Galois Shuffle Operation in Fusion Rings

• Galois shuffle is a method that redefines fusion rings by conjugating modular data, shifting conformal weights in CFTs.
• It employs a similarity transformation to modify fusion coefficients and Cardy states, affecting symmetry and physical interpretations.
• In cyclotomic double shuffle Lie algebras, the operation enacts a Galois action on formal alphabets, preserving crucial Hopf algebra properties.

The Galois shuffle operation is a mathematical procedure that systematically alters the algebraic and symmetry structures of a given object—most notably appearing in the paper of fusion rings in conformal field theory (CFT) and in the double shuffle Lie algebras governing multiple zeta values and polylogarithms. In both contexts, the Galois shuffle implements an action analogous to a similarity transformation, either within the modular data of CFTs (producing new fusion rules and symmetry topological field theories, or SymTFTs) or via Galois automorphisms in the theory of cyclotomic multiple zeta value relations. This operation is generically characterized by the Galois conjugation of algebraic data, such as modular matrices or formal alphabets, with profound consequences for the classification of symmetry-protected phases, renormalization group flows, and the arithmetic structures underlying special function identities.

1. Formal Definition and Algebraic Mechanism

In the setting of two-dimensional rational conformal field theory, the Galois shuffle arises as a specific similarity transformation labeled by a distinguished primary (often the field of minimal conformal weight other than the identity) denoted $o$. Given a fusion ring $A$ of a (possibly nonunitary) diagonal CFT—specified by a set of primary fields $\alpha\in I$, modular data $(S,T)$, central charge $c$, and conformal weights $h_\alpha$—one selects an alternative "effective vacuum" $o\in I$ such that $S_{o,\beta}\neq 0$ for all $\beta$. An invertible operator $\eta^{[o]}$ is defined that shifts the Virasoro zero-mode:

$$L_0^{[o]} - c^{[o]}/24 = \eta^{[o]} (L_0 - c/24) (\eta^{[o]})^{-1},$$

with

$$c^{[o]} = c - 24h_o, \qquad h_\alpha^{[o]} = h_\alpha - h_o.$$

This operation conjugates the projection operators and Cardy states, resulting in a deformation of both the T-matrix and the Verlinde fusion formula. The T-matrix acquires a phase $e^{-2\pi i h_o}$, while the S-matrix remains numerically unchanged but is inserted with $S_{o,\delta}$ in place of the traditional vacuum $S_{0,\delta}$ in the modified Verlinde formula. The new ("shuffled") fusion coefficients take the form

$$N_{\alpha\beta}^{[o]\,\gamma} = \sum_{\delta \in I} \frac{S_{\alpha\delta}S_{\beta\delta}\overline{S_{\delta\gamma}} S_{o,\delta}}{S_{0,\delta}}.$$

The symmetry operators transform as $$Q_\alpha^{[o]} = \sum_\beta S_{\alpha\beta}S_{o,\beta}P_\beta^{[o]}$$, and the ring $A^{[o]} = \{ Q_\alpha^{[o]} \}$ is a fusion ring with structure constants given by the shuffled $N_{\alpha\beta}^{[o]\,\gamma}$.

2. Galois Shuffle in Cyclotomic Double Shuffle Lie Algebras

Within the algebraic framework of multiple zeta values and cyclotomy, the Galois shuffle operation is understood as a cyclotomic Galois action on the underlying formal alphabet. Given Racinet’s cyclotomic double shuffle Lie algebra $dmr_0^{\mu_N}$—constructed from a formal alphabet $X = \{x_0\} \cup \{x_\zeta\mid\zeta\in\mu_N\}$ and corresponding completed $\mathbb{Q}$-algebra $\mathbb{Q}\langle\langle X\rangle\rangle$—the action of the absolute Galois group $$G_N = \text{Gal}(\mathbb{Q}(\mu_N)/\mathbb{Q}) \cong (\mathbb{Z}/N\mathbb{Z})^\times$$ is implemented as

$$\delta_\gamma: x_0 \mapsto x_0,\,\, x_\zeta \mapsto x_{\zeta^\gamma},\,\, \gamma \equiv k \mod N\quad (k \in (\mathbb{Z}/N\mathbb{Z})^\times).$$

This "Galois shuffle operation" preserves the primitive condition (the shuffle coproduct) due to the Hopf automorphism property, while conjugating cyclotomic data:

$$\psi(x_0 x_\zeta) + \psi(x_\zeta x_0) = 0 \quad \longrightarrow \quad \psi(x_0 x_{\zeta^\gamma}) + \psi(x_{\zeta^\gamma} x_0) = 0.$$

On alternative $\mathbb{Q}$-forms—such as the congruent version $dmr_0^{[N]}$ constructed with an alphabet indexed by $\mathbb{Z}/N\mathbb{Z}$—an analogous action is given by $$\tilde{\delta}_\gamma: \tilde{x}_\alpha \mapsto \tilde{x}_{\gamma\alpha}$$.

3. Construction of Nonlocal Symmetry TFTs via Shuffle

The Galois shuffle provides a bridge between local nonunitary CFTs and nonlocal but unitary CFTs. Specifically, starting with a local, possibly nonunitary CFT (with vacuum $0$ and effective vacuum $o$), one applies the similarity transformation $\eta^{[o]}$:

• Shift every conformal weight by $-h_o$.
• Conjugate primitive idempotents $P_\alpha$ and Cardy modules.
• Form the new symmetry operators $Q_\alpha^{[o]}$.
• The resulting fusion ring $A^{[o]}$ is realized in a nonlocal but unitary CFT.

This process defines an explicit ring isomorphism $\iota: A \to A^{[o]}$, rendering both fusion rings abstractly identical, though their representation-theoretic interpretations and boundary conditions diverge.

4. Physical and Mathematical Implications

The Galois shuffle operation bears extensive implications:

• RG Flows: Massless RG flows are preserved under the shuffle as commutative diagrams of ring homomorphisms, aligning patterns of broken charges in both original and shuffled fusion rings. Massive (gapped) flows do not preserve the NIM-rep property—the annulus partition function can acquire negative values in the nonlocal theory, signaling the breakdown of local boundary conditions. Cardy states must be interpreted as inner products of gapped ground states, not true boundary partition functions.
• Boundary/Domain-Wall Phenomena: The correspondence between gapped/gapless phases and domain-wall fusions is transferred isomorphically to the nonlocal fusion ring, with the crucial change that the nonlocal theory lacks ordinary local boundaries. The breakdown of Cardy positivity, $$\sum_\gamma N^{[o]}_{\alpha\beta} \chi_\gamma(-1/\tau)\geq 0$$, reflects the nonlocal character of the theory after the Galois shuffle.
• Ring Theoretic Isomorphism: All ring-theoretic aspects—fusion, modules, RG flow classification—are carried across the shuffle isomorphism, but only the local nonunitary theory realizes a NIM-rep structure.

5. Concrete Examples

Two archetypal examples demonstrate the operation:

Model	CFT Data	Fusion Before Shuffle	Fusion After Galois Shuffle
M(2,5) (“Lee–Yang”) vs. Osp(1,2)_1 WZW	$c=-22/5$, $h_o=-1/5$	$Q_o \times Q_o = Q_I + Q_o$	$Q_I^{[o]} \times Q_I^{[o]} = Q_o^{[o]} - Q_I^{[o]}$
M(2,7) vs. Osp(1,2)_2	$c=-68/7$, $h_\phi=-2/7$, $h_o=-3/7$	See appendix of (Fukusumi et al., 1 Dec 2025)	Fusion coefficients match those of Osp(1,2)_2 WZW

• For M(2,5), the nonunitary minimal model’s fusion ring (Fibonacci) is mapped by the Galois shuffle to the fusion ring of a nonlocal but unitary Osp(1,2)_1 WZW model, with conformal weights $h_I^{[o]}=1/5$, $h_o^{[o]}=0$, $c_{\mathrm{eff}}=2/5$.
• In M(2,7) and Osp(1,2)_2, the effective central charge after shuffle is $c_{\mathrm{eff}}=4/7$, and fusion matches the unitary model exactly.

6. Connections to Galois Theory and Motivic Structures

In the context of multiple zeta values, the Galois shuffle embodies the expected interplay between the arithmetic Galois action and the de Rham (motivic) realization. The two $\mathbb{Q}$-forms of the cyclotomic double shuffle Lie algebra—Racinet’s classical and the congruent version—are isomorphic under a transformation $\mathcal{F}$ matching the respective alphabets. Invariance under the Galois shuffle identifies each form as the fixed-point subspace under the relevant group action, see (Furusho et al., 2 Feb 2025). This provides a unifying perspective: the shuffle relations for level-$N$ congruent multiple zeta values and for multiple polylogarithms at roots of unity are Galois-conjugate, reflecting the underlying motive and its symmetry.

A plausible implication is that the Galois shuffle serves as a fundamental tool in exposing hidden symmetries of algebraic and physical structures, connecting nonunitary and unitary realizations, as well as disparate $\mathbb{Q}$-forms related by arithmetic descent.

7. Conceptual Summary and Outlook

The Galois shuffle operation is the algebraic implementation of a similarity or Galois conjugation that, in the CFT setting, maps the algebraic data of a fusion ring—through an explicit shift associated with a distinguished field—to a ring-theoretically isomorphic structure, potentially moving the system outside the class of unitary, local, or NIM-rep-realizable theories. In motivic and multiple zeta value theory, it organizes the passage between different rational structures as dictated by Galois symmetries. Across both areas, it reveals a new layer of structure within the classification of symmetry, RG phases, and the arithmetic of special functions, with ring-theoretic correspondence preserved but representation-theoretic and boundary data fundamentally altered (Fukusumi et al., 1 Dec 2025, Furusho et al., 2 Feb 2025).