A second handle asks harder questions

A torus partition function is an extraordinarily informative answer to a rather specific question. It tells us about a trace over states. It does not, by itself, hand us every interaction coefficient of the theory.

That is why adding a second handle is more than a request for a prettier picture. A genus-two surface can be assembled by gluing simpler pieces, and the gluing brings additional CFT data into play.

If the torus is an inventory, sewing asks whether the inventory can be put to work.

Cut, insert, join

Picture cutting a surface along a circle. To glue the two sides back together, a quantum theory inserts a complete set of intermediate states and sums over them. A long thin tube makes this especially intuitive: low-energy states dominate propagation along it.

Now use pieces with three circular boundaries, often called pairs of pants. The name is homely; the information is not. Their amplitudes are related to three-point data. Gluing many such pieces involves both the spectrum and the rules for combining operators.

There is more than one way to cut the same surface. A genuine theory must give compatible answers. A change of sewing channel cannot change the physical object merely because we used a different assembly manual.

This is the broader reason higher-genus amplitudes matter. They bring geometry, factorization limits, and operator products into one consistency problem.

The extremal test bench

In 2007, Gaiotto and Yin constructed genus-two partition functions for particular extremal-CFT candidates and studied stringent consistency tests.^[1] Yin also investigated gravitational partition functions and their relation to these boundary constructions, including handlebody contributions.^[2]

The bulk picture has a satisfying counterpart to sewing. A boundary Riemann surface can admit different fillings; in a handlebody filling, selected boundary cycles become contractible. Once again, knowing the boundary shape does not uniquely tell us which bulk geometry to choose.

But this episode comes with an important correction to an overly triumphant retelling. Gaberdiel, Keller, and Volpato showed that consistent genus-two vacuum amplitudes for the proposed extremal theories can exist for all \(k\), while not testing their consistency beyond what is already testable at genus one in that setting.^[3]

So “higher genus probes more structure in general” must not be turned into “this particular genus-two success proves the extremal theories exist.” The actual strength of a test depends on the data and constraints that enter it.

More checks are not automatically independent checks

This is a lesson that travels well outside gravity. Suppose a calculation passes ten tests, but nine are mathematical consequences of the first. You have excellent evidence that you implemented the first condition consistently. You do not have ten independent reasons to believe the entire theory exists.

The useful question is not how complicated a check looks. It is which new failure it could have detected.

For a proposed boundary dual, a convincing program therefore needs to keep track of exactly what has been constructed: a modular function, a collection of amplitudes, consistent operator products, or an actual theory satisfying the full requirements. These achievements are related, but the labels should not be silently exchanged.

One boundary, several holes; several boundaries, another puzzle

So far we have made one boundary surface more complicated. Another possibility is to ask for multiple disconnected boundaries. In the bulk, connected geometries may join them.

That shift changes the question dramatically. Instead of asking how one theory sews its amplitudes, we confront whether a connected gravitational contribution is compatible with the factorization expected from a fixed boundary theory. The next article follows that tension into the language of ensembles.