---
title: Crosscap States in Conformal Field Theory
url: https://www.emergentmind.com/topics/crosscap-states
type: topic
---

# Crosscap States in Conformal Field Theory

Crosscap states are conformal states that encode an orientation-reversing antipodal identification, and therefore organize quantum field theory on non-orientable manifolds such as the real projective plane and the Klein bottle. In two dimensions they are special conformal boundary states obeying parity-twisted gluing conditions; in higher dimensions they are associated with quotient geometries such as $\mathbb{RP}^3$ or, more generally, with $\mathbb{Z}_2$ quotient defects. Their physical content is carried by crosscap coefficients, one-point functions, non-orientable partition functions, and overlaps with energy eigenstates, and the concept now spans RCFT, integrable QFT, lattice models, holography, SPT diagnostics, and nonequilibrium dynamics [2305.16046, 2507.20005].

## 1. Geometric origin and symmetry structure

A crosscap is the local geometric operation that implements orientation reversal by identifying antipodal points. In two-dimensional CFT, inserting a crosscap turns the worldsheet into a non-orientable surface such as $\mathbb{RP}^2$ or the Klein bottle. On a spatial circle, the identification is $x\sim -x$, and in Euclidean language it is the non-orientable analogue of an ordinary boundary condition. In $(2+1)$-dimensional CFT, the corresponding construction places the theory on $\mathbb{RP}^3$, obtained by identifying antipodal points on the boundary $S^2$ of a three-ball; the crosscap is the local feature implementing that identification [2305.16046, 2507.20005].

This quotient viewpoint extends to general dimension. A recent formulation defines crosscap defects by quotienting spacetime with a $\mathbb{Z}_2$ automorphism that flips a subset of coordinates, leaving a $p$-dimensional fixed locus. The preserved conformal symmetry is then $SO(p+1,1)\times PO(d-p)$, and the case $p=-1$ reduces to ordinary CFT on $\mathbb{RP}^d$, with no fixed locus at all. In that sense, ordinary crosscap states are the codimension-maximal member of a broader family of quotient defects [2604.19868].

The essential point is that crosscaps are neither ordinary boundaries nor ordinary local operator insertions. They encode a global parity-type identification, so their natural observables are non-orientable partition functions, parity-weighted traces, and one-point functions that would vanish on orientable manifolds.

## 2. Algebraic construction in two-dimensional CFT

In unitary $2d$ CFT, the defining gluing condition for the stress tensor is
\[
\big(L_n-(-1)^n \bar L_{-n}\big)\,|C\rangle=0,
\]
with analogous relations for conserved chiral currents, for example
\[
\big(J_n+(-1)^n \bar J_{-n}\big)\,|C\rangle=0.
\]
The factor $(-1)^n$ distinguishes crosscaps from ordinary conformal boundary states and implements the orientation-reversing identification at the level of modes [2305.16046].

As in boundary CFT, one introduces crosscap Ishibashi states and expands the physical crosscap as
\[
|C\rangle=\sum_i \Gamma_i\,|C_i\rangle\rangle,
\]
where the coefficients $\Gamma_i$ are the crosscap coefficients. In RCFT these coefficients are constrained by Klein-bottle and Möbius-strip consistency, and the relevant modular transformation is governed by the matrix
\[
P=T^{1/2}ST^2ST^{1/2}.
\]
For diagonal theories, the Pradisi–Sagnotti–Stanev solution gives
\[
\Gamma_{0i}=\frac{P_{0i}}{\sqrt{S_{0i}}},
\]
and the corresponding crosscap entropy is controlled directly by $P_{u0}$ rather than by $S_{0i}$ alone [1811.07238].

A substantial generalization replaces the traditional simple-current labeling by a labeling with arbitrary Verlinde lines. The proposed states are
\[
|C_a\rangle=\eta_a\sum_i \frac{P_{ai}}{\sqrt{S_{1i}}}\,|i\rangle\rangle_C,
\]
with evidence from a generalized Cardy condition that incorporates topological defects. In this formulation, crosscaps are organized by the full Verlinde category, not only by invertible symmetries, so non-invertible symmetries enter the unoriented sector on the same footing as simple currents [2508.18357].

The $2d$ Ising theory furnishes an explicit duality-rich example. Two distinct crosscap states were proposed: one identifying Ising spins at antipodal points and one identifying dual spins, or domain walls. In the CFT basis they are
\[
|C_\pm\rangle
=
\sqrt{\frac{2+\sqrt2}{2}}\,|1\rangle\rangle_C
\pm
\sqrt{\frac{2-\sqrt2}{2}}\,|\epsilon\rangle\rangle_C,
\]
with vanishing coupling to $\sigma$. Kramers–Wannier duality acts as $\epsilon\to -\epsilon$ and exchanges these two crosscaps, so the distinction is physically meaningful and not merely a choice of sign convention [2409.11046].

## 3. Crosscap coefficients, one-point functions, and entropy

Crosscap data are extracted from overlaps between the crosscap state and primary states. In $(2+1)d$ radial quantization, the basic quantity is
\[
\mathcal A_i=\langle \mathcal C|\mathcal O_i\rangle,
\]
defined through state-operator correspondence and equal to the one-point function coefficient of the scalar primary on $\mathbb{RP}^3$. Conformal invariance fixes the $\mathbb{RP}^3$ one-point function up to this coefficient, so $\mathcal A_i$ is the intrinsic crosscap datum [2507.20005].

A notable recent result is that these coefficients can be extracted directly from microscopic realizations of the $(2+1)d$ Ising CFT. On the icosahedron, Bell-type antipodal entanglement produces numerical overlaps
\[
\mathcal A_{\mathbb I}=1.133,\qquad
\mathcal A_\epsilon=0.782,\qquad
\mathcal A_{\epsilon'}=1.293,
\]
while in the spherical Landau-level realization the thermodynamic extrapolation gives
\[
\mathcal A_{\mathbb I}=1.129,\qquad
\frac{\mathcal A_\epsilon}{\mathcal A_{\mathbb I}}=0.674,\qquad
\frac{\mathcal A_{\epsilon'}}{\mathcal A_{\mathbb I}}=0.898.
\]
These ratios agree closely with $\mathbb{RP}^3$ bootstrap values
\[
\frac{\mathcal A_\epsilon}{\mathcal A_{\mathbb I}}=0.667,\qquad
\frac{\mathcal A_{\epsilon'}}{\mathcal A_{\mathbb I}}=0.896,
\]
but the microscopic construction additionally yields absolute normalizations, which the bootstrap does not fix. In the Landau-level model there is also a parity selection rule: when $2Q+1$ is even, the crosscap state has even fermion parity, so overlaps with $\mathbb I$, $\epsilon$, and $\epsilon'$ are nonzero, whereas the overlap with $\sigma$ vanishes [2507.20005].

In $1+1$ dimensions, the analogous universal quantity is often called the crosscap entropy. In integrable QFT it is defined by
\[
s_{\mathcal C}=\log |p|,\qquad p\equiv \langle C|\Omega_L\rangle,
\]
with $\Omega_L$ the finite-volume ground state; unlike the boundary $g$-function, it does not require a subtraction by localized counterterms. In non-oriented RCFT one similarly has $s_C=\ln \Gamma_{u0}$, and one can rewrite it as
\[
s_C(u)=\hat S_u-S_{\text{top}(0)},\qquad \hat S_u=\ln P_{u0},
\]
which makes the $P$-matrix the non-orientable analogue of the modular data controlling boundary entropy [2111.09901, 1811.07238].

## 4. Integrable and microscopic realizations

Crosscap states have a particularly rigid form in integrable models. In $1+1$-dimensional integrable QFT and spin chains, the defining criterion is preservation of the commuting charges, equivalently
\[
T(\lambda)\,|\Psi_0\rangle=T(-\lambda)\,|\Psi_0\rangle
\]
for the transfer matrix. This implies that all odd local charges annihilate the state and that only parity-symmetric Bethe states have non-vanishing overlap with it [2305.16046].

In the SU$(2)$ XXX/XXZ chain, the canonical crosscap state entangles antipodal sites into Bell pairs, while in the Lieb–Liniger model it is the continuum pair-creation state
\[
|\mathcal C\rangle=\exp\left(\int_0^{\ell/2}dx\,\Phi^\dagger(x)\Phi^\dagger(x+\ell/2)\right)|\Omega\rangle.
\]
For both spin chains and Lieb–Liniger, the exact normalized overlap with an on-shell parity-symmetric Bethe state takes the universal form
\[
\frac{\langle \mathcal C|\boldsymbol\lambda_N\rangle}
{\sqrt{\langle \boldsymbol\lambda_N|\boldsymbol\lambda_N\rangle}}
=
\sqrt{\frac{\det G^+_{N/2}}{\det G^-_{N/2}}},
\]
with a trivial prefactor $\mathcal A_{\text{triv}}=1$. This absence of a model-dependent one-particle factor is one of the cleanest algebraic signatures of crosscaps among integrable initial states [2305.16046].

The XXX chain also admits a distinct antipodal singlet state, and the off-shell overlaps of both the Bell-pair and singlet constructions can be written using ABA scalar-product technology. Beyond rank one, integrable crosscaps were classified for rational $\mathfrak{gl}(N)$ chains: untwisted classes satisfy $K^2=\mathbf1$, while twisted classes satisfy $V K^t V=\pm K$, leading respectively to residual $\mathfrak{gl}(M)\oplus\mathfrak{gl}(N-M)$, $so(N)$, and $sp(N)$ structures [2207.12354, 2207.10598].

A separate development shows that crosscap eigenstates need not be restricted to maximally entangled Bell pairs. For a periodic chain of length $2L$, a tunable-entanglement antipodal product state
\[
|\Psi\rangle=\prod_{j=1}^{L}|\psi^{(2)}_{j,j+L}\rangle
\]
is an exact zero-energy eigenstate whenever the local density satisfies
\[
A^{(r)}h+h^T A^{(r)}=0.
\]
This applies to the XX model, the Bariev model, the folded XXZ model, and several non-integrable models with global $U(1)$ symmetry. Its entanglement entropy is
\[
S(\ell)=-\ell\,\mathrm{Tr}\!\left[A^T A\log(A^T A)\right],
\]
so the crosscap family interpolates continuously between product states and volume-law states [2503.15640].

## 5. Topological diagnostics, holography, and quench dynamics

Crosscap states are powerful diagnostics of parity-related anomalies. For reflection-protected $(2+1)d$ SPT phases, twisting reflection places the edge theory on an unoriented spacetime, typically a Klein bottle, and the resulting crosscap state can transform anomalously under additional symmetries. In bosonic examples this reproduces several $\mathbb Z_2$ classifications, while in fermionic BDI$+R_{++}$ the anomalous phase of the crosscap state yields the $\mathbb Z_8$ classification of topological crystalline superconductors protected by reflection and time reversal [1501.07285].

In holography, crosscap CFT on $\mathbb{RP}^2$ and the Klein bottle has been modeled by AdS$_3$ with a dS$_2$ end-of-the-world brane. The crosscap overlap with the vacuum,
\[
Z_{\mathbb{RP}^2}=\langle 0|C\rangle,
\]
is the holographic $p$-function, with semiclassical result
\[
p\equiv Z_{\mathbb{RP}^2}=\exp\!\left(-\frac{\eta_*}{4G_N}\right)
=\exp\!\left(-\frac{c}{6}\,\mathrm{arccoth}(-T)\right).
\]
The same construction reproduces the $\mathbb{RP}^2$ one-point function and the Klein-bottle partition function, and it leads to a holographic $p$-theorem under a null-energy condition on the brane matter [2405.03755].

A different holographic perspective identifies bulk local states at the AdS origin as superpositions of crosscap Ishibashi states,
\[
|\Phi_{\text{bulk}}(0)\rangle=\sum_\phi v_\phi\,|\phi)),
\]
but also shows a sharp tension: the crosscap bootstrap on $\mathbb{RP}^d$ and microscopic bulk causality are generically incompatible in the weak-gravity regime. In particular, the analytic structure required by crosscap crossing produces branch cuts for $\eta>1$ that obstruct a causal bulk-local interpretation, and in AdS$_3$ the Virasoro organization of crosscap states does not resolve the problem [1605.00334].

Crosscaps also define a distinctive nonequilibrium protocol. The crosscap quench starts from
\[
|\psi(t)\rangle=e^{-itH}e^{-\alpha H}|C\rangle.
\]
In $2d$ CFT, a single interval is already thermal at inverse temperature $\beta=4\alpha$ and remains time-independent in the thermodynamic limit, whereas an antipodal double interval begins with an area law, then shows linear growth, and finally saturates to the same thermal volume law as a single interval. The holographic dual is the $\mathbb{RP}^2$ geon, where the relevant HRT surface traverses the interior only for the antipodal double interval [2412.18610].

Lattice and circuit realizations display a related but not identical pattern. For crosscap initial states built from antipodal EPR pairs, chaotic dual-unitary or random circuits keep the bipartite entropy of a single interval constant while driving the mutual information between antipodal intervals to zero; integrable circuits and free or interacting fermion chains instead show delayed linear decrease and revivals. The appropriate quasiparticle counting is shifted from the usual local-pair form to a crosscap form involving $|2tv-L|$, reflecting antipodal rather than local pair creation [2412.04187].

## 6. Higher-dimensional generalizations and open problems

The most systematic higher-dimensional extension is the theory of crosscap defects. Their two-point functions admit three OPE channels—bulk, image, and defect—and the universal crossing condition is
\[
G(z,\bar z)=\pm G(-z,-\bar z).
\]
The defect-channel conformal blocks are identical to ordinary defect-CFT blocks after a redefinition of cross-ratios, but the transverse symmetry is $PO(q)$ rather than $O(q)$, so the local transverse sphere is effectively replaced by $\mathbb{RP}^{q-1}$. In the $O(N)$ model, explicit Gaussian and Wilson–Fisher data can be computed as functions of the defect dimension $p$; unlike standard defects, generic crosscap defects lack displacement and tilt operators, providing examples of defect conformal manifolds without exactly marginal operators [2604.19868].

There is also a classical integrable counterpart. For sigma models with Lax connection $A(\lambda)$, integrable crosscaps are encoded by a classical KT relation for half-monodromies,
\[
\kappa\,T^{(1)}(\lambda)=T^{(2),\beta}(-\lambda)\,\kappa,
\]
with untwisted classes obeying $\kappa^2=1$ and twisted classes obeying symmetry conditions such as $\kappa^t=\pm\kappa$ or their target-space analogues. This yields explicit crosscap classifications for principal chiral models, $O(N)$ models, AdS$_N$, $\mathbb{CP}^{N-1}$, and supercoset sigma models relevant to AdS/CFT [2210.02230].

Several limitations remain explicit in the current literature. In $(2+1)d$ microscopic extractions, finite-size scaling forms such as linear fits in $1/N_e$ or $1/N_e^2$ are empirical rather than derived. In integrable many-body systems, zero-rapidity sectors and finite-size corrections remain delicate. In RCFT, the systematic incorporation of non-invertible symmetry into crosscap constructions is recent and still relies on generalized Cardy checks in specific examples. And in holography, the incompatibility between bulk microscopic causality and the crosscap bootstrap remains an unresolved structural issue rather than a technical gap [2507.20005, 2305.16046, 2508.18357, 1605.00334].

Across these developments, the unifying feature is stable: a crosscap state is the state-theoretic encoding of a parity-twisted quotient. What varies from framework to framework is the precise observable it controls—$\Gamma_i$, $\mathcal A_i$, $s_{\mathcal C}$, anomalous symmetry phases, Gaudin-determinant overlaps, or defect OPE data—but in each case the crosscap organizes non-orientable information that is invisible on orientable backgrounds.

Source: https://www.emergentmind.com/topics/crosscap-states