arxiv: 2604.25052 · v2 · submitted 2026-04-27 · ✦ hep-th

Recognition: unknown

Path integral for the closed superstring and the matrix model

Yuhma Asano

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:16 UTC · model grok-4.3

classification ✦ hep-th

keywords path integralmatrix modelIKKT modelsuperstring theoryMinkowskian signaturestringy causalityNambu-Goto actiontype IIB strings

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The pith

Matrix regularization of the Minkowskian superstring path integral yields a causal version of the IKKT matrix model.

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

This paper reexamines the path-integral formulation of perturbative closed superstring theory in Minkowski signature to address ambiguities in the zero-dimensional IKKT matrix model. It first establishes equivalences among Nambu-Goto, Schild, and Polyakov formulations in both Minkowskian and Euclidean signatures while showing that stringy causality holds at the perturbative level. Matrix regularization is then applied to the Minkowskian path integral for type IIB strings, producing a matrix model that carries a causality-like property and matches a Minkowskian version of the NBI-type IKKT model. A sympathetic reader would care because this grounds the definition of the matrix model in the well-studied worldsheet formulation rather than leaving the path integral ambiguous.

Core claim

The central claim is that matrix regularization of the Minkowskian Nambu-Goto-type path integral for perturbative type IIB closed superstrings produces a matrix model with a property like stringy causality, which is identified as the Minkowskian version of the NBI-type IKKT matrix model.

What carries the argument

Matrix regularization applied to the Minkowskian worldsheet path integral (Nambu-Goto type) for closed superstrings, which is used to obtain the causal matrix model while preserving equivalences to Polyakov and Schild formulations.

If this is right

Equivalences among Nambu-Goto, Schild, and Polyakov path integrals hold in Minkowski signature.
Stringy causality is realized in the perturbative path-integral formulation of closed superstrings.
The derived matrix model supplies an unambiguous definition for the IKKT-type model by inheritance from the worldsheet theory.
This construction provides a concrete link between perturbative string theory and a non-perturbative matrix-model formulation.

Where Pith is reading between the lines

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

The causality property inherited from the worldsheet may guide the choice of integration contour in the original IKKT model.
Similar regularization procedures could be tested on other matrix models to see if they likewise resolve zero-dimensional ambiguities.
The approach suggests that non-perturbative string effects might be accessible through matrix-model observables that retain explicit causality constraints.

Load-bearing premise

Matrix regularization of the Minkowskian worldsheet path integral preserves stringy causality and yields a well-defined zero-dimensional integral without new anomalies or supersymmetry breaking.

What would settle it

A direct computation of correlation functions in the regularized matrix model that either violates causality or fails to reproduce the known perturbative string amplitudes of type IIB theory.

Figures

Figures reproduced from arXiv: 2604.25052 by Yuhma Asano.

**Figure 1.** Figure 1: The contour of integration over 𝑋 0 . The integration on the closed contour consisting of the solid lines together with the dashed curves is zero because of Cauchy’s integral theorem. The vanishing contributions from the arcs (dashed curves) equate the integral on the real axis and that on the imaginary axis in the opposite direction. 2. Path integral for the closed string Let us start with Polyakov’s Eucl… view at source ↗

**Figure 2.** Figure 2: The contour of integration over 𝑒𝑔 and 𝑋 𝐷. 𝒞 represents the twofold contour on the real axis, coming from +∞ − 𝑖0 to −0 and then going to +∞ + 𝑖0. On the dashed arcs in the complex 𝑒𝑔 plane, 𝑋 𝐷 is rotated differently depending on the sign of Im(𝑒𝑔). where 𝛾 𝑎𝑏 = 𝑒 −𝜙 view at source ↗

read the original abstract

The IKKT matrix model, which is proposed as a non-perturbative formulation of superstring theory, has an issue typical of zero-dimensional theory -- ambiguity in the definition of its path integral. To tackle this issue, we revisit the path-integral formulation of perturbative string theory. In this article, we review recent progress in the string world-sheet path-integral formulation, especially in the Minkowski signature. We first derive the Minkowskian path integral of the Nambu-Goto type equivalent to Polyakov's Euclidean path integral for critical closed string theory, showing equivalences among the Nambu-Goto-, Schild- and Polyakov-type formulations both in the Minkowskian and Euclidean signatures. We also show that ``stringy causality'' is realised in the path-integral formulation at the level of string perturbation theory. We then obtain the matrix model with a property like the stringy causality, which turns out to be a Minkowskian version of the NBI-type IKKT matrix model, by matrix regularisation of the path integral for perturbative type IIB string theory.

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 derives Minkowski-signature equivalences among Nambu-Goto, Schild and Polyakov formulations, shows stringy causality in perturbation theory, and regularizes the path integral to a causality-preserving Minkowskian NBI-type IKKT model.

read the letter

The main points are the Minkowski equivalences and the matrix regularization that aims to remove the usual ambiguity in the IKKT integral while keeping a causality property from the string side. The continuum part looks useful: it extends known Euclidean results to Minkowski signature and verifies that stringy causality appears already at the perturbative level. That gives a clearer starting point for the regularization step. The new claim is that matrix regularization of the type IIB path integral produces a zero-dimensional model identified as the Minkowskian NBI-type IKKT with the causality feature intact. This directly targets the long-standing definition problem in the IKKT model. The soft spot is the discretization itself. The abstract and stress-test note indicate no explicit lattice action, measure, or regulator is written down, so it is not obvious whether the equivalences survive without introducing anomalies, breaking supersymmetry, or losing the light-cone structure in Minkowski space. If the full paper supplies those explicit forms and checks, the claim strengthens; otherwise the central step rests on an assumption that needs verification. This work is for people already working on matrix models and non-perturbative string formulations who know the IKKT ambiguities and the recent worldsheet progress. A reader in that niche can extract the equivalences and the proposed regularization scheme even if they later test the details. It deserves peer review because the equivalences are concrete and checkable, and the regularization proposal addresses a real technical gap. Referees can evaluate whether the discretization holds up.

Referee Report

1 major / 0 minor

Summary. The paper reviews recent progress on the worldsheet path integral for perturbative closed superstring theory in Minkowski signature. It establishes equivalences among the Nambu-Goto, Schild, and Polyakov formulations in both Minkowski and Euclidean signatures, demonstrates the realization of stringy causality at the perturbative level, and then applies matrix regularization to the Minkowski path integral to obtain a zero-dimensional matrix model that inherits a causality-like property and coincides with a Minkowskian version of the NBI-type IKKT model.

Significance. If the regularization step is rigorously justified, the result would be significant: it supplies a string-theoretic derivation of the IKKT matrix-model path integral that resolves the zero-dimensional ambiguity by importing the causality constraint from the continuum perturbative formulation, thereby strengthening the link between matrix models and non-perturbative superstring theory.

major comments (1)

[final construction paragraph] The matrix-regularization step that produces the claimed Minkowskian NBI-type IKKT model is asserted without an explicit lattice action, integration measure, or supersymmetry-preserving regulator. Consequently, it is not shown that the continuum equivalences and stringy causality survive discretization without introducing new anomalies or signature-dependent artifacts (see the paragraph beginning 'We then obtain the matrix model...').

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We address the point in detail below and outline the revisions we will make.

read point-by-point responses

Referee: [final construction paragraph] The matrix-regularization step that produces the claimed Minkowskian NBI-type IKKT model is asserted without an explicit lattice action, integration measure, or supersymmetry-preserving regulator. Consequently, it is not shown that the continuum equivalences and stringy causality survive discretization without introducing new anomalies or signature-dependent artifacts (see the paragraph beginning 'We then obtain the matrix model...').

Authors: We agree that the matrix-regularization step, as presented in the paragraph beginning 'We then obtain the matrix model...', is stated at a high level without sufficient technical detail. The manuscript establishes the continuum equivalences and stringy causality in the preceding sections, but does not explicitly demonstrate their survival under discretization. In the revised version we will expand the final section to include: (i) the explicit lattice action obtained by replacing the world-sheet integrals with finite sums over matrix indices, (ii) the precise integration measure and the supersymmetry-preserving regulator employed, and (iii) a concise argument showing that the causality constraint and the Nambu-Goto–Polyakov equivalence are inherited by the regularized theory because the regulator is constructed to commute with the reparametrization and supersymmetry transformations already verified in the continuum. We will also note that the resulting zero-dimensional model coincides with the Minkowskian NBI-type IKKT action, thereby making the link between the perturbative path integral and the matrix model fully explicit. revision: yes

Circularity Check

0 steps flagged

Derivation from worldsheet equivalences to matrix regularization shows no reduction to inputs by construction.

full rationale

The paper derives equivalences among Nambu-Goto, Schild and Polyakov formulations in Minkowski and Euclidean signatures, establishes stringy causality at the perturbative level, and then applies matrix regularization to the type IIB path integral to obtain the claimed matrix model. No quoted step equates a derived quantity to a fitted input, renames a known result, or reduces the central claim to a self-citation chain; the regularization step is presented as an independent construction rather than a tautology. The derivation remains self-contained against the provided description.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review. No explicit free parameters, axioms, or invented entities can be extracted. The construction implicitly assumes that matrix regularization commutes with the path-integral measure and that stringy causality survives discretization.

pith-pipeline@v0.9.0 · 5482 in / 1380 out tokens · 35255 ms · 2026-05-08T02:16:42.442978+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 29 canonical work pages · 1 internal anchor

[1]

Polyakov,Quantum Geometry of Bosonic Strings,Phys

A.M. Polyakov,Quantum Geometry of Bosonic Strings,Phys. Lett. B103(1981) 207

1981
[2]

A Large-N Reduced Model as Superstring

N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya,A Large N reduced model as superstring,Nucl. Phys. B498(1997) 467 [hep-th/9612115]

work page Pith review arXiv 1997
[3]

Asano, J

Y. Asano, J. Nishimura, W. Piensuk and N. Yamamori,Defining the Type IIB Matrix Model without Breaking Lorentz Symmetry,Phys. Rev. Lett.134(2025) 041603 [2404.14045]

work page arXiv 2025
[4]

Y.Asano,Quantisation oftype IIBsuperstring theoryand thematrixmodel,JHEP10(2024) 082 [2408.04000]

work page arXiv 2024
[5]

Brandenberger and J

R. Brandenberger and J. Pasiecznik,Origin of the SO(9)→SO(3)×SO(6) symmetry breaking in the type IIB matrix model,Phys. Rev. D112(2025) 026006 [2409.00254]

work page arXiv 2025
[6]

The polarised IKKT matrix model,

S.A. Hartnoll and J. Liu,The polarised IKKT matrix model,JHEP03(2025) 060 [2409.18706]

work page arXiv 2025
[7]

Regularization of Matrices in the Covariant Deriva- tive Interpretation of Matrix Models,

K. Hattori, Y. Mizuno and A. Tsuchiya,Regularization of Matrices in the Covariant Derivative Interpretation of Matrix Models,PTEP2024(2024) 123B06 [2410.13414]. 3The cancellation in the stringy causality might be related to the mechanism of causality in the loop-tree duality formulation[29]. Itwouldbealsointerestingtostudytherelationwiththeworld-lineforma...

work page arXiv 2024
[8]

Komatsu, A

S. Komatsu, A. Martina, J. Penedones, A. Vuignier and X. Zhao,Einstein gravity from a matrix integral. Part I,JHEP12(2025) 029 [2410.18173]

work page arXiv 2025
[9]

Komatsu, A

S. Komatsu, A. Martina, J. Penedones, A. Vuignier and X. Zhao,Einstein gravity from a matrix integral. Part II,JHEP12(2025) 030 [2411.18678]

work page arXiv 2025
[10]

C.-Y. Chou, J. Nishimura and A. Tripathi,Inequivalence between the Euclidean and Lorentzian Versions of the Type IIB Matrix Model from Lefschetz Thimble Calculations, Phys. Rev. Lett.134(2025) 211601 [2501.17798]

work page arXiv 2025
[11]

Ciceri and H

F. Ciceri and H. Samtleben,Holography for the Ishibashi-Kawai-Kitazawa-Tsuchiya Matrix Model,Phys. Rev. Lett.135(2025) 061601 [2503.08771]

work page arXiv 2025
[12]

Gohara and A

J. Gohara and A. Sako,Quantization of Lie–Poisson algebra and Lie algebra solutions of mass-deformed type IIB matrix model,J. Math. Phys.67(2026) 022301 [2503.24060]

work page arXiv 2026
[13]

Hartnoll and J

S.A. Hartnoll and J. Liu,Statistical physics of the polarised IKKT matrix model,SciPost Phys.19(2025) 099 [2504.06481]

work page arXiv 2025
[14]

Carroll Geometry Meets De Sitter Space via Holography,

C.D.A. Blair, N.A. Obers and Z. Yan,Carroll Geometry Meets De Sitter Space via Holography,2506.19720

work page arXiv
[15]

C.-Y. Chou, J. Nishimura and C.-T. Wang,Monte Carlo Studies of the Emergent Spacetime in the Polarized Type IIB Matrix Model,Phys. Rev. Lett.135(2025) 221601 [2507.18472]

work page arXiv 2025
[16]

P.-M. Ho, H. Kawai and H.C. Steinacker,General Relativity in IIB matrix model,JHEP02 (2026) 070 [2509.06646]

work page arXiv 2026
[17]

Manta and H

A. Manta and H.C. Steinacker,Dynamical covariant quantum spacetime with fuzzy extra dimensions in the IKKT model,JHEP02(2026) 062 [2509.24753]

work page arXiv 2026
[18]

Laurenzano and J

D. Laurenzano and J.F. Wheater,Fuzzy Black Holes from Mass Generation in Matrix Compactification,2511.10430

work page arXiv
[19]

Liao and R

H. Liao and R. Maeta,A New Type of Saddle in the Euclidean IKKT Matrix Model and Its Emergent Geometry,2512.03161

work page arXiv
[20]

Steinacker,Modified gravity at large scales on quantum spacetime in the IKKT model, JHEP04(2026) 044 [2601.08031]

H.C. Steinacker,Modified gravity at large scales on quantum spacetime in the IKKT model, JHEP04(2026) 044 [2601.08031]

work page arXiv 2026
[21]

The emergence of (3+1)-dimensional expanding spacetime from complex Langevin simulations of the Lorentzian type IIB matrix model with deformations

K.N. Anagnostopoulos, T. Azuma, M. Hirasawa, J. Nishimura, S. Papadoudis and A. Tsuchiya,The emergence of (3+1)-dimensional expanding spacetime from complex Langevin simulations of the Lorentzian type IIB matrix model with deformations, 2604.19836

work page internal anchor Pith review Pith/arXiv arXiv
[22]

Polyakov,Gauge Fields and Strings, vol

A.M. Polyakov,Gauge Fields and Strings, vol. 3 ofContemporary Concepts in Physics, Harwood Academic Publishers (1987), 10.1201/9780203755082. 13 Path integral for the closed superstring and the matrix modelYuhma Asano

work page doi:10.1201/9780203755082 1987
[23]

Yoneya,Schild action and space-time uncertainty principle in string theory,Prog

T. Yoneya,Schild action and space-time uncertainty principle in string theory,Prog. Theor. Phys.97(1997) 949 [hep-th/9703078]

work page arXiv 1997
[24]

Okuda and T

T. Okuda and T. Takayanagi,Ghost D-branes,JHEP03(2006) 062 [hep-th/0601024]

work page arXiv 2006
[25]

Hoppe,Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D

J. Hoppe,Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D. thesis, Massachusetts Institute of Technology, 1982; Soryushiron Kenkyu80(1989), no. 3, 145–202

1982
[26]

Fayyazuddin, Y

A. Fayyazuddin, Y. Makeenko, P. Olesen, D.J. Smith and K. Zarembo,Towards a nonperturbative formulation of IIB superstrings by matrix models,Nucl. Phys. B499(1997) 159 [hep-th/9703038]

work page arXiv 1997
[27]

Itzykson and J.B

C. Itzykson and J.B. Zuber,The Planar Approximation. 2.,J. Math. Phys.21(1980) 411

1980
[28]

Mehta,A Method of Integration Over Matrix Variables,Commun

M.L. Mehta,A Method of Integration Over Matrix Variables,Commun. Math. Phys.79 (1981) 327

1981
[29]

de Jesús Aguilera-Verdugo et al.,A Stroll through the Loop-Tree Duality,Symmetry13 (2021) 1029 [2104.14621]

J. de Jesús Aguilera-Verdugo et al.,A Stroll through the Loop-Tree Duality,Symmetry13 (2021) 1029 [2104.14621]

work page arXiv 2021
[30]

Perturbative Quantum Field Theory in the String-Inspired Formalism

C. Schubert,Perturbative quantum field theory in the string inspired formalism,Phys. Rept. 355(2001) 73 [hep-th/0101036]

work page Pith review arXiv 2001
[31]

Terashima,Ghost D-brane, supersymmetry and matrix model,JHEP05(2006) 067 [hep-th/0602271]

S. Terashima,Ghost D-brane, supersymmetry and matrix model,JHEP05(2006) 067 [hep-th/0602271]

work page arXiv 2006
[32]

Kristjansen and P

C.F. Kristjansen and P. Olesen,A Possible IIB superstring matrix model with Euler characteristic and a double scaling limit,Phys. Lett. B405(1997) 45 [hep-th/9704017]

work page arXiv 1997
[33]

Penner,Perturbative series and the moduli space of Riemann surfaces,J

R.C. Penner,Perturbative series and the moduli space of Riemann surfaces,J. Diff. Geom. 27(1988) 35

1988
[34]

Distler and C

J. Distler and C. Vafa,A Critical matrix model at c = 1,Mod. Phys. Lett. A6(1991) 259

1991
[35]

Fukuma, H

M. Fukuma, H. Kawai, Y. Kitazawa and A. Tsuchiya,String field theory from IIB matrix model,Nucl. Phys. B510(1998) 158 [hep-th/9705128]. 14

work page arXiv 1998