Matrix String Theory Duality

• Matrix string theory duality is a framework connecting maximally supersymmetric 1+1 and 2D Yang–Mills theories with type II superstrings via matrix models and dualities.
• It employs T- and S-duality and open–closed triality to map gauge theory operators onto closed string amplitudes, providing precise operator correspondences.
• The framework enables practical insights into nonperturbative string interactions, discrete flux sectors, and holographic descriptions in varied string backgrounds.

Matrix string theory duality refers to the deep correspondence between certain supersymmetric gauge theories in the large-N limit—specifically, maximally supersymmetric 1+1 and 2D SU(N) Yang–Mills theories—and nonperturbative type II superstring theories, realized via matrix models and supported by open–closed duality, T- and S-duality, topological string constructions, and connections to M-theory. This duality underlies and unifies the S-matrix of type IIA/B superstring theory and the target-space dynamics of a 2D U(N) (or SU(N)) maximally supersymmetric Yang–Mills theory compactified on a circle, generalizing and refining earlier notions of gauge/string duality and nonperturbative string quantization.

1. Matrix String Theory Formulation and BFSS Limit

Matrix string theory (MST) emerges from the BFSS matrix model (0+1D U(N) supersymmetric Yang–Mills) after compactification and T-duality on a circle of radius $R_9$ to obtain a 1+1D U(N) SYM theory with coupling $g_{YM}^2 = g_s/\ell_s^2$ on a spatial $S^1$ of radius $R$:

$$S_{\text{MST}} = \frac{1}{4\pi g_s \ell_s^2} \int d^2\sigma\, \mathrm{Tr}\bigl[F_{\tau\sigma}^2 + 2 D_\alpha X^i D^\alpha X^i + g_s^2 \ell_s^4 [X^i, X^j]^2 + \cdots\bigr]$$

with $X^i$ the eight transverse scalar matrices, $F_{\tau\sigma}$ the curvature, and fermions included for maximal $(8,8)$ SUSY. In the $g_{YM}\to 0$ (or $g_s\to 0$) limit, the theory reduces to $N$ decoupled free strings; finite $g_{YM}$ encodes nonperturbative string interactions (splitting and joining) through the dynamics of off-diagonal matrices. The large-$N$ limit governs the sector of strings with quantized light-cone momentum $p^+ = N/R_-$ under discrete light-cone quantization (DLCQ), with $R_-$ the null circle and $W=N$ units of string winding after T-duality (Blair et al., 2023).

The equivalence between the S-matrix of type IIA strings in flat ten-dimensional spacetime and the S-matrix of 2D maximally supersymmetric SU(N) Yang–Mills (compactified on $S^1$) is precise in the strict large-$N$ limit, with the $U(1)$ center of mass decoupling. The parameter identifications are $\lambda = N g_{YM}^2$, $g_s = (2\pi g_{YM} R)^{-1}$, $\ell_s$ determined via $g_{YM}^2 = [2\pi g_s^2 R^2]^{-1}$ (Cho et al., 6 Jan 2026).

2. Open–Closed Duality and Topological Matrix-String Correspondence

The foundational mechanism underlying the gauge–string duality in the matrix context is open–closed duality, implemented through a chain:

$$\text{Matrix Model} \to \text{Open–Closed–Open Triality} \to \text{Imbimbo–Mukhi Matrix Model} \to \text{c=1 String} \to \text{Topological String Duals}$$

The open–closed–open triality asserts the co-existence of two inequivalent open string descriptions (compact vs. non-compact branes, e.g., different defect brane configurations) for a given closed string background. External single-trace insertions map to closed string vertex operators (V-type duality), and matrix faces (D-brane boundaries) map to closed string punctures (F-type duality), yielding a duality between the correlators of single-trace gauge-invariant operators and closed-string amplitudes (Gopakumar et al., 2022).

Explicitly, in the Hermitian one-matrix model (with arbitrary polynomial potential), the all-genus expansion of single-trace correlators matches precisely the genus expansion of closed-string amplitudes in both A-model and B-model topological strings. The A-model dual is a supersymmetric $SL(2,\mathbb{R})_1/U(1)$ Kazama–Suzuki coset, while the B-model is a Landau–Ginzburg theory with superpotential $W(Z)=1/Z + t_2 Z + ...$.

The operator map is

$$\frac{1}{N k} \mathrm{Tr}\, M^k \quad \longleftrightarrow \quad T_{-k}$$

where $T_{-k}$ denotes a tachyon operator of momentum $-k$ in the dual topological string (Gopakumar et al., 2022, Gopakumar, 2011).

3. Duality Chains, T/S-Duality, and Non-Lorentzian Backgrounds

Matrix string theory duality is embedded in a broader web of dualities:

Duality	Matrix Theory Side	String Theory Side
T-duality	BFSS $0+1$ U(N) QM on $S^1$	IIA $\leftrightarrow$ IIB, $P_9 \leftrightarrow$ winding
S-duality	(IIB) 1+1 U(N) SYM (F1/D1 exchange)	Type IIB $(p,q)$-string web

After T-duality, the discrete light-cone momentum $N$ becomes the fundamental string winding $W$ around the dual circle, supplying the DLCQ frame for the emergent worldsheet theory (Blair et al., 2023). Under S-duality, fundamental strings and D-strings are interchanged, which is reflected as the $\mathrm{SL}(2,\mathbb{Z})$ duality acting both on the $(p,q)$ winding data and the axio-dilaton background, leaving physical tensions and the worldsheet CFT invariant (0708.3484).

BPS decoupling limits lead to non-Lorentzian (Newton–Cartan) geometries, with matrix theory backgrounds characterized by longitudinal/transverse vielbeine, $B$-fields, and RR forms. T- and S-duality transformations in these backgrounds exhibit codimension jumps depending on the $B$-field rank, underpinning phenomena such as noncommutative geometry and Morita equivalence in matrix compactifications (Blair et al., 27 Feb 2025).

4. Open–Closed String Duality in Matrix Models: Topological Recursion

Matrix models with open–closed duality encapsulate the combinatorics of string perturbation theory via topological recursion. The (double-scaled) matrix model encodes the intersection theory of moduli spaces of punctured Riemann surfaces, with the loop equations reproducing recursion relations equivalent to the Eynard–Orantin framework (Lowenstein, 2024).

• The Wigner–Dyson class (β=2) matrix model describes closed string (compact) amplitudes, while the Altland–Zirnbauer class captures open + closed sectors.
• Closed-string insertions: $\sigma_k = \partial F_{\rm closed}/\partial t_k$; Macroscopic loop operators (Laplace exponentials) generate open-string boundaries.
• Open–closed correlators are constructed by the insertion of determinant/FZZT-brane operators or by employing suitable shift operators acting on closed-string parameters.
• Virasoro constraints for the tau function $\tau = \exp(-F/2)$ encode Ward identities leading to recursion kernels on the matrix model spectral curve.

This formalism establishes a rigorous dictionary between matrix model correlators, open/closed string topology, and moduli space volumes (e.g., Weil–Petersson), providing a robust algebraic and analytic underpinning to the open–closed duality in matrix string theory (Lowenstein, 2024).

5. (p,q)-Strings, Matrix Regularized Membranes, and $\mathrm{SL}(2,\mathbb{Z})$ Duality

Matrix regularization of the light-cone supermembrane on $T^2$ yields $(2+1)$-dimensional $U(N)$ SYM, which upon double-dimensional reduction yields the matrix string theory for $(p,q)$-strings of type IIB. This construction manifests the full $\mathrm{SL}(2,\mathbb{Z})$ S-duality at both the worldsheet and matrix levels:

• The $(p,q)$ winding data label charge vectors, coupling to the $\mathrm{SL}(2)$ doublet of NS--NS and R--R two-forms.
• $\mathrm{SL}(2,\mathbb{Z})$ acts naturally as modular transformations on the torus, mapping between different $(p,q)$-string sectors while transforming the background complex structure and the axio-dilaton as prescribed.
• At the matrix level, mode relabeling and cocycle/boundary conditions implement the same duality, preserving the spectrum and physical relations (0708.3484).

This formalism unifies the $(p,q)$ string web and its duality relations within the matrix string framework.

6. Flux Sectors, D0-brane Bound States, and Nonperturbative Tests

The Hilbert space of 2D SU(N) SYM decomposes into discrete flux sectors (superselection sectors labeled by electric flux $k$ around $S^1$), producing topologically distinct vacua with gapped spectra:

• Each $k$-flux vacuum is mapped to a bound state of $k$ D0-branes in IIA string theory.
• The spectrum above these vacua, including open-string excitation levels and semiclassical Regge trajectories, is captured in detail by the matrix string duality prediction:

$$M_{k, \ell} \sim \frac{k}{R} \oplus \frac{g_{YM}}{N} \sqrt{2\pi\ell}$$

• Level splittings and decay widths are suppressed as $1/(N R)$, rendering the states stable at large $N$.
• Verification of these discrete towers by Hamiltonian truncation, DLCQ, or classical bootstrap would provide direct nonperturbative confirmation of the matrix string paradigm (Cho et al., 6 Jan 2026).

7. Advanced Instances and Generalizations: Complex Liouville, Non-commutative Geometry, and Holography

Matrix string duality extends to more intricate string backgrounds:

• The complex Liouville string, defined by coupling two Liouville fields of central charges $c=13\pm i\lambda$, admits a dual description in terms of a double-scaled two-matrix integral with well-defined spectral curves and topological recursion, precisely matching string amplitudes and analytic structure (branch points, poles, discontinuities) (Collier et al., 2024).
• Non-commutative geometry arises in matrix theory compactifications on $T^2$ with background $B$-fields, linked to Morita equivalence through $SL(2,\mathbb{Z})$ actions on the non-commutativity parameter $\theta$; this is embedded within the broader duality web, including holography for noncommutative Yang–Mills (Blair et al., 27 Feb 2025).
• Open–closed–open triality, determinantal formulas, boundary operators, and topological recursion connect matrix string duality to moduli space volumes and to exact topological string/holographic computations for protected BPS subsectors (Gopakumar et al., 2022, Collier et al., 2024).

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Matrix string theory duality thus provides a unifying nonperturbative framework for type II string theory, matrix models, open–closed string correspondences, and gauge/gravity dualities, with rigorous confirmation provided by operator dictionary, all-genus correlator matching, nontrivial duality actions, and precise prediction for flux sector spectra and string interaction mechanisms (Gopakumar et al., 2022, Blair et al., 2023, Lowenstein, 2024, Blair et al., 27 Feb 2025, Cho et al., 6 Jan 2026, 0708.3484, Collier et al., 2024, Gopakumar, 2011).