Bulk Field Redefinitions and Contact Ambiguities

Local bulk field redefinitions change off-shell vertices while preserving separated-point boundary observables when every induced interaction and boundary term is retained, as formalized by the equivalence theorem Kamefuchi, O’Raifeartaigh, and Salam 1961. In AdS they can move contributions between exchange and contact diagrams. The invariant object is an equivalence class of renormalized correlators, not a unique list of Lagrangian coefficients.

Required background. Contact diagrams and exchange diagrams supply the two diagram classes being reshuffled. Local field redefinitions supplies the equivalence theorem.

Helpful background. Finite counterterms and multitrace schemes supplies boundary contacts. Curvature operators and field redefinitions supplies the gravitational extension.

Equation-of-motion operators

Consider

$$S=\int\sqrt g\left[\frac12\phi(-\Box+m^2)\phi +\frac g{3!}\phi^3+a\phi^2(-\Box+m^2)\phi\right].$$

The local change $\phi=\phi'-a(\phi')^2+O(a^2)$ removes the displayed equation-of-motion operator. Substitution into the kinetic and cubic terms generates quartic interactions, a Jacobian that is local in a perturbative regulator, and boundary contributions. Thus deleting the cubic term alone does not implement the field redefinition.

On an internal line, applying $(-\Box+m^2)$ collapses the propagator to a delta function. Diagrammatically the EOM exchange becomes a contact diagram. This explains how pole residues remain fixed while polynomial Mellin terms and position-space contacts move between representations. Boundary terms in supergravity reductions are required for the same reason Arutyunov and Frolov 1999, and explicit quartic AdS$_5\times S^5$ couplings exhibit the induced derivative structures Arutyunov and Frolov 2000.

First application: cubic removal and quartic induction

Compute a four-point function with one ordinary cubic vertex and one $a\phi^2E(\phi)$ vertex. Acting $E=-\Box+m^2$ on the internal propagator gives

$$E_XG(X,Y)=\frac{\delta^{d+1}(X-Y)}{\sqrt g},$$

so the exchange integral reduces to a four-kernel contact integral. In the redefined action the same term comes from the induced quartic vertex. Once source transformations and finite boundary contacts are matched, the separated-point four-point functions agree. Contact distributions at coincident boundary points may change with scheme.

Adversarial control: discard the induced term

Remove $a\phi^2E(\phi)$ but omit the quartic and boundary terms. The exchange pole may appear unchanged, yet the polynomial part of the correlator and its double-trace anomalous dimensions shift. This is not a violation of the equivalence theorem; it is a different action. The delta-function collapse above predicts the missing contact term exactly.

The evidence ceiling is equivalence of perturbative observables within a local, invertible field redefinition and fixed boundary conditions. Singular, nonlocal, or topology-changing transformations are outside the theorem. Correlators through finite order still leave higher-derivative contact coefficients undetermined. Reproducibility benchmarks test the invariant data directly.

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

• Arutyunov, G., and Frolov, S. (1999), “On the Origin of Supergravity Boundary Terms in the AdS/CFT Correspondence,” Nuclear Physics B 544, 576–589. arXiv:hep-th/9806216.
• Arutyunov, G., and Frolov, S. (2000), “Scalar Quartic Couplings in Type IIB Supergravity on AdS$_5\times S^5$,” Nuclear Physics B 579, 117–176. arXiv:hep-th/9912210.
• Kamefuchi, S., O’Raifeartaigh, L., and Salam, A. (1961), “Change of Variables and Equivalence Theorems in Quantum Field Theories,” Nuclear Physics 28, 529–549. doi:10.1016/0029-5582(61)90056-6.