Arrow of time and inter-sheet correlations in the emergent Big-Bounce cosmology of the IKKT matrix model

Karczmarek and Steinacker (J. Phys. A: Math. Theor. 56 (2023) 175401) studied a 1+1-dimensional, two-sheeted Lorentzian background of an IKKT-type Yang-Mills matrix model — a projection of the fuzzy hyperboloid interpretable as an open FLRW cosmology with a Big Bounce — and showed, at tree level, that a suitable i-epsilon regularization of the oscillatory matrix integral produces the standard Feynman propagator at late times, that fluctuation modes continue across the bounce, that the arrow of time selected by the Feynman prescription points toward the bounce on the pre-bounce sheet and away from it on the post-bounce sheet, and that smeared two-point functions exhibit correlations between points on opposite sheets that coincide in target space. The authors explicitly left open the dynamical or entropic origin of this arrow structure, the physical significance of the inter-sheet correlations (attributed to non-local string modes and possibly implying gravity-like inter-brane interactions), and whether the emergent locality, causality, and stability of the background survive interactions, loop corrections, and UV/IR mixing in the full maximally supersymmetric IKKT model. The open problem is to establish — or refute — a consistent physical interpretation of the emergent Big-Bounce cosmology along these three linked fronts.

Background

The IKKT matrix model (Ishibashi, Kawai, Kitazawa, Tsuchiya 1997) is a zero-dimensional reduction of ten-dimensional supersymmetric Yang-Mills theory: an integral over large Hermitian matrices with action built from commutators $[T^a,T^b]$, conjectured to serve as a non-perturbative definition of type IIB superstring theory. Unlike matrix quantum mechanics, it contains no space-time coordinates and, crucially, no a priori time variable; both space and time must emerge from suitable matrix backgrounds. A matrix background that solves the equations of motion can be interpreted as a noncommutative ('fuzzy') brane — a finite- or infinite-dimensional matrix algebra that quantizes a classical manifold, such as the fuzzy sphere built from angular-momentum matrices. Fluctuations around the background then behave as fields on this brane, governed by a kinetic operator whose leading symbol defines an effective metric determined by the commutator (semi-classically, Poisson) structure; this 'kinetic tensor' $\gamma^{\mu\nu}$ is closely related to the open-string metric that Seiberg and Witten derived for noncommutative D-branes carrying a B-field. Because the commutator structure is dynamical, the effective metric — including its signature — is a property of the background, not of the model.

Cosmological backgrounds of this type were constructed in earlier work: Steinacker (2018) found IKKT-type solutions describing quantized homogeneous and isotropic cosmologies with a resolved Big Bang, including an open ($k=-1$) FRW case built from two sheets of opposite chirality joined through a Euclidean pre-big-bang era, and Sperling and Steinacker (2019) constructed a covariant 3+1-dimensional quantum space-time with Big Bounce, whose fluctuation spectrum forms a higher-spin gauge theory containing the graviton. Karczmarek and Steinacker (2023) analyzed the simplest, 1+1-dimensional instance: projecting the fuzzy 2-hyperboloid $H^2_R$ onto two of its coordinates yields a two-sheeted space $M_{1,1}=M^+\cup M^-$ that is glued at a well-defined matrix configuration and carries the structure of a 1+1-dimensional open FLRW cosmology (FLRW = Friedmann-Lemaitre-Robertson-Walker, the standard homogeneous and isotropic cosmology; $k=-1$ denotes hyperbolic spatial slices) with a Big Bounce (BB) at $\eta=0$. The two sheets coincide in target space (the embedding space of the brane). Transverse matrix fluctuations act as scalar fields whose kinetic operator has the leading symbol of a $k=-1$ FLRW metric with time parameter $\eta$; the effective metric degenerates at the BB, but the underlying matrix geometry and all fluctuation modes remain well defined through it, which is why the singularity is called 'mild'.

To define the Lorentzian (oscillatory) matrix integral, the authors introduced an i-epsilon regularization — an infinitesimal complex rotation $T_0\to e^{-i\varepsilon}T_0$, $T_i\to e^{i\varepsilon}T_i$ of the matrices (the matrix analogue of the infinitesimal Wick rotation familiar from ordinary path integrals, where the $i\varepsilon$ prescription selects the causal, time-ordered boundary condition). Computing the two-point function of a transverse fluctuation, they recovered, at late times ($|\eta|\to\infty$), precisely the Feynman propagator of flat 1+1-dimensional field theory including the correct $i\varepsilon$ prescription, so that causal structure and an effectively unitary time evolution emerge from a model with no a priori time. The fluctuation modes stretch continuously across the BB, and Bogoliubov coefficients (the standard mode-matching coefficients relating the 'in' basis on the pre-bounce sheet to the 'out' basis on the post-bounce sheet) can be computed explicitly. The $i\varepsilon$/time-ordering prescription selects an arrow of time that points toward the BB on the pre-bounce sheet ($\eta<0$) and away from it on the post-bounce sheet ($\eta>0$), so that physical time runs continuously through the bounce while the two branches are oppositely oriented relative to it. Strikingly, the smeared two-point functions display correlations between points on opposite sheets, peaked at $(\eta,\chi)$ and $(-\eta,\chi)$ — points that coincide in target space — while the point-like propagators are ultraviolet-singular there. The authors attributed these inter-sheet correlations to non-local 'string modes' $|x\rangle_+\langle y|_-$ of the noncommutative theory, coherent-state bilinears connecting the two coincident brane sheets; such non-local string states are also the carriers of UV/IR mixing (the noncommutative phenomenon in which ultraviolet and infrared scales are coupled, so that the extreme UV sector is better described by extended, string-like degrees of freedom), which they argue is tamed only by maximal supersymmetry.

The analysis was, however, restricted to a free (tree-level) scalar theory on a truncated, 1+1-dimensional toy background, and the authors explicitly identified what remains open: the dynamical or entropic justification of the arrow structure; the physical significance of the inter-sheet correlations and their possible observable consequences in realistic 3+1-dimensional covariant solutions; the detailed stability of the background beyond the kinematic sign of the mass parameter; whether loop corrections and non-local string-like UV modes (with their UV/IR mixing) preserve the emergent locality and causality in the full interacting maximally supersymmetric IKKT model; and the precise relation between the matrix-derived kinetic tensor $\gamma^{\mu\nu}$ and the full open-string/effective metric in more realistic embeddings (the associated signature flip for time-like noncommutativity is already known to disappear on the covariant 3+1 backgrounds).

Problem Statement

The problem is to give the emergent Big-Bounce cosmology of the IKKT-type Yang-Mills matrix model a controlled physical interpretation beyond the free (tree-level), 1+1-dimensional analysis in which it was established. All three facets below share one objective — deciding whether the Big-Bounce space-time, its arrow of time, and its cross-bounce correlations are stable physical features of the interacting matrix model rather than artifacts of the free truncation — and they are linked: the arrow and the correlations are both kinematic consequences of the same tree-level mode structure on the two-sheeted background, whose physical status can only be settled together with the interacting-theory consistency of that background.

1. Arrow of time. On the two-sheeted, projected fuzzy-hyperboloid background, the $i\varepsilon$-regularized matrix integral yields the Feynman propagator at late times, and the selected arrow of time points toward the Big Bounce on the pre-bounce sheet ($\eta<0$) and away from it on the post-bounce sheet ($\eta>0$), so that physical time runs continuously through the bounce while the two branches are oppositely oriented relative to it. What is the dynamical or thermodynamic (e.g., entropic) origin of this structure? Is there a microscopic mechanism in a model with no a priori time that justifies the opposite-arrow interpretation, or is it an artifact of the chosen regularization/prescription?

2. Inter-sheet correlations. The free-theory propagator contains correlations between points on opposite sheets, peaked at target-space-coincident points, plausibly mediated by non-local string modes connecting the coincident brane sheets, with expected gravity-like inter-brane interactions at one loop. What is their physical significance? Do they survive interactions and loops — where non-local string-like UV modes and UV/IR mixing qualitatively alter the UV sector — and do they lead to observable signatures in realistic higher-dimensional (3+1) covariant cosmological solutions?

3. Consistency beyond tree level. Does the emergent locality, causality, and stability of the bounce background persist in the full interacting, maximally supersymmetric IKKT model? This requires a detailed stability analysis of the projected (squashed) fuzzy-hyperboloid background beyond the kinematic mass-sign argument, and control of loop corrections together with the non-local string-mode sector. (The related signature subtlety of time-like noncommutativity is already known to disappear on the covariant 3+1 backgrounds; the precise relation between the matrix-derived kinetic tensor $\gamma^{\mu\nu}$ and the full open-string/effective metric in realistic embeddings is background context for this facet rather than one of its deliverables.)

A complete answer establishes — or rules out — a consistent physical picture of the emergent Big-Bounce cosmology in which the arrow, the correlations, and the background itself are genuine, stable features of the interacting matrix model. The three facets above are independently assessable but jointly required: the first is settled by identifying a dynamical or thermodynamic origin for the opposite arrow or by showing that it is an artifact of the regularization; the second by fixing the physical role of the inter-sheet correlations in the interacting theory, whether they persist under loops and non-local string modes, and whether they carry observable consequences in a realistic 3+1-dimensional solution; the third by deciding whether emergent locality, causality, and stability persist beyond the kinematic mass-sign argument with the loop and string-mode sector under control. A partial answer that settles only one facet, or that leaves any facet at the level of the free tree-level analysis, resolves that facet alone and is not by itself a complete answer.

References

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