Quantum Gravity in Color

This week, during the first Swampland open mic event, there was a lot of discussion about the role of duality in quantum gravity, and in particular the appearance of light extended objects of various dimensions: particles, strings, and possibly membranes. Many facets of this question were discussed, including how to understand which objects appear at which infinite-distance limit in moduli space, the structure of theories with various numbers of supercharges, and light towers of objects which are not the lightest in the theory. I made some comments about where I think we should look for answers, but I'm sure those comments were close to unintelligible, as my thoughts are nowhere close to organized. In this blog post, I want to start describing the free associations that suggest, to me, the answers lie in chromatic homotopy theory.

If you haven't heard of chromatic homotopy theory, don't panic, since I'm mostly going to parrot words I've heard with no real comprehension.¹ The basic idea, as far as I understand, is to organize phenomena in algebraic topology according to a new filtration, called "chromatic height," or "chromatic level," and then break apart objects into their different chromatic pieces (like light through a prism, I guess). The best picture I've gotten is that chromatic height corresponds, loosely, to the dimension of the fundamental object in question, so height zero corresponds to stuff about particles, height one has to do with strings, and height two is all about membranes.² The tower continues all the way up to infinity, which should involve thinking about extended objects of higher and higher dimension. For the rest of this post, I'm going to try and show, through examples, what I mean by "stuff having to do with particles/strings/membranes," and perhaps in a future post I'll dig more deeply into the actual tools of chromatic homotopy theory.

The main phenomena I want to sort by chromatic height are the known dualities in string theory. The most simple duality, from the perspective of target-space physics, is electromagnetic duality, which exchanges a $p$-form flux with its $(d-p)$-form Hodge dual. Though the charged objects under such a gauge field are extended, the fluctuations of the gauge field itself are point particles, and so I would place electromagnetic duality at chromatic height zero. This is supported by the fact that the fluxes of the gauge fields are merely elements of ordinary cohomology, which also sits at height zero. In fact, most dualities we know about in quantum field theories likely occur at chromatic height zero, and are consequently "uninteresting" from the perspective of chromatic homotopy theory.³ An exception would likely be dualities involving the superconformal theories in six dimensions, where light strings play a key role.

Moving on up the chromatic tower, the next duality to discuss is T-duality, which relates theories compactified on the circle by exchanging winding strings with particle momentum around the dual circle. T-duality is an archetypical example of something at chromatic height one, as it requires the existence of light string states. I should note that while T-duality is very easy to see from the worldsheet, it's already much more mysterious from the perspective of the target space, as it acts non-geometrically on spacetime. Further evidence that T-duality is happening at chromatic height one comes from Type II string theory, where T-duality exchanges D-branes of different dimensions. This has the effect of inducing relationships between R-R fluxes of different degrees, leading to a proper quantization in K-theory, at chromatic height one! Though D-branes are extended objects of various dimensions, the salient point is that these features are induced by their coupling to fundamental strings.

What's next? How about S-duality, say in Type IIB? Indeed, S-duality forces us to look beyond a K-theoretic description of R-R fluxes, since it treats them on a separate footing from NS-NS fluxes, while S-duality exchanges them. It has been proposed that a properly S-duality invariant description of these fluxes requires something like elliptic cohomology, at chromatic height two. In what sense, though, is S-duality about membranes? It's a bit of a stretch, but one perspective comes from F-theory, where we think of Type IIB in the dual frame of M-theory on a shrinking elliptic curve, rendering S-duality geometric. M-theory is in some sense a theory of fundamental membranes (though this should be taken loosely), and so in this way S-duality is tied to a description involving membranes. Relatedly, if we look at the strong-coupling limits of Type IIA or Heterotic $E_8 \times E_8$, we also land on M-theory, and so in this sense S-duality is related to membranes, if somewhat tenuously.

To go further, we should find a duality not implied by T-duality or S-duality, and more mysterious than either. An example of such a duality is the duality between Type IIA on $K3$ and Heterotic on $T^4$, which acts to exchange wrapped fivebranes with unwrapped strings. At this point, my understanding is too murky to tell exactly what chromatic height we're dealing with, but it's certainly higher than two! In particular, this duality is intricately tied to the geometry of the $K3$ manifold, and a quick trip to our local nlab tells us that there is a notion of $K3$ cohomology that generalizes elliptic cohomology, and sits at higher chromatic heights. A related duality, if it acts as a physical duality and not merely a matching of charges, is the spherical T-duality that exchanges fivebranes wrapped on $S^3$ with unwrapped membranes. Interestingly, the manifolds $S^3$ and $K3$ are intimately related in stable homotopy theory, as $K3$ arises in spin bordism by killing off the class of $S^3$ in string bordism with the inclusion of fivebrane source charge (as $K3$ minus $24$ points is string).

Can we see even more color? To do so, we must truly leave Kansas, and start discussing undiscovered dualities acting on conjectural objects. Thankfully, this is a blog post, not a paper! The objects in question are the non-supersymmetric defects predicted in my paper with Cumrun on cobordism in quantum gravity, which would kill off the class of, say, F-theory on $K3$ (dually, Heterotic on $T^2$). In order for such objects to exist, something we have previously held sacred must break. One possibility would to make sense of F-theory on Calabi-Yau manifolds without an elliptic fibration! Indeed, for the case of F-theory on $K3$, as long as we require an elliptic fibration, the base $\mathbb{P}^1$ will represent a nontrivial class in spin$^c$ bordism. Indeed, passing from elliptically fibered Calabi-Yau threefolds to more general manifolds should allow us to construct analogues of elliptic cohomology or $K3$ cohomology at even higher chromatic height. A related possibility would be to perform the F-theory trick for the duality between Heterotic on $T^4$ and IIA on $K3$,⁴ and find a sense in which the ten-dimensional Heterotic theory has a description as a theory in fourteen dimensions reduced on $K3$, which might provide a new defect to kill Heterotic on $T^2$.

To wrap up this wild ride, I want to come back down to earth, and emphasize one extremely appealing aspect of chromatic homotopy theory, namely, its computability. That is to say, the tools of algebraic topology allow us to abstract away the irrelevant details and do good, honest calculations. Given a cohomology theory, you can break it apart into its pieces at different chromatic heights⁵ and analyze them piece-by-piece. Quantum gravity involves an enormous number of moving pieces, and it would be nice to be able to isolate the bits relevant for various swampland conjectures (for example, splitting off the membrane-like stuff in a theory with lighter strings). Further, passing from a geometric category to something like the category of stable motives may allow us to analyze quantum gravity on its own terms, as opposed to forcing it into the unnatural language of effective field theory. As we have seen in many different places, the natural description of quantum gravity is likely very different from that of quantum field theory, and is likely much more motivic, chromatic, and number-theoretic.

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1 Motive! Group scheme! Look ma, no hands!^↩
2 For experts, one concrete version of this is the result by Hopkins and Lurie that the Picard group of Morava E-theory at height $n$ looks a lot like a shift of the Anderson dual of the spheres, which by Freed-Hopkins is the space of framed invertible phases in $(n + 1)$ dimensions.^↩
3 In particular, they are much more differential-geometric than number-theoretic, and indeed we must go to positive chromatic height to see interesting behavior at nonzero primes.^↩
4 Instead of the duality between IIB on $S^1$ and M-theory on $T^2$. See also this paper, which contains a thirteen dimensional description of Type I', and a hint at the height of the corresponding $K3$ cohomology theory.^↩
5 And indeed, at different primes!^↩