Rigorous nonperturbative establishment of the 2+1-dimensional boson–fermion duality (3d bosonization) and its duality web

Seiberg, Senthil, Wang and Witten (2016) conjectured a web of infrared dualities in 2+1-dimensional quantum field theory, whose seed statement equates a free massless Dirac fermion with the O(2) Wilson–Fisher fixed point coupled to a dynamical U(1) gauge field with Chern–Simons level 1 and a background contact term; particle–vortex and fermion–fermion dualities follow from it. Their derivations were explicitly 'plausible (but not rigorous)', resting on unproven assumptions of existence and uniqueness of the infrared fixed points. A decade of subsequent work — exact lattice dualities that still assume universality, anomaly and global-structure systematization, large-N tests, conformal bootstrap, quantum Monte Carlo, and fuzzy-sphere numerics — has produced strong quantitative evidence for the web and has largely delivered the paper's requested numerical checks, but no rigorous nonperturbative proof of the seed duality or of the fixed-point assumptions exists. The problem is to prove or refute this duality nonperturbatively, thereby either establishing the derived duality web as a theorem or refuting the duality.

Background

Relativistic quantum field theories in $2+1$ dimensions (two spatial dimensions plus time) describe the universal long-distance physics of continuous quantum phase transitions in two-dimensional condensed-matter systems — superconductor–insulator transitions, quantum Hall plateau transitions, and magnetic transitions — as well as a rich family of conformal field theories (scale-invariant fixed points of the renormalization group, RG) studied in high-energy theory. A central organizing principle proposed for these theories is duality: two apparently different Lagrangians defining the same infrared (IR) physics, with a nontrivial map between their operators.

Two ingredients are needed. The Wilson–Fisher (WF) fixed point is the conformal field theory governing the critical point of the three-dimensional $O(2)$ (XY) universality class — the IR limit of a complex scalar field $\phi$ with quartic self-interaction; it is strongly interacting and not exactly solvable. A Chern–Simons (CS) term $\frac{1}{4\pi}b\,db$ is a topological gauge interaction for a $U(1)$ gauge field $b$ whose coefficient (the level) is quantized; coupling charged matter to a dynamical gauge field with a CS term attaches magnetic flux to charge and can transmute particle statistics, and 'gauging' a theory means promoting one of its global symmetries to a dynamical gauge field. A monopole operator $\mathcal{M}_b$ is a local, gauge-invariant disorder operator that inserts one unit of magnetic flux of $b$.

The oldest member of the web is particle–vortex duality, relating the critical complex scalar (XY model) to a gauged scalar theory (the abelian Higgs model). Unusually for a duality, it comes with essentially exact mappings between lattice regularizations of the two sides (Peskin 1978; Dasgupta–Halperin 1981), which set the standard of rigor against which later three-dimensional dualities are measured. Building on large-$N$ (many-flavor) Chern–Simons–matter dualities conjectured by Aharony, Seiberg–Senthil–Wang–Witten (2016) proposed the full duality web. Its seed is the 3d bosonization duality

$i\bar\Psi\gamma^\mu D_\mu^A\Psi \;\longleftrightarrow\; |D_b\phi|^2-|\phi|^4+\frac{1}{4\pi}\,b\,db+\frac{1}{2\pi}\,b\,dA,$

equating a free massless two-component Dirac fermion $\Psi$ coupled to a background $\mathrm{spin}_c$ connection $A$ (a slight generalization of a spin connection allowing charged matter to couple to background gauge fields on oriented manifolds without choosing a spin structure) with the $O(2)$ WF fixed point coupled to a dynamical compact $U(1)$ gauge field $b$ with CS level 1 and the displayed background contact term. The duality identifies the fermion with the gauge-invariant composite $\phi^\dagger\mathcal{M}_b$, and implies, by gauging and recombining, the particle–vortex duality (e.g. one-flavor QED3 — quantum electrodynamics in $2+1$ dimensions, a Dirac fermion coupled to a dynamical $U(1)$ gauge field — at half-integer CS level $-\tfrac{1}{2}$ is dual to the $O(2)$ WF theory) and fermion–fermion dualities, with condensed-matter applications to topological-insulator surfaces, the half-filled Landau level, and flux attachment (attaching Chern–Simons gauge flux to particles to change their statistics).

The global structure is subtle and load-bearing: fermions require a spin structure, half-quantized CS levels encode the parity anomaly (a single massless Dirac fermion in $2+1$ dimensions cannot be coupled to a gauge field while preserving both gauge invariance and time-reversal symmetry; the compensating half-integer Chern–Simons level is the anomaly's signature), and the dualities hold as stated only with the correct $\mathrm{spin}$ versus $\mathrm{spin}_c$ data, integer-quantized contact terms, and decoupled nearly-trivial topological quantum field theory (TQFT) factors. The source paper prescribes these choices but does not exhaust their classification. Its derivations are, by the authors' own description, 'plausible (but not rigorous)': they follow from qualitative assumptions about RG flows — in particular from 'assuming there is a unique fixed point controlling this transition' — and a $3+1$-dimensional electric–magnetic $S$-duality (a duality of four-dimensional gauge theories exchanging electric and magnetic variables and inverting the complexified coupling) interpretation that 'will not be a proof of the dualities'.

Subsequent work produced strong confirming evidence: exact Euclidean-lattice mappings that still assume universality of the strongly interacting continuum limit, systematization of the global structure and time-reversal anomalies, large-$N$ tests of three-sphere free energies, conformal-bootstrap (numerical constraints on operator dimensions from the crossing symmetry of four-point correlation functions) and Monte Carlo computations of monopole-operator scaling dimensions, and realizations on the fuzzy sphere (a finite lowest-Landau-level regularization of the two-sphere on which the critical theory appears as a transition between quantum Hall states, from whose spectra conformal data are extracted) of critical Chern–Simons–matter theories — all consistent with the web. What no later work supplies is a rigorous nonperturbative proof of the seed duality or of the fixed-point assumptions: an upgrade of the web from a consistency-checked conjecture to a theorem at, or beyond, the rigor of the original particle–vortex lattice mappings. That is the open problem.

Problem Statement

Prove — or refute — nonperturbatively and rigorously the seed boson–fermion (3d bosonization) duality of the $2+1$-dimensional duality web:

$i\bar\Psi\gamma^\mu D_\mu^A\Psi \;\longleftrightarrow\; |D_b\phi|^2-|\phi|^4+\frac{1}{4\pi}\,b\,db+\frac{1}{2\pi}\,b\,dA,$

where the left-hand side is a free massless two-component Dirac fermion coupled to a background $\mathrm{spin}_c$ connection $A$, and the right-hand side is the $O(2)$ Wilson–Fisher fixed point coupled to a dynamical compact $U(1)$ gauge field $b$ with Chern–Simons level 1 and the displayed background contact term. Specifically, establish (or disprove) that the infrared conformal field theory of the right-hand side — with the global data specified in the source work (choice of $\mathrm{spin}$ versus $\mathrm{spin}_c$ structures, integer-quantized contact terms, and the required nearly-trivial TQFT factors) — is identical as a conformal field theory to the free fermion: matching spectra and scaling dimensions of all local and monopole operators, correlation functions, global symmetries and their anomalies (including time-reversal), and background-field partition functions, under the operator identification $\Psi=\phi^\dagger\mathcal{M}_b$, where $\mathcal{M}_b$ is the monopole operator inserting one unit of magnetic flux of $b$. The proof must discharge, not assume, the dynamical assumptions on which the web's derivations rest: the existence, uniqueness, and stability of the infrared fixed points controlling the relevant transitions, and the absence of intervening phases or multicriticality. The proof must meet a standard of rigor matching or exceeding the original lattice mappings for particle–vortex duality (Peskin; Dasgupta–Halperin). From the seed duality, the derived web — the particle–vortex duality (one-flavor QED3 at Chern–Simons level $-\tfrac{1}{2}$ versus the $O(2)$ Wilson–Fisher theory) and the fermion–fermion dualities — must follow, with the gauging and recombination step exhibited for each member. A refutation of the duality is a discrepancy between the two sides in any observable or global datum that is proven, or reproducibly demonstrated with quantified control and shown not to arise from a convention mismatch or numerical error.

References

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