The matrix edge of holography

Abstract: The IKKT matrix model arises at the extremal $p= -1$ limit of holographic dualities based on D$p$-brane geometries. We review the one-dimensional maximal supergravity that governs bulk fluctuations dual to the lowest BPS multiplet of gauge-invariant operators in the IKKT model. We present the Killing spinor equations and discuss their general half-supersymmetric solutions within the $\rm{SO}(3)\times \rm{SO}(7)$-invariant subsector. The explicit uplift of these solutions to Euclidean IIB supergravity in ten dimensions is provided.

• The paper establishes a one-dimensional maximal supergravity theory that captures bulk duals of the IKKT model’s lowest BPS multiplet.
• It employs explicit derivation of Killing spinor equations and analytical 1/2-BPS solutions, leading to uplifted ten-dimensional Euclidean IIB supergravity backgrounds.
• The results provide a nonlinear holographic dictionary linking matrix model observables with supergravity fields, deepening our understanding of emergent spacetime.

Holographic Duality at the Matrix Edge: IKKT Model and One-Dimensional Supergravity

Introduction and Motivation

This paper investigates the extremal $p = -1$ case of the holographic duality realized between Type IIB string theory and maximally supersymmetric field theories. The field-theory side reduces to the Ishibashi–Kawai–Kitazawa–Tsuchiya (IKKT) matrix model—a zero-dimensional matrix integral proposed as a non-perturbative definition of Type IIB string theory [Ishibashi:1996xs], whose holographic interpretation has been long elusive. The study leverages recent progress on concrete mass-deformations (the polarized IKKT model), providing a tractable setting for the exploration of `timeless holography', i.e., holography without an explicit time direction [Hartnoll:2024csr, Komatsu:2024bop].

The main technical achievement is the construction of a one-dimensional maximal supergravity theory describing the bulk fluctuations dual to the lowest BPS multiplet of gauge-invariant operators in the IKKT matrix model. The authors analyze the Killing spinor equations for the theory and determine general half-supersymmetric ($1/2$-BPS) solutions within an $\mathrm{SO}(3)\times\mathrm{SO}(7)$-invariant subsector, explicitly uplifting these solutions to ten-dimensional Euclidean IIB supergravity.

IKKT Model: Holographic Principles and BPS Operator Structure

The IKKT model is defined by the action:

$$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$

where $X_a$ ($a=1,\ldots,10$) are bosonic and $\Psi^\alpha$ ($\alpha=1,\dots,32$) are fermionic $\mathfrak{su}(N)$-valued matrices, with $\Gamma_a$ the $1/2$0 gamma matrices. The model preserves sixteen supercharges and is interpreted as the dimensional reduction of ten-dimensional $1/2$1 SYM to zero dimensions.

The lowest BPS multiplet comprises $1/2$2 representations:

• $1/2$3: symmetric traceless bosonic operators, $1/2$4
• $1/2$5: spinorial operators, $1/2$6
• $1/2$7: antisymmetric combinations, $1/2$8

Global symmetries and supersymmetries subtract $1/2$9 degrees of freedom. The corresponding bulk dynamics are encoded in a one-dimensional maximal supergravity theory, yielding a complete and nonlinear description for the matrix model's BPS spectrum.

One-Dimensional Maximal Supergravity: Construction and Field Content

The one-dimensional theory is a gauged maximal supergravity with $\mathrm{SO}(3)\times\mathrm{SO}(7)$0 gauge group. The bosonic sector comprises: einbein $\mathrm{SO}(3)\times\mathrm{SO}(7)$1, dilaton $\mathrm{SO}(3)\times\mathrm{SO}(7)$2, $\mathrm{SO}(3)\times\mathrm{SO}(7)$3 gauge fields $\mathrm{SO}(3)\times\mathrm{SO}(7)$4, $\mathrm{SO}(3)\times\mathrm{SO}(7)$5 scalars $\mathrm{SO}(3)\times\mathrm{SO}(7)$6 parametrizing an $\mathrm{SO}(3)\times\mathrm{SO}(7)$7 matrix, and $\mathrm{SO}(3)\times\mathrm{SO}(7)$8 axions $\mathrm{SO}(3)\times\mathrm{SO}(7)$9. The fermionic sector contains gravitini, dilatini, and matter fermions, all $$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$0 spinors. The scalar currents and dressings are specified to realize the BPS operator structure.

The Lagrangian is:

$$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$1

where $$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$2 contains kinetic and Noether-type couplings, $$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$3 is a topological axion term, $$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$4 encodes fermion-scalar interactions, and $$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$5 is the scalar potential, all fully determined by maximal supersymmetry up to quadratic order in fermions.

Supersymmetry transformation rules are detailed for all fields. Importantly, in one dimension, gravitino and gauge couplings appear purely algebraic (as Lagrange multipliers), requiring a direct construction rather than a reduction from higher dimensions. Nevertheless, the model is expected to arise from a consistent truncation of Euclidean IIB supergravity on $$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$6, as demonstrated for the pure bosonic theory without axions.

BPS Solutions and Uplift to Euclidean IIB Supergravity

The authors analyze 1/2-BPS solutions preserving an $$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$7 symmetry, enabling a nontrivial scalar sector and the inclusion of a single axion field. The reduced Lagrangian and parametrization yield coupled first-order BPS equations:

$$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$8

together with a time evolution equation for $$S_{\text{IKKT}} = -\operatorname{Tr}\left[\frac{1}{4}[X_a,X_b][X^a,X^b] - \frac{1}{2}\bar{\Psi}\,\Gamma^a\,[X_a,\Psi]\right]$$9.

These equations are the Euclidean analogue of those describing BPS spherical branes, extrapolated to $X_a$0 [Bobev:2018ugk]. The solutions preserve half the supersymmetries via a specific projection operator on the Killing spinors. The SO(10)-invariant (D$X_a$1) instanton solution is recovered for $X_a$2, $X_a$3.

The uplift procedure produces explicit ten-dimensional metric, dilaton, axion, and fluxes in Euclidean IIB supergravity, specified for arbitrary profiles $X_a$4 and $X_a$5. The uplifted background matches established BPS instanton solutions for the SO(10)-invariant case [Gibbons:1995vg, Gubser:1996wt, Bergshoeff:1998ry, Ooguri:1998pf].

Moreover, the authors connect their uplifted solutions to the electrostatic potential formulation, previously established in [Komatsu:2024bop]. By mapping their coordinate system, they obtain the potential $X_a$6 governing regular ball distributions in the matrix model dual, confirming the equivalence of bulk and boundary descriptions for the polarized IKKT phase structure.

Implications and Outlook

The explicit construction relates the lowest BPS multiplet in the IKKT matrix model to a nonlinear, gauged, one-dimensional supergravity theory, bridging the gap between matrix-model quantum mechanics and ten-dimensional supergravity backgrounds. The results enable holographic computations of correlation functions and operator spectra in the matrix model, providing a concrete dictionary:

• Scalar $X_a$7 field corresponds to $X_a$8
• Axion $X_a$9 field corresponds to $a=1,\ldots,10$0

The identification of nonlinear uplift formulae and the rigorous solution space for the BPS sector of the matrix model stimulates further exploration of polarized vacua, non-conformal holography, and the statistical mechanics of emergent spacetime from quantum matrix dynamics [Hartnoll:2025ecj, Chou:2025rwy]. The structure also suggests the necessity for embedding the one-dimensional supergravity in the framework of exceptional field theory [Bossard:2022wvi, Bossard:2023jid], particularly when generalizing beyond the SO(3)$a=1,\ldots,10$1SO(7) sector and including nontrivial axion configurations.

Conclusion

The paper provides a technical and conceptual advance in understanding holographic dualities at the matrix edge, specifically for the $a=1,\ldots,10$2 case with the IKKT matrix model. The formulation of one-dimensional maximal supergravity, the precise identification of the BPS multiplet, and the explicit uplift of 1/2-BPS solutions to ten-dimensional Euclidean IIB supergravity, reinforce the holographic paradigm for zero-dimensional gauge theories. These results establish a robust foundation for further investigations into correlation functions, phase transitions, and the emergence of spacetime in the polarized IKKT model, with practical ramifications for both matrix quantum mechanics and string theory holography (2604.00355).