Chord Path Integral Formalism

• Chord Path Integral Formalism is a continuum coarse-graining approach that translates discrete chord-diagram expansions into a semiclassical bilocal field theory.
• It introduces a bilocal density field n(τ1,τ2) whose dynamics are governed by Liouville-type equations, linking combinatorics with thermodynamic behavior.
• Extensions to two-species models incorporate mixed crossing kernels to describe chaotic-integrable transitions and phase structures in SYK-type systems.

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 (Berkooz et al., 2024, Berkooz et al., 2024, Jia, 28 Mar 2025, Dai et al., 29 May 2026).

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 (Berkooz et al., 2024). 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 $(G\Sigma)$0, a $(G\Sigma)$1-multinomial for splitting outgoing chords, factors $(G\Sigma)$2 from reordering, and crossing weights $(G\Sigma)$3 whenever $(G\Sigma)$4 (Berkooz et al., 2024).

In the interpolating chaotic-integrable model, the moments of

$(G\Sigma)$5

are likewise sums over chord diagrams, now with two species: $(G\Sigma)$6-chords for $(G\Sigma)$7 and $(G\Sigma)$8-chords for $(G\Sigma)$9. Each $_2$0-chord contributes $_2$1, each $_2$2-chord contributes $_2$3, each $_2$4-$_2$5 or $_2$6-$_2$7 intersection contributes $_2$8, and each $_2$9-$$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$0 intersection contributes $$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$1 (Berkooz et al., 2024). 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 $$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$2 and $$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$3 while keeping rescaled occupation numbers finite. In the single-species construction one sets

$$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$4

and uses the $$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$5 expansions of $$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$6-Pochhammer symbols together with

$$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$7

The result is a functional integral

$$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$8

over a symmetric, nonnegative bilocal field $$Z(\beta)=\sum_{k=0}^\infty \frac{(-\beta)^k}{k!}\,\langle \mathrm{Tr}\,H^k\rangle =\sum_{k\text{-chord diagrams}} q^{\#\text{intersections}},$$9 (Berkooz et al., 2024). In the two-species case the same procedure yields

$q=e^{-\lambda}$0

with $q=e^{-\lambda}$1 and $q=e^{-\lambda}$2 interpreted as continuum densities of chaotic and integrable chords (Berkooz et al., 2024).

2. Bilocal fields, measure, and continuum action

The primary field is the chord density

$q=e^{-\lambda}$3

and similarly for $q=e^{-\lambda}$4 in the two-species theory (Berkooz et al., 2024, Berkooz et al., 2024). The measure is a flat functional measure on symmetric nonnegative fields. In the single-species formulation, the formal product $q=e^{-\lambda}$5 becomes $q=e^{-\lambda}$6, and the Jacobians arising from the $q=e^{-\lambda}$7-dependence cancel at one-loop (Berkooz et al., 2024). In the two-species presentation one writes

$q=e^{-\lambda}$8

with $q=e^{-\lambda}$9, periodicity on the $s$0-circle, and symmetry $s$1 (Berkooz et al., 2024).

For one species, the action decomposes into a quartic crossing term and an entropic term,

$s$2

with

$s$3

and

$s$4

Here $s$5 is a bi-quadratic crossing term over chord-intersection regions (Berkooz et al., 2024). In the contact-diagram literature the same leading functional form appears, often with the normalization

$s$6

which is the action used to evaluate probe correlators in the $s$7 regime (Jia, 28 Mar 2025).

For two species, the action acquires a mixed crossing kernel,

$s$8

or, in the normalization of the interpolating-model analysis,

$s$9

The absence of a $\beta_i=\beta/s$0 crossing term mirrors the weight $\beta_i=\beta/s$1 assigned to $\beta_i=\beta/s$2-$\beta_i=\beta/s$3 intersections in the discrete combinatorics (Berkooz et al., 2024, Berkooz et al., 2024).

A central structural feature is that $\beta_i=\beta/s$4 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 $\beta_i=\beta/s$5 have a direct combinatorial interpretation as chord densities (Berkooz et al., 2024).

3. Saddle structure and Liouville-type equations

Varying the one-species action yields the integral equation

$\beta_i=\beta/s$6

which is conveniently rewritten in terms of the bilocal potential

$\beta_i=\beta/s$7

The resulting equation of motion is the Liouville-type PDE

$\beta_i=\beta/s$8

with $\beta_i=\beta/s$9 whenever either argument hits $n_{ij}$0 or $n_{ij}$1 in the interval formulation (Berkooz et al., 2024).

In finite-temperature saddle notation, one finds

$n_{ij}$2

and in the low-temperature regime this reduces to

$n_{ij}$3

This conformal form is the kernel that subsequently controls probe crossing weights in contact correlators (Jia, 28 Mar 2025).

The relation to the standard large-$n_{ij}$4, fixed-$n_{ij}$5 bi-local action is precise at the level of equations of motion but not off shell. The conventional $n_{ij}$6 action can be written as

$n_{ij}$7

The chord action shares the same saddle equation but is otherwise “wildly different off-shell” and, unlike the standard form, is bounded below (Berkooz et al., 2024). 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,

$n_{ij}$8

providing a direct link between coarse-grained chord combinatorics and the thermodynamics of double-scaled SYK (Berkooz et al., 2024).

4. Two-species theory and chaotic-integrable transitions

For the interpolating Hamiltonian

$n_{ij}$9

the continuum description involves two chord densities, $i$0 and $i$1, corresponding to chaotic and integrable Wick contractions (Berkooz et al., 2024). Introducing

$i$2

the saddle equations take the coupled Liouville-type form

$i$3

in one normalization (Berkooz et al., 2024), or

$i$4

in the periodic-circle normalization of the parallel derivation (Berkooz et al., 2024). When $i$5, the system reduces to the one-species Liouville equation (Berkooz et al., 2024).

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 (Berkooz et al., 2024). The more explicit saddle analysis identifies a chaotic branch and a quasi-integrable branch: $i$6 and

$i$7

respectively (Berkooz et al., 2024). At low temperature their actions cross at

$i$8

which indicates a first-order line; at high temperature only one smooth solution exists, so the line ends at a critical point at finite $i$9 (Berkooz et al., 2024).

The phase distinction also appears in dynamical observables. The thermal two-point function $j$0 differs sharply between the two branches, and the Krylov complexity exponent $j$1 is maximal, $j$2, in the chaotic phase but tends to zero in the quasi-integrable phase (Berkooz et al., 2024). For more general deformations, the phase diagram can include a zero-temperature phase transition (Berkooz et al., 2024). 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$j$3 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 $j$4-chord with a probe chord is weighted by $j$5, where $j$6 is defined by the relative flux data, and subleading $j$7 contact contributions arise when a single index from the Fock sum is reused more than twice (Jia, 28 Mar 2025). In the path-integral language, a crossing of an $j$8-chord with a probe leg spanning $j$9 contributes

$(G\Sigma)$00

so time-ordered correlators reduce to expectation values of exponential functionals of $(G\Sigma)$01 or, equivalently, linear combinations of $(G\Sigma)$02 (Jia, 28 Mar 2025).

For three probes with $(G\Sigma)$03, saddle evaluation yields

$(G\Sigma)$04

which is exactly the unique form allowed by one-dimensional conformal invariance up to an overall OPE coefficient (Jia, 28 Mar 2025). For the simplest four-point contact function, in the frame $(G\Sigma)$05,

$(G\Sigma)$06

Some choices of $(G\Sigma)$07 match AdS$(G\Sigma)$08 contact Witten diagrams exactly, while in other cases the same functional form is obtained up to logarithmic deformations (Jia, 28 Mar 2025).

The later systematic construction generalizes this picture to arbitrary periodic lattice size $(G\Sigma)$09. A pure $(G\Sigma)$10-point contact diagram must satisfy the closure condition

$(G\Sigma)$11

where $(G\Sigma)$12 specifies the sign choice in each probe insertion, and flux averaging imposes

$(G\Sigma)$13

to avoid exponential suppression (Dai et al., 29 May 2026). Each allowed configuration $(G\Sigma)$14 carries conformal parameters $(G\Sigma)$15, subject to positivity constraints inherited from a Gaussian flux distribution. In particular,

$(G\Sigma)$16

Linear combinations

$(G\Sigma)$17

then generate generic bulk contact interactions (Dai et al., 29 May 2026).

This construction computes three- to six-point contact correlators and reproduces a broad class of AdS$(G\Sigma)$18 scalar contact Witten diagrams, including those with logarithmic singularities. Logarithmic terms arise by taking nearly degenerate exponents, for example

$(G\Sigma)$19

so that

$(G\Sigma)$20

A plausible implication is that the chord path integral supplies a microscopic basis of boundary functions from which local AdS$(G\Sigma)$21 contact data can be assembled (Dai et al., 29 May 2026).

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 $(G\Sigma)$22 carry a direct combinatorial interpretation as chord densities, while the conjugate field $(G\Sigma)$23 has a holographic “Crofton form” interpretation (Berkooz et al., 2024). The measure is flat in the chord-occupation variables, the dependence on the arbitrary slicing $(G\Sigma)$24 cancels at one-loop, and the construction extends to any multi-type-chord model, including RG flows between different $(G\Sigma)$25’s or Parisi-hypercube models, by changing the bilinear crossing kernels in the action (Berkooz et al., 2024). 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: $(G\Sigma)$26 In that context, the path-ordered exponential is a generating functional for iterated integrals of chord diagrams associated with braids and Vassiliev invariants (Gauthier, 2012). 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 (Berkooz et al., 2024, Berkooz et al., 2024, Jia, 28 Mar 2025, Dai et al., 29 May 2026).