De-quantizing a black hole

The world’s leading string-theory blogger recently suggested that NP might be in P after all, with a polynomial complexity of n^196883. The specific exponent was a joke, but let’s take it seriously…

The context was a discussion of E. Tang’s recommendation algorithm, which might be regarded as a de-quantized version of a specific algorithm from quantum machine learning. That number 196883 comes from the representation theory of the monster group. Within string theory, it is roughly the number of microstates of the lightest black hole in a special case of AdS3/CFT2.

It is conjectured that quantum black holes are “fast scramblers”, and that black hole evaporation encodes a cryptographic one-way function. So… perhaps the S-matrix of that lightest AdS3 black hole could be shown to be in NP, and then we could “de-quantize the black hole” and get a corresponding O(n^196883) algorithm?

The main problem I have with this scenario – the de-quantized black hole lacks any useful structure, it’s just a point singularity. But here I think of the “field with one element”. It’s not actually a field, but one can define more complex objects on top of it, analogous to constructions that elaborate on genuine fields.

In the same way, there could be an algebraic dressing of the classical black hole, which would provide the Tang-like de-quantization of AdS3 black hole formation and evaporation. Probably modular forms and other Langlands math would be involved.

Along the path

It’s almost four years since I created this blog. I have had little opportunity to develop this line of thought.

Meanwhile, there is a new theorem regarding the polynomial hierarchy. Its relationship to P vs NP is, according to the famous Scott A., like the relationship between the first attempts to prove Fermat’s Last Theorem, and the eventual proof by Andrew Wiles.

Wiles drew upon 20th-century innovations like those due to Grothendieck’s circle. I wonder if Mochizuki’s inter-universe theory can help GCT.

Within ten years

Last year I asked the Internet, “Is Witten’s new method of quantization useful for geometric complexity theory?” This was prompted by a rather vague notion that Ketan Mulmuley’s “geometric complexity theory”, which concerns itself, among other things, with certain algebraic varieties, might be able to profit by examining the loop spaces based on those varieties.

In the months since then, I learnt a lot more physics, but rather less complexity theory. A few days spent studying Ryan Williams’s celebrated incremental advance regarding circuit lower bounds was the high point of my complexity studies (unless you count an excursion into Conjecture 5 in the Aaronson-Arkhipov paper on noninteracting bosonic computation)… But a recent poll regarding when and how “P versus NP” will be decided (in particular, some rather silly responses) has prompted me to finally create my very own complexity-theory blog – perhaps to back up my own contention that the matter will be decided “within ten years”, by an extension of Mulmuley’s program to motivic mathematics.

Now that the blog has been created, it may languish, neglected, for a while. But so long as it’s there, I can keep going back to it. And perhaps, in time, it will amount to something.