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arxiv: 2606.30720 · v1 · pith:PN5YIUEOnew · submitted 2026-06-29 · ✦ hep-th · cond-mat.stat-mech· cond-mat.str-el

Complex Conformal Manifolds

Yuma Furuta , Wataru Harada , Yuya Kusuki , Yin Tang This is my paper

Pith reviewed 2026-07-01 02:07 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechcond-mat.str-el
keywords complex CFTsconformal manifoldsanalytic continuationmarginal deformationsrational CFTsconformal defectsIsing modelAdS/BCFT
0
0 comments X

The pith

Analytically continuing marginal couplings into the complex plane yields complex CFTs but confines rational points to the real regime.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a systematic way to construct complex conformal field theories by extending exactly marginal couplings from the real line into the complex plane. This continuation preserves conformal symmetry and applies equally to bulk theories, boundaries, and defects, generically making operator spectra and other data complex. A central result is that rational CFTs cannot be made genuinely complex this way; their special points stay on the real axis. The construction is worked out explicitly for the free boson manifold and for a family of defects in the Ising model, with predictions checked against lattice simulations of non-Hermitian chains. The approach supplies concrete examples and benchmarks for studying non-Hermitian criticality and complex renormalization-group flows.

Core claim

By analytically continuing exactly marginal couplings into the complex plane, one obtains complex CFTs with generically complex spectra while preserving conformal symmetry for bulk, boundary, and defect deformations. Using the compact free boson, the global structure of the complexified Gaussian conformal manifold is uncovered. More generally, genuinely complex rational CFTs do not exist: rational points remain confined to the real regime. For the Ising defect, exact expressions for the defect spectrum, energy transmission coefficient, and effective central charge are derived from the continuation and verified in lattice models via bulk-defect correlators, entanglement entropy, and complex e

What carries the argument

Analytic continuation of exactly marginal couplings into the complex plane, which preserves conformal symmetry across bulk, boundary, and defect deformations.

If this is right

  • The Gaussian conformal manifold acquires a specific global structure once its couplings are complexified.
  • Exact formulas for defect spectrum, energy transmission, and effective central charge follow directly from continuation in the Ising model.
  • Non-Hermitian Ising and free-fermion chains reproduce the predicted complex defect data through correlators, entanglement, and energy transport.
  • Complex boundary RG flows admit a holographic treatment via the AdS/BCFT dictionary.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same continuation procedure could be applied to other solvable CFTs with known marginal deformations to generate additional complex examples.
  • The restriction of rational points to the real axis may simplify classification efforts for fixed points in complex RG flows.
  • Holographic duals of these complex manifolds could provide new controlled settings for studying nonunitary gravity.

Load-bearing premise

The analytic continuation of exactly marginal couplings into the complex plane preserves conformal symmetry.

What would settle it

A lattice or numerical computation that produces a rational CFT with genuinely complex parameters whose data cannot be recovered by any real-axis continuation, or a mismatch between the continued Ising defect spectrum and direct measurements in a non-Hermitian chain.

Figures

Figures reproduced from arXiv: 2606.30720 by Wataru Harada, Yin Tang, Yuma Furuta, Yuya Kusuki.

Figure 1
Figure 1. Figure 1: Complex bulk conformal manifold in the R-plane. The shaded wedge denotes the consistent domain Re(R 2 ) > 0, where the torus partition function converges absolutely. The arc |R| = 1 (the black line) inside this wedge is the real locus, on which complex conjugation combined with T-duality implies Z(τ; R) ∈ R. Marked points on the positive real axis indicate rational loci with R 2 ∈ Q>0. 2.4 Distinguished re… view at source ↗
Figure 2
Figure 2. Figure 2: Complex defect conformal manifold in the complex [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical analysis of the bulk-defect correlator based on biorthogonal eigenstates and [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical analysis of the entanglement entropy scaling and the e [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Theoretical landscape of the complex effective central charge ceff over the complex coupling plane bϵ . The left and right panels show the real and imaginary parts of ceff, respectively, as given by the analytic continuation of Eq. (4.4). 4.4 Transmission coefficient on a complex Dirac fermion interface Finally, we study a different model which also hosts a continuous defect manifold: a critical free Dirac… view at source ↗
Figure 6
Figure 6. Figure 6: Spacetime evolution of the local energy density after a local quench in the presence of [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the numerical transmission coe [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
read the original abstract

Complex conformal field theories (CFTs) have recently emerged as essential frameworks for understanding non-Hermitian criticality, weakly first-order phase transitions, and walking renormalization group flows, while their general structures remain largely unknown. In this work, we propose a systematic construction of complex CFTs by analytically continuing exactly marginal couplings into the complex plane. This procedure applies uniformly to bulk, boundary, and defect deformations, preserving conformal symmetry while generically complexifying operator spectra and other universal data. Using the compact free boson as a solvable laboratory, we uncover the global structure of the complexified Gaussian conformal manifold. More generally, we demonstrate that genuinely complex rational CFTs do not exist: rational points remain confined to the real regime, providing a sharp distinction between real and complex theories. In the defect case, we investigate the one-parameter family of conformal defects in the Ising CFT and derive exact expressions for the defect spectrum, energy transmission coefficient, and effective central charge from analytic continuation. The theoretical predictions are precisely verified in non-Hermitian critical Ising and free fermion chains using bulk-defect correlators, entanglement entropy, and complex energy transport, providing concrete evidence for the complex defect conformal manifold. Finally, we study complex boundary renormalization-group flows through the AdS/BCFT correspondence. Our results establish complex conformal manifolds as a controlled bridge between solvable lattice models, complex CFTs, and holography, while providing stringent analytic benchmarks for the nonunitary conformal bootstrap.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proposes constructing complex CFTs by analytic continuation of exactly marginal couplings into the complex plane, uniformly for bulk, boundary, and defect cases while preserving conformal invariance. Using the compact free boson, it maps the global structure of the complexified Gaussian conformal manifold. It argues that genuinely complex rational CFTs cannot exist because analyticity of the spectrum combined with discreteness of the rationals confines rational points to the real axis. For the Ising defect, exact expressions are derived for the defect spectrum, energy transmission, and effective central charge via continuation; these are verified against non-Hermitian lattice chains. Boundary RG flows are studied via AdS/BCFT. The work positions complex conformal manifolds as a bridge between lattice models, complex CFTs, and holography.

Significance. If the central claims hold, the construction supplies a controlled, uniform method for generating complex CFT data and the no-go result on rational points supplies a sharp structural distinction between real and complex theories. Explicit lattice verifications for both bulk and defect cases, together with the AdS/BCFT analysis, provide reproducible benchmarks that strengthen the analytic continuation framework.

minor comments (3)
  1. [§2.2] §2.2 (Gaussian manifold): the statement that the radius squared remains the only modulus after complexification would benefit from an explicit listing of the preserved and broken symmetries to clarify the dimension of the manifold.
  2. The general no-go argument (analyticity plus discreteness) is stated clearly but the precise domain of analyticity in the complex coupling is not delimited; a short paragraph specifying the radius of convergence or branch cuts would remove ambiguity.
  3. Figure 4 (defect spectrum): the plotted points for complex g are difficult to distinguish from the real-axis data; a different marker style or inset would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central construction proceeds by analytic continuation of exactly marginal couplings, with explicit verification that conformal invariance is preserved in bulk, boundary, and defect sectors via the solvable Gaussian model and direct lattice computations (bulk-defect correlators, entanglement, energy transport). The claim that rational CFT points lie only on the real axis rests on the explicit global structure of the complexified Gaussian manifold plus a general analyticity-plus-discreteness argument; neither step reduces to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation. Lattice verifications are independent of the analytic continuation ansatz and provide external falsifiability. No enumerated circularity pattern is present.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based on abstract only; full paper likely contains additional parameters and assumptions not visible here.

free parameters (1)
  • complex marginal couplings
    Couplings continued into complex plane; no specific values or fitting procedure detailed in abstract.
axioms (1)
  • domain assumption Analytic continuation of exactly marginal couplings preserves conformal symmetry
    Stated as applying uniformly to bulk, boundary, and defect deformations.

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