Fortuity and fragility in supersymmetric SYK

The gist In the N = 2 supersymmetric SYK model, every BPS state is fortuitous: at fixed charge, it exists only over a finite range of N and allows for an uplift to arbitrarily large N along lattice walks inside the fortuity window.

Fortuitous and Monotonous States

The paper distinguishes between monotonous states, which descend from a supersymmetric state of the large-N theory, and fortuitous states, which exist only within a finite range of consecutive ranks. Monotonous states are naturally associated with smooth horizonless configurations such as graviton gases, while the exponentially numerous fortuitous states are the candidate microstates of supersymmetric black holes. The distinction has been studied in various models, including the N = 2 SYK model, a solvable single-matrix model, and higher-spin and bosonic vector models.

Cohomological Walks and Uplift Ambiguity

The paper shows that BPS classes may admit unobstructed cohomological extensions to arbitrarily large N along a walk inside the fortuity window. Uplifting is intrinsically ambiguous because the long exact sequence determines the uplift only as an element of a quotient of the rank-N+1 cohomology, and there is no canonical, purely algebraic map from harmonic states at N to harmonic states at N+1. The walk labels are not canonical, depending on choices of representatives and splittings of the exact sequence.

The Uplift Decoder and Metric Fragility

The paper introduces a decoder D whose spectrum quantifies exact uplift and its metric cost, calling the failure or increasing cost of uplifting harmonic representatives metric fragility. The decoder DT maps BPS states BpN' into the Hilbert space HpN of the smaller theory. Uplift fidelity FT is defined as the fraction of a state’s squared norm contained in the decoder image, and dressing cost κT captures the fractional norm overhead of the correction. In generic models, exact uplift eventually fails outright, FT < 1, which is called metric fragility because it depends on the inner-product geometry of the Hilbert space.

Solvable Models and Uplift Mechanisms

The paper evaluates these quantities in two analytically controlled settings. In the single-matrix model of Chen [4], one can uplift every BPS state of rank N to a BPS state of rank N+1 with no correction at all, yielding FT = 1 and κT = 0. In the symmetrized two-flavor model, the monotonous states lift exactly, FT = 1, with a nonvanishing correction fixed by symmetry and a dressing cost bounded by n/(N − n + 1).

BPS Chaos of the Decoder Spectrum

The operator constructed from the decoder D†T DT maps the BPS space of the enlarged theory at N + 1 to itself, exhibiting level statistics consistent with the Gaussian unitary ensemble. The nonzero eigenvalues determine visibility and dressing scales across system sizes, while their correlations provide a simultaneous probe of chaos within the enlarged BPS space. In generic models, the nontrivial decoder spectrum exhibits adjacent-gap statistics consistent with the unitary random-matrix class.

Metric Fragility as a Signature of Chaos

The contrast among the three models suggests that protected or integrable BPS families admit rigid maps across N, whereas in a chaotic BPS sector, exact uplift is noncanonical and increasingly costly. The decoder connects metric fragility with the spectral structure of the BPS sector; its nonzero eigenvalues determine metric uplift between system sizes, while correlations among them diagnose chaos within the enlarged BPS space.

Walk Counting of Hodge Sectors

The appendix shows that the dimensions of the three sectors of the Hodge decomposition are each counted by a mutually exclusive class of ±1 walks of length N. The ordinary BPS subspace Bp contains states corresponding to walks confined within the restricted fortuity strip for their entire length. The second space V−p is counted by bottom-exit walks, and the third space V+p is counted by walks first exiting through top boundary at +q.

Quantum Error Correction and Non-BPS States

The decoder also admits a quantum information theoretic interpretation, where passing from the enlarged system back to the original system amounts to removing additional fermionic modes, and metric fragility measures the recoverability of BPS information after this erasure. The analysis can be extended to non-BPS states by choosing any physically distinguished target subspace at system size N + 1.

Gravitational Interpretation

Since monotonous states are naturally associated with perturbative multigraviton states, while fortuitous states are candidate black hole microstates, it is natural to ask whether metric fragility has a bulk interpretation. The contrast between protected and chaotic uplift may distinguish semiclassical states from strongly mixed black hole microstates.

Summary of Findings

The paper demonstrates that while BPS classes can be algebraically continuable, the metric information provided by the decoder is necessary to determine if a harmonic state at size N occurs as a definite component of a harmonic state in the enlarged theory. The diagnostics FT (α) and κT (α) measure the visible fraction and norm amplification required to reconstruct the visible part. The operator D†T DT is of Lin–Maldacena–Rozenberg–Shan type, with its eigenvalues controlling visibility and reconstruction cost, while correlations among them probe chaos within the enlarged BPS space. This suggests that metric fragility provides a diagnostic of chaotic BPS dynamics across system sizes.

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QMUL-PH-26-29

Fortuity and fragility in supersymmetric SYK James Chryssanthacopoulos, David Vegh Centre for Theoretical Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom email: j.chryssanthacopoulos@qmul.ac.uk, d.vegh@qmul.ac.uk August 13, 2026 Abstract In the N = 2 supersymmetric SYK model with N fermions, every BPS state is fortuitous: at fixed charge, it exists only over a finite range of N and allows for an uplift to arbitrarily large N along lattice walks inside the fortuity window. However, there is no canonical uplift. We therefore introduce a decoder D whose spectrum quantifies exact uplift and its metric cost. We call the failure or increasing cost of uplifting harmonic representatives metric fragility. Chen’s single-matrix model and the protected tower of the two-flavor SYK model realize bare and genuinely dressed mechanisms of perfect uplift, respectively. In the generic one-flavor model, exact uplift eventually fails. The Lin–Maldacena–Rozenberg–Shan-type operator D†D exhibits level statistics consistent with the Gaussian unitary ensemble. Its eigenvalues quantify the metric continuity of BPS states across system size, while their correlations probe chaos within the BPS sector. Metric fragility and BPS chaos are thus encoded in complementary observables of the same operator, distinguishing chaotic BPS sectors from exactly solvable towers. arXiv:2608.12160v1 [hep-th] 12 Aug 2026

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Introduction A striking lesson of recent work on supersymmetric theories is that a large class of BPS microstates conjectured to be holographically dual to black holes exists only at finite N. In the classification of Chang and Lin [1], rooted in the supercharge cohomology approach to N = 4 super Yang–Mills [2], a BPS state of a family of theories labeled by the rank parameter N is monotonous if it descends from a supersymmetric state of the N → ∞ theory. Instead, a state is fortuitous if its supersymmetry relies on finite-N relations, so that the state exists only within a finite range of consecutive ranks. Monotonous states are naturally associated with smooth horizonless configurations such as graviton gases, while the exponentially numerous fortuitous states are the candidate microstates of supersymmetric black holes. The distinction has by now been studied in the N = 2 SYK model, in a solvable single-matrix model [4], and in D1–D5 system [5, 6, 7, 8], and in ABJM theory [12]. Why track individual microstates across system sizes? The question of whether a given black hole microstate retains any identity when the theory itself is enlarged is a sharp version of the question of what black hole microstates are. If fortuitous states simply flicker in and out of existence as N varies, microstate identity across theories is meaningless. If instead each state admits a controlled uplift, the family of theories carries a coherent tower of microstates, and one can ask quantitatively how much of a state survives the uplift to large N.

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Contents 1 Introduction 2 N = 2 supersymmetric SYK model 4 2.1 The Witten index...........5

1. Bold header: Metric Fragility Diagnosis

The improved AI system can diagnose whether a BPS state admits an exact uplift by calculating uplift fidelity FT, defined as the fraction of a state’s squared norm contained in the decoder image. This allows for the differentiation between fortuity and monotonous behavior in generic models, as exact uplift eventually fails outright (Page 4).

2. Bold header: Chaos Probe via Decoder Spectrum

The system can probe chaos within the BPS sector by analyzing the operator OT, which is of Lin–Maldacena–Rozenberg–Shan type. This involves examining the adjacent-gap statistics of its eigenvalues, which coincide with the nonzero spectrum of KT = DT D†T (Page 30).

3. Bold header: Cost Quantification for Uplift

The AI can quantify the cost of uplift by calculating dressing costs κT, defined as the fractional norm overhead of the correction, and comparing it to FT. This allows researchers to distinguish between exact-lift locus states, where FT = 1 and κT = 0, and approximate lifts where "FT < 1."

In the N=2 supersymmetric SYK model with N fermions, every BPS state is fortuitous: at fixed charge, it exists only over a finite range of N. Allowing the charge to increase, we show that BPS classes may instead be uplifted along lattice walks inside the fortuity window, potentially to arbitrarily large N. However, there is no canonical uplift. We therefore introduce a decoder D whose spectrum quantifies exact uplift and its metric cost. We call the failure or increasing cost of uplifting harmonic representatives metric fragility. Chen's single-matrix model and the protected tower of the two-flavor SYK model realize bare and genuinely dressed mechanisms of perfect uplift, respectively. In the generic one-flavor model, exact uplift eventually fails. The Lin-Maldacena-Rozenberg-Shan-type operator D D exhibits level statistics consistent with the Gaussian unitary ensemble. Its eigenvalues quantify the metric continuity of BPS states across system size, while their correlations probe chaos within the BPS sector. Metric fragility and BPS chaos are thus encoded in complementary observables of the same operator, distinguishing chaotic BPS sectors from exactly solvable towers.