---
title: Chord Path Integral Formalism
url: https://www.emergentmind.com/topics/chord-path-integral-formalism
type: topic
---

# Chord Path Integral Formalism

The chord path integral formalism is a continuum coarse-graining of chord-diagram expansions in double-scaled Sachdev-Ye-Kitaev-type models and related Fock-space constructions. Its basic dynamical variable is a bilocal, nonnegative chord-density field \(n(\tau_1,\tau_2)\), or a multi-component generalization when several chord species are present. In the one-species case, the formalism reproduces the same equations of motion as the bi-local \((G\Sigma)\) Liouville action while remaining otherwise different and, in particular, well defined; in two-species and probe-augmented versions it describes chaotic-integrable transitions, thermal phase structure, and contact correlators with conformal and AdS\(_2\) interpretations [2403.05980, 2403.01950, 2503.22619, 2605.30970].

## 1. Combinatorial origin in chord-diagram expansions

The starting point is the exact chord expansion of the annealed partition sum of a single-species double-scaled SYK-type Hamiltonian,
\[
Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle
=\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},
\]
with \(q=e^{-\lambda}\) in the single-species derivation [2403.05980]. The Euclidean circle is divided into \(s\) equal arcs of length \(\beta_i=\beta/s\), and for each pair of segments one introduces integers \(n_{ij}\), the numbers of chords with one endpoint in segment \(i\) and the other in segment \(j\). The exact discrete expression is built from four ingredients: a transfer-matrix factor \(\langle n_i|e^{-\beta_iT}|0\rangle\), a \(q\)-multinomial for splitting outgoing chords, factors \([n_{ij}]_q!\) from reordering, and crossing weights \(q^{n_{ij}n_{k\ell}}\) whenever \(i<k<j<\ell\) [2403.05980].

In the interpolating chaotic-integrable model, the moments of
\[
H(\kappa)=\nu H_{\rm chaos}+\kappa H_{\rm integ},\qquad \nu^2+\kappa^2=1,
\]
are likewise sums over chord diagrams, now with two species: \(n\)-chords for \(H_{\rm chaos}\) and \(z\)-chords for \(H_{\rm integ}\). Each \(n\)-chord contributes \((\nu^2J^2/\lambda)\), each \(z\)-chord contributes \((\kappa^2B^2/\lambda)\), each \(n\)-\(n\) or \(n\)-\(z\) intersection contributes \(q\equiv e^{-\lambda}\), and each \(z\)-\(z\) intersection contributes \(1\) [2403.01950]. This asymmetric crossing rule is the combinatorial origin of the distinct roles played by chaotic and integrable sectors in the continuum action.

The semiclassical continuum limit sends \(\lambda\to0\) and \(s\to\infty\) while keeping rescaled occupation numbers finite. In the single-species construction one sets
\[
\tilde n_{ij}=\lambda n_{ij},\qquad \tilde\beta_i=\sqrt{\lambda}\,\beta_i\ll1,\qquad \tilde n_{ij}\ll1,
\]
and uses the \(q\to1\) expansions of \(q\)-Pochhammer symbols together with
\[
[n]_q!\simeq \frac{n!}{(1-q)^n},\qquad
\langle n_i|e^{-\beta_iT}|0\rangle \simeq (-1)^{n_i}\lambda^{-n_i}\exp\!\Big[\frac{\beta_i^2}{2}-n_i\log(\beta_i)+O(\lambda)\Big].
\]
The result is a functional integral
\[
Z=\int [Dn]\,e^{-S[n]/\lambda}
\]
over a symmetric, nonnegative bilocal field \(n(\tau_1,\tau_2)\) [2403.05980]. In the two-species case the same procedure yields
\[
Z=\int Dn\,Dz\,e^{-S[n,z]/\lambda},
\]
with \(n(\tau_1,\tau_2)\) and \(z(\tau_1,\tau_2)\) interpreted as continuum densities of chaotic and integrable chords [2403.01950].

## 2. Bilocal fields, measure, and continuum action

The primary field is the chord density
\[
n(\tau_1,\tau_2)
=\lim_{\substack{s\to\infty,\;\lambda\to0}}
\frac{\lambda\,n_{ij}}{(\beta/s)^2},
\qquad \tau_i=i\,\beta/s,
\]
and similarly for \(z(\tau_1,\tau_2)\) in the two-species theory [2403.05980, 2403.01950]. The measure is a flat functional measure on symmetric nonnegative fields. In the single-species formulation, the formal product \(\prod_{i<j}dn_{ij}\) becomes \(\mathcal D n(\tau_1,\tau_2)\), and the Jacobians arising from the \(s\)-dependence cancel at one-loop [2403.05980]. In the two-species presentation one writes
\[
D n\,D z=\prod_{0<\tau_1<\tau_2<\beta} dn(\tau_1,\tau_2)\,dz(\tau_1,\tau_2),
\]
with \(\tau_i\in[0,\beta]\), periodicity on the \(\beta\)-circle, and symmetry \(n(\tau_1,\tau_2)=n(\tau_2,\tau_1)\) [2403.01950].

For one species, the action decomposes into a quartic crossing term and an entropic term,
\[
S[n]=S_4[n]+S_2[n],
\]
with
\[
S_4[n]
=
\frac14
\int_0^\beta d\tau_1\int_0^\beta d\tau_2
\int_{\tau_1}^{\tau_2} d\tau_3
\int_{\tau_2}^{\tau_1} d\tau_4\,
n(\tau_1,\tau_2)\,n(\tau_3,\tau_4),
\]
and
\[
S_2[n]
=
\frac12
\int_0^\beta d\tau_1\int_0^\beta d\tau_2\,
n(\tau_1,\tau_2)
\Bigl[\log\!\frac{n(\tau_1,\tau_2)}{\mathbf J^2}-1\Bigr],
\qquad \mathbf J^2\equiv \lambda \mathcal J^2.
\]
Here \(S_4\) is a bi-quadratic crossing term over chord-intersection regions [2403.05980]. In the contact-diagram literature the same leading functional form appears, often with the normalization
\[
S[n]
=
\frac14\int d\tau_a\,d\tau_b\,d\tau_c\,d\tau_d\,n(\tau_a,\tau_b)n(\tau_c,\tau_d)
+
\frac12\int d\tau_a\,d\tau_b\,n(\tau_a,\tau_b)\bigl[\ln n(\tau_a,\tau_b)-1\bigr],
\]
which is the action used to evaluate probe correlators in the \(q\to1\) regime [2503.22619].

For two species, the action acquires a mixed crossing kernel,
\[
\begin{aligned}
S[n,z]
&=
\frac14\!\!\int d^4\tau\,
\bigl[n(\tau_1,\tau_2)n(\tau_3,\tau_4)+2\,n(\tau_1,\tau_2)z(\tau_3,\tau_4)\bigr] \\
&\quad
+\frac12\!\!\int d^2\tau\,
\Bigl\{
n\Bigl[\log\!\frac{n}{\nu^2\mathbf J^2}-1\Bigr]
+
z\Bigl[\log\!\frac{z}{\kappa^2\mathbf J^2}-1\Bigr]
\Bigr\},
\end{aligned}
\]
or, in the normalization of the interpolating-model analysis,
\[
S[n,z]
=
\frac14\int (\cdots)\,[n\,n+2\,n\,z]
+\frac12\int d\tau_1\,d\tau_2\,
\left\{
n\bigl[\ln(n/\nu^2)-1\bigr]
+
z\bigl[\ln(z/\kappa^2)-1\bigr]
\right\}.
\]
The absence of a \(z\,z\) crossing term mirrors the weight \(1\) assigned to \(z\)-\(z\) intersections in the discrete combinatorics [2403.05980, 2403.01950].

A central structural feature is that \(S[n]\) is manifestly bounded below and arises from a sum of positive combinatorial weights. The same source also emphasizes that there are no gauge-like ambiguities, unlike the transfer-matrix convention, and that the fields \(n(\tau_1,\tau_2)\) have a direct combinatorial interpretation as chord densities [2403.05980].

## 3. Saddle structure and Liouville-type equations

Varying the one-species action yields the integral equation
\[
\int_{\tau_1}^{\tau_2} d\tau_3
\int_{\tau_2}^{\tau_1} d\tau_4\,
n(\tau_3,\tau_4)
+
\log\!\frac{n(\tau_1,\tau_2)}{\mathbf J^2}
=0,
\qquad \tau_1\neq\tau_2,
\]
which is conveniently rewritten in terms of the bilocal potential
\[
g(\tau_1,\tau_2)
=
-\int_{\tau_1}^{\tau_2} d\tau_3
\int_{\tau_2}^{\tau_1} d\tau_4\,n(\tau_3,\tau_4),
\qquad
n(\tau_1,\tau_2)
=
-\frac12\,\partial_{\tau_1}\partial_{\tau_2}g.
\]
The resulting equation of motion is the Liouville-type PDE
\[
\partial_{\tau_1}\partial_{\tau_2}g(\tau_1,\tau_2)
+2\,\mathbf J^2\,e^{g(\tau_1,\tau_2)}=0,
\]
with \(g=0\) whenever either argument hits \(0\) or \(\beta\) in the interval formulation [2403.05980].

In finite-temperature saddle notation, one finds
\[
e^{g(\tau_a,\tau_b)}
=
\frac{\cos^2\!\bigl(\frac{\pi v}{2}\bigr)}
{\cos^2\!\bigl[\frac{\pi v}{2}(1-2|\tau_b-\tau_a|/\beta)\bigr]},
\qquad
\beta=\frac{\pi v}{\cos(\frac{\pi v}{2})},
\]
and in the low-temperature regime this reduces to
\[
e^{g(\tau_a,\tau_b)}\approx \frac1{|\tau_a-\tau_b|^2}.
\]
This conformal form is the kernel that subsequently controls probe crossing weights in contact correlators [2503.22619].

The relation to the standard large-\(N\), fixed-\(p\) bi-local action is precise at the level of equations of motion but not off shell. The conventional \((G\Sigma)\) action can be written as
\[
I[g,\sigma]
=
\frac1\lambda
\left[
-\iint d^4\tau\,\sigma\,\sigma
+\iint d^2\tau\,\Bigl(i\sigma\,g-\frac{\mathbf J^2}{2}e^g\Bigr)
\right]
\longrightarrow
-\frac1{2\lambda}\iint d^2\tau\,
\Bigl[\frac14\partial g\partial g-\mathbf J^2 e^g\Bigr].
\]
The chord action shares the same saddle equation but is otherwise “wildly different off-shell” and, unlike the standard form, is bounded below [2403.05980]. This is one of the principal clarifications introduced by the formalism: agreement at the saddle does not imply equality of path-integral definitions.

Evaluating the one-species action on the saddle reproduces the known Schwarzian-density free energy,
\[
\log Z
\simeq
-\frac1\lambda
\left(
\frac{\pi^2v^2}{2}
-
2\pi v\tan\frac{\pi v}{2}
\right),
\]
providing a direct link between coarse-grained chord combinatorics and the thermodynamics of double-scaled SYK [2403.05980].

## 4. Two-species theory and chaotic-integrable transitions

For the interpolating Hamiltonian
\[
H=\nu H_{\rm chaotic}+\kappa H_{\rm integrable},
\]
the continuum description involves two chord densities, \(n(\tau_1,\tau_2)\) and \(z(\tau_1,\tau_2)\), corresponding to chaotic and integrable Wick contractions [2403.05980]. Introducing
\[
g_n=-\!\!\iint n,\qquad g_z=-\!\!\iint z,
\]
the saddle equations take the coupled Liouville-type form
\[
\partial_1\partial_2 g_n=-2\,\mathbf J^2\nu^2 e^{g_n+g_z},
\qquad
\partial_1\partial_2 g_z=-2\,\mathbf J^2\kappa^2 e^{g_n},
\]
in one normalization [2403.05980], or
\[
\partial_{\tau_1}\partial_{\tau_2}g_n=-4\nu^2e^{g_n+g_z},
\qquad
\partial_{\tau_1}\partial_{\tau_2}g_z=-4\kappa^2e^{g_n},
\]
in the periodic-circle normalization of the parallel derivation [2403.01950]. When \(\kappa=0\), the system reduces to the one-species Liouville equation [2403.05980].

The thermodynamic content is that the system has two distinct phases. One is continuously connected to the chaotic SYK Hamiltonian, and the other is continuously connected to the integrable Hamiltonian; at low temperature they are separated by a first-order phase transition [2403.05980]. The more explicit saddle analysis identifies a chaotic branch and a quasi-integrable branch:
\[
g_z=0,\qquad
g_n(\tau)=2\ln\!\left[\frac{\pi}{\beta\,\cos(\pi(1/2-\tau/\beta))}\right],
\qquad
S_{\rm chaos}=-2\beta+O(\kappa^4),
\]
and
\[
g_n=0,\qquad
g_z(\tau)=-(\kappa\beta)^2(\tau/\beta)(1-\tau/\beta),
\qquad
S_{\rm integ}=-\frac12(\kappa\beta)^2+O\!\left(\frac{\nu^4}{\kappa^2\beta^2}\right),
\]
respectively [2403.01950]. At low temperature their actions cross at
\[
\kappa_* \simeq \frac{2}{\sqrt{\beta}},
\]
which indicates a first-order line; at high temperature only one smooth solution exists, so the line ends at a critical point at finite \(\beta\) [2403.01950].

The phase distinction also appears in dynamical observables. The thermal two-point function \(e^{g_n+g_z}\) differs sharply between the two branches, and the Krylov complexity exponent \(2\alpha\) is maximal, \(2\pi/\beta\), in the chaotic phase but tends to zero in the quasi-integrable phase [2403.01950]. For more general deformations, the phase diagram can include a zero-temperature phase transition [2403.05980]. This suggests that the formalism is not merely a rewriting of the SYK saddle, but a framework for organizing chaotic-integrable competition in double-scaled models.

## 5. Probe insertions, contact diagrams, and AdS\(_2\) matching

A second major development uses the chord path integral to compute probe correlators and contact diagrams in Fock-space flux models with random Aharonov-Bohm phases. In these models each crossing of an \(H\)-chord with a probe chord is weighted by \(q^\Delta\), where \(\Delta\) is defined by the relative flux data, and subleading \(1/N\) contact contributions arise when a single index from the Fock sum is reused more than twice [2503.22619]. In the path-integral language, a crossing of an \(H\)-chord with a probe leg spanning \((\tau_a,\tau_b)\) contributes
\[
\exp\!\left[
-\Delta\!\int_{\tau_a}^{\tau_b}d\tau\int_{\tau_b}^{\tau_a}d\tau'\,n(\tau,\tau')
\right]
=
\exp\!\bigl[\Delta\,g(\tau_a,\tau_b)\bigr],
\]
so time-ordered correlators reduce to expectation values of exponential functionals of \(n\) or, equivalently, linear combinations of \(g_{ij}=g(\tau_i,\tau_j)\) [2503.22619].

For three probes with \(\tau_3>\tau_2>\tau_1\), saddle evaluation yields
\[
\langle O_3O_2O_1\rangle_{\rm contact}
=
\frac{1}{\sqrt N}\,
\frac{1}{
|\tau_{12}|^{\Delta_1+\Delta_2-\Delta_3}\,
|\tau_{13}|^{\Delta_1+\Delta_3-\Delta_2}\,
|\tau_{23}|^{\Delta_2+\Delta_3-\Delta_1}
},
\]
which is exactly the unique form allowed by one-dimensional conformal invariance up to an overall OPE coefficient [2503.22619]. For the simplest four-point contact function, in the frame \(\{\tau_1,\tau_2,\tau_3,\tau_4\}\to\{0,x,1,\infty\}\),
\[
I(x)
=
\lim_{\tau_4\to\infty}
|\tau_4|^{2\Delta_4}
\langle O_4(\infty)O_3(1)O_2(x)O_1(0)\rangle_{\rm contact}
=
\frac1N\,
x^{\Delta_{12}-\Delta_1-\Delta_2}
(1-x)^{\Delta_{23}-\Delta_2-\Delta_3}.
\]
Some choices of \((\Delta_i,\Delta_{ij})\) match AdS\(_2\) contact Witten diagrams exactly, while in other cases the same functional form is obtained up to logarithmic deformations [2503.22619].

The later systematic construction generalizes this picture to arbitrary periodic lattice size \(L\). A pure \(n\)-point contact diagram must satisfy the closure condition
\[
\sum_{i=1}^n \epsilon_i \equiv 0 \pmod L,
\]
where \(\epsilon_i=\pm1\) specifies the sign choice in each probe insertion, and flux averaging imposes
\[
\sum_{i=1}^n \epsilon_i\,\tilde F^{(i)} \in 4\pi\mathbb Z
\]
to avoid exponential suppression [2605.30970]. Each allowed configuration \(\mathcal C\) carries conformal parameters \(\{\Delta_i,\Delta_{ij},\Delta_{ijk},\dots\}\), subject to positivity constraints inherited from a Gaussian flux distribution. In particular,
\[
(\sqrt{\Delta_i}-\sqrt{\Delta_j})^2
\le
\Delta_{ij}
\le
(\sqrt{\Delta_i}+\sqrt{\Delta_j})^2.
\]
Linear combinations
\[
I_{\rm phys}=\sum_{\mathcal C} w_{\mathcal C}\,I_{\mathcal C},
\qquad
\sum_{\mathcal C}w_{\mathcal C}=1,
\]
then generate generic bulk contact interactions [2605.30970].

This construction computes three- to six-point contact correlators and reproduces a broad class of AdS\(_2\) scalar contact Witten diagrams, including those with logarithmic singularities. Logarithmic terms arise by taking nearly degenerate exponents, for example
\[
w_1=\frac{A}{\epsilon},\qquad w_2=-\frac{A}{\epsilon},
\]
so that
\[
w_1x^{a_1}+w_2x^{a_2}
=
\frac{A}{\epsilon}\,x^{a_1}(x^\epsilon-1)
\xrightarrow{\epsilon\to0}
A\,x^{a_1}\ln x.
\]
A plausible implication is that the chord path integral supplies a microscopic basis of boundary functions from which local AdS\(_2\) contact data can be assembled [2605.30970].

## 6. Interpretive features, extensions, and distinct meanings of the term

The formalism has several novel features that distinguish it from earlier collective-field descriptions. The fields \(n(\tau_1,\tau_2)\) carry a direct combinatorial interpretation as chord densities, while the conjugate field \(g\) has a holographic “Crofton form” interpretation [2403.05980]. The measure is flat in the chord-occupation variables, the dependence on the arbitrary slicing \(s\) cancels at one-loop, and the construction extends to any multi-type-chord model, including RG flows between different \(p\)’s or Parisi-hypercube models, by changing the bilinear crossing kernels in the action [2403.05980]. This suggests a controlled coarse-grained representation of double-scaled SYK-type models whose semiclassical limit reproduces, and in that sense UV-completes, the bi-local Liouville formula.

A common source of confusion is terminological rather than conceptual. The expression “chord path integral” also appears in a mathematically distinct setting, where the holonomy of the Knizhnik-Zamolodchikov connection on a bundle of chord-diagram algebras generates the Kontsevich integral:
\[
Z(\gamma)=P\exp\!\Bigl(\int_\gamma A\Bigr),
\qquad
A(z)=\frac1{2\pi i}\sum_{1\le i<j\le N}|ij\rangle\,d\log(z_i-z_j).
\]
In that context, the path-ordered exponential is a generating functional for iterated integrals of chord diagrams associated with braids and Vassiliev invariants [1202.5694]. Although both subjects involve chord diagrams and generating functionals, they concern different objects: the SYK-related formalism is a continuum functional integral over chord-density fields, whereas the Kontsevich/KZ construction is a holonomy in a flat connection on configuration space.

Within double-scaled SYK and its extensions, the chord path integral formalism therefore occupies a specific role: it converts exact discrete chord combinatorics into a bilocal semiclassical field theory, preserves the combinatorial meaning of the variables, clarifies the relation to Liouville-type saddle equations, and provides a unified language for thermodynamic transitions and contact correlators [2403.05980, 2403.01950, 2503.22619, 2605.30970].

Source: https://www.emergentmind.com/topics/chord-path-integral-formalism