A supergravity dual for IKKT holography (2511.23111v1)

Abstract: The IKKT matrix model, from the holographic perspective, arises at the p=-1 endpoint of the family of dualities relating type II supergravities on near-horizon Dp-brane geometries to (p+1)-dimensional super Yang-Mills theories with sixteen supercharges. In this work, we detail and expand results reported in a recent letter by establishing the holographic dictionary between gauge-invariant operators in the lowest BPS multiplet of the matrix model and the corresponding Kaluza-Klein fluctuations of Euclidean IIB supergravity compactified on the D(-1) instanton background. The full non-linear dynamics of these fluctuations can be encoded in a one-dimensional maximal gauged supergravity, which we construct explicitly. We provide its complete Lagrangian, up to second order in fermions, together with the corresponding supersymmetry transformations. We further discuss real forms of the theory for non-compact gauge groups and their embeddings into ten-dimensional supergravities with Lorentzian signature. From the analysis of the one-dimensional Killing spinor equations, we derive different classes of half-supersymmetric solutions, and discuss their uplifts as well as their relations to known solutions.

• The paper introduces an explicit one-dimensional supergravity dual for the IKKT matrix model, aligning its field content with the BPS operator spectrum.
• It constructs a consistent SO(10) gauged supergravity theory and demonstrates explicit 1/2-BPS solutions corresponding to varied D-instanton distributions.
• The work lays the groundwork for holographic renormalization, emergent spacetime insights, and systematic studies of nonperturbative matrix models.

Supergravity Duals and the Holography of the IKKT Matrix Model

Introduction

The IKKT matrix model, formulated as a zero-dimensional maximally supersymmetric Yang–Mills theory, has long been proposed as a fundamental nonperturbative definition of type IIB superstring theory. Its connections to the underlying structure of spacetime and holography have motivated extensive investigation, yet a detailed construction of its supergravity dual and a concrete holographic dictionary have remained underdeveloped. This paper provides a comprehensive treatment of the holographic duality between the IKKT model at large $N$ and supergravity on the D$(-1)$-brane (or instanton) background in Euclidean type IIB supergravity. The work constructs an explicit one-dimensional maximal gauged supergravity encoding the full non-linear dynamics of the lowest bulk fluctuating modes, aligns its field content with the single-trace BPS operator spectrum of the matrix model, and analyzes the structure of BPS solutions corresponding to D-instanton distributions.

The IKKT Matrix Model and Gauge-Invariant Operator Spectrum

The IKKT model is defined by a purely algebraic action involving ten bosonic $X_a$ and sixteen-component chiral fermionic matrices $\Psi^\alpha$, all transforming in the adjoint representation of $\mathfrak{su}(N)$:

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

The $X_a$ are real, and $\Psi^\alpha$ are complex Weyl spinors subject to $\Gamma_*$ chirality. The model is maximally supersymmetric ($\mathcal{N}=16$), admits SO(10) R-symmetry, and enjoys a scaling symmetry. Gauge-invariant, single-trace operators constructed from these matrices organize into infinite towers of long and short supermultiplets, corresponding to long and BPS representations of the supersymmetry algebra.

Of particular relevance for holography are the lowest-weight BPS (protected) multiplets, extracted from the spectrum by comprehensive Polya counting and explicit construction. The lowest such multiplet, labeled as $\mathcal{B}_2$, contains $129$ bosonic and $128$ fermionic degrees of freedom after accounting for SO(10) and supersymmetry gauge redundancies. Its representatives include symmetric traceless bilinears $\text{Tr}[X^a X^b]$, operators of the form $\text{Tr}[X^a\Psi]$, and cubic composites such as $\text{Tr}[X^a[X^b,X^c]]$ and $\text{Tr}[\bar{\Psi}\Gamma^{abc}\Psi]$, with rigid supersymmetry transformations relating the members of the multiplet.

Bulk Dual: One-Dimensional SO(10) Gauged Supergravity

The duality is realized at the supergravity level via IIB theory on the near-horizon geometry of coincident D$(-1)$-branes. This background is a flat Euclidean ten-dimensional spacetime equipped with a non-trivial dilaton/axion:

$$ds^2 = dt^2 + t^2 d\Omega_9^2, \quad e^\Phi = H(t), \quad H(t) = h + \frac{Q}{t^8}$$

where $Q$ encodes D-instanton charge. Upon a consistent truncation on $S^9$, the lowest bulk fluctuations are encoded in a one-dimensional $\mathcal{N}=16$ SO(10) gauged supergravity with scalar fields parametrizing the coset $SL(10)/SO(10)_K$, Lagrange-multiplier vector fields, and a sequence of axions corresponding to reductions of higher-dimensional p-forms.

Key structural features:

• Scalar Sector: The symmetric $T_{ij}$ matrix describes the $SL(10)/SO(10)$ coset; its diagonal parametrization reproduces the moduli for instanton distributions preserving various SO($n$) subgroups.
• Gauge Fields and Constraints: The SO(10) gauge fields act as Lagrange multipliers, enforcing constraints that reduce the true number of propagating fields, in direct correspondence with negative-multiplicity states in the operator spectrum.
• Fermionic Sector: The model contains a gravitino, a dilatino, and vector-spinors furnishing irreducible $\mathrm{SO}(s, t)_K$ representations, with all couplings and transformations fixed by maximal supersymmetry in one dimension.
• Scalar Potential and Interactions: The Lagrangian contains a non-trivial potential, constructed order-by-order via supersymmetry closure, and an axionic sector whose quartic interaction terms are rigidly dictated by the supersymmetry algebra.

This explicit construction demonstrates a precise matching between the field content (including negative degrees of freedom due to constraints) and the BPS operator spectrum of the IKKT model. The reduction ansatz is checked for consistency by examining the uplift of solutions and matching to known ten-dimensional instanton backgrounds.

BPS Solutions and Uplift

The Killing spinor equations of the one-dimensional supergravity admit explicit 1/2-BPS solutions corresponding to:

• Uniform D-instanton Distribution: All scalars equal, $T_{ij} = \delta_{ij}$, producing the SO(10) symmetric D$(-1)$-instanton solution with harmonic axion/dilaton. This is the bulk dual of the protected sector of the IKKT model.
• Spontaneous Symmetry Breaking: Multiple non-equal constants in the scalar sector correspond to distributions of D-instantons breaking SO(10) to smaller subgroups, directly reflecting predicted spontaneous symmetry breaking of the R-symmetry in the large $N$ limit of the matrix model.
• Axion Deformation (Spherical Brane Solutions): Switching on selected axions (e.g., $a_{8,9,10}$) breaks SO(10) to SO(7)$\times$SO(3). The resulting solutions uplift to ten-dimensional configurations involving nontrivial p-forms, precisely matching "spherical brane" solutions previously obtained in the context of type IIB$^*$ supergravity. The analysis distinguishes between solutions corresponding to IIB' and IIB$^*$, depending on the reality properties of the axions.

All these solutions preserve half the supersymmetries, with projective conditions on the spinors, and correspond to various vacua or deformations of the IKKT model at large $N$.

Implications, Outlook, and Generalizations

The explicit identification of the supergravity dual and holographic dictionary for the IKKT matrix model constitutes significant progress in the program of extending holography to non-conformal, non-local gauge theories and their large $N$ limits. Important implications include:

• Holographic Renormalization and Correlators: The explicit Lagrangian and uplift formulas enable systematic computations in the spirit of holographic renormalization for the IKKT matrix model, including the calculation of correlation functions and their comparison to matrix model results.
• Emergence of Spacetime: The one-dimensional supergravity encodes the non-linear dynamics underlying the conjectured emergent spacetime from the IKKT matrix model, offering a new perspective on how spacetime geometry arises from the dynamics of large $N$ matrices.
• BPS Operators and Bulk/Bdry Matching: The precise matching of multiplet content and symmetry-breaking patterns provides a concrete, structured framework for exploring both protected operator sectors and spontaneous symmetry breaking in large $N$ matrix models.

Potential future developments include:

• Extension to Lorentzian IKKT Holography: The analysis delineates the subtleties involved in constructing dual descriptions for the Lorentzian version of the IKKT model, related to type IIB$^*$ supergravity with internal $H^{9,1}$ hyperboloids.
• Exceptional Geometry and $E_{10(10)}$ Symmetry: The appearance of $SL(10)$ and its relations to the larger $E_{10(10)}$ group suggest connections with very extended symmetry algebras underlying M-theory and string theory at the nonperturbative level.
• Connections to Timeless Holography and Polarized Matrix Models: The framework sets the stage for systematic studies of mass deformations, "timeless" dualities, and polarized vacua recently studied in the context of deformed IKKT models.
• Numerical Tests: The explicit correspondence enables refined numerical tests in matrix model simulations, offering detailed predictions for the R-symmetry breaking pattern and spectrum of protected correlators.

Conclusion

By constructing the explicit SO(10) gauged one-dimensional maximal supergravity dual to the IKKT matrix model and rigorously deriving the holographic dictionary, this work provides a systematic and detailed foundation for the paper of holography in matrix models. The one-dimensional nature of the dual theory highlights unique aspects of bulk/boundary correspondence, the subtlety of degrees of freedom in minimal dimensions, and paves the way for both analytic and numerical exploration of the emergence of spacetime and quantum gravity from matrix degrees of freedom. The formalism and detailed computations established here will underpin future investigations into nonperturbative holography, symmetry breaking, and the role of exceptional symmetries in string theory.