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Mathematical Physics McKean-Vlasov Topological Field Theory Supersymmetry Dirac Operator Crypto · ETF v4 Update

Dompter le chaos des marchés volatils — modèles issus de la physique, de McKean-Vlasov à la théorie topologique des champs RESEARCH PREVIEW

Taming Chaotic Markets — Physics-Inspired Models from McKean-Vlasov to Topological Field Theory RESEARCH PREVIEW

May 24, 2026 48 min read HFThot Research Team

📖 Abstract

Crypto and ETF markets in stress regimes behave like physical systems with many weakly-constrained degrees of freedom — fertile ground for deterministic chaos. Classical tools (GARCH, HMM, copulas) partially capture the dynamics but fail to control phase transitions. This article lays out a physical roadmap: McKean-Vlasov mean-field interactions, dimensional reduction via the Wheeler-DeWitt minisuperspace ansatz, and topological invariants preserved under BRST/supersymmetry.

We show on 7 real underlyings (BTC, ETH, SOL, SPY, QQQ, TLT, GLD; 2024-01 → 2026-04) that the effective dimension of configuration space collapses sharply to $d_{eff} \approx 4$, that the discrete Dirac operator on the market graph yields an actionable η-invariant, and that persistent homology detects $H_1$ loops correlated with regime breaks. Some portions are adapted from the private WP-5 working paper in normalised, reproducible form.

v4 Update — this article was extended in May 2026 to fold in the v4 preprint (60 pp): geometric volatility index Φ-VIX, reparameterisation-invariant signature-Dirac $\eta^{\mathrm{Sig}}$, zigzag persistence under regime jumps, and the SUSY–Hawkes Witten index. The full preprint and the companion notebook (FR/EN) are downloadable at the bottom of the article.

1. What "chaos" means for a market

A chaotic market is not a noisy market. It is a deterministic market that is sensitive to initial conditions, with a positive Lyapunov exponent on the attractor of renormalised prices. On BTC/USD between 2020 and 2025, one measures $\lambda_1 \approx 0.05 \pm 0.01$ day⁻¹ — a predictability horizon of about 20 days, beyond which two trajectories initially separated by 0.1 % diverge past 100 %.

The observable signature is volatility clustering: calm periods interrupted by intense bursts, with a power-law duration distribution. This is the fingerprint of a non-equilibrium system near a phase transition, exactly as for the magnetisation near the critical point of a 2D Ising model.

Rolling annualised volatility for BTC, ETH, SPY showing regime clusters
Fig. 1 — Rolling annualised volatility (21 days) for BTC, ETH and SPY, Jan 2024 → Apr 2026. High-volatility clusters (>80 % on BTC) coincide with macro and microstructural event bursts, exactly as predicted by GARCH-multifractal models.

The problem: these models statistically describe chaos without providing a handle to control it. For that one needs a dynamical framework in which one can write an evolution equation for the portfolio manager and her environment. That is exactly what mean-field games provide.

2. McKean-Vlasov — agents that watch the crowd

A typical agent on a risky asset $X_t$ controls her position $\alpha_t$ to optimise a criterion $J(\alpha; \mu)$ where $\mu_t$ is the empirical distribution of all other positions. Her dynamics is a McKean-Vlasov SDE:

$$dX_t \;=\; b\bigl(t, X_t, \mu_t, \alpha_t\bigr)\,dt \;+\; \sigma\bigl(t, X_t, \mu_t\bigr)\,dW_t, \qquad \mu_t \;=\; \mathcal{L}(X_t).$$

The mean-field Nash equilibrium (Lasry-Lions 2007, Carmona-Delarue 2018) reduces to a coupled system of a backward Hamilton-Jacobi-Bellman for the value function and a forward Fokker-Planck for $\mu_t$. This is the first rung for modelling market reflexivity: each trader reacts to what others do, this modifies the distribution, which modifies future reactions. Chaos emerges from the instability of this fixed point.

2.1 Deriving the HJB–Fokker-Planck system

The representative agent minimises the cost functional $J(\alpha;\mu) = \mathbb{E}\!\left[\int_0^T f(X_s,\alpha_s,\mu_s)\,ds + g(X_T,\mu_T)\right]$. The value function is $v(t,x) = \inf_\alpha \mathbb{E}[\cdot\mid X_t=x]$. Bellman's dynamic-programming principle yields, formally, $v(t,x) = \inf_\alpha \mathbb{E}[f\,dt + v(t+dt, X_{t+dt})]$. Expanding by Itô:

$$0 \;=\; \partial_t v \;+\; \inf_{\alpha}\Bigl\{\, b(t,x,\mu_t,\alpha)\cdot\nabla_x v \;+\; \tfrac{1}{2}\,\mathrm{tr}\bigl(\sigma\sigma^\top\,\nabla^2_x v\bigr) \;+\; f(x,\alpha,\mu_t) \,\Bigr\}, \qquad v(T,x)=g(x,\mu_T).$$

The infimum, achieved at $\alpha^\star(t,x,\mu_t,\nabla_x v)$, supplies the optimal feedback control. Substituting it into the dynamics yields the forward Fokker-Planck PDE for $\mu_t(x)$:

$$\partial_t \mu_t \;+\; \nabla_x\!\cdot\!\bigl(b(t,x,\mu_t,\alpha^\star)\,\mu_t\bigr) \;-\; \tfrac{1}{2}\,\mathrm{tr}\bigl(\nabla^2_x(\sigma\sigma^\top \mu_t)\bigr) \;=\; 0, \qquad \mu_0 \text{ given.}$$

The backward HJB + forward FP couple forms the Lasry-Lions MFG system. The main theorem (existence and uniqueness under the Lasry-Lions monotonicity condition) guarantees a mean-field Nash equilibrium exists and coincides with this fixed point. Dimensional reduction and topological invariants are then layered on top of this system.

2.2 Closed-form linear-quadratic case

If dynamics are linear $dX_t = (aX_t + \bar X_t b + \alpha_t)\,dt + \sigma\,dW_t$ with $\bar X_t = \mathbb{E}[X_t]$ (first-moment mean-field interaction) and cost quadratic $f = \tfrac12 q\,x^2 + \tfrac12 r\,\alpha^2$, the HJB collapses to a Riccati equation for $P(t)$:

$$\dot P(t) \;+\; 2a\,P(t) \;-\; \tfrac{1}{r}\,P(t)^2 \;+\; q \;=\; 0, \qquad P(T)=q_T,$$

with explicit solution (assuming $a^2 + q/r > 0$, $\Delta = \sqrt{a^2 r + q\,r}$):

$$P(t) \;=\; \Delta \cdot \frac{(\Delta + a r)\,e^{2\Delta(T-t)/r} + (\Delta - a r)}{(\Delta + a r)\,e^{2\Delta(T-t)/r} - (\Delta - a r)}.$$

The optimal feedback control is then $\alpha^\star_t = -P(t)\,X_t/r$ — the MFG analogue of classical LQG. This closed form is the validation benchmark for the HJB-FP numerical solvers used in HFThot.

Why it matters: on a 5-ETF universe with 3 regimes, the coupled HJB-FP becomes an 8-equation system on a product space $\theta \times x$ — still tractable but already non-analytic. Beyond that, one must reduce dimension before hoping to solve.
Animation 1 (propagation of chaos) — empirical measure $\mu_t^N = \tfrac{1}{N}\sum_i \delta_{X_t^i}$ for $N \in \{20, 100, 500, 4000\}$, against the limit law $\mu_t$ (white dashed, reference $N = 2{\cdot}10^4$). Bottom: Wasserstein-2 distance $W_2(\mu_t^N, \mu_t)$ on a log scale — the $\mathcal{O}(1/\sqrt{N})$ decay makes the convergence of the particle system to the McKean-Vlasov dynamics visible.

3. Lagrangian, gauge fields and the arbitrage action

Ilinski (1997-2001, Physics of Finance) showed that a $U(1)$ gauge theory naturally describes arbitrage dynamics. The idea: associate to each asset pair $(i,j)$ a parallel-transport operator $U_{ij} \in \mathbb{R}_+^*$ — concretely, the exchange rate $S_{ij} = $ price of $i$ in units of $j$.

In the absence of arbitrage, the product around a closed loop equals 1 (law of one price). In its presence, the gauge curvature $F$ measures the deviation from 1 on a plaquette:

$$F_{ijk} \;=\; U_{ij}\,U_{jk}\,U_{ki} \;-\; 1, \qquad F_{ijk}=0 \;\Leftrightarrow\; \text{no triangular arbitrage on } (i,j,k).$$
Triangular gauge plaquette on (USD, BTC, ETH) and rolling loop curvature
Fig. 3 — Left: triangular plaquette on three currencies; the cyclic product $U_{12}\,U_{23}\,U_{31}$ equals 1 with no arbitrage. The curvature $F$ encodes the profitability of a cycle. Right: $F(t)$ signal on a USD-BTC-ETH-USD loop; positive spikes (green) are executable arbitrage windows.

3.1 Yang-Mills action and arbitrage Lagrangian

By analogy with electromagnetism, the Yang-Mills action on the market graph reads:

$$S[U] \;=\; \frac{1}{2g^2}\sum_{\text{plaquettes }(i,j,k)} \bigl|F_{ijk}\bigr|^2 \;-\; \sum_{\text{liens }(i,j)} \beta_{ij}\,\ln U_{ij}.$$

The first term penalises arbitrage opportunities (analogue of the magnetic energy $\frac14 F_{\mu\nu}F^{\mu\nu}$). The second is a source term representing exogenous order flow ($\beta_{ij}$ = demand elasticity). The equations of motion $\delta S/\delta U_{ij} = 0$ give a discrete version of Maxwell's equations for arbitrage:

$$\sum_{k} F_{ijk}\,U_{jk}\,U_{ki} \;=\; g^2\,\beta_{ij}.$$

This equation reads directly: aggregate arbitrage around each edge is exactly proportional to the in/out order-flow imbalance. As $g \to 0$ (perfectly liquid market), $F \to 0$ and the no-arbitrage condition is recovered as a deterministic limit. At finite $g$, one derives an arbitrage propagator that describes how a local perturbation (a large order on $i \to j$) diffuses across the graph and creates opportunities on apparently unrelated pairs.

3.2 Quantisation and arbitrage partition function

As in a statistical system, define the partition function $Z = \int \mathcal{D}U\,e^{-S[U]/\hbar_{\text{mkt}}}$ where $\hbar_{\text{mkt}}$ plays the role of a "market Planck constant" (the amplitude of tick-scale fluctuations). The correlation functions $\langle U_{ij}\,U_{kl}\rangle$ then yield the arbitrage covariances between asset pairs. The Wilson loop $\langle\prod_{\partial \mathcal{C}} U\rangle$ measures the average expected profitability of a strategy following the contour $\mathcal{C}$, and its exponential decay in loop size $\sim e^{-\sigma_{\text{mkt}} \mathrm{Area}}$ supplies an arbitrage surface tension $\sigma_{\text{mkt}}$ directly estimable from data.

Practical reading: in a "normal" regime, $\sigma_{\text{mkt}}$ is large and arbitrage loops die quickly. Approaching a stress event, $\sigma_{\text{mkt}}$ collapses: larger and larger arbitrage loops become profitable — this is the pre-break indicator derived from the Lagrangian.

4. How many dimensions does a market really have?

4.1 The naive count vs the effective count

Naively, a universe of $N$ underlyings observed over $T$ steps lives in an $NT$-dimensional space. For 7 assets and 850 days, that is 5 950 coordinates — astronomically large in the combinatorial sense. But as in any real physical system, this space is nearly empty: only a few directions are actually explored. The others remain frozen because of economic constraints (intra-class correlations, cross-market arbitrage, common monetary policy) which play the role of gauge relations.

The effective dimension $d_{eff}$ is the minimal dimension of a sub-manifold capturing essentially all of the data variance. The most robust estimator is PCA on the covariance matrix of normalised returns: keep the number of components needed to explain 90 % of variance (the spectral break is typically sharp).

Variance share of each principal component and cumulative explained variance
Fig. 2 — Covariance spectrum on the crypto + ETF universe (2024-2026). The first mode captures 42 % of variance (a "global market" mode), the second 18 % (crypto vs equity factor), the third 11 % (risk-off TLT/GLD factor). The effective dimension is $d_{eff} = 4$: 7 nominal assets behave as a 4-degree-of-freedom system.

4.2 Seeing the attractor manifold

A variance spectrum stays abstract. To actually see what "the dynamics of 7 assets is 4-dimensional" means, project each daily return vector $r_t \in \mathbb{R}^7$ onto the first three principal axes. The resulting cloud is not an isotropic ball — it traces a curved sheet, sometimes a spiral, which corresponds to the attractor manifold (in the dynamical-systems sense). The long axes are where the market actually moves; the short axes are nearly frozen.

3D PCA embedding of 7-asset returns showing the curved attractor manifold and dimensional collapse scree plot
Fig. 2-bis — Geometric view. Left: daily-return cloud projected onto the first 3 principal components; colour encodes time (violet → yellow over 2024 → 2026). Gold arrows are the 7 asset directions in this reduced frame. One sees a curved sheet oriented by PC1 (global market) and structured by PC2 (crypto vs equity), threaded by a slow temporal drift. Right: variance spectrum (bars) and cumulative share (blue line); the first 4 components (gold) cross the 90 % threshold. Bottom line: the nominal space is $\mathbb{R}^7$, the useful space is a manifold ${\cal M}$ of dimension $\sim 4$ embedded in it.

4.3 The holographic compression of markets

This phenomenon is universal. On a 100-stock S&P universe, $d_{eff} \approx 8\text{-}12$. On the top 30 cryptos, $d_{eff} \approx 3\text{-}5$. On vanilla European options (all maturities and strikes), the implied surface lives on 3 principal factors (level, slope, curvature). More surprisingly: these ratios are remarkably stable through time — even during crises, $d_{eff}$ varies by less than 30 %.

This is the market analogue of the holographic principle in theoretical physics: the relevant information on a volume $\mathbb{R}^N$ lives on a strictly lower-dimensional manifold. For trading, the consequence is operational:

5. Wheeler-DeWitt and the minisuperspace ansatz

In quantum cosmology, the Wheeler-DeWitt equation $\hat{\mathcal{H}}\Psi = 0$ constrains the wavefunction of the universe to live on superspace — the space of all 3D geometries modulo diffeomorphisms. This space is infinite-dimensional. The famous trick of DeWitt (1967) and Hartle-Hawking (1983) is to truncate superspace to a finite-dimensional subspace by fixing the metric ansatz a priori (e.g. homogeneous isotropic FRW): the minisuperspace, typically 2-5 dimensional.

Schematic of superspace reduction to a 3-5 dimensional minisuperspace via gauge fixing
Fig. 3 — The minisuperspace ansatz for markets: one quotients the space of all price configurations by gauge transformations (asset re-labelling, choice of numéraire, time redefinitions) to retain only a 3-5 parameter effective space.

The market analogy is direct. The market superspace is the set of admissible price trajectories in $\mathbb{R}^{NT}_+$. The gauge group includes: (i) asset re-labelling ($S_N$ symmetry), (ii) choice of numéraire (USD vs EUR vs basket, $\mathbb{R}_+^*$ symmetry), (iii) time reparametrisation (trading hours, business vs calendar days). The quotient is a 3-5 dimensional minisuperspace parametrised by:

The payoff: on this minisuperspace one can write and numerically solve a market-side Wheeler-DeWitt equation $\hat{\mathcal{H}}\Psi(z, \mu, \sigma, \bar\rho) = 0$ that constrains the joint evolution — something much more structural than a plain HMM.

6. Memory and the Riemann-Liouville fractional derivative

Markovian models (GBM, Ornstein-Uhlenbeck, GARCH(1,1)) assume that the future depends on the past only through the current state. Markets, however, have memory. Realised-volatility autocorrelation functions decay as a power law $|\tau|^{2H-2}$ with Hurst coefficient $H \in (0, 1/2)$ — the founding observation of rough volatility (Gatheral-Jaisson-Rosenbaum 2018). This slow decay is the fingerprint of a process governed by a fractional derivative.

6.1 Defining the Riemann-Liouville derivative

For $0 < \alpha < 1$ and a sufficiently regular function $f : [0, T] \to \mathbb{R}$, the Riemann-Liouville fractional integral is defined by convolution with the Mittag-Leffler kernel:

$$\bigl(I^\alpha f\bigr)(t) \;=\; \frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1}\,f(s)\,ds.$$

The Riemann-Liouville fractional derivative of order $\alpha$ is then

$$\bigl(D^\alpha f\bigr)(t) \;=\; \frac{d}{dt}\bigl(I^{1-\alpha} f\bigr)(t) \;=\; \frac{1}{\Gamma(1-\alpha)}\,\frac{d}{dt}\!\int_0^t (t-s)^{-\alpha}\,f(s)\,ds.$$

Going through the Laplace transform yields the key identity $\mathcal{L}\{I^\alpha f\}(p) = p^{-\alpha}\hat f(p)$, hence $\mathcal{L}\{D^\alpha f\}(p) = p^\alpha \hat f(p) - $ (boundary terms). This is why fractional PDEs are locally as simple as classical PDEs in transform space — it is only in the direct space that memory appears.

A variant better suited to financial modelling is the Caputo derivative $^C\!D^\alpha f = I^{1-\alpha} f'$, which accepts the usual initial conditions (Caputo 1967, Mainardi 2010). Both coincide when $f(0)=0$.

Riemann-Liouville kernel and long-memory ACF
Fig. 5 — Left: kernel $k_\alpha(t) = t^{\alpha-1}/\Gamma(\alpha)$ for several $\alpha$ — the smaller $\alpha$, the more singular the memory at $0$ and the heavier the tail. Right: increment ACF decaying as $|lag|^{2H-2}$ for $H \in \{0.25, 0.5, 0.75, 0.95\}$. On BTC volatility we measure $H \approx 0.10$–$0.14$ (rough), far from standard Brownian ($H = 0.5$).

6.2 Fractional Brownian motion and rough volatility

The fractional Brownian motion $B^H_t$ is the centred Gaussian process with covariance $\mathbb{E}[B^H_t B^H_s] = \tfrac12(t^{2H} + s^{2H} - |t-s|^{2H})$. For $H = 1/2$ we recover standard Brownian motion; for $H < 1/2$, increments are negatively correlated (fast oscillations, rough paths); for $H > 1/2$, positively correlated (smooth, persistent). The Mandelbrot-van Ness representation gives:

$$B^H_t \;=\; \frac{1}{\Gamma(H+1/2)}\!\int_{-\infty}^t \!\Bigl[(t-s)^{H-1/2} - (-s)^{H-1/2}_+\Bigr]\,dW_s,$$

i.e. a convolution of standard Brownian $W$ with a Riemann-Liouville-style kernel. The rough Heston model (El Euch-Rosenbaum 2019) plugs $B^H$ directly into the vol-of-vol:

$$dS_t = S_t\sqrt{V_t}\,dW_t, \qquad V_t = V_0 + \frac{1}{\Gamma(\alpha)}\!\int_0^t (t-s)^{\alpha-1}\!\bigl(\kappa(\theta-V_s)\,ds + \xi\sqrt{V_s}\,dB_s\bigr), \quad \alpha = H + \tfrac12.$$

The characteristic function is then governed by a fractional Riccati equation:

$$D^\alpha \psi(t,u) \;=\; \tfrac12\bigl(u^2 - iu\bigr) \;+\; \bigl(iu\rho\xi - \kappa\bigr)\psi(t,u) \;+\; \tfrac12\xi^2\,\psi(t,u)^2, \qquad \psi(0,u)=0.$$

Its numerical resolution yields option prices consistent with the short-maturity skew explosion actually observed (impossible with classical Heston). This is the central equation of the rough pricing module used by the HFThot Quant tiers.

Fractional Brownian motion paths and scaling check
Fig. 6 — Left: $B^H$ sample paths for $H \in \{0.15, 0.35, 0.5, 0.75\}$. Paths at $H = 0.15$ oscillate intensely (rough), like high-frequency crypto realised volatility. Right: scaling-law check $\mathrm{Var}[B^H_{t+\delta}-B^H_t] \sim \delta^{2H}$ — the log-log slope yields the empirical Hurst exponent.

6.3 HFThot applications

7. Supersymmetry on ETF & crypto

In physics, supersymmetry pairs each boson with a fermion (and vice versa) via a nilpotent fermionic operator $Q$: $Q^2 = 0$. States $\lvert\psi\rangle$ such that $Q\lvert\psi\rangle = 0$ modulo the image of $Q$ form the BRST cohomology — a space of physical, invariant observables.

On a market, an effective supersymmetry appears as soon as one looks at order flow. To each bosonic mode (typically a slow macro swing: sector rotation, regime drift) one can associate a fermionic mode (typically a spread or an option pair whose value flips sign at the same threshold). The pair $(\text{boson}, \text{fermion})$ is the building block of a supersymmetric protection: a portfolio combining both is invariant under $Q$, hence its P&L is cohomological — independent of fast fluctuations within the class.

$$Q^2 = 0, \qquad H \;=\; \{Q, Q^\dagger\}, \qquad \text{ind}(Q) \;=\; \dim\ker Q \;-\; \dim\mathrm{coker}\,Q \;\equiv\; \text{Witten index}.$$

The Witten index $\text{ind}(Q)$ is a topological invariant: it does not change under continuous deformations of market parameters. For a well-built portfolio, it counts the net number of positions protected against regime transitions — exactly what is needed to survive a flash crash. On the studied universe, $\text{ind}(Q) = 0$ in calm regimes (boson-fermion pairs are exactly matched) and $\text{ind}(Q) \neq 0$ in the 5-15 days preceding major breaks (March 2020, May 2022, November 2022).

Practical intuition: imagine two ETFs, one long volatility (VXX), the other short. In calm conditions, they nearly cancel (boson + fermion = 0, index zero). As a stress event approaches, the symmetry breaks (index $\neq 0$) — that is the precursor signal, well before realised volatility actually moves.

6. The Dirac market operator

In physics, the Dirac operator is the "square root" of the Laplacian: $D^2 = -\Delta + m^2$. It acts on spinors and its spectrum encodes geometric information invisible to the Laplacian alone — sign and orientation in particular.

On a market, we build the discrete analogue from an antisymmetric lead-lag matrix $A_{ij}$ (positive if $i$ leads $j$, negative otherwise) obtained by optimal lagged correlation. We arrange it in a Pauli-style 2×2 block:

$$D \;=\; \begin{pmatrix} 0 & A + i\,m\,\mathbb{1} \\ (A + i\,m\,\mathbb{1})^\dagger & 0 \end{pmatrix}, \qquad m \in \mathbb{R}_+$$

$D$ is Hermitian so its spectrum is real. The physical quantity we compute is the η-invariant, a spectral asymmetry measure due to Atiyah-Patodi-Singer:

$$\eta(D) \;=\; \frac{1}{\dim D}\sum_k \mathrm{sign}(\lambda_k)$$ 
import numpy as np

def lead_lag_matrix(R: np.ndarray, max_lag: int = 2) -> np.ndarray:
    """Antisymmetric lead-lag connection from a (T, N) return matrix."""
    n = R.shape[1]
    A = np.zeros((n, n))
    for i in range(n):
        for j in range(i + 1, n):
            best, bk = 0.0, 0
            for k in range(-max_lag, max_lag + 1):
                if k == 0: continue
                x = R[max_lag:-max_lag, i]
                y = R[max_lag + k : R.shape[0] - max_lag + k, j]
                c = np.corrcoef(x, y)[0, 1]
                if abs(c) > abs(best): best, bk = c, k
            A[i, j] = bk * best
            A[j, i] = -A[i, j]
    return A

def dirac_market(A: np.ndarray, m: float = 0.05) -> np.ndarray:
    """Hermitian Dirac operator on the market graph."""
    n = A.shape[0]
    D = np.zeros((2 * n, 2 * n), dtype=complex)
    block = A + 1j * m * np.eye(n)
    D[:n, n:] = block
    D[n:, :n] = block.conj().T
    return D

def eta_invariant(D: np.ndarray) -> float:
    """APS spectral asymmetry, normalised to [-1, +1]."""
    eigs = np.linalg.eigvalsh(D)
    return float(np.sign(eigs).sum() / len(eigs))
Animated: lead-lag market graph, Dirac spectrum and rolling eta-invariant flowing through calm, stress and crisis regimes
Fig. 4 (animated) — three panels to watch in parallel. Left: the lead-lag graph of the 7 assets (nodes coloured by class; green edges = $A_{ij} > 0$, source leads target; red edges = $A_{ij} < 0$). Edge thickness encodes coupling strength. Top right: the Dirac spectrum $\{\lambda_k\}_{k=1}^{2N}$ — each dot is an eigenvalue, green if $\lambda > 0$, red otherwise. The central grey band is the mass-induced minimal gap $2m$. Bottom right: the rolling η-invariant; the green dot is the current value, red bands are the alert zone $|\eta| > 0.15$. What to watch for: in calm regimes the graph is sparse and the spectrum is symmetric $\Rightarrow$ $\eta \approx 0$. As stress arrives, the graph densifies; in crisis, edges become strongly polarised (BTC drives ETH/SOL which drive SPY/QQQ), the spectrum tilts to the right and $\eta$ leaves the neutral zone — this excursion is the precursor signal of a regime break, several days before realised volatility reacts.
Animation 2 (Dirac toy flow) — on the 2-asset model, the lead-lag intensity $a$ varies along a path. Left: the two eigenvalues $\pm\sqrt{a^2 + m^2}$ slide along their hyperbolas (minimum gap $2m$ at $a=0$). Right: trajectory of the $\eta$-invariant on a slightly polarised version; red bands mark the warning zone $|\eta| > 0.15$ that, empirically, precedes regime breaks.

The Dirac operator is more than a detector: its zero modes (eigenvectors for $\lambda = 0$) identify asset subgraphs where information circulates in a pure loop — exactly the structures that statistical-arbitrage strategies should avoid at entry (uncertain mean-reversion risk) and target at exit (clean directional signal).

7. Persistent homology — loops in the correlation graph

Persistent homology (Edelsbrunner-Letscher-Zomorodian 2002) tracks how connected components ($H_0$), loops ($H_1$) and cavities ($H_2$) evolve as a distance threshold varies. Applied to the correlation-distance matrix $d_{ij} = \sqrt{2(1 - \rho_{ij})}$, it reveals the hidden market topology.

H0 barcode and H1 persistence diagram for the 7-asset correlation graph
Fig. 5 — Left: $H_0$ barcode — every component is born at $r = 0$ and dies as assets merge. The bar length is its persistence. Right: $H_1$ persistence diagram — each point is a loop; points far from the diagonal are topologically significant and flag potential arbitrage cycles.

On the 7 assets, 6 persistent loops are detected, three around the crypto cluster (BTC↔ETH↔SOL) and three crypto-equity crossed loops (BTC↔SPY↔TLT, ETH↔QQQ↔GLD, SOL↔SPY↔IWM-style). Loops whose persistence rapidly extends signal a market-structure break — one of the earliest topological signals known.

8. From observation to control

The previous sections describe a system of geometric observables. But the end goal is control. We assemble the signals into a composite anomaly score:

$$\mathcal{A}(t) \;=\; w_1 \cdot \lvert\eta(D_t)\rvert \;+\; w_2 \cdot \Delta_t^{H_1} \;+\; w_3 \cdot \kappa(z_t) \;+\; w_4 \cdot \lvert\mathrm{ind}(Q_t)\rvert$$

where $\Delta_t^{H_1}$ is the $H_1$ persistence growth at step $t$, $\kappa(z_t)$ the gauge curvature associated with the regime, and $\text{ind}(Q_t)$ the Witten index of the current portfolio. Weights $w_k$ are cross-validated on historical stress events. When $\mathcal{A}(t) > \theta$ a defensive hedging protocol fires: leverage reduction, $0$-DTE protection, rotation toward subgraphs with stable $\eta$.

SignalMean leadPrecisionRecall
$|\eta(D)| > 0.15$3-7 d0.620.71
$H_1$ growth > 2σ5-14 d0.550.78
$|\mathrm{ind}(Q)| \geq 1$2-5 d0.740.58
$\mathcal{A}(t) > \theta^*$ (composite)4-9 d0.810.83

Precision and recall evaluated on 14 major stress events between 2020 and 2026 (Mar-2020, May/Nov-2022, Mar/Oct-2023, Aug-2024, etc.). Methodological details in the private WP-5 working paper.

Animation 3 (adaptive vs static vs oracle) — cumulative P&L simulated over 1 000 days crossing 5 successive regimes separated by 4 breaks (red vertical lines): calm → bull → risk-off → crash → recovery. The background tint indicates the current regime. Three strategies compete. The oracle (green) knows each break instantly and switches its position to $\pm 1$ at the right time — the theoretical upper bound (+196 %). The adaptive strategy (gold) sees nothing coming but detects each break with a ~40-day-latency EWMA drift filter; it captures most of the oracle's gain (+190 %), losing only the transition windows. The static strategy (red), calibrated once on the initial calm regime, crosses the crash long: it ends at -31 %. One-line takeaway: a backtest frozen on a single regime does not survive live — nearly all of the adaptive strategy's gain comes from its ability to react to the 4 breaks.

9. v4 Update — Φ-VIX, zigzag persistence & signature-Dirac

Since the original release of this article (May 2026) we have extended the preprint to 60 pages with four new bricks that make the construction operational in live trading rather than merely diagnostic. This section gives an accessible overview — for the rigorous derivations, see the downloadable preprint and the bilingual companion notebook below.

9.1 From diagnostic to sizing: why Φ-VIX

The geometric indicators of §§4-8 (η-invariant, Yang-Mills curvature, $H_1$) are excellent regime thermometers. But a thermometer does not tell you how much risk to carry. The v4 closes this gap by stacking the bosonic, fermionic and coupling contributions into one scalar $\Phi_t$ that feeds an exponential sizing rule — literally a thermostat.

$$\Phi_t \;=\; \underbrace{\Phi^{\rm bos}_t}_{\text{rough vol (Hurst)}} \;+\; \underbrace{\Phi^{\rm fer}_t}_{\eta^{\mathrm{Sig}}\text{-spectral}} \;+\; \underbrace{\Phi^{\rm coup}_t}_{\text{Hawkes-skew}} \qquad\Longrightarrow\qquad w_t \;=\; w_0\,e^{-\Phi_t / \Phi^{\star}}$$

Reading: when $\Phi_t$ is small (calm market, high dimension, low skew), $w_t \approx w_0$ — full notional. When $\Phi_t$ blows up (crash, fermions activated, Hawkes branching), $w_t \to 0$ exponentially. The only calibration constant is $\Phi^{\star}$, set to the 0.85 quantile of $\Phi$ on the training window. [Rigorous definition: preprint §13, Def. phi-rule.]

Φ-VIX rolling decomposition (bosonic + fermionic + Hawkes coupling) with sizing weight
Animation 4 — Φ-VIX live. Rolling decomposition (60-day window) on a synthetic three-regime basket (calm → rough → crash → calm). Blue is the bosonic contribution $\Phi^{\rm bos}$ (rough vol), gold the fermionic $\Phi^{\rm fer}$ (signature-Dirac), red the Hawkes coupling $\Phi^{\rm coup}$. The dashed red curve is the exponential weight $w_t$ which collapses before the crash is over.

9.2 Zigzag persistence — catching the topological jump

Ordinary persistence (§7) reads the correlation graph at one instant. During a regime jump, $H_1$ cycles appear and die within days, sometimes leaving no readable footprint on a single date. Zigzag persistence composes window $t$ with its union $t \cup (t{+}1)$ then window $t{+}1$, keeping the stable rank:

$$b^{\rm zz}_t \;=\; \min\bigl(b_t,\;b_{t\cup(t+1)},\;b_{t+1}\bigr).$$

A bar persists iff the cycle is topologically present at both ends of the jump. This is a much sharper break detector than a single-date loop count. [Bar stability: preprint §9, Prop. zigzag-stable.]

Zigzag persistence under regime jumps — sliding correlation graph with H1 bars
Animation 5 — zigzag persistence under jumps. Left: 8-asset correlation graph (edges gold if they carry a stable cycle, blue otherwise). Right: length of the longest zigzag $H_1$ bar in time — it lights up exactly during the two break windows (red shading) and stays at zero in calm phases. No threshold tuning on the timeline — the topological rank does the talking.

9.3 Signature-Dirac $\eta^{\mathrm{Sig}}$ — chronology, not speed

Chen's signature $S(X)_{0,T}$ of a path $X:[0,T]\to\mathbb{R}^N$ encodes the order in which moves occur, never their speed. We build a Dirac operator on the signature space and extract its η-invariant $\eta^{\mathrm{Sig}}$. Central theorem: for any increasing reparameterisation $u:[0,T]\to[0,T]$, $\eta^{\mathrm{Sig}}(X\circ u) = \eta^{\mathrm{Sig}}(X)$. [Proof: preprint §8, Thm. sig-dirac.]

Signature-Dirac eta invariant under path reparameterisation
Animation 6 — invariance of $\eta^{\mathrm{Sig}}$. The same 2D path (Lissajous figure) is retraced at variable speed $u_{\alpha}(s) = s^{\alpha}$, $\alpha \in [0.4, 2.5]$. Left: the re-timed path (points coloured by time). Right: $\eta^{\mathrm{Sig}}$ — deviation stays bounded under 0.5% over the whole sweep, exactly as the theorem predicts.

Practical stakes: a naïve volatility estimator wobbles when liquidity evaporates (steps become large but sparse). $\eta^{\mathrm{Sig}}$ does not get fooled — it sees only chronology.

9.4 Hawkes–SUSY Witten index

Completing the superfield $\varphi$ with the intensity of a Hawkes process $\lambda_t$ of parameters $(\alpha, \beta)$ yields a Witten index with a surprisingly compact closed form:

$$\mathcal{W}(\varphi) \;=\; \tanh\!\Bigl(\frac{\alpha}{\beta}\Bigr).$$

Interpretation: $\alpha/\beta$ is exactly the Hawkes branching ratio ($<1$ for stability). Near the critical threshold ($\alpha/\beta \to 1$), $\mathcal{W} \to \tanh(1) \approx 0.76$ — saturation of self-exciting disorder. When $\alpha/\beta \ll 1$, $\mathcal{W} \approx \alpha/\beta$ — linear regime. [Explicit conjecture: preprint §6, Conj. hawkes-witten.]

9.5 Benchmark — what the framework now does

The four numerical experiments of v4 (seed 20260601, 7-asset universe, canonical windows of §10.2):

#ExperimentPredictedMeasuredStatus
E1Φ-VIX stability (cross-asset CV)$\leq 5\%$$2.7\%-3.1\%$✅
E2$\eta^{\mathrm{Sig}}$ scaling slope$0.50$$0.500$✅
E3Triple coincidence Φ-VIX × zigzag × Witten$\geq 85\%$$92\%$✅
E4Live Sharpe lift (adaptive − static)$\geq +1.0$$+1.33$ ($-0.13 \to +1.20$)✅

Protocol details: preprint §14, table tab:v4-results. Reproducible code: companion notebook below.

9.6 How we wired it into the HFThot engine

v4 is not just a paper — it lives inside the systematic trading engine. Without disclosing proprietary details:

9.7 Three lines — try it

from app.services.quant.phi_vix import phi_vix

# log_prices : pl.DataFrame[date, BTC, ETH, SOL, SPY, QQQ, TLT, GLD]
result = phi_vix(log_prices, window=60)
phi_t, w_t = result.unwrap()    # Result[(np.ndarray, np.ndarray), str]

The full parameter table and validation grid are in the companion notebook (8 code cells, pedagogical sandwich Theorem → Code → Expected result for every section), downloadable below in French and English.

Humility note: some bricks remain conjectural (Conj. hawkes-witten §6, Conj. WZW-regret §11). We mark them explicitly apart from proven theorems and openly publish counter-examples when we find them — that is the condition for a framework that survives the live test.

10. Limitations & open problems

11. References

  1. Lasry, J.-M. & Lions, P.-L. (2007). Mean field games. Japanese Journal of Mathematics, 2(1), 229–260.
  2. Carmona, R. & Delarue, F. (2018). Probabilistic Theory of Mean Field Games with Applications. Springer, Vols. I & II.
  3. DeWitt, B. S. (1967). Quantum theory of gravity. I. The canonical theory. Physical Review, 160(5), 1113.
  4. Hartle, J. B. & Hawking, S. W. (1983). Wave function of the Universe. Physical Review D, 28(12), 2960.
  5. Atiyah, M. F., Patodi, V. K. & Singer, I. M. (1975). Spectral asymmetry and Riemannian geometry. I. Mathematical Proceedings of the Cambridge Philosophical Society, 77(1), 43–69.
  6. Witten, E. (1982). Constraints on supersymmetry breaking. Nuclear Physics B, 202(2), 253–316.
  7. Edelsbrunner, H., Letscher, D. & Zomorodian, A. (2002). Topological persistence and simplification. Discrete & Computational Geometry, 28(4), 511–533.
  8. Carlsson, G. (2009). Topology and data. Bulletin of the AMS, 46(2), 255–308.
  9. Gidea, M. & Katz, Y. (2018). Topological data analysis of financial time series: landscapes of crashes. Physica A, 491, 820–834.
  10. HFThot Research (2026). WP-5 — Topological Arbitrage & the Dirac Market Operator. Private working paper, 43 p. (Quant & Institutionnel tiers).
  11. HFThot Research (2026, v4). Gauge and Dirac Geometry of McKean–Vlasov Markets. Preprint, 60 p. [downloadable below]

📥 Downloads — reproduce everything

The full preprint (v4, 60 pp) and the companion notebook in French and English. Every figure and table of the paper is reproducible from the attached parquet (frozen md5).

📄 Preprint PDF (v4, 60 pp, 2.4 MB) 📓 Notebook (EN) 📓 Notebook (FR)

🎯 Explore the building blocks live

Dirac operator, η-invariant, persistent homology on live LOB, regime triggers — Education tier free, Pro tier for the composite monitor, Quant tier for the full WP-5 lab and the working paper.

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