Consistency of gauging discrete symmetries by field-space identification: conical fixed-point singularities in the continuum Krauss-Wilczek construction

Krauss and Wilczek (1989) showed that a discrete symmetry can be a local (gauge) symmetry of a four-dimensional continuum quantum field theory without introducing any new gauge field: the symmetry is implemented as a prescription on the functional integral in which the fields take values in the quotient of field space by the discrete group action. They derived striking consequences - Aharonov-Bohm observables, discrete gauge 'quantum hair' on black holes, and protection of gauged discrete symmetries against violation by wormholes or black-hole evaporation - and noted one unanalyzed caveat about the construction itself: nontrivial elements of the discrete group have fixed points in field space, so the quotient has conical, orbifold-like singularities; in their words, 'it is conceivable that these singularities lead to subtle problems that have not yet been discerned'. The other questions raised in or around the source paper - which discrete symmetries are anomaly-free and gaugeable, whether black holes can carry discrete gauge charge, and whether discrete symmetries in string-derived effective theories are gauge remnants - have since been taken up elsewhere (discrete-anomaly machinery and its modern cobordism classification; the quantum-hair literature; the no-global-symmetry results in quantum gravity). The one piece left genuinely untouched is the consistency of the quotient construction itself: the modern definition of gauging a finite symmetry (coupling to a discrete gauge field and summing over sectors, with its BF-type topological description) bypasses rather than analyzes it. The problem posed here is to settle that question: formulate the field-space-quotient prescription precisely, determine whether the conical fixed-point singularities induce renormalization-theoretic inconsistencies in the four-dimensional continuum theory, and establish whether the prescription defines the same quantum field theory as the standard gauging.

Background

A symmetry is called global if it is a physical invariance of the laws (the same transformation applied everywhere, yielding selection rules such as baryon-number conservation), and local or gauged if it is a redundancy of the description: only gauge-invariant quantities are meaningful, and no physical process whatsoever - not even black-hole evaporation or quantum-gravitational tunneling - can violate a gauged symmetry. A discrete symmetry is one whose transformations form a finite group, the simplest example being the cyclic group $\mathbb{Z}_p$ of $p$ elements. In the late 1980s it was argued, notably in Gilbert's analysis of Euclidean wormholes (hypothetical narrow tunnels connecting distant or 'baby-universe' regions of spacetime, which would induce arbitrary symmetry-violating local interactions in the low-energy effective theory), that quantum-gravitational effects violate every global symmetry, including discrete ones such as the $\mathbb{Z}_2$ 'R-parity' that protects the proton in supersymmetric extensions of the Standard Model. This threatened rapid proton decay in otherwise viable models.

Krauss and Wilczek (1989) pointed out that a discrete symmetry can secretly be a local symmetry of an underlying gauge theory, in which case it is absolutely protected. Their canonical model is a $U(1)$ gauge theory with two complex scalar fields, $\eta$ of electric charge $pe$ and $\xi$ of charge $e$. When $\eta$ condenses at a high mass scale $M$ (spontaneously breaking the $U(1)$), the low-energy theory of $\xi$ alone retains the invariance $\xi \to e^{2\pi i/p}\xi$: a residual $\mathbb{Z}_p$. Low-energy probes see only the selection rules of this $\mathbb{Z}_p$ and cannot tell whether it is global or local - but the local nature is observable. The broken phase contains stable strings (vortices) threaded by magnetic flux $2\pi/(pe)$, and a $\xi$ quantum encircling such a string acquires an Aharonov-Bohm phase (the quantum phase $e^{i\theta}$, with $\theta$ = charge times flux, picked up by a charged particle transported around a confined magnetic flux). The flux strings can be made arbitrarily heavy by raising $M$, so the local discrete symmetry has a sensible limit as $M \to \infty$ in which no new light gauge field appears: implementing a local discrete symmetry requires no modification of the Lagrangian. The difference between the global and local versions is instead a prescription for the functional integral: in the local version the field is taken to take values not in the complex plane but in the quotient of the complex plane by the $\mathbb{Z}_p$ action, and this difference, as the authors note, shows up only for large fluctuations of the fields. Krauss and Wilczek drew out the consequences: in the Higgs phase the gauge charge is conserved only modulo $pe$ (the condensate screens charge in multiples of $pe$), and this discrete charge is measured by the observable $\exp(2\pi i Q/pe)$, which by Gauss's law can be rewritten as a surface integral of the electric field over a sphere at infinity and so cannot be affected by the gravitational collapse of whatever lies inside; black holes can therefore carry discrete gauge charge - a form of nonclassical 'quantum hair' measurable through Aharonov-Bohm scattering, evading the classical no-hair theorem (the statement that a stationary black hole is characterized only by its mass, angular momentum, and continuous gauge charges). And since gauged symmetries cannot be violated by anything, promoting phenomenologically needed discrete symmetries to local ones protects matter stability against wormholes.

The authors flagged two caveats about the construction. First, some discrete symmetries carry 't Hooft anomalies - topological obstructions, detectable by studying a theory on nontrivial background gauge fields, that make gauging inconsistent - and identifying such anomalies is, in their words, 'a difficult but well developed art' into which they did not enter. Second - the question at issue here - the quotient construction is not locally trivial: in the most interesting cases the discrete transformations have fixed points in field space, so the quotient has conical (orbifold-like) singularities there. (An orbifold is the quotient of a space by a discrete group action; its fixed points are generically singular, like the tip of the cone obtained by identifying the plane under a rotation.) In the model above the origin $\xi = 0$ is fixed by every group element and the quotient field space $\mathbb{C}/\mathbb{Z}_p$ is a cone. They wrote that 'it is conceivable that these singularities lead to subtle problems that have not yet been discerned', noting the resemblance to orbifold constructions in string theory. The concern is dimension-specific: in two spacetime dimensions scalar fields are dimensionless, so arbitrary functions of them - including restrictions to exotic field spaces - still give renormalizable theories, and orbifold target spaces are under complete control there; in four dimensions scalar fields carry mass dimension one, a theory whose fields are constrained to a curved or singular submanifold is a nonlinear sigma model, and renormalization generates infinitely many counterterms (extra local terms added to the Lagrangian, with free coefficients, that absorb the divergences) built from the geometry of the field space. Whether the singular geometry at the cone tip produces a genuine consistency problem in four dimensions was left open.

The modern understanding takes a different route. Gauging a finite symmetry is now defined by coupling the theory to a discrete ($\mathbb{Z}_p$-valued) gauge field and summing over gauge sectors; this adds no propagating degrees of freedom, and at long distances the discrete gauge sector is described by a topological BF theory (a topological field theory of a one-form gauge field $A$ and a two-form field $B$ with action $\int B \wedge dA$). Banks and Seiberg (2011) showed that such a $\mathbb{Z}_p$ gauge theory carries an emergent $\mathbb{Z}_p$ one-form symmetry - a generalized symmetry acting on extended line operators rather than on local points - and argued, as did Harlow and Ooguri (2018) using the AdS/CFT correspondence (the equivalence between certain gravitational theories in anti-de Sitter space and nongravitational field theories), that quantum gravity admits no exact global symmetries at all, discrete ones included, so every exact discrete symmetry in a consistent quantum theory of gravity must be gauged in this standard sense. This modern definition never quotients field space. Whether Krauss and Wilczek's field-space-identification prescription - the 'no gauge field needed' picture in which the fields simply live on the quotient - defines the same quantum field theory as this standard gauging, and whether its conical fixed-point singularities are renormalization-theoretically benign, is the part of the 1989 construction that was never analyzed.

Problem Statement

Determine whether the Krauss-Wilczek realization of a local (gauged) discrete symmetry in a four-dimensional continuum quantum field theory - implemented not by modifying the Lagrangian but by requiring the fields to take values in the quotient of field space by the discrete group action, with the functional integral carried out over that quotient - is a fully consistent definition of the gauged theory, given that the quotient has conical, orbifold-like singularities at the fixed points of nontrivial group elements; in the source's words, whether 'these singularities lead to subtle problems that have not yet been discerned'. Concretely, the problem is to: (1) formalize the prescription precisely for a general four-dimensional theory with a finite symmetry group, at minimum the canonical case of a complex scalar with $\xi \to e^{2\pi i/p}\xi$ so that field space is the cone $\mathbb{C}/\mathbb{Z}_p$ - specifying the configuration space, the functional-integral measure, the allowed sectors (including configurations winding around the conical locus), and a regularization; (2) analyze the renormalization of the resulting four-dimensional theory, determining which local counterterms the singular quotient requires - in particular counterterms associated with the fixed-point locus, where the quotient geometry is singular and the gauge-invariant variables have vanishing Jacobian - and whether any genuine obstruction arises, such as an unabsorbable divergence, a failure of unitarity or of reflection positivity (the requirement that Euclidean correlators, analytically continued back to real time, define a unitary theory with nonnegative probabilities), or an ambiguity in a physical observable, that is absent both in the ungauged theory and in the standardly gauged theory; and (3) assess the prescription against the theory it purports to define - the $M \to \infty$ discrete-gauge limit of the broken-$U(1)$ model, formulated in modern language as the standard gauging of the same finite symmetry by coupling to a discrete gauge field and summing over gauge sectors. The source question is settled either way: a demonstration that the quotient theory is consistent and a demonstration that the conical singularities do imply a subtle problem - an internal inconsistency of the quotient theory such as an unabsorbable divergence, a failure of unitarity or reflection positivity, or an ambiguity in a physical observable - both resolve it, even if no pairwise observable disagreement with the standard gauging is exhibited.

References

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