The gravitational index and allowable complex metrics

Technical Summary: The Gravitational Index and Allowable Complex Metrics

Problem Statement
The gravitational path integral (GPI) is a fundamental tool for studying quantum gravity, where observables are computed by summing over gravitational field configurations weighted by their Euclidean action. In the semiclassical limit, this sum is dominated by saddle points corresponding to classical solutions. However, a significant ambiguity arises regarding which complex metrics should be included in this sum. Specifically, supersymmetric rotating black holes, when analytically continued to Euclidean time to compute supersymmetric indices, generically yield complex metrics. The central problem addressed is determining the validity of these complex saddles: are they "allowable" contributions to the gravitational path integral, or should they be excluded?

The authors investigate the Kontsevich–Segal–Witten (KSW) criterion, a proposal by Witten based on Kontsevich and Segal's work, which provides a condition for the allowability of complex metrics. The paper seeks to compare the KSW criterion against two other physical requirements in the context of supersymmetric black hole microstate counting:

1. Geometric Consistency: Conditions derived from the smoothness of Euclidean sections and the causal structure of the corresponding Lorentzian analytic continuation (e.g., existence of horizons, absence of superluminal frames).
2. Microscopic Convergence: The requirement that the dual microscopic trace (the supersymmetric index in the boundary theory) converges, imposing constraints on complex chemical potentials.

Methodology
The authors apply the KSW criterion to complex saddle points found in the gravitational path integrals corresponding to three specific supersymmetric indices in different asymptotic backgrounds:

1. Asymptotically Flat (AF) space in 4D ($AF_4$): The supersymmetric index (helicity supertrace) in $N=2$ ungauged supergravity.
2. Asymptotically AdS (AAdS) space in 4D ($AAdS_4$): The topologically twisted index in minimal gauged supergravity.
3. Asymptotically AdS space in 4D ($AAdS_4$): The superconformal index in minimal gauged supergravity.

For each case, the authors:

• Construct the relevant complex saddle point solutions by imposing supersymmetry on the gravitational equations of motion and performing necessary analytic continuations of parameters (such as angular velocity and electric potential) to satisfy boundary conditions for fermions.
• Analyze the KSW criterion, which requires that at every point on the manifold, the metric induces a complex quadratic form such that the real part of the square root of the determinant times the contraction of the metric with any real non-zero $q$-form is positive. This is mathematically equivalent to requiring the sum of the absolute values of the arguments of the eigenvalues of the metric (in a real basis) to be less than $\pi$.
• Derive the constraints on the parameter space imposed by geometric consistency (smoothness of the instanton, absence of velocity-of-light surfaces in the Lorentzian continuation).
• Derive the constraints imposed by the convergence of the microscopic trace (ensuring the real part of the inverse temperature $\beta$ is positive and the imaginary part of the chemical potential $\tau$ is positive).

Key Contributions and Results

1. AF4 Space (Supersymmetric Index):

   • The saddle points correspond to supersymmetric solutions in the Israel–Wilson–Perjes family.
   • To satisfy supersymmetry and boundary conditions, the angular velocity must be analytically continued to be purely imaginary.
   • This continuation renders the metric real (Riemannian). Consequently, the metric trivially satisfies the KSW criterion.
   • The condition for the metric to be smooth (requiring the radial coordinate to be real and positive) is found to be identical to the condition required for the convergence of the microscopic trace.

2. AAdS4 Space (Topologically Twisted Index):

   • The relevant saddles are static dyonic black holes with a Riemann surface horizon.
   • Similar to the AF case, satisfying supersymmetry requires an analytic continuation of the electric charge, resulting in a real Euclidean metric.
   • The KSW criterion is trivially satisfied.
   • The condition for the regularity of the solution ($r_+ > r_*$) is shown to be exactly equivalent to the condition for the convergence of the microscopic trace ($\text{Re}(\beta) > 0$).

3. AAdS4 Space (Superconformal Index):

   • This case is distinct because the supersymmetric saddle points are intrinsically complex; they cannot be made real via analytic continuation of parameters.
   • The metric depends on complex chemical potentials (angular velocity $\Omega$ and electric potential $\Phi_e$).
   • The authors perform a detailed analysis of the KSW criterion for this complex metric:
     • Near the horizon: The criterion reduces to requiring the real part of the derivative of the radial function $\Delta_r$ to be positive, which implies $r_+ > r_*$.
     • Asymptotic region: The criterion requires the absence of velocity-of-light surfaces ($|\Omega| < 1$) and the positivity of $\text{Re}(\beta)$.
     • Interior region: Numerical checks on $10^6$ random points in the parameter space confirm that the KSW criterion holds if and only if the geometric and convergence constraints hold.
   • Result: The region in parameter space allowed by the KSW criterion is precisely identical to the region required for the convergence of the microscopic trace and the geometric consistency of the Lorentzian solution.

Significance and Claims
The paper claims that in all situations considered (AF and AAdS spaces, various indices), the three distinct criteria for selecting saddle points in the gravitational path integral yield the exact same region of validity:

1. The KSW criterion (a condition on the allowability of complex metrics for defining quantum field theories).
2. Geometric consistency (smoothness of the Euclidean section and causal well-behavedness of the Lorentzian continuation).
3. Microscopic convergence (the requirement that the dual field theory trace is well-defined).

The authors conclude that these three spaces "cut out" by the different criteria are exactly the same. This agreement provides a non-trivial justification for including complex saddles in the gravitational path integral when computing supersymmetric indices. It suggests that the KSW criterion is not merely a mathematical consistency condition but is deeply linked to the physical requirements of thermodynamic stability and the existence of a well-defined dual microscopic description.

The paper notes that while the KSW criterion has been studied in non-supersymmetric contexts (e.g., spectral form factors, no-boundary proposals), this work specifically validates its consistency with supersymmetric black hole entropy and microstate counting. The authors also acknowledge open questions regarding the generalization to non-minimal supergravity theories (with flavor refinements) and the application of the criterion to higher-dimensional uplifts in string/M-theory, where complex gauge fields may complicate the definition of allowability.