Add up the geometries. Do you get a theory?

A Euclidean path integral offers an appealing instruction: hold the boundary fixed, fill it in every allowed way, and add the answers.

For a torus boundary, the instruction almost sounds manageable. A torus has two basic cycles. A solid-torus filling can make one of them contractible in the interior. Changing which cycle shrinks changes the bulk interpretation. Thermal AdS and Euclidean black-hole geometries enter the same conversation.

At last, perhaps the simplicity of three dimensions will pay the entire bill.

The boundary keeps the score

The torus has a complex shape parameter \(\tau\). Different choices of its basic cycles are related by modular transformations. A sensible boundary partition function must behave consistently under those changes of description.

One can therefore begin with the thermal-AdS contribution and organize other fillings as modular images. The vacuum descendants we met earlier appear in the seed. At one loop, the relevant factor includes

Giombi, Maloney, and Yin developed heat-kernel methods for these one-loop gravitational partition functions.^[2] The result has a satisfying boundary interpretation: precisely the sort of vacuum-descendant tower we expected.

There is a catch before we even ask about states. The sum over images needs a regularization. “Sum all the fillings” is not a complete definition until we explain how that sum is made meaningful.

Symmetry is a demanding friend, but not the only one

Maloney and Witten evaluated the regulated sum of the known saddle contributions and found that the resulting answer did not have the desired physical interpretation.^[1] In particular, the spectrum inferred from this prescription was not that of an ordinary compact, unitary CFT with a discrete set of states and acceptable degeneracies.

This is a useful moment to pause. The problem is not that the expression looks ugly. It can satisfy an important symmetry requirement while failing other requirements of a quantum theory.

For a conventional discrete spectrum, a thermal trace is a sum over actual states. At each energy, the multiplicity counts something. A negative multiplicity cannot be repaired by admiring modular invariance. And a continuous spectral density needs a different physical interpretation from the discrete compact-CFT target.

Keller and Maloney’s later analysis of Poincaré series sharpens the interplay between modular completions and spectral data.^[3] It also helps make clear why specifying a low-energy beginning and demanding modularity need not settle every ambiguity.

What failed, exactly?

There is a bad way to retell this episode: “People calculated quantum gravity and proved it does not exist.” That conclusion is much broader than the calculation.

The result tests a prescription: a class of contributions, their weights and corrections, and a way of regularizing their sum. If it misses the intended answer, possibilities include missing contributions, an inadequate prescription, or an incorrect expectation about the target. Distinguishing those possibilities is research, not a footnote.

Imagine assembling a machine from every part listed on a diagram. It does not run. You have learned something decisive about that assembly. You have not yet established that no machine performing the task can exist.

A failure worth keeping

The calculation raises the standard for every future proposal. A credible construction must do more than reproduce a classical saddle or a leading entropy. It must explain how the complete answer becomes a legitimate quantum object.

And even a perfect torus partition function would not finish the job. A spectrum tells us who is present. To learn how the states interact, we need to ask the theory a more elaborate question. The simplest next move is to give the boundary another handle.