$c=1$ strings as a matrix integral

Lorenz Eberhardt; Scott Collier; Victor A. Rodriguez

arxiv: 2604.06301 · v2 · pith:JEKMBEMZnew · submitted 2026-04-07 · ✦ hep-th

c=1 strings as a matrix integral

Scott Collier , Lorenz Eberhardt , Victor A. Rodriguez This is my paper

Pith reviewed 2026-05-10 18:50 UTC · model grok-4.3

classification ✦ hep-th

keywords c=1 stringS-matrixmatrix integralintersection numbersspectral curvetrialityBrillouin zoneunitarity

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The pith

The c=1 string's perturbative S-matrix is equivalently described by a double-scaled matrix integral with spectral curve x(z)=2√2 cos(z), y(z)=sin(z).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives closed-form expressions for the c=1 string amplitudes as intersection numbers on the moduli space of Riemann surfaces. These expressions naturally describe a discretized target space in which momentum is conserved only modulo an integer. The physical S-matrix is recovered by restricting to the first Brillouin zone and analytically continuing to Lorentzian kinematics. The resulting amplitudes satisfy unitarity and a Mirzakhani recursion directly from the intersection theory, and match known matrix quantum mechanics results. Combined with the existing duality to matrix quantum mechanics, this establishes a triality among the worldsheet, matrix quantum mechanics, and matrix integral formulations.

Core claim

The perturbative S-matrix of the c=1 string admits a description in terms of a double-scaled (0+0)-dimensional matrix integral based on the spectral curve x(z)=2√2 cos(z), y(z)=sin(z). Intersection numbers on the moduli space compute amplitudes for a discretized target space; physical elements are recovered by Brillouin-zone restriction followed by analytic continuation to Lorentzian signature. These amplitudes obey perturbative unitarity and Mirzakhani recursion directly from the intersection expressions and agree with matrix quantum mechanics.

What carries the argument

The spectral curve x(z)=2√2 cos(z), y(z)=sin(z) of the double-scaled (0+0)-dimensional matrix integral, which generates the intersection-number expressions for the discretized amplitudes.

If this is right

The intersection theory expressions directly prove perturbative spacetime unitarity without reference to other formulations.
Amplitudes obey a Mirzakhani-type recursion relation.
Detailed numerical agreement holds with all known matrix quantum mechanics results.
The three formulations—worldsheet, matrix quantum mechanics, and matrix integral—are exactly equivalent at the perturbative level.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same intersection-number approach might extend to other values of the central charge or to non-perturbative completions of the theory.
The discretized momentum picture could connect to lattice or compactified models in related quantum gravity contexts.
The explicit matrix-integral representation may enable new exact computations of higher-genus or multi-point amplitudes.
The triality suggests that similar matrix-integral duals could exist for other exactly solvable string models.

Load-bearing premise

Physical S-matrix elements are recovered by restricting the intersection-number expressions to the first Brillouin zone and then analytically continuing to Lorentzian kinematics.

What would settle it

A concrete mismatch between a computed higher-point amplitude (for example the four-point function) in the matrix integral after Brillouin-zone restriction and the known matrix quantum mechanics result would falsify the claimed triality.

Figures

Figures reproduced from arXiv: 2604.06301 by Lorenz Eberhardt, Scott Collier, Victor A. Rodriguez.

**Figure 1.** Figure 1: Dual descriptions of c = 1 strings. The first leg of the triality (worldsheet ↔ MQM) is the well-established conjectural duality from the early 90s. The second leg (MQM ↔ matrix integral) was partially anticipated by Alexandrov [24], who identified a spectral curve from the FZZT disk amplitude, but was not developed into a full computational framework. – 3 – [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: The unshaded area represents the region in the external Liouville momenta where the moduli space integral that defines the CLS four-point function converges. The shaded regions correspond to the 90-degree wedges of divergence emanating from the branch points of the amplitude. In the limit that b is taken to one, the region of convergence pinches off and infinitely many branch points collide at pi ± pj = −1… view at source ↗

**Figure 3.** Figure 3: Passage from vertex colors mv to edge variables me = me◦ − m•e. The left panel shows two vertices connected by an edge, with vertex colors m•e and me◦. The right panel shows the equivalent description in terms of the edge variable me, which can be thought of as a discrete difference of positions. – 16 – [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: The three distinct ways of embedding a pair of pants with a distinguished external cuff (labeled by p1 above) into a surface. These correspond to the three classes of terms in the Mirzakhani-type recursion (3.35) for the c = 1 string amplitudes ˆsg,n(p). In the c = 1 string case there is a notion of ingoing versus outgoing momenta flowing through the diagram, although here the recursion is formulated for E… view at source ↗

read the original abstract

We study the perturbative $S$-matrix of the $c=1$ string and show that it admits a description in terms of a double-scaled (0+0)-dimensional matrix integral based on the spectral curve $\mathsf{x}(z) = 2\sqrt{2}\cos(z)$, $\mathsf{y}(z)=\sin(z)$. Combined with the famous duality to matrix quantum mechanics, this establishes a triality between three formulations of the theory: the worldsheet description, matrix quantum mechanics, and a matrix integral. Starting from the intersection number expressions for the complex Liouville string, we derive closed-form Feynman rule expressions for the $c=1$ amplitudes as intersection numbers on the moduli space of Riemann surfaces. The intersection theory naturally computes amplitudes corresponding to a discretized target space where momentum is conserved only modulo an integer. The physical $S$-matrix elements are recovered by restriction to the first `Brillouin zone' and analytic continuation to Lorentzian kinematics. We prove that these amplitudes satisfy perturbative spacetime unitarity directly from the intersection theory expressions, and show that they satisfy a Mirzakhani-type recursion relation. We show detailed agreement with the known matrix quantum mechanics results, providing strong evidence for the triality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit matrix integral for the c=1 perturbative S-matrix derived from intersection numbers, plus a triality claim, but the Brillouin-zone restriction and analytic continuation to Lorentzian kinematics is the step that still needs the closest look.

read the letter

The core claim is that the c=1 string S-matrix can be written as a double-scaled matrix integral with spectral curve x(z) = 2√2 cos(z), y(z) = sin(z). This sits alongside the worldsheet and matrix quantum mechanics descriptions to make a triality. They start from intersection numbers on the complex Liouville string, extract closed-form Feynman rules for the c=1 amplitudes, prove perturbative unitarity straight from the intersection expressions, and recover a Mirzakhani-style recursion. They also report detailed numerical matches with existing matrix quantum mechanics results. That agreement is concrete and useful evidence; it shows the construction is not just formal but reproduces known quantities. The derivations from intersection theory look reproducible on their own terms and avoid obvious circularity by treating the matrix integral as an independent equivalent formulation. The soft spot is exactly where the stress-test note flags it: the physical Lorentzian S-matrix is recovered by restricting the discretized (momentum mod 1) amplitudes to the first Brillouin zone and then analytically continuing. The paper asserts this map preserves poles and residues correctly, but the justification rests on the numerical checks and the formal rules rather than an independent derivation of the continuation. If that step has any ambiguity, the triality would be weaker than claimed. Minor issues like the usual care needed with double-scaling limits are present but not load-bearing. This is for people working on 2d string theory, matrix models, and intersection theory applications in physics. Anyone who already uses the c=1 model as a testbed will get immediate computational value from the Feynman rules if the continuation holds up. It deserves peer review because the explicit rules, unitarity proof, and MQM agreement are solid enough to merit referee time even if the continuation needs extra scrutiny.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that the perturbative S-matrix of the c=1 string admits a description as a double-scaled (0+0)-dimensional matrix integral with spectral curve x(z)=2√2 cos(z), y(z)=sin(z). Starting from intersection-number formulas for the complex Liouville string, it derives closed-form Feynman rules for c=1 amplitudes, proves perturbative unitarity directly from the intersection expressions, establishes a Mirzakhani-type recursion, and reports detailed numerical agreement with matrix quantum mechanics results. Physical Lorentzian S-matrix elements are recovered by restricting the discretized (modulo-1 momentum) intersection numbers to the first Brillouin zone followed by analytic continuation.

Significance. If the Brillouin-zone restriction and analytic-continuation map is shown to preserve the correct pole structure and residues, the result would be significant: it supplies a new matrix-integral formulation for c=1 string amplitudes, closes a triality with the worldsheet and matrix-quantum-mechanics descriptions, and furnishes closed-form Feynman rules together with a direct intersection-theoretic proof of perturbative unitarity. These features would constitute a concrete advance in the computational toolkit for non-critical strings.

major comments (2)

[the section deriving the physical S-matrix from intersection numbers] The central step that recovers physical Lorentzian S-matrix elements from the discretized intersection-number expressions (via restriction to the first Brillouin zone and analytic continuation) is load-bearing for the triality claim yet receives only schematic justification. It is not demonstrated that this map unambiguously preserves the locations and residues of all poles that appear in the known matrix-quantum-mechanics results; a concrete check for at least the four-point amplitude would be required.
[the section reporting numerical agreement with matrix quantum mechanics] The assertion of 'detailed agreement' with matrix quantum mechanics is stated in the abstract and introduction but is not accompanied by an explicit table or equation-by-equation comparison of the first few amplitudes (e.g., the two- and four-point functions) that would allow the reader to verify that the continued intersection numbers reproduce the known residues and imaginary parts.

minor comments (1)

Notation for the spectral curve variables x(z) and y(z) is introduced without an immediate reminder of the standard double-scaling conventions used in the matrix-model literature; a brief comparison sentence would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below and will revise the manuscript to incorporate the requested clarifications and explicit checks.

read point-by-point responses

Referee: The central step that recovers physical Lorentzian S-matrix elements from the discretized intersection-number expressions (via restriction to the first Brillouin zone and analytic continuation) is load-bearing for the triality claim yet receives only schematic justification. It is not demonstrated that this map unambiguously preserves the locations and residues of all poles that appear in the known matrix-quantum-mechanics results; a concrete check for at least the four-point amplitude would be required.

Authors: We agree that the current justification for the Brillouin-zone restriction followed by analytic continuation is schematic and requires strengthening to rigorously support the triality. In the revised version we will expand this section with a step-by-step derivation showing how the map acts on the momentum labels and preserves the pole locations and residues of the matrix-quantum-mechanics amplitudes. We will also include an explicit four-point computation that verifies the continued intersection numbers reproduce the known poles, residues, and imaginary parts. revision: yes
Referee: The assertion of 'detailed agreement' with matrix quantum mechanics is stated in the abstract and introduction but is not accompanied by an explicit table or equation-by-equation comparison of the first few amplitudes (e.g., the two- and four-point functions) that would allow the reader to verify that the continued intersection numbers reproduce the known residues and imaginary parts.

Authors: We acknowledge that the manuscript currently states agreement without providing the explicit numerical or symbolic comparisons needed for verification. We will add a dedicated subsection (or appendix) containing side-by-side expressions and a table for the two- and four-point amplitudes, showing the intersection-number results after Brillouin-zone restriction and analytic continuation against the known matrix-quantum-mechanics formulas, with explicit matching of residues and imaginary parts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from intersection theory

full rationale

The paper begins with established intersection-number formulas from the complex Liouville string (external to this work), derives closed-form Feynman rules for c=1 amplitudes as intersection numbers on moduli space, applies a Brillouin-zone restriction plus analytic continuation to obtain physical Lorentzian S-matrix elements, proves perturbative unitarity directly from those intersection expressions, verifies a Mirzakhani recursion, and reports numerical agreement with independent matrix-quantum-mechanics results. The matrix-integral formulation with spectral curve x(z)=2√2 cos(z), y(z)=sin(z) is then shown to reproduce the same continued amplitudes, establishing the triality as an equivalence rather than a redefinition. No step reduces by construction to its own inputs, no fitted parameters are relabeled as predictions, and no load-bearing claim rests solely on self-citation chains. The restriction/continuation map is an explicit additional step whose validity is supported by unitarity proofs and MQM matching, not assumed tautologically.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard framework of intersection theory on moduli spaces and the known duality to matrix quantum mechanics; the new element is the identification with the double-scaled matrix integral.

axioms (2)

domain assumption Intersection numbers on the moduli space of Riemann surfaces compute the amplitudes corresponding to a discretized target space with momentum conserved modulo an integer.
Invoked to obtain the closed-form Feynman rules for the c=1 amplitudes.
domain assumption Restriction to the first Brillouin zone plus analytic continuation to Lorentzian kinematics recovers the physical S-matrix.
Required to connect the intersection-number expressions to the physical amplitudes.

pith-pipeline@v0.9.0 · 5514 in / 1503 out tokens · 55546 ms · 2026-05-10T18:50:01.303301+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

c=1 amplitudes as intersection numbers on the moduli space of Riemann surfaces ... sum over stable graphs

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