Virasoro Symmetry, Vacuum Blocks, and Semiclassical Gravity

A Virasoro conformal block resums every stress-tensor descendant exchanged in a fixed channel. At large central charge this resummation can behave like propagation in a classical AdS3 geometry, but only after the external weights, OPE channel, and dominance assumptions are specified. The universal object is the block; its interpretation as the leading correlator is an additional dynamical statement.

Required background. The Virasoro Algebra and the Stress Tensor defines the blocks and Ward identities, while AdS3/CFT2 and the Brown–Henneaux Central Charge identifies $1/c$ with the gravitational loop parameter.

Helpful background. Liouville Theory and the Virasoro Bootstrap develops exact Virasoro data, and Heavy States, Coherent States, and Semiclassical Geometries states when one-point data support a geometric interpretation.

Semiclassical Virasoro limits

There are two useful but distinct limits. In the semiclassical limit, $c\to\infty$ with all ratios $h_i/c$ fixed, a block exponentiates,

$$\mathcal V(c,h_i,h_p;z) \sim \exp\!\left[-\frac{c}{6}f\!\left(\frac{h_i}{c},\frac{h_p}{c};z\right) +O(c^0)\right].$$

In the heavy–light limit, two operators have $h_H/c$ fixed while the probe weights $h_L$ remain $O(1)$. The heavy insertions determine a classical stress-tensor background; the light operators probe it without leading backreaction. These limits answer different questions and need not commute with Lorentzian continuation or a late-time limit.

For spinless heavy insertions at zero and infinity, define

$$\alpha=\sqrt{1-\frac{24h_H}{c}}.$$

The coordinate $w=z^\alpha$ uniformizes the leading heavy-state stress tensor. Below the threshold $h_H=c/24$, real $\alpha$ describes conical-defect kinematics; above it, imaginary $\alpha$ gives thermal identifications characteristic of BTZ. In the vacuum block, a pair of light operators transforms approximately as a two-point function in $w$,

$$\mathcal V_{\rm vac}(z_1,z_2) \simeq \frac{[w'(z_1)w'(z_2)]^{h_L}} {[w(z_1)-w(z_2)]^{2h_L}},$$

up to the operator placement, antiholomorphic factor, and normalization chosen for the four-point function.

Monodromy and bulk propagation

The semiclassical block can be found by solving

$$\psi''(z)+T_{\rm cl}(z)\psi(z)=0,$$

where $T_{\rm cl}$ has singular terms fixed by external weights and accessory parameters fixed by the desired monodromy around the exchanged pair. Integrating the accessory parameter reconstructs $f$. On the gravity side the same saddle may be represented by a worldline or geodesic network in the heavy geometry. Agreement of its action with the block exponent tests the dictionary at leading order.

First application. Compute a heavy-light vacuum block in the semiclassical limit and compare its singularities with propagation in a conical or BTZ background. The useful comparison is not merely the final formula: the branch of $\alpha$, the Euclidean-to-Lorentzian continuation, the chosen OPE channel, and whether the probe crosses a horizon must match on both sides.

When the vacuum block is not the correlator

A conformal block is one term in an OPE decomposition. Vacuum-block dominance requires the weighted sum of nonvacuum blocks to be small in the stated kinematic region. Large $c$ alone does not ensure this; a sparse light spectrum and bounds on OPE coefficients are typical additional inputs. After analytic continuation, a term subleading in Euclidean kinematics can become important, and at sufficiently late times a single semiclassical block cannot reproduce a discrete finite-entropy spectrum.

Adversarial control. Add a light primary with an unsuppressed heavy–light OPE coefficient. Its block competes at the same order and can change the singularities attributed to the background. Alternatively cross a Stokes or OPE-channel boundary: the monodromy saddle selected in one region need not dominate in another. The geometric interpretation then remains valid for the individual vacuum block, but not as a universal approximation to the full correlator.

Evidence ceiling

Virasoro symmetry fixes the vacuum block once weights and central charge are given. A semiclassical geometry efficiently represents particular asymptotics of that block. Neither fact establishes vacuum dominance, identifies a unique bulk state, or controls finite-$c$ recurrences and nonperturbative terms.

The heavy–light semiclassical vacuum block and its thermal-looking background interpretation are developed in Fitzpatrick, Kaplan, and Walters 2015; vacuum-block dominance remains an additional condition on the full correlator.

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

• Fitzpatrick, A. Liam, Jared Kaplan, and Matthew T. Walters. “Virasoro Conformal Blocks and Thermality from Classical Background Fields.” Journal of High Energy Physics 2015, no. 11 (2015): 200. DOI; Open PDF.
• Hijano, Eliot, Per Kraus, Eric Perlmutter, and River Snively. “Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks.” Journal of High Energy Physics 2016, no. 1 (2016): 146. DOI; Open PDF.
• Zamolodchikov, Alexander B. “Conformal Symmetry in Two-Dimensional Space: Recursion Representation of Conformal Block.” Theoretical and Mathematical Physics 73 (1987): 1088–1093. DOI.