Covariant OPE expansions
Organize local operators and derivatives using tensors and geometric displacement expansions instead of coordinate-dependent Taylor coefficients.
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04 / Research theme
Conformal field theory on curved spacetime
How does geometry change the way nearby quantum observables combine?
The idea
What is the question?
The operator product expansion, or OPE, expresses two nearby observables as a sum of local operators and their derivatives. In flat-space CFT, conformal symmetry tightly organizes these derivative contributions. On a curved background, the same expansion must also know about the surrounding geometry.
My work asks how to write this local expansion in a generally covariant form: one that does not depend on arbitrary coordinate choices. For conformally flat Euclidean backgrounds, local Weyl rescalings provide a route from the flat-space OPE to an algorithmic construction of the scalar descendant tower.
Explore an idea
A little physics to play with.
Change the controls and watch what follows.
Interactive geometry · Three insertions
A triangle remembers the curvature.
Start at x₁ and follow two geodesics of lengths d and r, separated by an angle γ. Does the third side obey the familiar cosine rule? Compare the curved triangle with the same two vectors in the tangent plane, then shrink the configuration towards the local OPE limit.
Reduced motion is enabled. The sliders and geometry buttons provide the same comparisons without animation.
Exact geodesic side c
Flat side c₀
Leading expansion cexp
Relative error in c²exp
Angle sum minus 180°
Local curvature scale |K|r²
Positive curvature changes the separation between the endpoints.
From exact geometry to a local series
The curved cosine rule.
The flat answer is the first term. At fixed triangle shape, the first curvature correction is quartic in the radial lengths:
\(c^2 = \underbrace{d^2+r^2-2dr\cos\gamma}_{c_0^2} - \frac{K}{3}d^2r^2\sin^2\gamma + O(K^2r^6).\)
The chart keeps d/r and γ fixed while changing the overall size. The solid line uses the exact spherical or hyperbolic cosine law; the dotted line keeps the displayed curvature correction. The marker follows your size slider.
Exact geometryFlat referenceLeading expansion
Shrink the triangle to see the local expansion improve.
Where the OPE enters
Think of x₁, x₂ and x₃ as three operator insertions. In my work with Anatoly Konechny and Naveen B. Umasankar, the short-distance test uses x₂ = expx₁(d t) and x₃ = expx₁(r n), with d ≪ r ≪ Rcurv. The triangle expansion translates pairwise geometric quantities into local tensors at x₁. Try reducing d/r first, then shrinking the whole patch.
What is exact here? The distances and angles belong to a two-dimensional constant-curvature model, with K > 0 for a sphere and K < 0 for hyperbolic space. Lengths share an arbitrary unit; K has inverse-square units. The remainder shown is for this constant-curvature family at fixed d/r and γ. A general background also brings curvature derivatives and tensor structure. This geometric example illustrates an ingredient of the curved-space OPE; it does not compute its coefficients or the full ambient-space construction. On constant curvature, the angle excess equals K times the triangle’s area.
Research connection: Das, Konechny & Umasankar · §4 and Appendix A ↗. Geometric background: Brewin · Riemann normal coordinate expansions ↗.
On more general backgrounds, covariance alone is not the whole story. Together with Anatoly Konechny and Naveen B. Umasankar, I test whether proposed correlation functions actually admit a local covariant OPE. Ambient-space geometry gives powerful building blocks, while short-distance consistency reveals curvature-dependent corrections that must accompany them.
The OPE is one of quantum field theory's basic organizing tools. Understanding its curved-space form clarifies which effects are dictated by geometry and symmetry, and which require additional information about the theory or state. It also provides a concrete consistency test for proposed curved-background correlators.
The toolkit
How I approach it.
Weyl and tractor methods
Track how operators respond to local changes of scale and construct curved descendant combinations from flat-space data.
Ambient-space geometry
Use an auxiliary higher-dimensional geometry to build conformal invariants, then test their short-distance behavior against locality.
Research contributions
What the work shows.
- Constructed the scalar-channel descendant expansion to arbitrary order on conformally flat Euclidean backgrounds in D ≥ 3.
- Separated geometric conversion maps from the coefficients already fixed by flat-space CFT data.
- Tested the pairwise ambient continuation of an operator-family contribution against local OPE consistency.
- Determined short-distance trilocal corrections governed by Cotton and Weyl curvature, to the order studied.
Where these results apply. The general-background result is a short-distance analysis at linear order in conformal curvature, not a full correlator on an arbitrary geometry. A global completion and extensions to further operator sectors remain open.
Read further
Papers in this direction.
OPE and correlation functions in a generally covariant form
Key results4 highlights
- Constructs the scalar-channel descendant expansion of the OPE to arbitrary order on conformally flat backgrounds in Euclidean dimensions D ≥ 3.
- Organizes the construction through Weyl transformations, covariant displacement expansions and a map from flat derivatives to curved-space derivatives.
- Tests a pairwise ambient-space contribution against the requirement of a local covariant OPE on backgrounds that are not conformally flat.
- Identifies the first short-distance terms of an additional trilocal correction, controlled by Cotton curvature in three dimensions and by Weyl curvature and its derivatives in higher dimensions.
Scope. The general-background result is a local expansion at linear order in conformal curvature. It is not a complete global three-point function, and does not claim that the earlier ambient ansatz represented every operator sector.
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