Modular Berry Transport and Bulk Connections

Adiabatically varying a region or state changes its modular Hamiltonian and the basis used to identify modular eigenoperators. Because modular zero modes can be freely redefined, this comparison carries a Berry-like connection. In symmetric holographic examples its curvature has a bulk normal-frame interpretation; in generic states that geometric interpretation is a proposal, not a unique local connection.

Required background. Modular Berry Transport and Holonomy supplies the abstract connection. Modular Flow Reconstruction and Bulk Modular Evolution supplies the holographic modular map.

Helpful background. Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities supplies the bundle language. Shape Deformations and Displacement Response supplies the region variation.

First application. Compute modular Berry curvature for a family of ball regions or states with a controlled semiclassical bulk.

Zero-mode gauge freedom

Let $K(\lambda)$ be a smooth family of modular Hamiltonians. An infinitesimal unitary frame change generated by $V_\lambda$ decomposes the variation as

$$\partial_\lambda K =[V_\lambda,K]+P_0[\partial_\lambda K],$$

where $P_0$ projects onto operators commuting with $K$. The off-diagonal part determines $V_\lambda$ only up to a zero mode $Q_0$ with $[Q_0,K]=0$. Choosing that zero-mode component is a gauge choice on the bundle of modular frames.

A connection $\Gamma_\lambda$ specifies how zero modes are transported. Under a zero-mode frame transformation it changes inhomogeneously, while the curvature

$$\mathcal F_{\lambda\kappa} =\partial_\lambda\Gamma_\kappa -\partial_\kappa\Gamma_\lambda +[\Gamma_\lambda,\Gamma_\kappa]$$

transforms covariantly. Holonomy around a closed loop is meaningful only after identifying the zero-mode group and representation.

Vacuum balls and a bulk frame

For ball-shaped regions in a CFT vacuum, modular flow is geometric. In AdS, the corresponding RT surfaces form a family with a normal two-plane. Varying the ball changes the surface and its local boost frame. In AdS$_3$/CFT$_2$, symmetry fixes the modular Berry connection and relates its holonomy to bulk geometric transport Czech et al. 2018, §§3–5.

The comparison works because three structures align:

• the boundary modular generator is known;
• the bulk surface and boost Killing field are known;
• the code-subspace modular dictionary identifies their action.

For excited states or arbitrary regions, none of these need be explicit. A formal modular connection can still be defined, but interpreting it as the Levi-Civita or normal-bundle connection of a unique bulk geometry requires additional reconstruction and semiclassical assumptions.

A curvature calculation

Choose a two-parameter family of nearby balls. Solve the commutator equation for the off-zero-mode transport generator in a fixed gauge. Transport a modular zero mode around a small parameter-space rectangle. The mismatch is

$$U_\square=\mathbf 1+\mathcal F_{12}\,\delta\lambda^1\delta\lambda^2+O(\delta\lambda^3).$$

On the bulk side, parallel transport the corresponding normal frame around the family of RT surfaces. Agreement of the curvatures checks the zero-mode identification and orientation. It does not reconstruct the full bulk Riemann tensor unless the chosen family samples enough independent planes and the dictionary is injective.

Adversarial controls

Fix eigenvectors without quotienting zero modes. The resulting connection changes arbitrarily under commuting unitary rotations. Only covariant curvature or specified holonomy is comparable.

Cross a degeneracy. When modular eigenvalues collide or the QES branch changes, the bundle rank and zero-mode group can change. Adiabatic transport through that point is not licensed by the smooth-family formula.

Infer unique geometry from one loop. Distinct connections can share selected holonomies. Reconstruction needs a sufficiently rich region/state family and independent dictionary data.

Evidence ceiling

Modular Berry transport is well defined for a smooth modular family after fixing its zero-mode gauge structure. Symmetric holographic examples match it to bulk normal-frame transport. This does not provide a unique local geometric connection for arbitrary states, cross surface transitions, or reconstruct complete bulk curvature from a sparse set of loops.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Czech, Bartlomiej, Lampros Lamprou, Samuel McCandlish, and James Sully. “Modular Berry Connection.” Journal of High Energy Physics 2018, 175 (2018). DOI; arXiv:1712.07123.