One-Loop Partition Function in AdS3 Gravity

• The one-loop partition function is the quantum correction to the classical action, obtained by integrating out fluctuations via heat kernel techniques.
• It encapsulates contributions from scalars, gauge fields, and gravitons, employing regularization and ghost subtractions to yield finite determinants.
• This method connects bulk geometrical spectral data with boundary CFT structures, exemplifying the AdS3/CFT2 duality through Virasoro symmetry characters.

The one-loop partition function is the quantum correction to the classical action in a field theory or gravity theory, evaluated by integrating out free fluctuations around a given background. In three-dimensional (3D) gravity with negative cosmological constant (locally AdS$_3$ backgrounds), the one-loop partition function for free quantum fields—scalars, (Abelian) gauge fields, and gravitons—encapsulates crucial information about both bulk quantum corrections and the structure of the boundary conformal field theory (CFT). This partition function is computed via heat kernel techniques, which efficiently encode the spectral data of the kinetic operators associated with each field and connect directly to the representation theory of the asymptotic symmetry algebra.

1. Euclidean Path Integral and Heat Kernel Representation

The starting point for the one-loop analysis is the Euclidean quantum path integral for a (free) field ϕ on a fixed background manifold $\mathcal{M}$: $Z = \int \mathcal{D}\varphi\, e^{-S[\varphi]}.$ Expanding around a classical saddle $\varphi^{(0)}$ yields a semiclassical expansion: $$\log Z = -g^{-2} S^{(0)} + S^{(1)} + g^2 S^{(2)} + \cdots$$ where $S^{(1)}$ is the one-loop quantum correction. The one-loop partition function for an operator $A$ governing linearized fluctuations (e.g., $A = -\nabla^2 + m^2$ for a free scalar) is

$$S^{(1)} = - \log\det A = \int_0^{\infty} \frac{dt}{t} \, \text{Tr}\, e^{-tA}.$$

The key tool is the heat kernel $K(t; x, x')$, satisfying

$$\left(\partial_t + A_x \right) K(t; x, x') = 0, \quad K(0; x, x') = \delta(x,x').$$

This formulation allows one to capture the entire (continuous) spectrum of $A$ using the geometry of the background.

2. Explicit Heat Kernel Computations in $\mathrm{AdS}_3$ for Scalars, Gauge Fields, and Gravitons

2.1 Scalar Fields

For a scalar of mass $m$, the kinetic operator is $A = -\nabla^2 + m^2$ in the hyperbolic $\mathbb{H}_3$ metric $ds^2 = (dy^2 + dz d\bar{z})/y^2$. The scalar heat kernel depends only on the geodesic distance $r$: $$K_{\mathbb{H}_3}(t, r) = \frac{e^{-(m^2 + 1) t}}{(4\pi t)^{3/2}} \frac{r}{\sinh r} \exp\left( -\frac{r^2}{4t} \right).$$

2.2 Vector Fields

For a $U(1)$ gauge field (in Feynman gauge), the operator $A_\mu{}^\nu$ leads to a bitensor heat kernel. By symmetry on $\mathbb{H}_3$, it reduces to a combination of derivatives of the chordal distance $u(x, x')$: $$K_\mu{}^{\nu'}(t; x, x') = F(t,u) \partial_\mu \partial^{\nu'} u + S(t,u) \partial_\mu u \partial^{\nu'} u,$$ with $F,S$ determined via differential equations. Ghost determinants from the Faddeev–Popov procedure (scalar Laplacians) cancel unphysical longitudinal modes.

2.3 Gravitons

For graviton fluctuations $h_{\mu\nu}$, the Euclidean action includes a cosmological constant, and a gauge-fixing plus York decomposition is imposed. The physical content is in the traceless-transverse part; the heat kernel for the TT-tensor incorporates up to six independent tensor structures. The spectral problem for the kinetic operator is reduced to calculating traces over the appropriate TT-sector, after inclusion of ghost and trace contributions.

3. Determinant Evaluation and Regularization

The zeta-regularized determinant for a generic field is rank-dependent; for the scalar case, for example,

$$S^{(1)}_\text{scalar} = -\log\det(-\nabla^2 + m^2) = \int_{0}^{\infty} \frac{dt}{t} \int d^3x \sqrt{g} K_{\mathbb{H}_3}(t, x, x).$$

The vector and graviton cases require appropriate subtraction of unphysical and ghost contributions. Regularization of the $t$-integral typically involves analytic continuation, tantamount to evaluating a Gamma function with negative argument. Divergences correspond to renormalizations of background couplings (e.g., cosmological constant or Newton's constant).

For gravitons, the one-loop determinant yields: $$e^{-k S^{(0)} + S^{(1)}} = |q|^{-2k} \prod_{n=2}^{\infty} \frac{1}{|1 - q^n|^2},$$ where $q = \exp(2\pi i \tau)$ with $\tau$ being the conformal modulus.

4. Quotient Geometries: Thermal $\mathrm{AdS}_3$, BTZ, and Higher Genus

To obtain the thermal partition function (or handle BTZ black hole backgrounds), $\mathbb{H}_3/\mathbb{Z}$ is considered; the heat kernel is built via the method of images: $$K_{\mathbb{H}_3/\mathbb{Z}}(t; x, x') = \sum_{\gamma \in \mathbb{Z}} K_{\mathbb{H}_3}(t; r(x, \gamma x')).$$ The spectrum of thermal AdS$_3$ is encoded by the partition function: $$Z_{1-\text{loop}}(T, \theta) \propto \operatorname{Tr}\left( q^{L_0} \bar q^{\bar L_0} \right),$$ identifying $L_0,\,\bar L_0$ as boundary CFT generators.

For higher genus Riemann surfaces, quotienting by more general discrete subgroups $\Gamma$ leads to hyperbolic manifolds with corresponding method-of-images heat kernels. In these cases, the partition function may be related to the Selberg trace formula.

5. Structure of the Partition Function and the Role of Asymptotic Symmetries

The gravitational one-loop partition function takes the form: $$Z_\text{gravity}(T, \theta) = |q|^{-2k} \prod_{n=2}^{\infty} \frac{1}{|1 - q^n|^2}$$ This is interpreted as the character of the vacuum representation of the Virasoro algebra, highlighting that quantum states above empty AdS$_3$ are generated by Virasoro descendants: $$Z = \operatorname{Tr}_\text{vac} (q^{L_0} \bar q^{\bar L_0}), \quad \text{with states } L_{-n_1}\ldots L_{-n_k} |0\rangle,\ n_i \geq 2.$$ This structure realizes the Brown–Henneaux result that the asymptotic symmetry group of AdS$_3$ gravity is two copies of the Virasoro algebra, with $|q|^{-2k}$ being the classical action contribution. The product over $n\geq2$ reflects the absence of $L_{-1}, L_0$ descendants that are pure gauge.

In the canonical ensemble, the same expression provides the one-loop correction to the Euclidean action and thus to the free energy and entropy of the BTZ black hole, matching black hole thermodynamics in three dimensions.

6. Extensions, Applications, and Higher Genus Partition Functions

The explicit heat kernel machinery generalizes beyond the BTZ background to any quotient $\mathbb{H}_3/\Gamma$ (including higher genus surfaces), though algebraic complexity increases. The general product structure persists, with the partition function encoding information about operator product expansions in the dual CFT and placing constraints from topologically non-trivial bulk backgrounds.

The tabular summary for fields and relevant formulas is:

Field	Heat Kernel $K(t, r)$	Determinant Contribution
Scalar	$$\frac{e^{-(m^2+1)t}}{(4\pi t)^{3/2}} \frac{r}{\sinh r}e^{-r^2/4t}$$	$-\log\det(-\nabla^2+m^2)$
U(1) Gauge	Bitensor in terms of $u(x,x')$, ref. structure above	Ghost subtraction, bitensor trace
Graviton	6-tensor structure, involved solution	$|q|^{-2k}\prod_{n=2}^\infty |1-q^n|^{-2}$

The gauge-fixing, correct subtraction of non-physical modes, and careful regularization are all essential to obtain the result matching boundary CFT expectations.

7. Significance and Relation to CFT Structure

The explicit computation of one-loop determinants in AdS$_3$ (and quotients) demonstrates that physical excitations of quantum gravity correspond precisely to Virasoro descendants of the vacuum rather than to bulk propagating modes. This bridges semiclassical gravity and CFT operator algebras, substantiates the AdS$_3$/CFT$_2$ duality at the one-loop level, and enables precise calculations of subleading corrections to black hole thermodynamics. The methods developed provide a brute force derivation that aligns with previous indirect arguments, and the explicit product structure confirms the picture of AdS$_3$ gravity as a boundary-driven, CFT-like quantum system (0804.1773).