Complex Conformal Field Theory (CCFT)

• Complex conformal field theory is a framework where conformal data such as scaling dimensions and central charges become genuinely complex while preserving symmetry.
• It emerges from analytic continuations of real critical points and finds applications in non-Hermitian systems, lattice models, and exactly marginal deformations.
• Innovative observables like biorthogonal entanglement entropy and pseudo-entropy provide crucial insights into the universality and boundary phenomena of CCFT.

Searching arXiv for papers on complex conformal field theory, non-Hermitian criticality, and related conformal manifolds. Complex conformal field theory (CCFT) denotes a conformal field theory whose conformal data become genuinely complex while conformal symmetry is retained. In the modern usage developed for non-Hermitian criticality and complex renormalization-group fixed points, CCFT is distinguished both from ordinary unitary CFT and from more familiar nonunitary theories with negative but real central charge. A CCFT may arise from fixed points that move into the complex plane of couplings, from non-Hermitian microscopic Hamiltonians whose universal scaling is governed by complex central charges, or from analytic continuation along exactly marginal conformal manifolds into complex coupling space. The subject now spans lattice models, continuum field theory, entanglement observables, defect and boundary conformal systems, and holographic constructions, while remaining conceptually entangled with several distinct uses of the acronym “CCFT,” including contracted conformal field theory, Carrollian conformal field theory, celestial conformal field theory, chiral conformal field theory, and quaternionic conformal field theory (Haldar et al., 2023, Shimizu et al., 4 Feb 2025, Furuta et al., 29 Jun 2026).

1. Definitions, scope, and competing meanings of “CCFT”

In the modern statistical-mechanical and non-Hermitian literature, CCFT refers to complex conformal field theory: a conformal theory with genuinely complex conformal data. One paper states the distinction explicitly by separating three cases: ordinary unitary CFT with positive real central charge $c>0$, nonunitary CFT with real negative central charge $c<0$, and CCFT with central charge

$c\in\mathbb{C},$

typically associated with fixed points that have moved off the real coupling axis into the complex plane (Shimizu et al., 4 Feb 2025). Another formulation defines complex CFTs as theories “whose conformal data—such as scaling dimensions, OPE coefficients, transmission coefficients, or effective central charges—become genuinely complex after analytic continuation, even though the theory still satisfies conformal Ward identities and remains at a fixed point” (Furuta et al., 29 Jun 2026).

This usage must be distinguished from several other meanings of “CCFT” that coexist in the literature. In flat-space holography, “CCFT” may mean contracted conformal field theory, obtained by contraction of an ordinary $2d$ CFT so that its symmetry algebra matches the asymptotic symmetry algebra of asymptotically flat spacetime (Fareghbal et al., 2014). In Carrollian field theory, “CCFT” may instead mean Carrollian conformal field theory, whose $1+1$-dimensional symmetry algebra is isomorphic to $\mathrm{BMS}_3$ (Saha, 2022) and whose higher-dimensional representation theory is governed by the Carrollian conformal algebra (Chen et al., 2021). In the celestial literature, one paper uses “CCFT” for celestial conformal field theory, with logarithmic soft sectors at null infinity (Bissi et al., 2024), while another studies exact “celestial/complex conformal observables” from Mellin transforms of integrable $1+1$-dimensional $S$-matrices (Kapec et al., 2022). Elsewhere, “CCFT” may mean chiral conformal field theory (Fukusumi, 2024), and in a quaternionic context “CCFT” is used for two-dimensional complex-parametrized conformal field theory as the model for a quaternionic analogue (Giardino, 2015).

A plausible implication is that any encyclopedia treatment of CCFT must begin with disambiguation. In what follows, the primary sense is complex conformal field theory in the modern non-Hermitian and complex-fixed-point sense, with separate discussion of these other uses where they have influenced the broader discourse.

2. Origins in complex fixed points and hidden criticality

A central route to CCFT is the collision of real fixed points followed by their displacement into the complex plane. The two-dimensional $O(n)$ loop model gives a direct microscopic example. For $-2\le n\le 2$, the model has two real critical branches

$c<0$0

At $c<0$1 the branches meet, and for $c<0$2,

$c<0$3

so the critical points move into the complex $c<0$4-plane as a conjugate pair (Haldar et al., 2023). The corresponding Coulomb-gas conformal data are analytically continued through

$c<0$5

with central charge

$c<0$6

and thermal and magnetic dimensions

$c<0$7

For $c<0$8, these quantities become complex, and the two branches form a complex-conjugate CCFT pair (Haldar et al., 2023).

The paper interprets these as “hidden critical points”: the real theory has no ordinary critical point on the real axis, yet its physics is controlled by complex fixed points in the complexified temperature plane (Haldar et al., 2023). This picture is explicitly tied to “weakly first-order” behavior and “walking” renormalization-group flows, where the real-axis theory passes near complex-conjugate fixed points and displays approximate scale invariance over an extended range (Haldar et al., 2023). A related statement appears in the study of the non-Hermitian five-state Potts chain, where the $c<0$9 transition is described as weakly first-order in the Hermitian theory but directly realized at the complex fixed point by a non-Hermitian deformation (Shimizu et al., 4 Feb 2025).

The same mechanism is extended in the two-dimensional $c\in\mathbb{C},$0 nonlinear sigma model. One recent work argues that the standard asymptotically free $c\in\mathbb{C},$1 NLSM, usually regarded as having no nontrivial finite-coupling fixed point for real positive coupling, does possess a pair of complex fixed points once the coupling is allowed to be complex,

$c\in\mathbb{C},$2

and that these fixed points are described by a CCFT (Yang et al., 5 Jan 2026). The conformal data are inherited by analytic continuation from the $c\in\mathbb{C},$3 loop-model branches: $c\in\mathbb{C},$4 with

$c\in\mathbb{C},$5

For $c\in\mathbb{C},$6, the paper quotes

$c\in\mathbb{C},$7

for one branch (Yang et al., 5 Jan 2026). It further argues that loop crossings are irrelevant because the singlet $c\in\mathbb{C},$8 operator satisfies

$c\in\mathbb{C},$9

for all $2d$0, with $2d$1 at $2d$2, so the CCFT is generic for non-Hermitian $2d$3-symmetric systems once a single complex parameter is tuned (Yang et al., 5 Jan 2026).

These examples establish a recurring structural picture. CCFTs arise not merely as formal continuations of isolated formulas, but as conformal endpoints of analytically continued theory families, often in complex-conjugate pairs, with complex central charges and scaling dimensions encoding spiraling linearized RG flow: $2d$4 This suggests that complex scaling exponents are not incidental pathologies but the natural universal data of fixed points located off the real coupling plane (Haldar et al., 2023, Yang et al., 5 Jan 2026).

3. Conformal data, spectra, and complex conformal manifolds

A systematic construction of CCFTs from ordinary conformal manifolds has recently been proposed by analytically continuing exactly marginal couplings into the complex plane (Furuta et al., 29 Jun 2026). The general principle is to begin with a bulk, boundary, or defect CFT possessing an exactly marginal deformation parameter $2d$5, then continue

$2d$6

If the beta function vanishes on the real manifold and is analytic, the identity theorem extends

$2d$7

to the connected complex domain (Furuta et al., 29 Jun 2026). In this construction, conformal symmetry survives while operator dimensions and related observables generically become complex.

The compact free boson provides the main solvable example. With

$2d$8

or equivalently $2d$9 with

$1+1$0

the real Gaussian conformal manifold is parametrized by $1+1$1. The complexified manifold is

$1+1$2

and the vertex operators

$1+1$3

have momenta

$1+1$4

with dimensions

$1+1$5

For generic complex $1+1$6, $1+1$7 is complex, while the spin remains integral,

$1+1$8

and mutual locality survives because

$1+1$9

independently of $\mathrm{BMS}_3$0 (Furuta et al., 29 Jun 2026).

The two-point function then acquires the characteristic log-periodic dependence of a complex scaling dimension: $\mathrm{BMS}_3$1 so if $\mathrm{BMS}_3$2,

$\mathrm{BMS}_3$3

This is one of the clearest universal signatures of CCFT behavior (Furuta et al., 29 Jun 2026).

The torus partition function of the complexified Gaussian model,

$\mathrm{BMS}_3$4

converges absolutely only in the domain

$\mathrm{BMS}_3$5

Within this wedge the theory retains modular invariance and T-duality

$\mathrm{BMS}_3$6

and there is a distinguished real locus on the unit circle,

$\mathrm{BMS}_3$7

where $\mathrm{BMS}_3$8 is real despite nonunitarity because $\mathrm{BMS}_3$9 implies $1+1$0 (Furuta et al., 29 Jun 2026).

The same work makes a broader structural claim: genuinely complex rational CFTs do not exist. Under the standard notion of rationality as a finite sum of chiral characters, Vafa’s equations force the conformal weights and central charge to be rational; the argument does not assume reality in advance, so it excludes truly complex rational points (Furuta et al., 29 Jun 2026). This yields a sharp distinction between complex conformal manifolds and the rational points familiar from real RCFT: rational loci remain confined to the real regime.

A plausible implication is that complex conformal manifolds give a controlled source of CCFTs without producing a parallel theory of “complex RCFT.” Complexification preserves conformal symmetry and often locality, but not the finite algebraic closure characteristic of rational models (Furuta et al., 29 Jun 2026).

4. Non-Hermitian realizations and complex entanglement entropy

The strongest microscopic realization of CCFT to date is the non-Hermitian five-state Potts chain studied with biorthogonal density-matrix renormalization group (Shimizu et al., 4 Feb 2025). The Hermitian $1+1$1-state quantum Potts chain is

$1+1$2

with

$1+1$3

The non-Hermitian perturbation is

$1+1$4

and the full Hamiltonian is $1+1$5 (Shimizu et al., 4 Feb 2025). For the five-state case the critical couplings are quoted as

$1+1$6

The predicted complex central charge is

$1+1$7

while previous spectroscopy had found

$1+1$8

(Shimizu et al., 4 Feb 2025).

The paper’s conceptual advance is to identify the correct entanglement observable for CCFT: not the ordinary von Neumann entropy of a Hermitian reduced density matrix, but a complex entanglement entropy built from biorthogonal left and right eigenstates. For a non-Hermitian Hamiltonian,

$1+1$9

the generalized density matrix is

$S$0

and the entropy is

$S$1

with generalized Rényi entropy

$S$2

Because $S$3, its spectrum is generally complex, so $S$4 is generically complex (Shimizu et al., 4 Feb 2025).

The scaling law is the direct CCFT generalization of the ordinary $S$5-dimensional result: $S$6 for periodic boundary conditions, and

$S$7

for open boundaries (Shimizu et al., 4 Feb 2025). Thus both

$S$8

in the periodic normalization.

The numerical results under periodic boundary conditions are close to the field-theory prediction. The extracted values include

$S$9

at $O(n)$0, compared with

$O(n)$1

(Shimizu et al., 4 Feb 2025). By contrast, alternative Hermitian entanglement measures fail to reproduce the same data. Using

$O(n)$2

or an SVD-based positive operator yields fitted “central charges”

$O(n)$3

for $O(n)$4, $O(n)$5, neither of which matches the CCFT value as well as the biorthogonal entropy does (Shimizu et al., 4 Feb 2025).

The open-boundary case is markedly subtler. The real part appears roughly logarithmic, but the fit gives

$O(n)$6

far from the periodic and field-theory values, while the imaginary part does not follow the CCFT prediction (Shimizu et al., 4 Feb 2025). The paper interprets this as genuine boundary physics rather than numerical inaccuracy, linking it to the sensitivity of non-Hermitian systems to boundary conditions and to phenomena such as the non-Hermitian skin effect (Shimizu et al., 4 Feb 2025).

This suggests that complex entanglement entropy is not a mere diagnostic convenience but the appropriate universal observable for CCFT in non-Hermitian many-body systems. At the same time, the paper explicitly notes that its physical interpretation remains unsettled and that it does not retain all standard entropy properties, such as strong subadditivity (Shimizu et al., 4 Feb 2025).

5. Defects, pseudo-entropy, and holographic extensions

CCFT has also entered defect and holographic contexts. The one-parameter family of conformal defects in the Ising CFT provides an exact complex defect conformal manifold (Furuta et al., 29 Jun 2026). The exactly marginal defect perturbation

$O(n)$7

admits analytic continuation to complex $O(n)$8, or equivalently to complex

$O(n)$9

The defect scaling dimensions become

$-2\le n\le 2$0

the transmission coefficient becomes

$-2\le n\le 2$1

and the effective central charge continues to a complex multi-sheeted function with a logarithmic branch point at the factorizing defect $-2\le n\le 2$2, where

$-2\le n\le 2$3

(Furuta et al., 29 Jun 2026). The paper verifies these predictions in non-Hermitian Ising and free-fermion chains by bulk-defect correlators, entanglement entropy, and complex energy transport (Furuta et al., 29 Jun 2026).

Another line of work argues that what had been called “entanglement entropy” in flat-space holographic CCFT should in fact be interpreted as pseudo-entropy (Fareghbal et al., 6 Nov 2025). In that setting, “CCFT” means Carrollian conformal field theory, but the result is relevant to broader nonunitary conformal systems because the paper identifies a genuinely complex entropy-like observable associated with spacelike and timelike extremal curves. For two states $-2\le n\le 2$4, the transition matrix

$-2\le n\le 2$5

defines

$-2\le n\le 2$6

Because $-2\le n\le 2$7 is generally non-Hermitian, $-2\le n\le 2$8 is generically complex (Fareghbal et al., 6 Nov 2025).

In the Einstein-gravity case the previously known real contribution

$-2\le n\le 2$9

is reinterpreted as only the real part, with an equally large imaginary contribution: $c<0$00 up to regulator-dependent terms (Fareghbal et al., 6 Nov 2025). The paper argues that the flat-space limit of both AdS and dS holographic pseudo-entropy yields the same complex answer, and concludes that the relevant dual CCFTs are non-unitary (Fareghbal et al., 6 Nov 2025).

Although this work uses “CCFT” in the Carrollian sense rather than the complex-fixed-point sense, it is conceptually adjacent to the lattice result on complex entanglement entropy: in both cases, the natural entropy-like observable of a non-Hermitian or nonunitary conformal system is complex, and the imaginary part carries universal information rather than being a removable artifact (Shimizu et al., 4 Feb 2025, Fareghbal et al., 6 Nov 2025).

6. Broader context: other CCFT traditions and methodological relevance

The acronym “CCFT” is historically overloaded, and several neighboring traditions have supplied tools or conceptual contrasts relevant to complex conformal field theory.

In flat-space holography, “CCFT” as contracted conformal field theory was developed as the boundary dual of asymptotically flat spacetimes through an ultra-relativistic contraction of a parent $c<0$01 CFT (Fareghbal et al., 2014). In that framework the Virasoro generators combine into the $c<0$02 algebra,

$c<0$03

with central charges obtained by contraction of $c<0$04 (Fareghbal et al., 2014). That literature is not about complex central charges or complex fixed points, but it established a recurring association between non-Lorentzian symmetries, nonunitarity, and modified entropy formulas.

In higher-dimensional Carrollian conformal field theory, the representation theory is governed by the Carrollian conformal algebra and is markedly non-semisimple. Finite-dimensional highest-weight representations are “generically reducible but indecomposable,” organized into multiplets with chain or net structures, and 2-point functions can be polynomials in time rather than simple power laws (Chen et al., 2021). This is not complex CFT in the modern sense, but it offers a useful contrast: nonunitarity and indecomposability can arise from spacetime symmetry alone, whereas in complex CFT the hallmark is genuinely complex conformal data (Chen et al., 2021).

The celestial literature adds a different kind of “complexity.” Exact Mellin transforms of integrable $c<0$05-dimensional $c<0$06-matrices yield celestial correlators that are real tempered distributions of the conformal weights, with asymptotics controlled by the mass spectrum and three-point couplings of the underlying QFT (Kapec et al., 2022). A separate work identifies a logarithmic pair in the soft gravitational sector associated with supertranslations, with weights

$c<0$07

and standard rank-two logarithmic two-point functions

$c<0$08

(Bissi et al., 2024). These works do not define complex CFT via complex central charge, but they reinforce the broader point that complexified or nonunitary conformal observables often appear most naturally as distributions, logarithmic multiplets, or non-Hermitian entropy-like quantities.

Finally, the rigorous lattice-level construction of Virasoro symmetry from discrete complex analysis in the discrete Gaussian free field and Ising model is not a CCFT result in the modern sense—the central charges are $c<0$09 and $c<0$10, and no complex fixed points are involved—but it is methodologically relevant. It shows how chiral algebra structures can be extracted directly from lattice observables, a strategy that may prove useful for future microscopic constructions of complex CFTs if suitable non-Hermitian or analytically continued lattice observables are identified (Hongler et al., 2013).

7. Status, limitations, and open problems

Several conclusions now appear robust. First, CCFT in the modern sense is a genuine universality class rather than an ad hoc analytic continuation. Hidden critical points in the $c<0$11 $c<0$12 model, the non-Hermitian five-state Potts chain, the complexified Gaussian conformal manifold, and the $c<0$13 nonlinear sigma model all exhibit the same core structure: conformal symmetry survives, but central charges, scaling dimensions, and related observables become genuinely complex (Haldar et al., 2023, Shimizu et al., 4 Feb 2025, Furuta et al., 29 Jun 2026, Yang et al., 5 Jan 2026).

Second, complex entanglement-like observables are emerging as primary diagnostics. In the lattice Potts model, only the biorthogonal entropy reproduces the complex central charge quantitatively (Shimizu et al., 4 Feb 2025). In flat-space holography, the entropy of intervals is argued to be fundamentally pseudo-entropy with both real and imaginary parts (Fareghbal et al., 6 Nov 2025). This suggests that information-theoretic observables in CCFT must often be reformulated from the outset.

Third, complex conformal manifolds provide a controlled bridge between solvable real CFTs and genuinely complex theories (Furuta et al., 29 Jun 2026). This route is especially attractive because it preserves exact marginality and often keeps the central charge fixed, allowing exact calculations in bulk, boundary, and defect settings.

At the same time, important limitations remain. The complex fixed-point route is best understood in two-dimensional models with exact or nearly exact control; higher-dimensional complex CFTs remain largely undeveloped. The microscopic interpretation of complex entropy and pseudo-entropy is still unsettled (Shimizu et al., 4 Feb 2025, Fareghbal et al., 6 Nov 2025). Open-boundary non-Hermitian criticality does not yet fit the simplest CCFT entanglement formulas (Shimizu et al., 4 Feb 2025). Even where exact conformal data are known, such as the complex defect manifold in Ising, the full nonunitary bootstrap structure remains to be clarified (Furuta et al., 29 Jun 2026).

The distinction between local and global consistency is also unresolved in general. The compact boson shows that local conformal data may be analytically continued well beyond the region where the Euclidean torus partition function converges; only the wedge

$c<0$14

supports a conventional torus sum (Furuta et al., 29 Jun 2026). This suggests that some CCFTs are best understood as analytic continuations of local operator data rather than as fully convergent Euclidean path integrals.

A plausible implication is that the next stage of the subject will depend on unifying three perspectives that are currently only partially connected: exact lattice realizations of non-Hermitian criticality, analytic constructions from complex conformal manifolds and continued fixed points, and bootstrap or holographic frameworks capable of handling complex spectra without positivity. The available evidence already indicates that CCFT is not a peripheral deformation of ordinary CFT, but a mathematically distinct and physically consequential extension of conformal theory (Shimizu et al., 4 Feb 2025, Haldar et al., 2023, Furuta et al., 29 Jun 2026, Yang et al., 5 Jan 2026).