Symmetry Transmutation and Anomaly Matching

TL;DR

The paper introduces symmetry transmutation, a mechanism by which a UV zero-form global symmetry can act in the IR as a higher-form symmetry, with exact anomaly matching achieved through an extended map Φ between UV background fields and IR higher-form backgrounds. It generalizes the traditional symmetry-homomorphism picture to include central extensions classified by H^2(G,K), linking transmutation to symmetry fractionalization and providing concrete gauge-theory realizations across 1+1, 2+1, and 3+1 dimensions, including lattice models and QCD-like theories. Across multiple examples (e.g., 1+1d QED, two-flavor QED3, and one-flavor QCD4), UV zero-form symmetries that do not act faithfully in the IR transmute into one-form (or higher-form) IR symmetries, with the UV 't Hooft anomalies matched by IR anomalies of these emergent higher-form symmetries. The work also clarifies the relationship between symmetry transmutation and symmetry fractionalization, provides geometric and defect-based pictures of the transmutation map, discusses emergent anomalies, and outlines general conditions for when transmutation occurs, highlighting its potential relevance to Higgs-confinement continuity and SP T-like phenomena. Overall, symmetry transmutation offers a unifying lens to understand how UV constraints propagate into IR topological and phase structures via higher-form symmetry data.

Abstract

We explore a situation where a global symmetry of the ultraviolet (UV) theory does not act faithfully on the local infrared (IR) degrees of freedom, but instead acts effectively as a higher-form symmetry. We refer to this phenomenon as symmetry transmutation, where the UV symmetry is "transmuted" into a higher-form symmetry in the IR. Notably, unlike emergent (accidental) symmetries, which are approximate, these symmetries are exact. We illustrate the ubiquity of this phenomenon in various continuum and lattice systems and provide examples where the 't Hooft anomalies of the UV symmetry are matched by those of the new higher-form symmetry in the IR. We also show that in certain phases and for certain energies, the UV baryon-number symmetry of one-flavor QCD is transmuted into a discrete one-form global symmetry. Finally, we compare our symmetry transmutation to the well-known phenomenon of symmetry fractionalization.

Figures (4)

• Figure 1: The configuration of the spatial circle with a (topological) charge conjugation defect at $x=0$ (the blue point) and a non-topological defect of charge $q$ at $x=x_0$ (the red point). The latter is given by $W$ of \ref{['Wqdef']}. The electric field between the two defects is $f_{01} = e^2(Q-\frac{\theta}{2\pi}) = \pm e^2 \frac{q}{2}$. Note that for $\theta=0$ and even $q$, the Hilbert space has a single state. And for odd $q$, the Hilbert space is empty. For $\theta=\pi$, there are no states unless we add a defect with an odd charge $q$.
• Figure 4: The action of the UV zero-form symmetry $G^{(0)}$ on an IR line defect $L$. The symmetry operators $U_g$, for $g \in G$, can act projectively on the $L$-defect Hilbert space. This is labeled by the phase $\gamma_L(g,h)$, which is determined by the IR one-form symmetry via $[\gamma] \in H^2(G,K)$. Intuitively, the phase $\gamma_L(g,h)$ arises from the region of the intersection of $L$ with the symmetry operators. It is associated with a contact term there. Here, we limit ourselves to $U_g$ that do not change the anyon type. The more general case, involving permutation of lines is mentioned in the text.
• Figure 5: The topological junction ${\cal O}$ between the ($d-1$)-form symmetry operators/defects $W$ and $W^{-1}$. The point operator ${\cal O}=\phi^2$ is charged under the zero-form symmetry operator $U$, i.e., $U{\cal O} = - {\cal O}U$.
• Figure 7: Defects demonstrating the transmutation of a $\mathbb{Z}_2^X \times \mathbb{Z}_2^Y$ zero-form symmetry to a $\mathbb{Z}_2^{(1)}$ one-form symmetry. In the middle panel, the $\mathbb{Z}_2^X \times \mathbb{Z}_2^Y$ zero-form symmetry defects $U_X$ and $U_Y$ are inserted along two spheres that intersect each other along the circle $S^1$. They also intersect the line operator $W$ at the points $x_1,x_2,x_3,x_4$. The zero-form $\mathbb{Z}_2^X \times \mathbb{Z}_2^Y$ symmetry could act projectively on $W$, i.e., $U_XU_YU_X^{-1}U_Y^{-1}=\gamma=\pm 1$. As can be seen in the left panel, the phase $\gamma$ can be determined by shrinking the defects. The right panel configuration describes the IR perspective of it, where the UV zero-form symmetries are transmuted into a $\mathbb{Z}_2^{(1)}$ one-form symmetry, whose line operator $L(S^1)$ is localized at the intersection circle $S^1$. Shrinking the one-form symmetry line operator should also lead to $\gamma$. Therefore, $\gamma$ is the result of braiding $W$ and $L$, i.e., the action of the one-form symmetry on $W$.