Hartle–Hawking Chord State Overview

• Hartle–Hawking chord state is a family of representations that express the traditional Hartle–Hawking state in conjugate variables or chord-number bases.
• In minisuperspace, it appears as the Fourier dual of the Chern–Simons (Kodama) state, yielding a uniform distribution in the Hubble (b) variable.
• In double-scaled and supersymmetric SYK models, it manifests as a discrete wavefunction encoding chord numbers and exact BPS degeneracies.

“Hartle-Hawking chord state” is not a uniformly standardized term across the current literature. In FRW minisuperspace, the phrase does not appear in the paper that proves the relevant result, but that paper does provide a mathematically natural analogue: the Hartle–Hawking state represented in the connection/Hubble variable $b$, where it becomes the minisuperspace Chern–Simons wavefunction (Magueijo, 2020). In double-scaled SYK, by contrast, the Hartle–Hawking wavefunction is explicitly defined in the chord-number basis as $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$ (Okuyama, 2022). In the $\mathcal N=2$ supersymmetric extension, the zero-temperature Hartle–Hawking chord state is the unique normalizable zero-energy state of the super-chord transfer matrix in a fixed $U(1)_R$ sector, interpreted as a supersymmetric Einstein–Rosen bridge (Boruch et al., 2023). A further, non-native use is suggested by de Sitter bra-ket wormhole constructions, where the natural object is a Wigner distribution on minisuperspace phase space; a chord representation would then arise by Fourier transformation, although that terminology is not used in the paper (Fumagalli et al., 2024).

1. Terminological scope

The phrase designates different but related constructions depending on context. In one line of work it means a Hartle–Hawking state written in variables conjugate to the standard metric variable. In another it means the Hartle–Hawking wavefunction expressed in a literal chord-number basis. The literature therefore does not support a single universal definition.

Setting	State object	Status of the term
FRW minisuperspace	HH state in the conjugate $b$-representation	Analogue; term not used natively
Double-scaled SYK	$$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$	Native chord-basis usage
$\mathcal N=2$ SYK with chords	Zero-temperature supersymmetric wormhole wavefunction	Native “Hartle–Hawking chord state” usage
dS bra-ket wormholes	Wigner distribution on phase space	Chord interpretation is only a plausible extrapolation

This distinction matters because the minisuperspace and SYK constructions are not merely different realizations of one formalism; they use different underlying Hilbert spaces and different meanings of “chord.” In FRW minisuperspace, the relevant duality is between $a^2$ and $b$. In SYK, the label $\ell$ counts open chords crossing a Euclidean time slice and is interpreted as a discretized bulk geodesic length (Magueijo, 2020, Okuyama, 2022).

2. Minisuperspace Fourier duality and the “chord-like” Hartle–Hawking state

In FRW minisuperspace, the starting point is the minisuperspace reduction of the Einstein–Cartan action

$$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$0

The relevant canonical pair is $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$1, not $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$2, with

$$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$3

This exact canonical conjugacy underlies the duality between the metric and connection/Hubble representations.

In the $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$4-representation one has

$$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$5

and the Hamiltonian constraint becomes first order: $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$6 Its general solution is

$$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$7

The paper identifies this as exactly the minisuperspace reduction of the Chern–Simons/Kodama state. In the $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$8-representation, by contrast, the Wheeler–DeWitt equation is second order and admits the Hartle–Hawking and Vilenkin branches. The transform between the two representations is the Fourier kernel determined by the commutator, and after substitution of $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$9 the integral reduces to the standard Airy representation with

$\mathcal N=2$0

The corresponding metric-representation solutions are

$\mathcal N=2$1

The core result is that the minisuperspace Chern–Simons state is the Fourier dual of the Hartle–Hawking and Vilenkin wavefunctions, with the contour in the complex $\mathcal N=2$2-plane selecting which state is represented. For Hartle–Hawking, the simplest contour is the entire real line,

$\mathcal N=2$3

whereas for Vilenkin the contour is

$\mathcal N=2$4

The paper is explicit that if one insists on $\mathcal N=2$5 being real, only the Hartle–Hawking wavefunction can be dual to the Chern–Simons state. This is why the most natural “Hartle–Hawking chord state” in this minisuperspace sense is the real-$\mathcal N=2$6 Chern–Simons representation of the HH state. The same paper also emphasizes that this equivalence is exact only within the minisuperspace model; it is not a proof of a global full-theory identification between the full Hartle–Hawking state and the full Chern–Simons state (Magueijo, 2020).

A further consequence is probabilistic. Since $\mathcal N=2$7 is a pure phase on the real contour for the HH choice, the paper concludes that the Hartle–Hawking state predicts a uniform distribution in $\mathcal N=2$8 over the whole real line. A plausible implication is that, if “chord state” is understood as a conjugate-variable representation, the Hartle–Hawking chord state is not a new state but the HH state viewed in a representation adapted to its canonically conjugate Hubble/connection variable (Magueijo, 2020).

3. Hartle–Hawking wavefunctions in the chord basis of double-scaled SYK

In double-scaled SYK, the terminology is literal. The model is defined by

$\mathcal N=2$9

and the disorder-averaged dynamics is encoded by a transfer matrix acting on a chord Hilbert space spanned by orthonormal states $U(1)_R$0. The state $U(1)_R$1 is the state with $U(1)_R$2 open chords crossing a fixed Euclidean time slice, so $U(1)_R$3 is the vacuum chord-number state and $U(1)_R$4 has exactly $U(1)_R$5 chords threading the slice.

The transfer matrix is

$U(1)_R$6

with

$U(1)_R$7

The Hartle–Hawking state is not one distinguished basis vector $U(1)_R$8; it is the Euclidean-prepared state

$U(1)_R$9

and its wavefunction in the chord basis is

$b$0

This amplitude is interpreted as the Hartle–Hawking wavefunction of the bulk gravitational theory in a basis labeled by chord number, and the discrete label $b$1 is interpreted as a discretized geodesic length.

The computation proceeds by diagonalizing $b$2 with $b$3-Hermite polynomials: $b$4 This yields the spectral integral

$b$5

which can then be rewritten as an explicit Bessel/$b$6-Pochhammer sum. The final formula given in the paper is

$b$7

This wavefunction is then used as a building block for un-crossed matter correlators. For example, the two-point function takes the gluing form

$b$8

where the factor $b$9 counts intersections between the matter chord and the $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$0 background $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$1-chords. In this precise DSSYK usage, the Hartle–Hawking chord state is therefore a Euclidean-prepared state whose components in the chord-number basis encode a bulk wavefunction in a discrete length variable (Okuyama, 2022).

4. The supersymmetric Hartle–Hawking chord state

The most explicit use of the phrase appears in the $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$2 supersymmetric SYK-with-chords construction. There the Hartle–Hawking chord state is the exact zero-temperature realization of the two-sided supersymmetric wormhole. The starting relation is the fixed-$$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$3 version of the two-sided partition function,

$$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$4

so the $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$5 limit projects onto the zero-energy state of the chord transfer matrix in that charge sector.

The super-chord Hilbert space is built from basis states

$$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$6

where $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$7 is the chord number. The paper improves the earlier construction by introducing an extra label $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$8, interpreted as the net effect of previously closed chords on future friend/enemy factors. In the empty-wormhole sector, the transfer matrix enjoys an enhanced $$\Psi_{\rm HH}^{(\beta)}(\ell)=\langle \ell|e^{-\beta T}|0\rangle$$9 supersymmetry: $\mathcal N=2$0 with $\mathcal N=2$1. This allows the Hartle–Hawking state to be characterized as the unique normalizable zero-energy state annihilated by all supercharges.

In the sector $\mathcal N=2$2, only the $\mathcal N=2$3 and $\mathcal N=2$4 components remain, and the BPS ansatz is

$\mathcal N=2$5

The conditions

$\mathcal N=2$6

lead to the recursion relations

$\mathcal N=2$7

$\mathcal N=2$8

The explicit solution is

$\mathcal N=2$9

$a^2$0

This is the exact supersymmetric Hartle–Hawking chord wavefunction.

The norm is also exact: $a^2$1 and the normalized summand defines the probability $a^2$2 that the supersymmetric wormhole has chord number $a^2$3, interpreted as wormhole length. The most important quantity is the zero-length probability. The paper proves that

$a^2$4

so the probability that the supersymmetric Einstein–Rosen bridge has vanishing length equals the fraction of one-sided states at charge $a^2$5 that are exact supersymmetric ground states. The resulting exact formula is

$a^2$6

In this supersymmetric setting, the Hartle–Hawking chord state is therefore a genuine two-sided bound-state wavefunction whose short-length tail computes exact BPS degeneracies, including non-perturbative corrections (Boruch et al., 2023).

5. Wigner-space cosmology and chord-space extrapolation

A different but phase-space-adjacent development appears in de Sitter bra-ket wormhole models. There the central claim is that, once one allows connected geometries producing bra and ket simultaneously, the natural object is a Wigner distribution rather than a wavefunction. For the boundary-size variable $a^2$7, the density matrix is

$a^2$8

and the corresponding Wigner transform is

$a^2$9

With an inflaton clock, the phase space enlarges to $b$0, with

$b$1

In this framework, the disconnected Hartle–Hawking contribution remains present and can be Wigner-transformed explicitly, while the connected bra-ket wormhole becomes a genuine semiclassical saddle only in the Wigner representation. For pure gravity the paper finds

$b$2

whereas with an inflaton field one obtains a Gaussian-type Wigner distribution peaked on a classical phase-space constraint surface. The same paper states that in the regime of large universes the connected geometry dominates over the Hartle–Hawking saddle and gives a distribution with a meaningful probabilistic interpretation for local observables, but not a normalizable probability measure on the entire phase space.

A plausible implication is that a “Hartle–Hawking chord state” could be defined here by Fourier transforming the Wigner distribution over phase space into the corresponding characteristic-function or chord representation. That step is not performed in the paper itself, and the term is not used there. The nearest formal analogue is therefore a phase-space transform of the Hartle–Hawking or bra-ket wormhole Wigner distributions, not a named object already introduced in the paper (Fumagalli et al., 2024).

A related non-metric representation appears in the microcanonical thermofield-double literature, where the Hartle–Hawking wavefunction of an eternal black hole is treated in a fixed-$b$3 basis as the overlap

$b$4

computed semiclassically by a wedge or “pacman” geometry. That paper does not use the term chord state, but it realizes the same general idea of expressing a Hartle–Hawking state in a nonstandard basis of conjugate or reduced variables (Chua et al., 2023).

6. Conceptual status, neighboring literatures, and limits

The term should therefore be used with care. In the quantum-cosmology literature around the Chern–Simons/Kodama state, “Hartle–Hawking chord state” is at most an analogical description of the HH state in a conjugate $b$5-representation. The paper proving the relevant equivalence explicitly does not embed the discussion in a standard chord/Wigner phase-space formalism, and it is equally explicit that its exactness is minisuperspace-exact rather than a full-theory equivalence (Magueijo, 2020). In DSSYK and supersymmetric SYK, by contrast, the chord basis is literal and native: the Hartle–Hawking state is a wavefunction on a Hilbert space spanned by chord-number states, and in the $b$6 model the zero-temperature state is a precise supersymmetric wormhole wavefunction (Okuyama, 2022, Boruch et al., 2023).

This context dependence also separates the term from the standard black-hole quantum-field-theory notion of the Hartle–Hawking or Hartle–Hawking–Israel state. In that literature the state is typically described either as the double KMS state at the Hawking temperature, perturbatively equivalent to the Euclidean Hartle–Hawking state in real-time formalism, or as the unique pure Hadamard extension of the exterior thermal state across a bifurcate Killing horizon (Higuchi et al., 2021, Gérard, 2018). Those papers do not use chord terminology. Likewise, string-theoretic analyses of the Hartle–Hawking state for the $b$7 eternal black hole describe the state as a left-right entangled kernel deformed by an energy-dependent phase, not as a chord state (Ben-Israel et al., 2015).

Two recurring limitations follow from this survey. First, the phrase does not pick out one universally accepted object. Second, whenever it is used or suggested, it denotes a representation of an underlying Hartle–Hawking state rather than a separate boundary-condition proposal. In FRW minisuperspace the relevant representation is Fourier-dual in $b$8; in DSSYK it is the chord-number basis $b$9; in de Sitter bra-ket wormhole models it is, at most, a prospective phase-space transform of a Wigner distribution. The most precise encyclopedia-level conclusion is therefore that a Hartle–Hawking chord state is not a unique concept but a family of representation-dependent constructions whose best-defined realizations are the $\ell$0-space Fourier dual of the minisuperspace HH wavefunction and the chord-basis Hartle–Hawking wavefunctions of double-scaled SYK (Magueijo, 2020, Okuyama, 2022, Boruch et al., 2023, Fumagalli et al., 2024).