ER = EPR is Descent Data

Last Friday, I had the opportunity to give a very fun talk at the MIT student journal club on our Baby Universe Hypothesis. A major theme of the talk, and the resulting discussion, was that holography isn't free! In many ways, this is the heart of the Swampland: if you just throw whatever junk you want into the Euclidean path integral, there's really no reason to expect anything but junk out the other end. This is strikingly different from quantum field theory, where (up to canceling anomalies) you are free to write down whatever you wish. In this post, I want to share some observations that make this difference between gravity and other quantum systems more obvious, as well as point the way towards the type of mathematical structure we need in order to guarantee holography.

The miracle of holography, and the true heart of the issue, is not that a bulk gravitational system becomes a quantum system on the boundary, but rather that it becomes a local quantum system on the boundary. Why do I call this a miracle? For quantum field theories, it's no miracle, as locality follows from the very structure of the path integral. Indeed, suppose we have a composable pair of cobordisms:

\[ X_1 : M_1 \to M_2, \quad X_2 : M_2 \to M_3. \]

The path integral over the composite $X_1 \sqcup_{M_2} X_2$ can be broken up into the separate path integrals over $X_1$ and $X_2$, with the boundary condition that the fields agree on their mutual boundary $M_2$. Thus, we learn that the output of the path integral is functorial, i.e., defines a local quantum field theory.

Why can't we apply this same argument to the quantum gravitational path integral? Naively, there's no issue: just replace the dynamical fields with a dynamical bulk spacetime! Importantly, we cannot fix the topology of the bulk, since the manifold itself is dynamical. However, this is our downfall: we must now consider wormhole geometries connecting different regions of the boundary. Including wormholes immediately breaks the above argument, since it's no longer true that bulks with boundary $X_1 \sqcup_{M_2} X_2$ can be cut into bulks with boundary $X_1$ and $X_2$ and glued over $M_2$. Put differently, the assignment

\[ X \quad \rightsquigarrow \quad \left\{ \text{Bulk manifolds}\ Y\ |\ \partial Y = X \right\}, \]

of bulk spacetimes $Y$ with boundary $X$ is not a sheaf!

If you are not familiar with the mathematical notion of a sheaf, the key property it captures is descent. What this means is that if we have some data depending on a topological space $T$, we might hope that the data can be restricted to local data defined on open subsets $U \subset T$. Further, given data on two subsets $U_1, U_2$, we can reconstruct the data on $U_1 \cup U_2$ as pairs of data on $U_1$ and $U_2$ that agree on the overlap $U_1 \cap U_2$. Importantly, if the data is subject to gauge redundancies, we shouldn't demand strict equality on the overlap, but instead ask for a specified gauge equivalence.

Now, in any local quantum field theory, the assignment

\[ X \quad \rightsquigarrow \quad {\rm Fields}(X), \]

defines a sheaf! For example, scalar fields on $X$ may be restricted to subsets, as well as glued back together when they agree on an overlap. Similarly, gauge fields form a sheaf: by specifying prescribed gauge transformations on the overlaps, we may glue gauge fields together over multiple patches. Note that the sheaf condition is meaningful at the level of the prequantum field theory (the input to the path integral), and ensures that the output of the path integral defines a local quantum field theory.

As described above, however, the possibility of wormholes shows that fields in the gravitational path integral do not form a sheaf. In fact, it's not even clear how to restrict a bulk manifold to a subset of the boundary. Only when the bulk is disconnected can we attempt to make sense of this restriction, and even then it's not clear what to do with pieces of the bulk that are disconnected from the boundary: which subset of the boundary should we assign them to? It is exactly this tension, between the bulk path integral and the locality required by holography, that has played a central role in the ongoing discussion about Euclidean wormholes, baby universes, and the ensemble.

Not all hope is lost, though: recognizing this failure as the lack of descent data points a clear way forwards. What if, in some more subtle way, we could think of the collection of bulk geometries as a sheaf? What could it mean for this assignment to satisfy descent? In fact, we know where to look: the answer is some version of ER = EPR! In the case of the two-sided AdS Schwarzschild wormhole, ER = EPR tells us that a connected geometry equivalent to an entangled superposition of disconnected geometries. Thus, in quantum gravity, we may ontologically replace certain wormhole geometries with a collection of disconnected geometries. While I do not know how to realize this in any concrete example, I want to put forward the following proposal:

The proper understanding of ER = EPR is descent data

for the quantum gravitational path integral.

I want to wrap up by emphasizing that the existence of such data is not at all automatic, and instead defines a Swampland condition! In particular, requiring that the result of gluing two bulks together be unique implies that there cannot be any ambiguity around whether to include extra baby universes or not, and thus implies the Baby Universe Hypothesis! This is not so surprising, as it also follows from demanding that the holographic dual be a local quantum field theory. Further, demanding restrict any bulk to subsets of the boundary implies that we must be able to cap off any bulk wormhole, which is equivalent to the triviality of cobordism in quantum gravity! Thus, we see that multiple Swampland conditions follow from demanding that the gravitational path integral satisfy descent in some more subtle sense. This descent data, though we do not currently understand it, would be a precise realization of ER = EPR, and would likely tell us quite a bit about the non-perturbative structure of quantum gravity.