta=\pi$8 implementing $\theta=\pi$9, parity and other invertible symmetries act projectively; for example,

$\theta=\pi p/N$00

The source describes this as the $\theta=\pi p/N$01-dimensional manifestation of a mixed anomaly between parity and the non-invertible duality symmetry (Seiberg et al., 2024). This is not a literal non-invertible time-reversal operator, but it is closely related because the symmetry algebra involving parity/time-reversal is realized projectively in the presence of the defect.

4. Anomalies, indicators, and hidden anti-unitary structure

In $\theta=\pi p/N$02 dimensions, anomalies of non-invertible symmetries can be studied by $\theta=\pi p/N$03-dimensional bulk topological quantum field theories built from Abelian two-form gauge theories with a $\theta=\pi p/N$04-form permutation symmetry (Cordova et al., 2023). Gauging the $\theta=\pi p/N$05-form symmetry produces the inflow theory for the non-invertible symmetry. Two levels of anomalies appear: the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and the Frobenius–Schur indicator of the non-invertible symmetry may be incompatible with trivial boundary dynamics (Cordova et al., 2023).

Within this framework, defects associated with ordinary $\theta=\pi p/N$06 symmetry host worldvolume theories with time-reversal symmetry $\theta=\pi p/N$07 obeying

$\theta=\pi p/N$08

with $\theta=\pi p/N$09 a unitary charge-conjugation symmetry (Cordova et al., 2023). Since $\theta=\pi p/N$10, one may equivalently write

$\theta=\pi p/N$11

Fusion with the orientation-reversed defect gives

$\theta=\pi p/N$12

which makes non-invertibility explicit (Cordova et al., 2023).

The anomalies of this algebra in $\theta=\pi p/N$13 dimensions are classified by $\theta=\pi p/N$14 in the bosonic case and by $\theta=\pi p/N$15 in the fermionic case (Cordova et al., 2023). After gauging charge conjugation $\theta=\pi p/N$16, one can define two indicators,

$\theta=\pi p/N$17

and

$\theta=\pi p/N$18

with $\theta=\pi p/N$19 the topological spin and $\theta=\pi p/N$20 the quantum dimension (Cordova et al., 2023). According to the source, if $\theta=\pi p/N$21, $\theta=\pi p/N$22 is an ordinary invertible symmetry with no ’t Hooft anomaly; if $\theta=\pi p/N$23 but $\theta=\pi p/N$24, gauging $\theta=\pi p/N$25 forces $\theta=\pi p/N$26 into a $\theta=\pi p/N$27-group extension; and if $\theta=\pi p/N$28, gauging $\theta=\pi p/N$29 makes $\theta=\pi p/N$30 genuinely non-invertible (Cordova et al., 2023). The paper describes this as a higher-dimensional analogue of the $\theta=\pi p/N$31-dimensional Frobenius–Schur obstruction.

These anomaly discussions connect directly to the hidden time-reversal symmetry on defect worldvolumes studied in self-dual $\theta=\pi p/N$32 theories (Apte et al., 2022). There, the worldvolume TQFT $\theta=\pi p/N$33 has anyon spins

$\theta=\pi p/N$34

and the anti-unitary action $\theta=\pi p/N$35 flips spins $\theta=\pi p/N$36 precisely when $\theta=\pi p/N$37 (Apte et al., 2022). This provides an explicit realization of anti-unitary structure tied to self-duality and condensation, rather than to a group-like involution.

5. Constraints on phases and dynamics

One of the central uses of non-invertible time-reversal and related defect symmetries is to constrain infrared dynamics. In $\theta=\pi p/N$38-dimensional self-dual theories with $\theta=\pi p/N$39 one-form symmetry, a symmetry-preserving vacuum state with a gapped spectrum is often forbidden (Apte et al., 2022). The source states the obstruction theorem as follows: a self-dual theory with $\theta=\pi p/N$40 one-form symmetry is gapless or spontaneously breaks the self-duality symmetry unless

$\theta=\pi p/N$41

and a symmetry-preserving gapped phase exists iff this condition holds (Apte et al., 2022). When it does fail, the combined non-invertible symmetry enforces either gaplessness or spontaneous symmetry breaking.

The same paper gives explicit examples. For $\theta=\pi p/N$42, a $\theta=\pi p/N$43 TQFT preserving duality is allowed. For $\theta=\pi p/N$44, a duality-invariant SPT exists but no non-trivial TQFT. For $\theta=\pi p/N$45, no suitable factorization exists, so at self-duality the lattice model spontaneously breaks duality in a first-order transition (Apte et al., 2022). The source also notes that for small $\theta=\pi p/N$46 the duality or triality defect is spontaneously broken at the self-dual point, whereas for large $\theta=\pi p/N$47 the system remains symmetric but gapless in the Coulomb phase (Apte et al., 2022). This places non-invertible anti-unitary structure within the general program of symmetry-enforced gaplessness beyond ordinary ’t Hooft anomalies.

A related constraint appears in $\theta=\pi p/N$48 dimensions for the exact lattice non-invertible Kramers–Wannier symmetry $\theta=\pi p/N$49 on a tensor-product Hilbert space. Any finite-range Hamiltonian commuting with $\theta=\pi p/N$50 must satisfy either: the system is gapless, or it is gapped and its global symmetry $\theta=\pi p/N$51 is spontaneously broken (Seiberg et al., 2024). In the gapped case, the number of superselection sectors is a multiple of three (Seiberg et al., 2024). The source explicitly compares this with Lieb–Schultz–Mattis-type obstructions. Although the operator $\theta=\pi p/N$52 is not itself labeled as time reversal, the work states that the symmetry algebra involving parity/time-reversal is realized projectively in the presence of the defect (Seiberg et al., 2024). This suggests a broader pattern in which non-invertible defects obstruct trivially gapped symmetric phases and force either criticality or symmetry breaking.

In gauge theory, the rational-$\theta=\pi p/N$53 non-invertible time-reversal similarly constrains RG flows. Massive QED at $\theta=\pi p/N$54 cannot trivially gap without saturating the non-invertible constraints (Choi et al., 2022). A plausible implication is that the anti-linear topological defect should be regarded as an RG-invariant datum that organizes admissible infrared phases, much as ordinary anomalies do.

6. Open quantum systems and non-invertibility through $\theta=\pi p/N$55

A distinct notion of non-invertible time-reversal appears in the study of open quantum systems (Guff et al., 2023). There the starting point is a system $\theta=\pi p/N$56 coupled to a bath $\theta=\pi p/N$57 with total Hamiltonian

$\theta=\pi p/N$58

and an anti-unitary time-reversal operator $\theta=\pi p/N$59 acting on $\theta=\pi p/N$60 such that

$\theta=\pi p/N$61

Thus the microscopic dynamics is time-reversal symmetric (Guff et al., 2023). Correlation functions of bath forces obey

$\theta=\pi p/N$62

so the bath correlations are stationary and even in time difference (Guff et al., 2023).

The key claim is that the Markov approximation does not imply a violation of time-reversal symmetry. Rather, if one keeps track of the sign of time, the secularized Markovian master equation takes the form

$\theta=\pi p/N$63

valid for all $\theta=\pi p/N$64 (Guff et al., 2023). The factor $\theta=\pi p/N$65 renders the dynamics non-invertible at $\theta=\pi p/N$66: one cannot run the Lindblad-type evolution continuously through the origin. Yet every term is chosen so as to commute appropriately with time reversal, and the equation satisfies

$\theta=\pi p/N$67

that is, full time-reversal invariance (Guff et al., 2023).

Projecting onto the energy basis yields a time-symmetric Pauli master equation,

$\theta=\pi p/N$68

with Golden-Rule rates

$\theta=\pi p/N$69

Detailed balance holds in each temporal direction:

$\theta=\pi p/N$70

The paper concludes that both the Lindblad and Pauli equations predict approach to Gibbs equilibrium for $\theta=\pi p/N$71 and $\theta=\pi p/N$72, producing two opposing arrows of thermodynamic time pointing away from $\theta=\pi p/N$73 (Guff et al., 2023).

The associated dynamical map,

$\theta=\pi p/N$74

has the semigroup property on each temporal half-axis but not across $\theta=\pi p/N$75 (Guff et al., 2023). The source therefore proposes a time-symmetric definition of Markovianity based on divisibility for pairs of times with the same sign (Guff et al., 2023). This use of “non-invertible time-reversal symmetry” differs from the topological-defect notion: the non-invertibility is not the failure of a topological defect to have an inverse, but the impossibility of continuously extending the dissipative evolution through the time origin while preserving the time-symmetric form.

7. Conceptual synthesis and common distinctions

The literature now supports several technically precise meanings of non-invertible time-reversal symmetry. In gauge theory and topological-defect language, it denotes a conserved anti-linear codimension-one operator whose fusion is not group-like. At rational $\theta=\pi p/N$76, this operator is $\theta=\pi p/N$77 and satisfies $\theta=\pi p/N$78 (Choi et al., 2022). In defect-worldvolume settings, anti-unitary symmetry can obey $\theta=\pi p/N$79 or $\theta=\pi p/N$80, so non-invertibility is encoded by fusion into multiple sectors (Apte et al., 2022, Cordova et al., 2023). In generalized Wigner form, non-invertible antiunitary symmetry appears as a partial isometry $\theta=\pi p/N$81 on an extended gauged Hilbert space (Ortiz et al., 29 Sep 2025). In open-system dynamics, non-invertibility refers to the inability to continue a time-symmetric Markovian semigroup through $\theta=\pi p/N$82 despite full microscopic $\theta=\pi p/N$83-invariance (Guff et al., 2023).

A common misconception is that non-invertible time-reversal is merely ordinary time reversal at a special parameter value with some anomaly attached. The rational-$\theta=\pi p/N$84 construction shows something sharper: the operator is conserved and anti-linear, but it cannot be implemented by an anti-unitary operator because its adjoint times itself is a condensation defect rather than the identity (Choi et al., 2022). Another misconception is that anti-unitary symmetry must always be invertible by Wigner’s theorem. The generalized result states that once bijectivity is relinquished, probability-preserving non-invertible symmetries are realized as partial isometries on an extended gauged Hilbert space (Ortiz et al., 29 Sep 2025). Conversely, in open quantum systems, the presence of a sign function in the generator does not by itself signal broken time-reversal symmetry; the cited work argues that the Markov approximation itself does not break microscopic time-reversal symmetry, and that the apparent arrow of time arises only when one restricts to $\theta=\pi p/N$85 or $\theta=\pi p/N$86 alone (Guff et al., 2023).

Taken together, these results indicate that non-invertible time-reversal symmetry is not a single universal algebraic object but a family of related anti-linear structures appearing in QFT, topological order, lattice models, quantum mechanics, and open-system dynamics. What unifies them is the replacement of group-like inversion by condensation, projection, or half-axis semigroup structure, together with new Ward identities, anomaly indicators, and infrared constraints that are absent for ordinary invertible time reversal.