Suppose you want a quantum theory of gravity with no extra matter. What does “no extra matter” look like from the boundary?
It cannot mean “no states.” We already have boundary excitations and black holes. The more interesting thought is to leave the low-energy spectrum as empty as possible, apart from the states that gravity itself seems to require.
This turns minimalism into a mathematical challenge. How much emptiness will consistency tolerate?
The states you cannot remove
The vacuum of a two-dimensional CFT does not sit alone. Virasoro generators produce descendants: states organized by the symmetry algebra. For the gravitational interpretation, these vacuum descendants are the boundary-graviton sector.
A new primary is different. It is not obtained merely by acting on the vacuum with those generators. If we are trying to avoid additional light bulk particles, delaying such primaries is a natural idea.
Witten’s 2007 proposal explored this logic with a strong additional assumption: holomorphic factorization. In the chiral extremal problem, the central charge is \(c=24k\), with positive integer \(k\), and no non-vacuum primary appears through weight \(k\).[1] The first possible new primaries occur at weight \(k+1\).
This is a special ansatz for the boundary theory, not a consequence that follows from every possible definition of Einstein gravity.
A partition function with very little freedom
The opening of the chiral partition function is fixed by the vacuum descendants:
Read the formula as a constraint on the expansion through the constant term. The product starts at two because the level-one vacuum descendant is absent. Modular invariance then severely restricts how this beginning can be completed.
The attraction is immediate: instead of searching blindly through all CFTs, we can write down sharply constrained candidate partition functions. At \(c=24\), the Monster CFT is an actual constructed theory, not merely a plausible series of coefficients.[2] It makes the proposal feel tantalizingly tangible.
A guest list is not a party
But a partition function mostly tells us how many states of each energy are invited. It does not tell us how they interact.
A CFT needs operator products, consistency under different ways of combining operators, and amplitudes that fit together on different surfaces. Nonnegative integer degeneracies are necessary for a conventional discrete Hilbert-space interpretation. They do not construct those additional structures.
This is the trap of an especially elegant candidate. When modularity leaves almost no freedom, the resulting formula can look inevitable. Yet uniqueness of an answer to one restricted consistency problem is not existence of a solution to the full problem.
Higher-genus work soon pursued additional tests of extremal candidates.[3] We will return to what those tests do—and do not—establish. For now, keep the distinction between “a candidate partition function” and “a constructed CFT” in view.
Two ways to search
The extremal route begins on the boundary: demand a sparse spectrum and see what survives. Another route begins in the bulk: sum the geometries that seem to belong in a gravitational path integral, and ask what spectrum the answer implies.
The two routes aim at related territory without making identical assumptions. In particular, the usual nonholomorphic gravitational saddle sum is not simply the extremal holomorphic ansatz in disguise.
The next article takes the bulk route. The striking result will be that a sum can be beautiful, computable, and symmetric—and still fail to look like the partition function of the quantum theory we wanted.
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Original sources behind this story. The dates follow the work’s first appearance; publication dates are included where useful.