Transgression Effect in AdS Gravity

• Transgression effect is a two-connection formulation replacing Chern–Simons forms, ensuring strict gauge invariance and a well-defined action in AdS gravity.
• It naturally produces finite conserved charges and regularizes boundary anomalies by incorporating a built-in regulator without extra counterterms.
• This framework extends to Lovelock and higher-curvature gravities, providing a unifying approach to manage divergences and maintain anomaly consistency.

Searching arXiv for the specified paper and closely related work on transgression forms to ground the article. arXiv search query: id:(Mora, 2014) OR title:"Gauge Symmetries and Holographic Anomalies of Chern-Simons and Transgression AdS Gravity" The transgression effect denotes the characteristic set of consequences that follow when a Chern–Simons action is replaced by a two-connection transgression form. In the AdS-gravity setting, this replacement yields a manifestly gauge-invariant bulk action whose boundary term automatically regularizes the theory, produces a well-defined action principle, and renders conserved charges and holographic anomalies finite without further regularization (Mora, 2014). In this sense, the effect is not a separate dynamical principle, but a structural consequence of working with transgression forms rather than a single Chern–Simons form; in the formulation emphasized for Chern–Simons AdS gravity, it restores strict invariance under the full AdS gauge group, supplies the correct boundary terms, regularizes divergences, and yields finite Noether charges and black-hole thermodynamics (Mora, 2014).

1. Formal definition

In $2n+1$ dimensions, a Chern–Simons form for a gauge connection $A$ is a $(2n+1)$-form $C_{2n+1}(A)$ such that

$$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$

with $\langle\cdots\rangle$ an invariant symmetric trace on the Lie algebra (Mora, 2014).

A transgression form depends on two connections $A$ and $\bar A$. It admits the equivalent representations

$$T_{2n+1}(A,\bar A)=C_{2n+1}(A)-C_{2n+1}(\bar A)-d\,B_{2n}(A,\bar A),$$

and

$$T_{2n+1}(A,\bar A) =(n+1)\int_0^1 dt\,\langle \Delta A\cdot (F_t)^n\rangle,$$

where

$A$0

Under a simultaneous gauge transformation

$A$1

the transgression form is strictly gauge-invariant (Mora, 2014).

This two-connection structure is the formal core of the transgression effect. The exact term $A$2 is not ancillary: it is precisely the part that later controls boundary finiteness, anomaly descent, and the relation between gauge and diffeomorphism anomalies. A plausible implication is that transgression reorganizes quasi-invariant Chern–Simons data into a strictly invariant bulk-plus-boundary object.

2. Action principles and automatic finiteness

For AdS gravity in odd $A$3 dimensions, one chooses the action

$A$4

with gauge group the AdS group, $A$5 a manifold with boundary, and $A$6 a second connection playing the role of a regulator or background (Mora, 2014).

Two distinct variational regimes are central. When both $A$7 and $A$8 are treated as dynamical and varied in $A$9, gauge invariance is manifest and no anomalies arise. When $(2n+1)$0 is held fixed and only $(2n+1)$1 is varied, gauge invariance under $(2n+1)$2 is broken at $(2n+1)$3, and one obtains finite anomalies directly (Mora, 2014).

The same mechanism controls Noether charges. For the gauge variation $(2n+1)$4, the current satisfies

$(2n+1)$5

Because the transgression boundary term $(2n+1)$6 was engineered to render $(2n+1)$7 finite, all would-be divergences cancel without further regularization (Mora, 2014). In the corresponding AdS-gravity action principles, the transgression construction supplies the boundary terms needed to render the action finite and the variational principle well-posed under Dirichlet boundary conditions, with no ad hoc holographic counterterms required (Mora, 2014).

This is the operational content of the transgression effect in AdS gravity: the regulator $(2n+1)$8 is built into the action itself, rather than appended afterward as a subtraction prescription.

3. Finite holographic gauge and diffeomorphism anomalies

When only $(2n+1)$9 is varied, the transgression form varies as

$C_{2n+1}(A)$0

with

$C_{2n+1}(A)$1

This boundary quantity is identified as the consistent gauge anomaly in the presence of background $C_{2n+1}(A)$2 (Mora, 2014).

Two limiting cases are canonical. Setting $C_{2n+1}(A)$3 reproduces the classic Wess–Zumino–Wu–Zee anomaly without background. Setting $C_{2n+1}(A)$4 yields the covariant anomaly (Mora, 2014). Thus the transgression framework interpolates between consistent and covariant forms through the choice of background connection.

For the “Backgrounds” action principle, $C_{2n+1}(A)$5 is chosen to be the AdS vacuum, which solves the field equations so that $C_{2n+1}(A)$6. The resulting AdS gauge anomaly is expressed in terms of the Levi–Civita symbol $C_{2n+1}(A)$7, the boundary covariant derivative $C_{2n+1}(A)$8, and the boundary field strengths $C_{2n+1}(A)$9, all evaluated at $$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$0; all possible divergences cancel automatically in the trace (Mora, 2014).

Diffeomorphism anomalies arise analogously by treating only $$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$1. The same structure reappears with the effective gauge parameter

$$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$2

This establishes the relation between gauge anomalies associated to the AdS gauge group and diffeomorphism anomalies (Mora, 2014).

For the alternative “Kounterterms” action, $$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$3 is chosen with trivial vielbein but nontrivial spin-connection. The resulting anomaly analysis yields no Lorentz anomaly, while the Weyl anomaly reduces to

$$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$4

namely the Euler density of the boundary (Mora, 2014).

4. Regularization of Lovelock AdS gravity

The transgression effect is not restricted to Chern–Simons gravities. Lovelock theories in $$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$5 have bulk lagrangians built from

$$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$6

and, as shown in $$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$7, precisely the same boundary term $$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$8 that renders the Chern–Simons AdS action finite also regularizes Lovelock AdS gravity (Mora, 2014).

The mechanism follows directly from

$$d\,C_{2n+1}(A)=\langle F^{n+1}\rangle, \qquad F=dA+A^2,$$9

When $\langle\cdots\rangle$0 is chosen to match asymptotic AdS in each Lovelock sector, the $\langle\cdots\rangle$1 term automatically supplies exactly the Gibbons–Hawking–type boundary term plus the minimal set of “Kounterterms.” In particular, $\langle\cdots\rangle$2 cancels all divergences in the Lovelock action without extra counterterms, and one finds a fully finite on-shell action and finite boundary stress tensors (Mora, 2014).

This suggests that transgression is an organizing principle for higher-curvature AdS gravities, not merely a special device tied to the Chern–Simons point.

5. Wess–Zumino consistency and representative cases

With $\langle\cdots\rangle$3 held fixed, the quantum effective action may be denoted

$\langle\cdots\rangle$4

and its anomaly is

$\langle\cdots\rangle$5

Because $\langle\cdots\rangle$6 arises as the exterior derivative of $\langle\cdots\rangle$7, it automatically satisfies the Wess–Zumino consistency condition

$\langle\cdots\rangle$8

Thus consistency is built into the transgression construction itself (Mora, 2014).

The special cases displayed in the AdS-gravity analysis illustrate the descent pattern. For $\langle\cdots\rangle$9-dimensional Chern–Simons gravity with $A$0-dimensional boundary, one recovers the well-known $A$1D Weyl and Lorentz anomalies proportional to the boundary curvature and torsion. For $A$2-dimensional Chern–Simons gravity with $A$3-dimensional boundary, the anomaly reduces to the cubic trace of the boundary Riemann tensor, in agreement with $A$4. The distinction between consistent and covariant form is recovered by setting $A$5 at the end (Mora, 2014).

The appearance of the boundary Euler density as the Weyl anomaly in odd bulk dimensions is one of the most characteristic outputs of this framework. A plausible implication is that the transgression prescription fixes not only finiteness, but also the cohomological type of the boundary anomaly.

6. Broader uses of transgression in mathematical physics

The term transgression is used in several adjacent literatures, all involving the passage from a bulk invariant to boundary, loop-space, or higher-categorical data. The following examples exhibit that wider pattern.

Context	Role of transgression	Source
AdS Chern–Simons gravity	Finite action, finite charges, finite holographic anomalies	(Mora, 2014, Mora, 2014)
Topological-insulator interfaces	Gauge-invariant bulk-plus-boundary action; edge states written in terms of bulk connections	(Açık et al., 2013)
Anomalous hydrodynamics	$A$6 and $A$7 bulk-boundary terms from anomaly inflow identified as transgression forms	(Abanov et al., 17 Jun 2026, Jain, 2015)
Loop spaces and gerbes	Transgression/regression relates manifold cohomology or gerbes to loop-space bundles and fusion structures	(Kottke et al., 2013, Waldorf, 2010, Waldorf, 2012, Nikolaus et al., 2011)
Higher gauge theory and supersymmetric path integrals	Exact higher gWZW terms; Chern–Simons transgression current for metric variation	(Song, 12 Apr 2026, Boldt et al., 2021)

At interfaces of topological insulators, transgression field theory is proposed as a gauge-invariant effective action for two bulk theories sharing a common boundary, and the boundary term $A$8 encodes the interface dynamics in terms of the bulk gauge fields alone (Açık et al., 2013). In anomalous hydrodynamics, the relevant four- and five-dimensional bulk-boundary terms owing to anomaly inflow are identified as transgression forms involving a dynamical field and a background field; restricted variations then lead to four-dimensional equations of motion from the five-dimensional transgression terms (Abanov et al., 17 Jun 2026). In Galilean fluids on torsional Newton–Cartan backgrounds, the transgression form directly yields closed-form expressions for the anomaly-induced gauge current, energy flux, and spatial spin current (Jain, 2015).

On loop spaces, loop-fusion cohomology is isomorphic to the cohomology of the manifold, and the classical transgression homomorphism factors through that isomorphism (Kottke et al., 2013). For gerbes with connection, transgression and regression establish an equivalence with fusion bundles with superficial connection over free loop space (Waldorf, 2010). The same transgression-regression technique is used to construct String$A$9 from the basic gerbe on a compact, simple, simply connected Lie group, and non-abelian transgression relates degree-one non-abelian cohomology on a manifold to degree-zero non-abelian cohomology on the free loop space, with applications to string structures and spin structures on loop space (Waldorf, 2012, Nikolaus et al., 2011).

In strict higher gauge theory based on Lie crossed modules, higher transgression forms yield canonical higher WZW and gauged WZW terms; for the symmetric invariant polynomial associated with differential crossed modules, the pure-gauge higher WZW term vanishes identically, whereas the higher gWZW term is exact, so that on manifolds with boundary all gauge dependence is encoded in boundary terms (Song, 12 Apr 2026). In the supersymmetric path-integral setting on a closed even-dimensional spin manifold, variation of the metric induces a Chern–Simons current satisfying

$\bar A$0

thereby isolating a differential topological invariant essentially stemming from the $\bar A$1-genus (Boldt et al., 2021).

Taken together, these examples suggest that the “transgression effect” is best understood as a recurrent structural pattern: the difference of two characteristic constructions is represented by a globally defined transgression object whose boundary or loop-space contribution carries the physically or geometrically relevant data.