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Urs Schreiber


String theory and the real world. Over on PhysicsOverflow somebody is asking for the "dictionary" which translates string theory to particle physics.

One decent book that compiles an introduction to a bit of this dictionary is this here:

Michael Douglas, Elias Kiritsis et. al. (eds.), "String theory and the real world", Les Houches Session LXXXVII 2007

But of course the REAL answer is: that dictionary is taking the limit which sends a 2d SCFT to... a spectral triple:
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You're welcome! As Niels Bohr used to say ``I say this not to criticize, but only just to learn''-it'll be interesting to see what Planck comes up with-and in the meantime, hopefully, the ``maps'' will become  a priority.
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p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf) A question I asked at Physics.SE

please see there for a hyperlinked version of the text to follow.

First some background:

The very last words of the seminal article

Edward Witten, Elliptic Genera And Quantum Field Theory, Commun.Math.Phys. 109 525 (1987) 

on the partition function of the superstring -- now called the Witten genus -- were the following:

"A properly developed theory of elliptic cohomology is likely to shed some light on what string theory really means."

One of the breakthrough results in pure mathematics originally motivated by this was the construction of tmf and of the string orientation of tmf by Michael Hopkins and collaborators:

where the original Witten genus is a homomorphism

Zsuperstring : ΩString → MF∙

from the cobordism ring of spacetime manifolds with String structure (meaning: Green-Schwarz anomaly-free spacetimes) to the ring of modular forms (which are the possible 1-loop correlators, modular invariant up to the relevant anomaly), the string orientation of tmf (or "topological Witten genus") is a homomorphism of coherently homotopy-commutative ring spectra


from the String-structure Thom spectrum of cobordism cohomology theory to topological modular forms.

This is a homotopy-theoretic refinement of the Witten genus, the latter is the "decategorification" of σ in that it is reproduced on homotopy groups:


a result due to

Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf).

This is maybe noteworthy in view of the quote above, since tmf and its String-orientation have a fairly "god-given" origin right at the foundations of stable homotopy theory ("chromatic stable homotopy theory").

By some magic (rigorously proven magic, though) this fundamental abstract math knows the modular invariant of the heterotic superstring including its Green-Schwarz anomaly cancellation. And it knows something more, since the Witten genus is only the shadow of this on homotopy groups.

Now to come to my question, one thing that is maybe noteworthy here is that where the Witten genus Zsuperstring is built from ordinary string worldsheets being ordinary genus-1 Riemann surfaces, hence elliptic curves over the complex numbers, it's homotopy theoretic refinement σ is built from all elliptic curves in the sense of algebraic geoemetry, hence elliptic curves over general base rings.

Moreover, in the standard construction of tmf via artihmetic fracture squares it is explicitly built from a piece over the rational numbers and one piece which for each prime number p receives a contribution from p-adic elliptic curves.

Here an elliptic curve over the p-adic integers Zp is in a precise sense a genus-1 closed string worldsheet, but not over the complex numbers, but over the p-adic integers.

This of course now reminds one of what is called p-adic string theory. This is a field driven by the observation that the integrals over the real numbers which give the scattering amplitudes of the open string -- where the real line parameterize its boundary -- still make sense and still are very interesting when one replaces the real numbers by the p-adic numbers Qp.

This means to regard the boundary of the open string as an object in p-adic geometry. In traditional literature on p-adic string theory it is usually stated that the generalization of the p-adic theory to closed strings remains unclear, since it remains unclear which adic version of the complex numbers (parameterizing the interior of the string worldsheet) one should use.

But now by the above, in algebraic geometry there is in fact an obvious concept at least of adic genus-1 closed string worldsheets: these are just the elliptic curves over the p-adics. And moreover, since the Witten genus, being the partition function, is a particular case of a string scattering amplitude, it follows by the above story of the String-orientation of tmf that making this identification indeed does relate closed p-adic string worldsheets to actual physical closed string scattering (at least to the 1-loop vacuum amplitude).

In conclusion, it seems that there should be a close interrelation between p-adic string theory and the refinement of the superstring partition function to the string orientation of tmf.

My question, finally, is: has anyone made this close interrelation more explicit? Is there anything in the literature that makes this connection? Or else, if unpublished, has anyone seriously thought about this?
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I hope someone answers your question!  Thinking a bit more, it seems worthwhile to try adelic versions of many physical theories, from the quantum harmonic oscillator, to general free field theories, to topological quantum field theories or perturbative interacting quantum field theories.  I think I've seen some work on an adelic quantum harmonic oscillator, but it didn't look very exciting.   Since the Chern-Simons TQFT built starting from a simple Lie group is a wonderful collection of interesting algebraic ideas, and adelic versions of algebraic groups seem to be important in the Langland's program, there might be nice things to learn from a version where the complex numbers are replaced by adeles.  Maybe someone is already studying this....
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Esoteric questions of Computability of fundamental Physics have recently been in the mass media a bit. The mathematical underpinning is (or should be) computability theory viz the realizability interpretation of constructive mathematics.

(See the beautiful lecture notes by +Andrej Bauer  behind the above link.)

But looking through the literature, I see only a single document from rather recently that begins to seriously address what seems to be the central foundational issue here: the "computable analysis" (or not) in the foundations of quantum mechanics. That document is

Thomas Streicher, "Computability Theory for Quantum Theory", talk at Logic Seminar Univ. Utrecht in July 2012 (

The following is what might be regarded as a brief lead-in to that. Pointers to relevant literature that I am missing are welcome.

What follows is taken from an nLab entry that I am writing:

[ Thanks to the following discussion, the nLab entry now contains a good bit more information. Better see there. ]


The following idea or observation or sentiment has been expressed independently by many authors. We quote from section 2 of

Matthew Szudzik, "The Computable Universe Hypothesis", in Hector Zenil (ed.) "A Computable Universe: Understanding and Exploring Nature as Computation", World Scientific, 2013, pp. 479-523_ (

"The central problem is that physical models use real numbers to represent the values of observable quantities, [...] Careful consideration of this problem, however, reveals that the real numbers are not actually necessary in physical models. Non-negative integers suffice for the representation of observable quantities because numbers measured in laboratory experiments necessarily have only finitely many digits of precision."

Diverse conclusions have been drawn from this. One which seems useful and well-informed by the theory of computability in mathematics is the following (further quoting from Szudzik 10, section 2)

"So, we suffer no loss of generality by restricting the values of all observable quantities to be expressed as non-negative integers — the restriction only forces us to make the methods of error analysis, which were tacitly assumed when dealing with real numbers, an explicit part of each model."

There are two main kinds of computable functions in computabiliy theory

In type-I computability the computable functions are partial recursive functions and in view of this some authors conclude (and we still quote Szudzik 10, section 2) for this:

"To show that a model [ of physics ] is computable, the model must somehow be expressed using recursive functions."

However, in computability theory there is also the concept of type-II computable functions used in the field of “constructive analysis”, “computable analysis”.

This is based on the idea that for instance for specifying computable real numbers as used in physics, an algorithm may work not just on single natural numbers, but indefinitely on sequences of them, producing output that is in each step a finite, but in each next step a more accurate approximation.

This concept of type-II computability is arguably closer to actual practice in physics.

Of course there is a wide-spread (but of course controversial) vague speculation (often justified by alluding to expected implications of quantum gravity on the true microscopic nature of spacetime and sometimes formalized in terms of cellular automata, e.g. Zuse in 67) that in some sense the observable universe is fundamentally “finite”, so that in the end computability is a non-issue in physics as one is really operating on a large but finite set of states.

However, since fundamental physics is quantum physics and since quantum mechanics with its wave functions, Hilbert spaces and probability amplitudes invokes (functional) analysis and hence “non-finite mathematics” even when describing the minimum of a physical system with only two possible configurations (a “qbit”) a strict finitism perspective on fundamental physics runs into severe problems and concepts of computable analysis would seem to be necessary for discussing computability in physics.

This issue of computable quantum physics has only more recently been considered, in

Thomas Streicher, "Computability Theory for Quantum Theory", talk at Logic Seminar Univ. Utrecht in July 2012 (

where it is shown that at least a fair bit of the Hilbert space technology of quantum mechanics/quantum logic

sits inside what is called the function realizability topos
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I just assume QRNGs produce sequences that are "truly random" (no deterministic program can make them), so that's why I called them "trivial".

What could be more of interest is finding uncomputable tasks:

In the "some uncomputable quantum mechanical tasks" paper above, these are:
· Ground energies are uncomputable
· Halflives are uncomputably large
· There is no computable bound on the adiabatic time

"These are the first examples we know of where uncomputably large numbers arise naturally in theoretical physics."
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Reimagining the Foundations of Algebraic Topology There were nice talks at the MSRI meeting a few days back, and well-done video recordings are available on the website:

Thomas Nikolaus recalled the new cohesive foundations of differential cohomology∞,1)-topos


David Gepner recalled the general theory of integration in twisted generalized cohomology


 Goncalo Tabuada recalled the theory of noncommutative motives


David Ben-Zvi spoke on ideas about formulating the 6d field theory on the M5-brane,0)-supersymmetric+QFT


André Joyal recalled how to think of homotopy type theory as a definition of elementary infinity-toposes,1)-topos


and much more. 
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As you've stressed on another occasion, Urs,
there are anomalies that are not easy to see. The relation between the Hamiltonian and time evolution and the Lagrangian and the transtion amplitudes hinges, essentially, on this issue, which involves the measure on the space of fields, beyond the classical equations of motion.
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Kriz-Sati conjecture on modular equivariant charges in F-theory 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?

This question equipped with hyperlinks that lead to more background information on all keywords used I have posted to PhysicsOverflow here:

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 (←the B-field) KR-theory ("real K-theory"). A comprehensive statement of this has been given fairly recently in 

Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)
with techncial details in 

Daniel Freed, Lectures on twisted K-theory and orientifolds, lecures at K-Theory and Quantum Fields, ESI 2012 (pdf)

What happens here mathematically is ​that the complex K-theory spectrum KU has a canonical Z2-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  Z2-action, such that in computing cocycles both these  Z2-actions are intertwined -- just as known from orientifold strings. 

Moreover, a recent observation in the appendix of 

Tyler Lawson, Niko Naumann, Strictly 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  Z2-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

Ashoke Sen, F-theory and Orientifolds (arXiv:hep-th/9605150)

Ashoke Sen, Orientifold Limit of F-theory Vacua (arXiv:hep-th/9702165)

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

Igor Kriz, Hisham Sati, Type II string theory and modularity, JHEP 0508 (2005) 038 (arXiv:hep-th/0501060)
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 SL2(Z/2Z) 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

Michael Hill, Tyler Lawson, Topological modular forms with level structure (arXiv:1312.7394)

Moreover, by theorem 9.3 there, expanding on the earlier

Tyler Lawson, Niko Naumann, Strictly 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 Bandos, Superembedding 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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New PhysicsOverflow forum for research-level physics discussion 

A new site for online discussion of research-level physics is now up and running here:

PhysicsOverflow is meant to be some kind of a rebirth of the untimely passed away Theoretical Physics SE ( and a little physics brother of the very successful MathOverflow

Compared to the former theoretical physics site, the organizers have slightly lowered the bar to ask questions to graduate-level upward and broadened the scope to include experimental physics and phenomenology.

Apart from the high-level Q&A, PhysicsOverflow will offer in the future a Reviews section

dedicated to discussion and peer review (mostly ArXiv but other sources can be considered too) papers publicly and "in real time".

(The above text is slightly adapted but essentially copied from the announcement here: )
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Dear Urs,

thanks again very much for announcing PhysicsOverflow here !

+Igor Khavkine Khavkine : The administrators are not that important. At present, they are just the small group of people who have technically set up the site. In the long run, they can be considered as technical stuff that helps people with technical issues, such as loging in etc ...

Good moderators are still wanted, in an optimal world they might be Moshe / Piotr Migdal / and Joe Fitzsimons , etc ...

As Urs said elsewhere, the most important thing now is to get a good community and nice physics discussions seriously started.

Any stories etc  that happend outside PhysicsOverflow elsewhere should stay there, they are of no relevance for the new site.

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Back In 1998 I wrote and programmed... something. That something has now been archived by the German Literature Archive in Marbach:
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Here is the entry point more explicitly:

Use your mouse cursor to explore, click on stuff.

If you feel you got stuck, go to the radar and proceed from there

All at your own risk.
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A Workshop on Higher Gauge Theory and Higher Quantization will take place this June 26/27 in Edinburgh, organized by +Christian Saemann  , see here

(higher gauge theory:

(higher quantization:
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Hi Urs, thanks for posting this! I keep forgetting about social media.
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