**Orbifolds by R-parity could be neat, too**

*Optimized for the mobile template*

Senior particle phenomenologist John Ellis was heard as saying the proposition about supersymmetry (SUSY) in the title and your humble correspondent agrees with that, whether or not he has actually said it.

If it is turning out that SUSY is broken near the Planck scale, it's just fine with me. I would still find a theory with such a high-scale SUSY breaking to be more likely than a theory with no SUSY at all. String theory simply requires SUSY at "some level". Even if lots of apparent fine-tuning (in the Higgs mass and the cosmological constant) remains when SUSY is broken at a higher scale, SUSY still improves the situation relatively to no SUSY. And SUSY makes it more likely that the coupling unification is achieved. The dark matter candidate is another reason.

Sometimes, I was trying to prove or disprove the assertion that a consistent theory of quantum gravity – or any fully consistent string vacuum – restores SUSY at energy scales above the Planck scale. This claim may seem potentially vacuous but it does say something about the spectrum of the black hole microstates. They should see a restored SUSY in some way.

At the same moment, experiments indicate that the superpartners are harder to see. It just happens that within a supersymmetric standard model, we have seen the particles with the R-parity \(P_R=+1\) while we have seen no particles with \(P_R=-1\). An unusual idea that I would often find intriguing would say that there exists a straightforward stringy SUSY breaking that makes all \(P_R=-1\) particles significantly heavier than the known \(P_R=+1\) particles.

Adopting an extreme rhetoric, we might say that we want to project out all the \(P_R=-1\) superpartners from the light spectrum. Do some solutions of string theory do such a thing? If they do, how?

A canonical way to eliminate one-half of the particle spectrum is the orbifolding. In the case of SUSY breaking, the canonical example is the Scherk-Schwarz compactification. You assume that the spacetime has a compact, circular spatial direction and you make all spacetime fermionic fields antiperiodic:\[

\psi(x^9+2\pi R) = -\psi(x^9)

\] while the bosonic fields remain periodic. That twist breaks SUSY. In the language of the momenta \(p^9\), we may say that the bosonic fields' momenta belong to \(\ZZ/R\) while the fermionic fields' momenta belong to \((\ZZ+1/2)/R\). The minimum half-integer is \(1/2\) which is strictly separated from zero, so \(1/R\) is the scale of SUSY breaking. SUSY gets broken at lower energies than that; it is restored at higher energies. We may say that the SUSY current still exists but it is antiperiodic in the \(x^9\) dimension which prevents us from defining the global supercharge by integration.

The Scherk-Schwarz mechanism eliminates all fermions. But to describe the observed particles, we don't really want to do it. We want to eliminate all the \(P_R=-1\) particles where the R-parity is\[

P_R = (-1)^{3(B-L)+2s}

\] where \(s\) is the spin. If the difference between the baryon and lepton number \(B-L\) were omitted, we would get the usual "Grassmann parity" that is relevant in the Scherk-Schwarz mechanism. But here, we want to modify the rules by \(B-L\). The hypothesis is that the real world could be a \(\ZZ_2\) orbifold of a nicely SUSY-preserving supersymmetric standard model by \(P_R\). In string theory, where this orbifolding breaks SUSY at the string scale, the states non-invariant under the group i.e. the states with \(P_R=-1\) get projected out.

But string theory adds another subtlety to the orbifolded theory, namely the twisted sector. The boundary conditions for the world sheet fields are twisted by the action of \(P_R\):\[

\lambda(\sigma+\pi) = P_R \lambda(\sigma) P_R^{-1}

\] So those fields \(\lambda(\sigma)\) that commute with the R-parity are periodic in the twisted sector; those that anticommute with \(P_R\) are antiperiodic.

In the case of the \(E_8\times E_8\) heterotic string in the RNS formulation, I think that the \(\psi^\mu(\sigma)\) right-moving fermions may effectively be assigned \(s=1/2\) due to the standard GSO projection acting on these fields. So the \(P_R\)-antiperiodic fields will include all 10 real right-moving fermions \(\psi^\mu(\sigma)\) as well as 6 real left-moving fermions \(\lambda^{1\dots 6}(\sigma)\); note that in total, we have 16 real fermionic fields that are "active" in the twisted sector. Why the latter six?

Well, the heterotic string has 32 real left-moving fermions \(\lambda(\sigma)\) that are divided to groups 16+16 assigned to the two \(E_8\) factors. And those 16 responsible for the visible \(E_8\) are divided to 10+6 for \(SO(10)\times SO(6)\) where the latter \(SO(6)\) is hidden. But we want to split the GUT \(SO(10)\) group to \(SO(6)\times SO(4)\) and it is the last \(SO(6)\) that mentions, one containing the QCD's \(SU(3)\), that produces the antiperiodic fermions.

What the light/massless spectrum exactly depends on the role played by the 6 "Calabi-Yau" compactified bosonic dimensions. But let's ignore them and try to extract the massless spectrum in 10 dimensions. What states that are "free to live in 10 dimensions" will be found at the massless level?

It may be fun to play some games with the "sum of integers". Recall that\[

1+2+3+4+\dots = -\frac{1}{12}

\] as we have shown in roughly 6 different ways. It's actually useful to generalize the identity. We also have\[

(n)+(n+1)+(n+2)+\dots = -\frac{1}{12} + \frac{n-n^2}{2}.

\] Some of the methods are sufficient to derive this more general identity, too. This generalized identity is very useful for the calculation of ground state energies in the presence of twisted free bosons or fermions. You may see that \(n-n^2\) vanishes for \(n=1\) as well as \(n=0\) so we actually also have\[

0+1+2+3+\dots= -\frac{1}{12}

\] which sounds consistent with the formula where the leading zero is omitted. Also, if we substitute \(n=m+1\), the sum \((m+1)+(m+2)+\dots\) may be calculated in two different ways. We either substitute \(n=m+1\) to the right hand side, the result, to get\[

(m+1)+(m+2)+\dots = -\frac{1}{12} +\frac{ (m+1)-(m+1)^2 }{2} =-\frac{1}{12}+\frac{-m-m^2}{2}

\] or we realize that the sum is the same thing as the original \((n)+(n+1)+\dots\) minus the initial \(n=m+1\) term, with \(n\) renamed as \(m\) at the end. Using this method, we get\[

(m+1)+(m+2)+\dots = -\frac{1}{12} + \frac{m-m^2}{2} - m

\] which is the same thing! So both results agree. It's useful to mention that the minimum of the original sum \((n)+(n+1)+\dots\) is realized for \(n=1/2\) where we learn that\[

\frac 12+ \frac 32 +\frac 52 +\dots = +\frac{1}{24}.

\] This particular sum may also be calculated as "one-half of the sum of odd numbers" which is "one-half of the sum of all positive integers" minus "one-half of the sum of all even integers". But the term we subtract is nothing else than the "sum of positive integers" so we learn that the "sum of odd half-integers" is \((1/2-1)\) times the "sum of integers" which is indeed \(+1/24\).

Note that the difference between the (twisted) "sum of odd half-integers" and the original "sum of integers" is\[

+\frac{1}{24} - \zav{ -\frac{1}{12} } =\frac{3}{24} = \frac{1}{8}

\] and it is no coincidence that this result, \(+1/8\), is also twice the dimension \(+1/16\) of the spin field for a single fermion in a CFT (or the twist field for a single boson). The spin fields and twist fields are the operator capable of mapping the periodic sector to the antiperiodic sector so they must increase the energy appropriately.

So let me return to the ground state energies in the heterotic string and the R-parity orbifolds.

In the light-cone gauge and the fermionic description, the heterotic string has 8 left-moving (transverse) bosons and 32 real fermions that produce the \(E_8\times E_8\) or \(SO(32)\) symmetry. If the fermions are periodic, the ground state energy is\[

\frac 12\times \zzav{ 8\times \zav{-\frac {1}{12} } -32 \times \zav{ -\frac{1}{12} } } = +1

\] where the minus sign in front of 32 was there because the fermions' ground state energy has the opposite sign. The ground state is already massive (the result is positive) so we don't get any massless states.

What about the sectors with some antiperiodic \(\lambda\) fields? Well, everytime we switch a periodic fermion to an antiperiodic one, we subtract \(1/16\). So if all fermions are antiperiodic, we will get to \(-1\). If one-half i.e. 16 out of the 32 real fermions are antiperiodic, we get exactly to the massless level, \(0\). You may see that if you choose just 6 antiperiodic fermions, you get to bizarre ground state energies such as \(+1-6/16=5/8\) etc. If you switch the periodicity of the fermions to the opposite one relatively to what I just mentioned, the ground state energy is \(-5/8\). That's dangerous because one might get a tachyon at \(-1/8\) by a single excitation of a \(1/2\) mode of a fermion.

I leave it as homework for you to find out what would be the representation in which these states would be transforming. Note that in general, perturbative string theory only allows tachyons if they're scalar particles; this may also be derived from some consistency considerations in quantum field theory. Can these "slight" tachyons at \(-1/8\) be stabilized? Aren't they the GUT-breaking Higgs fields?

At the level of string theory, such orbifolds would break SUSY "completely", just like type 0 string theories. But the SUSY breaking is still "at the string scale" which may be parameterically smaller than the Planck scale (if the string coupling is much smaller than one). There should exist a way to choose the orbifolds so that the SUSY breaking is in some sense "much weaker" than in type 0 string theories but it's just a speculation to suggest that something like that could be true.

I want to return to the claims that don't depend on unproven mathematical conjectures: SUSY theories and non-SUSY theories should be assigned comparable prior probabilities and with this unbiased perspective, the evidence still favors the SUSY theories even in the absence of observations that directly exclude the non-SUSY theories (direct observation of superpartners etc.).

*This blog post won't be proofread because I don't expect too many people to seriously read it.*

**A different topic: Two papers on emergent black-hole interior**

In recent days, two new papers on the Raju-Papadodimas strategy to study the black hole interior have been released. Lowe and Thorlacius are of course well-known names in the research of quantum black hole physics. I sort of sympathize with their words but I find the amount of maths – especially maths of quantum physics – in their paper to be too low to understand what they exactly mean. It's mostly words about the classical spacetime diagrams etc. So if they say something that goes beyond e.g. Papadodimas-Raju, I am not quite able to decode it. They agree that the interior local description is approximate and emergent which is OK but not too new. Thorlacius in particular may have written some ideas long before others but the formulations may have been vague enough to decide whether he was really the originator of the ideas that may be formulated sharply – but only using mathematics.

Lashkari and Simon seem to be much more specific. There are lots of equations, quantum states, and density matrices in the paper. Again, I sympathize and their claim seems to be somewhat novel: local operators are exactly those that are unable to distinguish pure black hole microstates from the maximally entangled states. Those that can distinguish them are either requiring huge energies or long times so they violate the locality of the experiments, kind of. Generally, it's fine. I have some problems with the "maximally"; a "maximally balanced" mixture of black hole microstates is a particular state, the empty black hole of a sort, isn't it? So I feel that they prohibited the local, large, coarse-grained perturbations by local operators as well. It seems to me that they have thrown the baby out with the bath water but maybe I don't understand something.

At any rate, I think that the general philosophy in both of these papers is right. There's some Hilbert space of black hole microstates and the question is whether the operator algebras for the local black hole field operators may be embedded into the Hilbert space, how it acts, how it's state-dependent, how accurately different descriptions like that agree with each other, and so on. I would creadit Papadodimas and Raju with this general strategy to approach the black-hole information problems.

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