Universal Deformations

TL;DR

The paper introduces universal deformations for quantum field theories with local topological operators, showing that adding dimension-0 perturbations can exactly shift the vacuum energy densities of distinct universes labeled by these operators, thereby controlling phase structure and confinement in 2d gauge theories. It develops a general framework for invertible LTOs and computes explicit energy shifts, with a detailed treatment of the charge-N Schwinger model where a concrete LTO realizes a 1-form symmetry and a deformation via U_1(x) tunes vacuum energies and breaks chiral symmetry without generating a fermion mass. The authors extend the construction to non-invertible LTOs in 2d YM, analyze lattice regularizations, and demonstrate that suitable deformations can deconfine multiple Wilson-loop representations. They also discuss deep implications for symmetry breaking, Coleman–Mermin–Wagner-type expectations for generalized symmetries, and an apparent violation of EFT naturalness, along with possible resolutions via non-invertible symmetries and emergent scales.

Abstract

QFTs with local topological operators feature unusual sectors called "universes," which are separated by infinite-tension domain walls. We show that such systems have relevant deformations with exactly-calculable effects. These deformations allow one to dial the vacuum energy densities of the universes. We describe applications of these deformations to confinement in 2d gauge theories, as well as a curious violation of the effective field theory naturalness principle.

Figures (5)

• Figure 1: Each sketch above shows the effective potential energies in two universes as a function of the vacuum expectation value of some scalar field. The three sketches are related by dialing the coefficient of an appropriate "universal deformation" by a local topological operator. On the left, the deformation is turned off, and the two universes have distinct vacuum energies. If we dial the coefficient of the deformation, we can make their vacuum energies coincide (middle figure), or change which universe has the lower vacuum energy density (right figure).
• Figure 2: Two static test particles with charge $\pm 1$ are held at a fixed large distance from each other in a 2d abelian gauge theory where the dynamical fields have charge $N>1$. If the universe in between the particles (shown in red at a fixed time) has a higher vacuum energy density than the universe outside, the particles will be confined by a linear potential.
• Figure 3: The blue and gold curves in the plots above, with minima at $0$ and $\pi$ respectively, are effective potential energy densities in the $k=0$ and $k=1$ universes of the charge-$2$ Schwinger model, displayed as a function of the bosonized scalar $\varphi$ with six different choices of the mass parameter $m$ and universal deformation parameter $\Lambda$. Here we have chosen $\theta=\delta=0$, and therefore $\chi=0$. The only effect of the deformation is to shift the relative potential energy density curves in the two universes, without affecting their shape. At $m=0$, the deformation explicitly breaks chiral symmetry, but no fermion mass term is generated.
• Figure 4: A sketch of the phase diagram of the charge-$2$ Schwinger model as a function of the fermion mass term and the local topological operator deformation, with coefficients $m$ and $\Lambda^2$ respectively. Here we have chosen $\theta=\delta=0$, and therefore $\chi=0$. The theory is in a confined phase in the bulk of the plot, but it is deconfined (with spontaneously broken $\mathbb{Z}_2$$1$-form symmetry) along the red curve.
• Figure 5: A sketch of some energy scales in 2d $U(1)$ gauge theory with gauge coupling $e$ with a charge $N$ fermion $\psi$ with zero mass in the bare action. We also add a charge $1$ field $Q$ with a large mass $M \gg e$. The scale $e$ can be thought of as a kind of "meson" ($\overline{\psi}$-$\psi$ bound state) mass scale. Imagine we add a relevant deformation by an approximately-topological local operator with a coefficient $\Lambda$, with the hierarchy $e \ll \Lambda \ll M$, which is charged under the chiral symmetry $(\mathbb{Z}_N)_A$ of the charge $N$ fermion. Normally one would expect that this would induce a mass term of size $\Lambda$. But for reasons discussed in the text, the mass term has to vanish if $M \to \infty$, so $m_{\psi}$ has to be proportional to a positive power of $\Lambda/M$. In the sketch we show the most naive possibility $m_{\psi} \sim e \Lambda/M$.