---
title: 'IIB Matrix Model: Emergent Spacetime'
url: https://www.emergentmind.com/topics/iib-matrix-model
type: topic
---

# IIB Matrix Model: Emergent Spacetime

The IIB matrix model, also known as the IKKT model after Ishibashi, Kawai, Kitazawa, and Tsuchiya, is a zero-dimensional large-\(N\) matrix model proposed as a nonperturbative definition of type IIB superstring theory in ten dimensions. Its basic premise is that spacetime is not an input manifold but an emergent structure encoded in ten Hermitian matrices and their fermionic superpartners. In one standard interpretation, eigenvalues of commuting or approximately commuting matrix configurations are read as spacetime coordinates; in another, the matrices are identified with covariant derivatives on a manifold. Both viewpoints are central to current work on emergent geometry, spontaneous symmetry breaking of rotational symmetry, and cosmological dynamics in the Lorentzian theory, including evidence for a dynamically generated \((3+1)\)-dimensional expanding universe [1405.5904][2404.14045].

## 1. Matrix formulation, field content, and symmetries

The model is obtained by dimensional reduction of ten-dimensional \(\mathcal{N}=1\) super Yang–Mills theory to a point. In Lorentzian signature, the partition function is written as
\[
Z = \int d A \, e^{iS[A]} \, {\rm Pf}\,{\cal M}[A] \ ,
\]
with bosonic matrices \(A_\mu\) (\(\mu=0,\dots,9\)) and a fermionic Pfaffian \({\rm Pf}{\cal M}[A]\). A standard form of the bosonic action is
\[
S[A] = - \frac{1}{4} N \,\eta_{\mu\lambda} \eta_{\nu \rho} \, \mathrm{tr} \big([A_\mu , A_\nu][A_\lambda , A_\rho]\big) \ ,
\]
while the reduced-supersymmetry notation used in reviews writes
\[
S_{\rm b} = - \frac{1}{4 g^2}  \Tr [ A_\mu, A_\nu ] [ A^\mu, A^\nu ] ,
\qquad
S_{\rm f} = - \frac{1}{2 g^2} \Tr\left(\Psi_\alpha ({\cal C} \Gamma^\mu)_{\alpha\beta} [ A_\mu, \Psi_\beta ] \right) .
\]
The matrices are traceless Hermitian \(N\times N\) matrices, the fermions are Majorana–Weyl matrix variables, and the model retains ten-dimensional Lorentz symmetry together with maximal supersymmetry in the reduced sense [2404.14045][1405.5904].

Two symmetries organize essentially all later interpretations. The first is the \(SU(N)\) or \(U(N)\) adjoint action,
\[
A_\mu' = U A_\mu U^\dagger \ ,
\]
which implies that any extraction of spacetime information must be formulated in a gauge-consistent way. The second is Lorentz symmetry,
\[
A_\mu' = {\cal O}_{\mu\nu} A_\nu \ , \quad {\cal O} \in \mathrm{SO}(9,1) \ .
\]
A further structural feature emphasized in later studies is the shift symmetry \(A_\mu \to A_\mu + \alpha_\mu \mathbf{1}\), which underlies the interpretation of matrix eigenvalues as coordinates in an emergent target space [2404.14045][2201.13200].

In the commuting limit the coordinate interpretation is direct:
\[
A_\mu = U\, \mathrm{diag}(x_\mu^{(1)},\dots,x_\mu^{(N)})\, U^{-1},
\]
so the eigenvalues \(x_\mu^{(i)}\) are interpreted as points in ten-dimensional spacetime. In the full quantum model the matrices are generically noncommuting, and the relevant geometry is correspondingly noncommutative or only approximately classical in suitable large-\(N\) regimes [2008.01058].

## 2. Euclidean and Lorentzian theories

The Euclidean and Lorentzian versions are not merely computational variants; they encode different notions of spacetime and lead to different conceptual problems. In the Euclidean model one Wick-rotates the temporal matrix and obtains an \(\mathrm{SO}(10)\)-invariant theory with positive-definite bosonic action and weight \(e^{-S_{\rm E}}\). This version is finite for all \(N\), is amenable to Monte Carlo methods, and has yielded evidence for spontaneous symmetry breaking of \(\mathrm{SO}(10)\) to \(\mathrm{SO}(3)\) in large-\(N\) analyses. The Lorentzian model keeps the Minkowski signature, retains \(\mathrm{SO}(9,1)\), and is the version relevant for cosmological interpretation [1405.5904][2404.14045].

The Lorentzian partition function is not absolutely convergent. One longstanding regularization strategy introduced infrared cutoffs that distinguish temporal and spatial directions, for example through conditions of the form
\[
\frac{1}{N}\Tr (A_0)^2  \le \kappa \, \frac{1}{N}\Tr (A_i)^2 ,
\qquad
\frac{1}{N}\Tr (A_i)^2   \le  L^2 .
\]
This made simulations possible and led to the early expanding-universe results, but it also explicitly broke Lorentz symmetry [1405.5904]. A later analysis argued that the divergence of the Lorentzian partition function is tied to the noncompact Lorentz group itself and that Lorentz-breaking infrared cutoffs produce an unphysical “classicalization” of Lorentz-invariant observables. To remove the infinite Lorentz volume without breaking symmetry, the Lorentz group was treated as a gauge redundancy and fixed by minimizing \(\mathrm{tr}(A_0^2)\) along each Lorentz orbit, leading to the conditions
\[
\mathrm{tr}(A_0 A_i) = 0 \quad (i=1,\dots,9)
\]
and a Faddeev–Popov determinant
\[
\Delta_{\rm FP}[A] = \det \Omega , \qquad
\Omega_{ij} = \mathrm{tr} (A_0)^2 \delta_{ij} + \mathrm{tr} (A_i A_j) .
\]
The resulting gauge-fixed Lorentzian partition function is
\[
Z_{\rm g.f.} = \int d A \, \Delta_{\rm FP}[A] \, \prod_{i=1}^9\delta\big( \mathrm{tr} (A_0 A_i)\big) \, e^{iS[A]} \, {\rm Pf}{\cal M}(A) \ ,
\]
which was proposed as a nonperturbative definition of the Lorentzian theory without Lorentz-symmetry-breaking artifacts [2404.14045].

A separate line of work on bosonic late-time truncations clarified the relation between Euclidean and Lorentzian formulations in a different sense. In a deformed bosonic model studied with the complex Langevin method, the unconstrained Lorentzian and Euclidean theories were shown to be analytically equivalent up to fixed complex phases in observables. Introducing either a Lorentz-invariant mass term or a constraint on the largest eigenvalue of \(A_0\) breaks this equivalence and allows a dynamical transition in which the time direction becomes effectively Lorentzian at late times while early-time behavior remains Euclidean-like [2112.15368][2201.13200]. This suggests that the status of “Lorentzian time” depends not only on the bare signature of the action but also on how the large-\(N\) configuration space is defined and probed.

## 3. Emergent spacetime and cosmological dynamics

The best-known numerical evidence for spacetime emergence in the Lorentzian IIB matrix model comes from the standard procedure of diagonalizing \(A_0\),
\[
A_0 | t_a \rangle = t_a | t_a \rangle, \quad t_1 < t_2 < \dots < t_N,
\]
and then analyzing the spatial matrices \(A_i\) in that basis. Empirically, the \(A_i\) become band-diagonal with respect to the ordered \(A_0\) eigenbasis, which supports a notion of locality in the emergent time direction. One defines \(n\times n\) blocks
\[
\bar{A}_i^{(ab)}(t) := \langle t_{\nu+a}| A_i | t_{\nu+b}\rangle,
\qquad
t = \frac{1}{n} \sum_{a=1}^n t_{\nu+a},
\]
and from them the spatial tensor
\[
T_{ij}(t) = \frac{1}{n}\tr\{\bar{A}_i(t)\bar{A}_j(t)\},
\qquad
R^2(t) = \frac{1}{n}\tr \bar{A}_i(t)^2 = \sum_{k=1}^9 \lambda_k(t) .
\]
The eigenvalues \(\lambda_k(t)\) measure the time-dependent extent of space in each direction [1405.5904].

In this framework the Lorentzian model displays spontaneous symmetry breaking
\[
\mathrm{SO}(9)\to \mathrm{SO}(3)
\]
after a critical time \(t_{\rm c}\): three eigenvalues of \(T_{ij}(t)\) become large while six remain small. The resulting \(R(t)\) shows an early-time exponential growth in the original Lorentzian simulations, while simplified and quenched late-time models exhibit a crossover to
\[
R^2(t)\propto t \qquad \Rightarrow \qquad R(t)\propto t^{1/2},
\]
which matches the scale-factor law of a radiation-dominated Friedmann–Robertson–Walker universe. Classical solutions based on \(\mathrm{SU}(1,1)\) representations were then used to model still later times and to obtain a time-dependent effective equation-of-state parameter interpolating from \(w\approx -1\) to \(w\to -1/3\) [1405.5904].

Klinkhamer proposed a complementary large-\(N\) extraction procedure that starts directly from an assumed Lorentzian master field rather than finite-\(N\) simulation snapshots. One first diagonalizes and orders \(\widehat A^0\),
\[
\widehat{A}^{0} = \mathrm{diag}\big[\overline{\alpha}(1),\ldots,\overline{\alpha}(N)\big],
\]
then partitions each matrix into \(K\) non-overlapping \(n\times n\) blocks with \(N=Kn\), and finally defines coarse-grained coordinates by averaging eigenvalues inside corresponding blocks:
\[
\widehat{x}^{0}(\sigma) = \frac{1}{n}\sum_{l=1}^{n}\overline{\alpha}_{(k-1)n + l},
\qquad
\widehat{x}^{1}(\sigma) = \frac{1}{n}\sum_{l=1}^{n} \overline{\beta}^{1}_{(k-1)n + l},
\qquad
\sigma = k/K.
\]
The same block label is used for all \(\mu\), producing a sequence of correlated spacetime points \((\widehat t(\sigma),\widehat x^i(\sigma))\) [2008.01058].

That analysis emphasized two structural requirements. First, the spatial matrices should be band-diagonal in the \(A^0\)-diagonal basis, with a bandwidth \(\Delta N\), so that \(n\gtrsim \Delta N\) defines a meaningful coarse-graining scale. Second, the master field may require a more refined “fine-structure” along the diagonal. Test master fields with parabolic spatial envelopes and alternating sign or amplitude modulations produce “strands” in the \((t,x^1)\) plane, and these strands separate as \(|t|\) grows. The paper interprets such structures as evidence that the true large-\(N\) master field may need nontrivial microscopic modulation in order to yield the macroscopic point set of an expanding universe [2008.01058].

## 4. Master field, metric reconstruction, and singularity regularization

The master-field program seeks a sharper statement than “spacetime is encoded in eigenvalues.” Large-\(N\) factorization implies that gauge-invariant single-trace observables can be evaluated directly on a single master field \(\widehat A^\mu\):
\[
\langle w^{\mu_1\cdots \mu_m}\rangle \stackrel{N}{=} \text{Tr}\big(\widehat A^{\mu_1}\cdots \widehat A^{\mu_m}\big),
\qquad
w^{\mu_1\cdots \mu_m} \equiv \text{Tr}\big(A^{\mu_1}\cdots A^{\mu_m}\big) .
\]
In this sense the large-\(N\) master field is the unique candidate repository for any emergent classical spacetime [2007.08485].

In Klinkhamer’s formulation, the master field determines not only a discrete set of emergent points but also an effective metric through a scalar hopping action on those points. The inverse metric is written schematically as
\[
g^{\mu\nu}(x) \sim \int d^{10}y \; \langle\langle \rho(y)\rangle\rangle (x-y)^\mu (x-y)^\nu f(x-y) r(x,y),
\]
or, in the explicitly four-dimensional cosmological truncation,
\[
g^{\mu\nu}(x) \sim \int_{\mathbb{R}^{4}} d^{4}y\; \rho_\text{av}(y) \; (y-x)^{\mu}\,(y-x)^{\nu}\;f(y-x)\;r(x,\,y).
\]
Here \(\rho\) is the emergent point density, \(f\) is an effective hopping kernel, and \(r(x,y)\) is a density-correlation function [2007.08485][2009.06525]. This shifts the problem of metric emergence from diagonalizing matrices to identifying the density and correlation structure extracted from the master field.

The same framework was then used to ask whether the Lorentzian IIB matrix model could yield a degenerate metric relevant to a regularized big bang. The target line element was
\[
ds^{2} = - \frac{ t^{2}}{b^{2}+ t^{2}}\,d t^{2} + a^{2}( t )\;\delta_{ij}\,dx^{i}\,dx^j,
\]
with
\[
g^{00}(t)= -\frac{t^2+b^2}{t^2},
\qquad
g_{00}(t)= -\frac{t^2}{t^2+b^2},
\]
so that the metric becomes degenerate at \(t=0\) while energy density and curvature remain finite. By choosing suitable Ansätze for \(\rho_{\rm av}\), \(h(z)\), and \(\overline r(x,z+x)\), Klinkhamer argued that the emergent inverse metric can take the form
\[
g^{\mu\nu}\,\Big|^\text{(final-result)} \sim
\begin{cases}
- {\displaystyle \frac{t^{2}+c_{-2}}{t^{2}}}, & \mu=\nu=0,\\[2mm]
1 + c_{2}\,t^{2} + c_{4}\,t^{4} + \dots , & \mu=\nu=m\in \{1,2,3\},\\[2mm]
0, & \text{otherwise},
\end{cases}
\]
with
\[
b_\text{eff}^{2} \sim c_{-2}\,\ell^{2}.
\]
This suggests that the length scale regulating the big bang singularity could be set by the intrinsic IIB-matrix-model scale \(\ell\) [2009.06525].

Progress toward the actual master field remains limited. The bosonic master-field equation was recast as an algebraic system using a quenched stochastic-quantization ansatz, and explicit numerical work for \((D,N)=(10,4)\) showed that the simplified bosonic equation admits almost exact solutions with strong band-diagonal structure after diagonalizing one matrix. In the full supersymmetric \((10,4)\) problem, only approximate solutions are currently available, but they already suggest that fermions may change which directions display pronounced band-diagonality [2110.15309][2203.15779]. A plausible implication is that the physically relevant \(3+1\)-dimensional split may be encoded in a subtle interplay between bosonic quasi-locality and fermion-induced structure.

## 5. Derivative interpretation, quantized curved spaces, and model extensions

A distinct interpretation identifies the matrices not as coordinates but as covariant derivatives on a manifold. In this “derivative interpretation,” one takes
\[
A_a =
\begin{cases}
i \nabla_{(a)}, & a = 0,\dots, D-1,\\
0, & a = D,\dots,9,
\end{cases}
\]
with \(\nabla_{(a)}\) an endomorphic covariant derivative acting on functions over the principal spin bundle. Substituting this ansatz into the bosonic equations of motion yields the vacuum Einstein equation \(R_{ab}=0\). In this framework the low-energy effective action factorizes as
\[
S_{\rm eff} = \sum_i c_i s_i + \sum_{i,j} c_{ij} s_i s_j + \sum_{i,j,k} c_{ijk} s_i s_j s_k + \cdots,
\qquad
s_i = \int d^D x\, \sqrt{-g(x)}\, O_i(x),
\]
and for block-diagonal backgrounds the factorized terms admit a wormhole or multiverse interpretation in which the blocks represent different universes coupled by products of local diffeomorphism-invariant functionals [1205.1468].

A recent development in the same geometric direction regularizes the covariant-derivative interpretation by finite-size matrices using Berezin–Toeplitz quantization. For quantizable Kähler manifolds, one constructs finite matrices \(\mathcal P_{(a)}\) that satisfy
\[
\mathcal P_{(a)} T(\varphi) = T(-i\nabla_{(a)}\varphi)+O(1/p),
\]
so that the infinite-dimensional operators \(-i\nabla_{(a)}\) are approximated by finite matrices acting on Dirac zero modes. The cases of \(T^{2n}\) and \(S^2\) were worked out explicitly, including the large-\(p\) recovery of the expected flat or \(\mathfrak{su}(2)\)-type covariant-derivative algebra [2606.24577]. This provides a direct bridge between curved-spacetime backgrounds and finite-\(N\) matrix configurations.

Several algebraic extensions enlarge the model without changing its type-IIB supersymmetry. A Lie 3-algebra extension introduces 3-brackets and a scalar \(\Phi\),
\[
S = \Big\langle - [\Phi, X^M, X^N]^2  + \bar{\Theta}\,\Gamma^M [\Phi, X_M, \Theta] \Big\rangle,
\]
and for the minimal 3-algebra including \(u(N)\) it has at least two classical phases: one reduces to the original IIB matrix model, and the other to a simpler supersymmetric action [1304.4796]. A Lie 4-algebra extension adds two scalar matrices \(\Phi_1,\Phi_2\), interpreted as two extra dimensions characteristic of F-theory, and yields three phases: reduction to the original IIB matrix model, reduction to a simpler supersymmetric model, and a pure torus sector [1304.7904].

Further deformations illustrate how specific emergent geometries can be engineered. In the orientifolded \(USp\) matrix model, each spacetime point is accompanied by a mirror image with respect to a four-dimensional plane, and the two-loop effective action of the bosonic truncation implies that eigenvalues distribute in a tubular region around that plane, with characteristic scale \(\sqrt{g}N^{1/4}\) [1009.1695]. In a holographic construction based on double non-Abelian T-duality of \(AdS_5\times S^5\), the resulting type IIB background was interpreted as dual to a type IIB matrix model with fuzzy-sphere vacua characterized by partitions of the numbers of D1 and D3′ branes [1711.03419]. These constructions do not define the canonical IKKT dynamics, but they show how orientifolding, higher brackets, and duality transformations reorganize the same underlying matrix idea.

## 6. Outstanding issues and current research directions

Several technical and conceptual questions remain unresolved. The most immediate is the status of the genuine Lorentzian large-\(N\) master field. Existing numerical studies access finite \(N\), while the master-field program requires a reliable approximation to the \(N\to\infty\) object itself. Coarse-graining prescriptions, band-diagonal diagnostics, and emergent-point constructions have been formulated, but the actual fine structure of the Lorentzian master field is still unknown [2008.01058][2203.15779].

A second issue concerns the definition of the Lorentzian theory. The Lorentz-symmetric gauge-fixing proposal modifies the weight by a Faddeev–Popov determinant and changes the saddle-point structure; commuting configurations that are saddle points in the ungauge-fixed model need not remain so once \(\Delta_{\rm FP}[A]\) is included. This means that previous conclusions based on Lorentz-breaking regularizations may require reexamination in the properly gauge-fixed theory [2404.14045]. At the same time, complex-Langevin studies of bosonic truncations indicate that the treatment of the complex phase can qualitatively alter the emergent spacetime, including whether the system appears Euclidean-like or Lorentzian-like at late times [2201.13200][2112.15368].

A third issue is the emergence of a full metric theory rather than only a point cloud or extent-of-space observables. Point extraction from \(A_0\)-ordered blocks is a first step, but the harder problem is to derive \(g_{\mu\nu}(x)\), curvature, and effective dynamics from the matrix model in a way that is robust under gauge choice, coarse-graining scale, and large-\(N\) limiting procedures [2008.01058][2009.06525]. This includes the open problem of whether Einstein equations, or some controlled deformation of them, arise dynamically from the matrix degrees of freedom.

Finally, the physically decisive question remains the robustness of the \((3+1)\)-dimensional universe scenario. The Lorentzian simulations, the master-field constructions, and several deformed or simplified models all suggest that three spatial directions are special, but the mechanisms differ: spontaneous symmetry breaking in the Lorentzian model, phase structure in algebraic extensions, orientifold-induced localization around a four-dimensional plane, and signature change in bosonic effective theories. This suggests convergence rather than contradiction, but it also indicates that the route from the ten matrix directions to a unique macroscopic spacetime is not yet fully understood [1405.5904][1009.1695][2201.13200]. A plausible inference is that the IIB matrix model is less a single solved construction than a family of closely related nonperturbative frameworks whose shared core problem is to identify the precise large-\(N\) structure from which classical spacetime, local field theory, and cosmology emerge.

Source: https://www.emergentmind.com/topics/iib-matrix-model