The IKKT renormalization group flow is IIB: toward zero-d holography

Technical Summary: The IKKT Renormalization Group Flow is IIB: Toward Zero-d Holography

Problem Statement
The paper addresses the challenge of extending the holographic principle to D-instantons ($p = -1$) in Type IIB string theory. In standard holographic dualities involving D$p$-branes with $p \neq 3$, the Renormalization Group (RG) flow of the $(p+1)$-dimensional Super Yang-Mills (SYM) coupling is geometrized as a radial direction in the bulk supergravity solution, tied to a non-constant dilaton profile. However, the worldvolume theory of D-instantons is the Euclidean IKKT matrix model, which lacks space and time dimensions. Consequently, the standard notion of RG flow (evolution with energy scale) is ill-defined, making it difficult to interpret how a radial coordinate and a running dilaton emerge from the matrix model to reproduce the IIB supergravity background.

Methodology
The authors propose two distinct methods to reduce the rank $N$ of the IKKT matrix model, interpreting this reduction as a matrix RG flow that mimics the emergence of the radial direction:

1. Coulomb Branch Integration: The authors induce a change in rank by placing the system on the Coulomb branch, splitting a stack of $N+M$ D-instantons into two separated stacks ($N$ and $M$) by a distance $\Delta$. They integrate out the heavy open strings (represented by off-diagonal matrix elements) stretching between the stacks. Unlike the standard field theory context where the Vacuum Expectation Value (VEV) is a fixed boundary condition, in the IKKT model, this separation is implemented by hand without integrating over the relative position $\Delta$ in the path integral. The leading correction to the effective action of the remaining $N$-stack is computed and compared to the supergravity harmonic function.
2. Brézin-Zinn-Justin (BZJ) Matrix RG Flow: The authors apply the BZJ method, which reduces the matrix rank from $N$ to $N-1$ by integrating out a single row and column. This approach differs from the Coulomb branch method as it effectively integrates over the Coulomb branch position. Since the IKKT action lacks a Gaussian term (preventing a direct perturbative application of BZJ), the authors utilize the exact partition function $Z(N, g_s)$ derived from D-instanton physics. They impose a Callan-Symanzik (CS) equation on this partition function, $N \frac{d}{dN} Z = 0$, to derive the flow of the coupling constants with respect to $N$.

Key Contributions and Results

• Emergence of the Radial Profile via Coulomb Branch:
  By separating the D-instanton stacks and integrating out the massive modes, the authors derive a leading correction to the IKKT action of the form $\tilde{c} \text{Tr}[Y, Y]^4$. They demonstrate that the coefficient $\tilde{c}$ is inversely proportional to the eighth power of the separation distance ($\tilde{c} \propto \Delta^{-8}$).
  Comparing this to the supergravity description of a two-centered D-instanton solution, the constant term $c$ in the harmonic function $H = c + Q_N/r^8$ is identified with the separation distance of the second stack ($c \propto \Delta^{-8}$). This establishes a direct map: the coefficient of the irrelevant operator in the matrix effective action corresponds to the constant deformation of the harmonic function in the dual geometry. This allows the extraction of the dilaton profile $e^\phi \sim H^{(3-p)/4}$ directly from the matrix model.

• Running Coupling via BZJ Flow:
  The authors analyze the $N$-dependence of the IKKT partition function, which includes a normalization factor involving the divisor function $\sigma_{-2}(N)$. By coarse-graining $\sigma_{-2}(N)$ (showing its derivative vanishes at leading order for large $N$) and applying the CS equation, they derive the flow of the string coupling $g_s$ and the RR axion $C_0$ with respect to the matrix rank $N$.
  The results show:

  • $g_s(N) \propto N$ (in the large $N$ limit).
  • $C_0(N) \propto N^{-1}$.
    These scalings exactly match the $N$-dependence of the dilaton and RR axion profiles in the near-horizon supergravity solution for a stack of $N$ D-instantons ($e^\phi \sim N/r^8$).

• Normalization and Physical Meaning:
  The paper clarifies the normalization of the IKKT partition function in the context of D-instanton physics, specifically how the tension term and the $R^4$ coupling fix the dependence on $N$ and $g_s$. This ensures that the "vacuum energy" term in the matrix action correctly reproduces the physical weight of D-instantons in Type IIB string theory.

Significance and Claims
The paper claims to provide a conceptual framework for "0-dimensional holography," demonstrating how the radial coordinate and the running of couplings in the bulk IIB supergravity can emerge from the IKKT matrix model without pre-existing spacetime.

• Conceptual Meaning: The work suggests that the radial direction in the bulk is geometrized through the reduction of degrees of freedom (rank reduction) in the matrix model.
• Dualities: It confirms that the matrix RG flow (via both Coulomb branch integration and BZJ rank reduction) reproduces the specific $N$-dependence and harmonic structure of the IIB supergravity background sourced by D-instantons.
• Limitations and Modesty: The authors are modest regarding the scope of their results. They note that while the Coulomb branch method recovers the harmonic deformation, the BZJ flow applied to the partition function does not explicitly track the generation of the full tower of "irrelevant operators" in the effective action, though they suspect maximal supersymmetry protects the flow (analogous to BPS bounds). They acknowledge that the validity of the flow derived from the partition function for expectation values remains an open question.

The paper concludes that these methods offer a starting point for deeper checks of the duality, such as deforming the background with sources (e.g., $B_2$ fields) or exploring fixed points in the flow corresponding to D(−1)/D7 bound states.