Gauge-Invariant Schwarzian Action

• Gauge-invariant Schwarzian action is a class of one-dimensional boundary functionals defined via projective invariance in gravitational theories.
• It emerges from BF and Chern–Simons formulations, linking reparametrization modes to flat connection reductions in AdS2/CFT1 setups.
• Generalizations to affine, local SL(2,R) gauging, and matrix frameworks extend its role in quantum gravity, holography, and non-abelian theories.

Gauge-invariant Schwarzian action denotes a family of one-dimensional boundary functionals in which the Schwarzian derivative, or a projective invariant that generalizes it, is fixed by residual gauge symmetry, by a bulk variational principle, or by an explicit gauging of projective symmetry. In the standard AdS$_2$/CFT$_1$ setting, the renormalized boundary action reduces to the Schwarzian functional $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$, invariant under residual $SL(2,\mathbb{R})$ Möbius transformations of the boundary reparametrization mode (Fukuyama, 30 May 2026). In BF and Chern–Simons formulations, the same structure appears as a boundary reduction of flat gauge fields (Özer et al., 13 Jun 2026, Chirco et al., 14 May 2026). Other usages are more specialized: in flat JT gravity, gauge consistency requires a BMS-Schwarzian corrected by a dilatonic zero-mode (Afshar et al., 2021), while a fully local gauging of the Schwarzian promotes global $SL(2,\mathbb{R})$ to a local gauge symmetry and replaces ordinary derivatives by covariant ones (Pinzul et al., 5 Jul 2025).

1. Projective invariance and the standard Schwarzian functional

The Schwarzian derivative of a monotonic function $f$ is

$$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$

It is invariant under Möbius transformations $f\mapsto (af+b)/(cf+d)$ with $ad-bc=1$, so the thermal configuration space is naturally $\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$ (Mertens, 2018). In this sense, the ordinary Schwarzian action already carries a gauge-type redundancy: the physical mode is the reparametrization modulo projective transformations.

In the gauge-theory-of-gravity formulation of AdS$_1$0/CFT$_1$1, the boundary curve is embedded in the Poincaré patch

$_1$2

with $_1$3 and $_1$4, and the induced boundary metric fixed as $_1$5. The extrinsic curvature expands as

$_1$6

so the renormalized action

$_1$7

reduces, for $_1$8, to

$_1$9

The same construction gives the boundary stress tensor $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$0 and an emergent Virasoro algebra with central charge $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$1; in the normalization where $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$2 is a dimensionless circle coordinate, $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$3 (Fukuyama, 30 May 2026).

The coadjoint-orbit interpretation is the natural algebraic counterpart of this geometric derivation. The Schwarzian functional is the geometric action on the Virasoro orbit $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$4, and its invariance reflects the vanishing of the Schwarzian derivative for Möbius maps. This identifies the standard Schwarzian as a projective connection rather than merely a higher-derivative mechanical term (Fukuyama, 30 May 2026).

2. BF and Chern–Simons origins of the gauge-invariant boundary action

A bulk-first derivation starts from two-dimensional BF gravity with an $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$5 connection $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$6 and an adjoint scalar $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$7,

$S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$8

The equations of motion are $S_{\mathrm{Sch}}=-C\int du\,\{f,u\}$9 and $SL(2,\mathbb{R})$0. After Drinfeld–Sokolov reduction in highest-weight gauge,

$SL(2,\mathbb{R})$1

the associated linear problem becomes the Hill equation

$SL(2,\mathbb{R})$2

If $SL(2,\mathbb{R})$3 are independent solutions and $SL(2,\mathbb{R})$4, then

$SL(2,\mathbb{R})$5

and the reduced boundary action is

$SL(2,\mathbb{R})$6

In this formulation, gauge-invariant boundary actions arise directly from bulk flatness and companion-form ODEs rather than from an independent boundary prescription (Özer et al., 13 Jun 2026).

The three-dimensional Chern–Simons derivation makes the same point in a higher-dimensional parent theory. For $SL(2,\mathbb{R})$7 gravity on $SL(2,\mathbb{R})$8 with toroidal boundary, the $SL(2,\mathbb{R})$9 connection is reduced by imposing $SL(2,\mathbb{R})$0 and decomposing

$SL(2,\mathbb{R})$1

The curvature splits as

$SL(2,\mathbb{R})$2

so flatness becomes the two-dimensional BF system $SL(2,\mathbb{R})$3, $SL(2,\mathbb{R})$4. The reduced action is

$SL(2,\mathbb{R})$5

and on the flat subsector the bulk term vanishes, leaving the universal one-dimensional boundary action

$SL(2,\mathbb{R})$6

This is the common origin of both the standard Schwarzian sector and its affine deformation (Chirco et al., 14 May 2026).

These constructions are gauge-invariant in a precise sense: the one-dimensional action is inherited from a gauge-invariant BF or Chern–Simons parent theory, and the remaining boundary dynamics are determined by residual gauge symmetry together with a well-posed variational principle. The Schwarzian is therefore not an arbitrary boundary ansatz but the projective data of a reduced flat connection.

3. Drinfel'd–Sokolov, affine deformation, and current dressing

The reduced Chern–Simons theory admits two inequivalent boundary sectors. In the Drinfel'd–Sokolov sector,

$SL(2,\mathbb{R})$7

the universal action becomes

$SL(2,\mathbb{R})$8

Using the highest-weight basis and the Gauss decomposition of the boundary group element, the DS constraints imply

$SL(2,\mathbb{R})$9

so

$f$0

The residual symmetry is $f$1 (Chirco et al., 14 May 2026).

In the generalized affine sector,

$f$2

with $f$3 and $f$4, the same boundary action yields a deformed Schwarzian. Choosing $f$5 and imposing highest-weight conditions gives

$f$6

while the boundary functional becomes

$f$7

Its residual symmetry is $f$8 rather than $f$9, and the parabolic choice of $$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$0 is associated with a Rindler or non-extremal regime (Chirco et al., 14 May 2026).

A crucial structural point is that these two one-dimensional theories are not gauge fixings of a single boundary mechanics. They arise from inequivalent boundary subspaces of the same universal action and therefore define distinct variational problems. The DS sector gives the standard projective orbit, while the affine sector gives a deformed orbit with reduced symmetry (Chirco et al., 14 May 2026).

The same work also shows that the second chiral sector can be reorganized as loop-valued currents dressed by the reparametrization mode, producing semidirect-product actions of the form $$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$1. This yields current-dressed Virasoro–Kac–Moody extensions in both the Schwarzian and deformed Schwarzian sectors, with levels fixed by the Chern–Simons data and by the boundary prescription for the second chiral copy (Chirco et al., 14 May 2026).

4. Local gauging of $$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$2

A stronger notion of gauge-invariant Schwarzian action arises when the global $$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$3 symmetry of the fractional-linear field $$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$4 is promoted to a local gauge symmetry. The construction begins with the composite adjoint field

$$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$5

which transforms linearly under global $$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$6 and obeys $$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$7. Introducing a gauge potential $$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$8, one defines the covariant derivative on the fractional-linear representation,

$$\{f,t\}=\frac{f'''(t)}{f'(t)}-\frac{3}{2}\left(\frac{f''(t)}{f'(t)}\right)^2.$$9

the covariant composite

$f\mapsto (af+b)/(cf+d)$0

and the adjoint covariant derivative

$f\mapsto (af+b)/(cf+d)$1

The gauge-invariant analogue of the Schwarzian derivative is then

$f\mapsto (af+b)/(cf+d)$2

with action

$f\mapsto (af+b)/(cf+d)$3

For $f\mapsto (af+b)/(cf+d)$4, this reduces to the ordinary Schwarzian $f\mapsto (af+b)/(cf+d)$5 (Pinzul et al., 5 Jul 2025).

This formulation preserves local $f\mapsto (af+b)/(cf+d)$6 by construction. At first order in the gauge field, the action couples universally to the Noether vector

$f\mapsto (af+b)/(cf+d)$7

through the term $f\mapsto (af+b)/(cf+d)$8. The framework also admits locally invariant couplings to additional fields; for fermions in the fundamental representation, one can use

$f\mapsto (af+b)/(cf+d)$9

This extends the usual Schwarzian mechanics by explicit gauge potentials and matter couplings (Pinzul et al., 5 Jul 2025).

The topological content becomes nontrivial on $ad-bc=1$0. On a topologically trivial domain such as $ad-bc=1$1, the gauge field can be gauged away and the action reduces to the standard Schwarzian. On $ad-bc=1$2, gauge transformations split into homotopy classes labeled by the winding number $ad-bc=1$3, and the holonomy

$ad-bc=1$4

distinguishes sectors. A simple representative is $ad-bc=1$5; for $ad-bc=1$6, global $ad-bc=1$7 is broken to $ad-bc=1$8, and different $ad-bc=1$9 label distinct vacua connected by large gauge transformations (Pinzul et al., 5 Jul 2025).

This local gauging differs conceptually from the standard Schwarzian quotient by $\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$0. The ordinary theory removes projective redundancy; the gauged theory introduces an actual one-dimensional gauge field and a covariant projective structure.

5. Flat JT gravity, BMS-Schwarzian, and one-dimensional quantum gravity

In flat JT gravity in Bondi gauge, the asymptotic symmetry is an extension of the warped Virasoro group denoted BMS$\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$1, with group law

$\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$2

The BMS-Schwarzian action is defined as the zero-mode of the transformed stress tensor,

$\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$3

However, the pure BMS-Schwarzian is not by itself the gauge-consistent boundary action of flat JT gravity. The improved bulk variational principle requires fixing the zero-mode

$\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$4

and in group variables this produces an additional zero-mode term proportional to $\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$5. The gauge-consistent combination is

$\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$6

which is invariant under constant BMS$\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$7 translations generated by right-invariant vector fields. The pure $\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$8 has no saddle points for real non-zero $\mathrm{Diff}(S^1)/\mathrm{PSL}(2,\mathbb{R})$9 and $_1$00; the added zero-mode restores saddle points and makes the thermal partition function one-loop exact (Afshar et al., 2021).

A different gauge-theoretic recasting appears in one-dimensional conformal quantum gravity. There the Schwarzian partition function is computed by a local path integral on the base circle, with $_1$01 realized on the base space rather than on the target reparametrization field. The exact fixed-modulus Schwarzian partition function is

$_1$02

and the construction uses a local quantum measure, with non-localities tied only to fixed gauge-invariant moduli. When all degrees of freedom fluctuate, the model develops an emergent Planck length

$_1$03

and exhibits UV-finiteness properties that differ from a standard Hilbert-space interpretation at short scales (Anninos et al., 2021).

These two examples use “gauge-invariant Schwarzian” in distinct but compatible senses. In flat JT gravity, gauge consistency means compatibility with the bulk variational principle and with constant BMS$_1$04 translations. In one-dimensional quantum gravity, it means a local gauge-theory representation whose path integral directly computes the Schwarzian partition function.

6. Matrix, higher-rank, and non-abelian generalizations

The bulk-first BF perspective extends beyond $_1$05. For $_1$06, the Drinfeld–Sokolov companion form yields the third-order ODE

$_1$07

and the boundary data are captured by the Wilczynski invariants $_1$08 and $_1$09. A generalized gauge-invariant Schwarzian action is then

$_1$10

This is the higher-rank analogue of the ordinary Schwarzian and is organized by the same reduced BF phase space, companion ODE, monodromy data, and Casimir sectors (Özer et al., 13 Jun 2026).

Colored Jackiw–Teitelboim gravity provides a matrix-valued extension based on $_1$11 BF theory. The zero-temperature boundary action contains a matrix Schwarzian term

$_1$12

together with a spin-1 sector involving $_1$13. The full colored theory is not stable: the spin-1 mode has the wrong overall sign in the quadratic action, and for smooth saddles with $_1$14 the Lyapunov exponent can exceed $_1$15. A stable truncation is obtained by dropping the spin-1 sector, leaving a manifestly stable matrix-Schwarzian theory (Alkalaev et al., 2022).

A different non-abelian extension is formulated on Riemann surfaces through the canonical matrix ODE

$_1$16

with non-abelian Schwarzian

$_1$17

Under gauge transformations, $_1$18 transforms by conjugation, and its traceless part

$_1$19

is a genuine $_1$20-valued quadratic differential. The paper does not present a full variational principle, but it identifies $_1$21 and $_1$22 as the natural gauge-covariant projective curvatures; a quadratic action built from $_1$23 is proposed there as the natural gauge-invariant functional (Ajoodanian, 4 Mar 2026).

Beyond AdS/JT, conformally invariant FLRW cosmology proposes a full $_1$24-invariant Schwarzian construction on proper time. One form is

$_1$25

whose invariance follows from the transformation of the Schwarzian and the compensating conformal variation of the $_1$26 kinetic term. In that setting, the extra kinetic term for $_1$27 upgrades the residual Möbius symmetry of the standard Schwarzian to full $_1$28 acting on proper time (Achour et al., 2020).

Taken together, these developments show that the gauge-invariant Schwarzian action is not a single object but a class of projective boundary functionals. In the simplest case it is the Möbius-invariant Schwarzian on $_1$29; in BF and Chern–Simons gravity it is the induced action of a reduced flat connection; in flat JT it is the BMS-Schwarzian corrected by a required zero-mode; in the fully gauged construction it is a covariant density built from a local $_1$30 gauge field; and in higher-rank or matrix settings it becomes a projective-curvature functional controlled by residual gauge symmetry and bulk monodromy data.