# Kriz-Sati conjecture on modular equivariant charges in F-theory

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Below I'll ask a question about existing evidence (or counter-evidence, as it may be) in F-theory that a recent result in homotopy theory, which I'll recall below, is the answer to an old conjecture by Kriz and Sati on how to F-theoretically refine D-brane charges in K-theory as charges in elliptic cohomology : the universal modular equivariant elliptic cohomology (see there for most of what I am going to say here) which Kriz-Sati conjectured to govern F-theory has now been constructed mathematically. How does it compare to physical evidence?

Here is some more background, meant to make more clear what I am thinking of here.

To see the impact of this conjecture and result, first recall that on the most general orientifold backgrounds of type II string theory (hence including type I string theory) D-brane charges are cocycles  in super line-2-bundle-twisted ($\leftarrow$the B-field) KR-theory ("real K-theory"). A comprehensive statement of this has been given fairly recently in

with techncial details in

What happens here mathematically is ​that the complex K-theory spectrum KU has a canonical $\mathbb{Z}_2$-involution as a ring spectrum which makes it a "genuinely equivariant spectrum" in the sense of equivariant stable homotopy theory (a real-oriented cohomology theory, in fact), and this implies that it produces a cohomology theory on spaces equipped themselves with a  $\mathbb{Z}_2$-action, such that in computing cocycles both these  $\mathbb{Z}_2$-actions are intertwined -- just as known from orientifold strings.

Moreover, a recent observation in the appendix of

• Tyler LawsonNiko NaumannStrictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2, Int. Math. Res. Not. (2013) (arXiv:1203.1696)

amplified that this  $\mathbb{Z}_2$-action on complex K-theory is in a precise sense indeed the worldsheet parity operator. This is just as is well familiar in string theory, but the point here being, for what comes next, that one can discover this from just the mathematics of chromatic stable homotopy theory. This fundamental piece of math natively knows a remarkable lot of detail about superstrings...

This is relevant for the question to follow: by the famous insight in

it is known that as one considers the "M-theory lift" of type II string theory in the guise of F-theory, then the target space involution of orientifolds is identified as the inversion involution inside the S-duality modular group acting on the elliptic curves that constitute the axio-dilaton elliptic fibration in F-theory. Taken together, this raises an obvious question:

should the rest of the modular S-duality also be accompanied by modular operations on the worldsheet?

Back in

it was conjectured/speculated (see p. 3 and pages 17-18) that indeed there should be a modular equivariant version of universal elliptic cohomology (tmf) and that with respect to that at least for the quotient group $\mathrm{SL}_2(\mathbb{Z}/2\mathbb{Z})$ the answer is: yes, the charges of F-theory really live in "modular equivariant universal elliptic cohomology".

Of course when this was conjectured in 2005, such a theory had not been constructed yet, mathematically. But now recently it has. This is the content of theorem 9.1 in

Moreover, by theorem 9.3 there, expanding on the earlier

• Tyler LawsonNiko NaumannStrictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2, Int. Math. Res. Not. (2013) (arXiv:1203.1696)

this modular equivariant universal elliptic cohomology is in a precise sense exactly the generalization of orientifold KR-theory from the string to one dimension up, in that the latter is precisely the "point particle limit" of the former, in the sense that it is the restriction of the modular elliptic theory as one approaches the nodal compactification point.

And so this means that just from the mathematics alone, it follows that there is a modular-group equivariant elliptic cohomology theory which evaluates on spaces with modular group action (such as F-theory elliptic fibrations) and whose cohomology classes are computed by accompanying target space modular transformations with certain modular actions on genus-1 string worldsheets equipped "with level structure". And, to repeat,  all this in such a way that in the degeneration limit this comes down to being precisely the general KR-theory of orientifold type II strings.

So this does provide a good bit of further support to, or at least motivation of, the Kriz-Sati conjecture, it would seem.

But here is finally the question that I am wondering about: which plausibility checks from string theory exist that would give a physical interpretation to this combined S-duality target space/worldsheet "generalized orientifolding"-transformation which we know now to exist mathematically?

The following is what I know of, but possibly there is more along such lines, and that's what I am asking for here:

Namely, of course it is well known that S-duality in type IIB also acts on the worldsheet theory: after all, in type IIB the strings are really (p,q)-strings and the S-duality modular group of course acts on the pairs of in the canonical way, mixing the "F1-brane" witth the D1.

And indeed, this mixing does go along with some conformal compensating readjustment on the worldsheet, of roughly the kind that the above mathematical story suggests. This has been amplified once in

• Igor BandosSuperembedding Approach and S-Duality. A unified description of superstring and super-D1-brane, Nucl.Phys.B599:197-227,2001 (arXiv:hep-th/0008249)

This is qualitatively/conceptually what the Kriz-Sati conjecture suggests and what the Hill-Lawson modular equivariant universal elliptic cohomology produces. I haven't tried to check yet if it also matches more in detail.

But I suspect if it does, then there is something already known along these lines in the literature?

What (further) evidence in string theory/F-theory exists that would make plausible, or else make implausible, that the Kriz-Sati conjecture on F-theory would be realized by Hill-Lawson's modular equivariant universal elliptic cohomology?

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