Discovering hidden non-perturbative effects — negative tension branes, wall crossing, and doubly exponential contributions — in a new family of string theories and their connection to 3-dimensional gravity.
The Big Picture
In physics, we often compute things as long series of corrections — each term adding more precision. But some effects are invisible to these series. They're non-perturbative: exponentially small contributions that standard calculations miss entirely, yet are crucial for a complete picture.
This paper uses a powerful mathematical framework called resurgence to uncover these hidden effects in the Virasoro Minimal String (VMS) — a recently constructed toy model of string theory — and shows how they relate to 3d gravity and black holes.
Background
Physicists compute quantities like the "free energy" as a sum over increasingly complex diagrams (genus expansion). Each term involves surfaces of higher genus (more handles = more loops). But these series are divergent — the coefficients grow like (2g)! (factorially), meaning the sum doesn't converge.
This divergence is actually a feature, not a bug. It's a signal that there are non-perturbative effects — exponentially small corrections like e−A/gs that the series is trying to tell us about.
The Tool
Resurgence is a mathematical technique that recovers the full, non-perturbative answer from the divergent perturbative series. Here's how it works:
Start with the genus expansion F = Σ Fg gs2g-2
Turn the divergent series into a convergent one in a new "Borel plane"
Singularities at ±A in the Borel plane reveal hidden exponentials
Laplace transform back to recover: perturbative + e−A/gs + e+A/gs
A key insight: because the free energy is a genus expansion (only even powers of gs), the Borel plane singularities are always symmetric: they come in pairs (A, −A). This pairing is called resonance.
The Virasoro Minimal String
The Virasoro Minimal String (VMS) is a 2D worldsheet string theory built from two Liouville CFTs plus ghost fields. Parameterized by central charge c ≥ 25, it generalizes older "minimal string" models and reduces to JT gravity in the limit c → ∞.
Its physics is encoded in a spectral curve — a function y(E) with a square-root branch cut that has two sheets (physical and non-physical). The saddle points of this curve determine the non-perturbative instanton actions.
Key Discoveries
Eigenvalues tunnelling to saddle points of the spectral curve produce the familiar falling exponential corrections e−A/gs, corresponding to ZZ-branes. These match previous results.
The non-physical sheet of the spectral curve produces growing exponentials e+A/gs. These are the "anti-eigenvalues" or negative tension D-branes — new to the VMS! They come with a relative phase of i.
Every falling instanton has a growing sibling: (A, −A) pairs in the Borel plane. This "resonance" is a hallmark of genus expansions and ensures consistency.
FZZT-brane actions depend on energy E. As E varies, they can pass through branch cuts of ZZ-branes in the Borel plane and "switch off" — a wall crossing phenomenon.
All non-perturbative contributions are assembled into a Zak transform — a single closed-form expression for the complete, non-perturbative VMS partition function.
Via the VMS ↔ 3d gravity map, summing over genus in 3d gravity with end-of-the-world branes produces doubly exponential contributions in the central charge c.
Interactive Visualization
In the Borel plane, singularities reveal non-perturbative physics. For the VMS, these singularities come in symmetric pairs — a direct manifestation of resonance. Adjust b to see how they move.
Black Holes & Stokes Transitions
In the matrix model, the eigenvalue density changes its character at the edge of the distribution (E = 0): from exponentially decaying (E < 0) to oscillating (E > 0). This paper identifies this change as a Stokes transition of FZZT-branes.
For 3d gravity, this edge corresponds to the onset of black hole behavior — where the density starts exhibiting Cardy growth. The oscillations in the density are universal: they appear regardless of the choice of non-perturbative completion.
Stokes Geometry
Near the branch point E = 0, the FZZT instanton action behaves as Ã(E) ~ (−E)3/2. This creates a characteristic pattern of Stokes lines (where exponentials can turn on/off) and anti-Stokes lines (where they change dominance between damped and oscillatory).
Connection to 3d Gravity
Recent work showed that the VMS describes a subsector of 3d gravity on thickened surfaces Σg,n × I with end-of-the-world (EOW) branes. This paper uses this map to study what happens when you sum over genus in 3d gravity.
ZprimaryΣg,n×I = e−β/24 Zg,n
The gravitational path integral on Σg,n × I equals the VMS partition function
on Σg,n, with 1/gs = Z(a)disk Z(b)disk.
Summing over genus produces doubly exponential effects in c:
exp(−(n−m) Ak,±/gs)
where gs itself is exponentially small in c. Both positive and negative instantons appear,
confirmed by asymptotic checks to 3 digits.
Summary
This paper demonstrates that resurgence is a powerful and practical tool for uncovering non-perturbative physics in string theory and gravity. Key achievements:
Predicted & verified negative tension brane contributions in the VMS through large-order asymptotics.
Constructed the full non-perturbative partition function as a Zak transform, encoding all instanton sectors.
Found doubly exponential non-perturbative effects in 3d gravity by summing over genus.
Identified the onset of black hole behavior as a Stokes/anti-Stokes transition of FZZT-branes.