Large-Charge Bootstrap in CFT

• Large-charge bootstrap is a CFT method that analyzes sectors with high global charge using controlled macroscopic limits and heavy-light correlators.
• It employs crossing symmetry and finite Regge trajectory analysis to constrain excitation energies and OPE coefficients in a systematic expansion.
• Extensions to monopole and current probes in QED₃ connect effective field theory predictions with numerical bootstrap bounds.

Searching arXiv for recent and foundational papers on the large-charge bootstrap. arXiv search query: "large charge bootstrap conformal bootstrap U(1) heavy light current probes monopole" Large-charge bootstrap is a conformal-bootstrap analysis of CFT data in sectors of large global charge $Q$, exploiting the fact that correlation functions in such sectors admit a controlled macroscopic limit and a simplified spectrum. In a standard $U(1)$ setup, $\mathcal O_Q$ or $\Phi_Q$ denotes the lightest operator of charge $Q$, and one studies heavy-light four-point functions with two large-charge insertions and two light probes. Under assumptions such as a well-defined large-$Q$ expansion, existence of a nontrivial macroscopic limit, and organization of the spectrum into finitely many Regge trajectories at leading nontrivial order, crossing symmetry can be rewritten as algebraic constraints on excitation energies and OPE data. The one-trajectory solution is the conformal superfluid Goldstone EFT, while current and stress-tensor probes impose additional constraints that force a Goldstone trajectory and sharply reduce the allowed solution space (Jafferis et al., 2017, Kiaee et al., 23 Dec 2025, Kiaee et al., 10 Jul 2026). In gauge-theory settings, the same charged-sector logic appears in monopole correlators of QED$_3$, where the relevant charge is topological rather than flavor charge (Chester et al., 2016).

1. Large-charge sector, heavy states, and macroscopic limit

The basic setting is a unitary CFT in $d>2$ with a continuous global symmetry, often specialized to $U(1)$. For each integer charge $Q$, one considers the lightest operator $U(1)$0 with scaling dimension $U(1)$1. The large-charge regime assumes that the charge dependence of CFT data is smooth enough to admit an expansion in inverse powers of $U(1)$2, and that the cylinder state $U(1)$3 created by $U(1)$4 has a thermodynamic interpretation. Keeping charge density and energy density fixed on $U(1)$5 leads to the scaling

$U(1)$6

and, in $U(1)$7, the Goldstone EFT gives

$U(1)$8

These are the characteristic large-charge asymptotics assumed throughout the program (Jafferis et al., 2017).

The canonical observable is a heavy-heavy-light-light correlator. One frequently writes

$U(1)$9

or, in the notation of later work, heavy-light correlators with $\mathcal O_Q$0, $\mathcal O_Q$1, and neutral probes. On the cylinder one introduces $\mathcal O_Q$2 and $\mathcal O_Q$3 through

$\mathcal O_Q$4

and defines a macroscopic limit by sending $\mathcal O_Q$5, $\mathcal O_Q$6, $\mathcal O_Q$7, with $\mathcal O_Q$8 and $\mathcal O_Q$9 held fixed. In this regime the heavy state behaves as a macroscopic medium, such as a superfluid, and correlation functions admit an asymptotic expansion in powers of $\Phi_Q$0 (Kiaee et al., 23 Dec 2025).

A central simplification is that the relevant exchanged operators lie close to the heavy ground state: $\Phi_Q$1 In the large-$\Phi_Q$2 limit, descendants in the heavy-light channel are parametrically suppressed, so the leading nontrivial crossing problem is governed by a finite-energy sector above $\Phi_Q$3. This is the sector in which Regge trajectories, excitation energies, and their OPE weights become the primary bootstrap variables (Jafferis et al., 2017).

2. Crossing equations, smoothness, and polynomial classification

The scalar large-charge bootstrap studies the first nontrivial correction to the heavy-light correlator and assumes that, at that order, only a finite number $\Phi_Q$4 of Regge trajectories contribute. For $\Phi_Q$5, the spectral data are encoded by residual energies $\Phi_Q$6 and weights $\Phi_Q$7. Smoothness at the boundary between the $\Phi_Q$8- and $\Phi_Q$9-channel representations, combined with the existence of the macroscopic limit, yields moment-like equations

$Q$0

with $Q$1 polynomial in $Q$2. The general solution is that the $Q$3 are the roots of a degree-$Q$4 polynomial

$Q$5

whose coefficients $Q$6 are degree-$Q$7 polynomials in $Q$8; the weights are then fixed by Cramer’s rule (Jafferis et al., 2017).

The current-probe formulation recasts the same structure in terms of

$Q$9

together with trajectory-dependent coefficients $Q$0 and $Q$1 extracted from scalar and current OPE data. For each $Q$2, crossing and the macroscopic singularity analysis lead to three Vandermonde-like systems,

$Q$3

all governed by the same characteristic polynomial

$Q$4

Thus scalar and current probes see the same Regge trajectories, but currents relate different polynomial sectors nontrivially (Kiaee et al., 23 Dec 2025).

The additional information carried by conserved currents is summarized by a constraint absent in the scalar-only analysis: $Q$5 In the current-probe paper this condition is derived from low-spin consistency and Ward identities, and is interpreted as the algebraic manifestation of the zero mode associated with Goldstone shift symmetry. This suggests that current probes do not merely refine scalar bootstrap solutions numerically; they select the subset compatible with a conserved current of a Goldstone mode (Kiaee et al., 23 Dec 2025).

3. Goldstone trajectory and effective-field-theory realization

For one Regge trajectory, the scalar large-charge bootstrap has a unique unitary solution. In $Q$6 it is

$Q$7

and the corresponding correlator is the Goldstone propagator on $Q$8. The associated EFT is a conformal superfluid with a single Goldstone mode, and the excitation spectrum above $Q$9 is

$_3$0

At leading nontrivial order, the heavy-light three-point coefficient scales as

$_3$1

which is precisely the scaling needed for a nontrivial macroscopic limit (Jafferis et al., 2017).

With current probes, the $_3$2 system becomes even more rigid: $_3$3 This is exactly the conformal superfluid EFT spectrum, and the paper states that with one Regge trajectory, including currents uniquely selects the EFT solution. For $_3$4, the surviving current-compatible families can be realized by local quadratic EFTs containing the Goldstone plus one additional light scalar, vector, or tensor; in particular, the current bootstrap renders EFT interpretations available even for scalar-only solutions that previously appeared non-quasiparticle-like (Kiaee et al., 23 Dec 2025).

The three-dimensional spinning large-charge bootstrap strengthens this conclusion further. Restricting to parity-even exchanged operators and using scalar, current, and stress-tensor probes, it derives the tensor-tensor constraint

$_3$5

which implies that any trajectory coupling to the stress tensor must satisfy

$_3$6

The paper then shows that at least one such trajectory must exist, and under additional non-degeneracy assumptions it is unique; all non-Goldstone trajectories contribute only to the scalar-scalar channel at this order (Kiaee et al., 10 Jul 2026).

4. Current and stress-tensor probes

The extension from scalar probes to conserved-current probes introduces new tensor structures, new Ward identities, and new parity constraints in cylinder variables. For mixed scalar-current correlators one obtains functions $_3$7 and $_3$8 satisfying

$_3$9

while current-current correlators require five functions $d>2$0 with even or odd parity in $d>2$1 fixed by crossing. Their macroscopic singularities are more severe than in the scalar case: $d>2$2 These singularities generate additional moment equations and make low-spin consistency much more restrictive than with scalar probes alone (Kiaee et al., 23 Dec 2025).

Stress-tensor probes in $d>2$3 add a further universal layer. The spinning parity-even analysis assumes: a unique lowest scalar $d>2$4 at fixed charge; a macroscopic limit implying $d>2$5; a finite number of analytic Regge trajectories at first subleading order; and parity-even exchanged operators only. Within that framework, all channels involving $d>2$6 or $d>2$7 are controlled by the Goldstone trajectory once the stronger non-degeneracy assumption is imposed. In that case,

$d>2$8

for the Goldstone trajectory, while the couplings of non-Goldstone trajectories to $d>2$9 and $U(1)$0 vanish. This leaves purely scalar correlators as the only place where extra light fields can contribute at that order (Kiaee et al., 10 Jul 2026).

A closely related heavy-light methodology uses the Lorentzian inversion formula in a regime where the heavy operator dimension scales as the central charge, $U(1)$1. That work studies multi-stress-tensor exchange in heavy-light four-point functions and proposes a back-and-forth inversion algorithm between HHLL and HLLH channels. The paper explicitly describes this regime as structurally extremely close to what one does in the large-charge bootstrap, with the heavy state defining a semiclassical background and the inversion formula extracting universal large-spin data (Li, 2019). This suggests a direct conceptual bridge between large-charge EFT inputs and inversion-based determination of multi-current or multi-stress sectors.

5. Monopole sectors and the bootstrap of topological charge

In QED$U(1)$2 with $U(1)$3 massless two-component complex fermions, the global symmetry is

$U(1)$4

and monopole operators $U(1)$5 are disorder operators creating magnetic flux

$U(1)$6

Via the state-operator correspondence, $U(1)$7 equals the ground-state energy on $U(1)$8 in a background flux $U(1)$9. The large-$Q$0 dimensions quoted in the paper include

$Q$1

From the perspective of fixed $Q$2 and large topological charge $Q$3, monopoles are described there as prototypical large-charge operators (Chester et al., 2016).

The bootstrap problem uses the four-point function of the basic charge-$Q$4 monopole,

$Q$5

where $Q$6 are auxiliary $Q$7 indices encoding $Q$8, and $Q$9 are antisymmetric $U(1)$00 flavor indices. The OPE decomposes into topological sectors $U(1)$01, with $U(1)$02 corresponding to charge $U(1)$03, and into flavor irreps

$U(1)$04

The doubly charged monopole $U(1)$05 is the lowest scalar in the $U(1)$06, $U(1)$07, $U(1)$08 channel, and bounding $U(1)$09 is the main spectral target (Chester et al., 2016).

Numerically, the paper obtains upper bounds on $U(1)$10 as a function of $U(1)$11, with and without spectral gaps in neutral sectors motivated by large-$U(1)$12 data. As these gaps are increased, the bound develops a kink. For $U(1)$13, the large-$U(1)$14 extrapolated values of $U(1)$15 lie very close to the kink line, and for a particular neutral gap they essentially coincide with the kink; for $U(1)$16, adding further reasonable gaps in other neutral sectors shifts the kink line downward and brings it closer to the large-$U(1)$17 prediction. With two assumptions for $U(1)$18—a large gap in the neutral $U(1)$19 sector and a gap above the lowest doubly charged monopole—the allowed region becomes a peninsula around the kink (Chester et al., 2016).

The paper does not work out the large-$U(1)$20 asymptotics $U(1)$21, but it explicitly frames monopole operators as the natural charged operators for probing non-perturbative dynamics in QED$U(1)$22. This suggests that monopole bootstrap is a gauge-theory realization of the broader large-charge program: the charge label is now topological charge $U(1)$23, and crossing relates sectors with $U(1)$24 in a way directly analogous to fixed-charge bootstrap in theories with ordinary global $U(1)$25 symmetry (Chester et al., 2016).

6. Analytic functionals, numerical methods, and open directions

A complementary analytic framework for charged correlators on the line develops two functional bases: a “simple” basis, essentially a direct sum of uncharged analytic functionals across irreducible-representation channels, and a GFF-dual basis yielding a charged Polyakov bootstrap and a crossing-symmetric dispersion relation. The charged crossing equation can be projected onto crossing-symmetric and crossing-antisymmetric scalar equations, and one obtains channel-by-channel sum rules together with asymptotic OPE-density bounds. The paper is not explicitly a large-charge analysis, but it states that from a large-charge bootstrap perspective this delivers a representation-decomposed functional framework for charged sectors and a charged Polyakov machinery compatible with GFF scaling dimensions in each irrep (Ghosh et al., 2021).

On the numerical side, accurate conformal-block approximation has become part of the practical infrastructure of large-parameter bootstrap. A paper develops approximations of conformal blocks as positive functions times polynomials and argues that accuracy should be measured in an error norm related to the asymptotics of dispersive functionals. The interpolation nodes minimizing this norm have an optimal density governed by a force-balance equation for charges in one dimension, solvable by standard large-$U(1)$26 matrix-model techniques. The same optimal-density nodes can also improve condition numbers inside SDPB, leading to more accurate bootstrap bounds with fewer computational resources. The paper explicitly notes that the construction is natural to import into large-parameter or large-charge bootstrap settings (Chang et al., 17 Sep 2025).

The current state of the subject therefore contains both rigid universal statements and explicit limitations. Scalar-only large-charge bootstrap permits many multi-trajectory solutions; current probes impose $U(1)$27; stress-tensor probes force a Goldstone trajectory and, under additional assumptions, isolate it in all spinning channels (Jafferis et al., 2017, Kiaee et al., 23 Dec 2025, Kiaee et al., 10 Jul 2026). At the same time, several open directions are stated explicitly across the literature: inclusion of parity-odd exchanges and conformal solids; higher-spin probes and the stress tensor beyond the presently analyzed orders; systematic classification for $U(1)$28 and higher trajectories; higher-charge monopole correlators and mixed correlators in QED$U(1)$29; direct extraction of EFT coefficients such as $U(1)$30 and $U(1)$31; and the question of UV completeness, since not every algebraic bootstrap solution is guaranteed to correspond to an actual CFT (Kiaee et al., 23 Dec 2025, Kiaee et al., 10 Jul 2026, Chester et al., 2016).

Taken together, these developments define large-charge bootstrap as a program that combines heavy-light crossing, macroscopic limits, analyticity in spin, and charged-sector Ward identities to constrain spectra and OPE coefficients above large-charge ground states. Its central universal statement is that, when the assumptions of the modern spinning analyses apply, the conformal superfluid Goldstone trajectory is not merely an EFT expectation but a consequence of crossing symmetry.