Non-Invertible Chiral Symmetry in Lattice QFT

• Non-invertible chiral symmetry is a generalized quantum field theory symmetry generated by non-invertible topological defect operators, distinguishing it from conventional chiral symmetry.
• It arises in lattice gauge theories through formulations like Ginsparg–Wilson and overlap fermions, where renormalization group blockings yield an infinite, non-commuting group of chiral transformations.
• This framework addresses key challenges such as fermion doubling and CP symmetry ambiguities, ensuring proper anomaly matching and a consistent mapping between lattice and continuum theories.

Non-invertible chiral symmetry refers to a generalization of conventional chiral symmetries in quantum field theory, where the symmetry is generated not by invertible group actions but by topological defect operators whose fusion rules fail to admit inverses. This structure naturally arises in several key settings, including lattice gauge theories with Ginsparg-Wilson fermions, continuum renormalization group flows, and models exhibiting lattice artifacts such as @@@@1@@@@. The phenomenon is especially prominent in the overlap formulation of lattice fermions, where the abundance of equivalent but non-commuting lattice chiral symmetries reflects the structure of non-invertible chiral symmetry.

1. Ambiguity and Multiplicity of Lattice Chiral Symmetry

The Ginsparg–Wilson (GW) relation,

$\gamma_5 D + D \gamma_5 (1 - D) = 0,$

admits infinitely many algebraically equivalent formulations on the lattice. For overlap fermions, diverse choices of "left" and "right" chiral operators (e.g., $\gamma_L^{(1)} = \gamma_5$ and $\gamma_R^{(1)} = \gamma_5 (1-D)$, or $\gamma_L^{(-1)} = (1-D) \gamma_5$ and $\gamma_R^{(-1)} = \gamma_5$) lead to an infinite group of chiral symmetries with non-commuting generators. The combination

$$(\gamma_L^{(-1)} \gamma_L^{(1)})^n D - D (\gamma_R^{(-1)} \gamma_R^{(1)})^n = 0$$

demonstrates that the lattice admits a non-abelian, non-invertible group structure, with the generators differing by their action on the Hilbert space and corresponding to distinct conserved currents and lattice Ward identities. All these symmetries collapse in the continuum limit ($a \to 0$) to the same $\gamma_5$ chiral symmetry, but on the lattice they encode a richer, non-invertible symmetry structure characteristic of the overlap construction (Cundy et al., 2011).

2. Renormalization Group Blocking and Origin of Symmetry Excess

The origin of the infinite group of lattice chiral symmetries is explained via renormalization group (RG) blocking from the continuum. Blocking is performed with operators $B^{(\eta)}$ and $\bar{B}^{(\eta)}$,

$$B^{(\eta)} = D_0^{-((1+\eta)/2)} Z D^{((1+\eta)/2)}, \qquad \bar{B}^{(\eta)} = D^{((1-\eta)/2)} Z^\dagger D_0^{-((1-\eta)/2)}$$

for tunable parameter $\eta$ and a unitary operator $Z$ ensuring locality. Varying $\eta$ produces distinct, equivalent blockings and correspondingly different lattice chiral symmetry generators. This explains the Mandula group of overlap chiral symmetries: each blocking defines its own mapping from continuum to lattice fields, resulting in an overabundance of symmetry at finite lattice spacing, all of which reduce to the correct continuum symmetry. The group multiplication of the lattice generators (for instance, $$(\gamma_L^{(\eta_1)} \gamma_L^{(\eta_2)}) = \gamma_L^{(\eta_1-\eta_2+1)}\gamma_5$$) is non-abelian and does not admit an overall inverse, characteristic of non-invertible categorical symmetry (Cundy et al., 2011).

3. Technical and Conceptual Challenges: Fermion Doublers and CP

A conceptual challenge arises due to the spectrum of the lattice Dirac operator: in addition to physical zero modes, there exist doubler modes with eigenvalue $2/a$ which do not have continuum counterparts. This disrupts the mapping between continuum and lattice eigenmodes, especially in constructing overlap operators and defining lattice $\mathcal{CP}$ symmetry. To resolve the issue, an additional "doubler field" is introduced into the continuum action: $$S = \bar{\psi}_0 D_0 \psi_0 + \bar{\psi}_d (D_d + 2/a) \psi_d$$ where $D_0$ is the standard Dirac operator and $D_d$ is its CP-transformed version. The heavy doubler field mimics the lattice doublers' contribution, enabling a correct RG mapping from continuum to lattice. Applying the blocking to this extended action yields extra GW-type relations and additional non-commuting chiral symmetries. This stratagem ensures that lattice ambiguities (arising from the lack of doubler analogues in the continuum) are compensated, and the "excess" of chiral symmetry is properly interpreted as a renormalization- and regularization-dependent effect (Cundy et al., 2011).

4. Extended Ginsparg–Wilson Relations and Operator Group Structure

With both physical and doubler branches included, the overlap operator is decomposed: $$a D_1 = 1 + \gamma_5 \operatorname{sign}[\gamma_5(\frac{a^2 D_0}{1 - a^2 D_0^\dagger D_0 / 4} - m)], \qquad a D_2 = 1 - \gamma_5 \operatorname{sign}[\gamma_5(\frac{a^2 D_d}{1 - a^2 D_d^\dagger D_d / 4} - m)]$$ where $m$ is chosen so $0 < m < 2$ for locality. Each block satisfies its own GW relation. The chiral symmetry operators generated via RG blocking have combinatorial structure, e.g. for the left chiral generator,

$$\gamma_L^{(1+2n)} = \gamma_L^{(1)} (\gamma_L^{(-1)} \gamma_L^{(1)})^n$$

and similar for $\gamma_R$. The non-invertibility is explicit: different symmetry transformations act on different eigenspaces, their composition forms a non-abelian infinite group, and at doubler eigenvalues the relations degenerate to distinct behavior compared with the physical sector. Thus, the non-invertible chiral symmetry contains a richer algebraic structure than an ordinary group symmetry (Cundy et al., 2011).

5. Mathematical Structure: Rotation Matrices and Ward Identities

Mathematically, the operators generated by blocking can be expressed in terms of $\mathrm{SO}(2)$ rotation matrices $R(\cdot)$: $$\gamma_L^{(\eta)} = R((1-\eta)(\varphi - \alpha)) \gamma_5, \qquad \gamma_R^{(\eta)} = \gamma_5 R((1+\eta)(\varphi - \alpha) + 2\alpha - \pi)$$ where $\varphi$ is a function of the Dirac operator's eigenvalues and $\alpha$ is defined via $\tan 2\alpha = (m/2) \tan 2\varphi$. These formulas compactly summarize the action of each chiral symmetry generator on the (finite and doubler) eigenstates. The explicit implementation of conserved currents and lattice Ward identities is dependent on the choice of blocking—reflecting non-invertibility at the algebraic and functional level. Relations like

$$\gamma_L^{(\eta_1)} \gamma_L^{(\eta_2)} = \gamma_L^{(\eta_1-\eta_2+1)}\gamma_5$$

emphasize the groupoid structure of the lattice chiral symmetry algebra. In the continuum limit, all such symmetries merge into the standard $\gamma_5$ symmetry, but at finite lattice spacing the non-invertible structure is essential to the correct encoding of chiral properties (Cundy et al., 2011).

6. Physical and Conceptual Implications

The existence of infinitely many non-commuting lattice chiral symmetry transformations (and corresponding excess conserved currents) has important implications for defining lattice anomalies, currents, and the treatment of CP symmetry. Non-invertibility at the operator level underlies the necessity of carefully choosing renormalization/blocking schemes when connecting lattice and continuum quantities—especially for calculating anomaly coefficients and constructing symmetry-respecting observables. This formalism explains the Mandula "abundance" of lattice chiral symmetries as a categorical structure stemming from renormalization freedom. Importantly, it clarifies that in the correct continuum (zero lattice spacing) limit, physical observables must be independent of the blocking choice, ensuring universality while the off-shell structure remains non-invertible at finite lattice cutoff (Cundy et al., 2011).

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By elucidating the multiplicity and structure of lattice chiral symmetries via RG blockings and doubler field extensions, the framework establishes non-invertible chiral symmetry as a generic property of regularized fermionic systems, with significance for anomaly matching, CP transformations, and the analytic continuation between continuum and lattice formulations in non-perturbative gauge theories.