The Standard Model from SO(32) superstrings?

[lpetrich]

There are 5 types of supersymmetric string or superstring: I, IIA, IIB, HO, and HE.

Their low-energy limits have forms of supergravity, and some of them also have gauge fields. Here are their gauge symmetries:

IIA, IIB: No gauge field
I, HO: SO(32)
HE: E8*E8

The gauge fields are all in the algebras' adjoint representations:
SO(32): 496
E8*E8: (248,1) + (1,248)

Of these, E8*E8 has gotten a lot of study as a superset of the Standard Model's symmetry -- there are several paths from E8*E8 to the Standard Model.

But has there been much work on getting the Standard Model out of SO(32)?

I've tried doing that with my Lie-algebra code, and I find it difficult to get it in a fashion comparable to the E8*E8 breakdown. There, the first step is
E8 -> E6*SU(3)
248 -> (78,1) + (1,8) + (27,3) + (27*,3*)
where the E6 gauge fields are associated with a SU(3) singlet and the chiral multiplets with SU(3) triplets.

I can get the Standard Model if I skip this condition, however. I can also get Georgi-Glashow SU(5) or Pati-Salam SO(6)*SO(4), though not SO(10) or E6 -- I can't get a SO(10) spinor, only a vector and an adjoint.

 

[arivero]

SO(16), SO(18) etc were considered long time ago in the context of expanding SO(10) with some family symmetry, to explain generations. oF course then you do not need the SU(3) or the usual arguments for families from compactification.

In any case, I think that to use E8xE8 or SO(32) as gauge groups is not the right way. One you are in M-theory, you have enough dimensions (7 extra) to get the SM groups out from plain KK compactification, as it was seen by Witten from quotientin SU(3)xSU(2)xU(1)/SU(2)xU(1)xU(1) i.e. by a maximal subgroup.

SO(32) seems a kind of extra quantiszation step; it is sometimes written as SO((D/2)^2), and the fundamental object under the carpet seems to have D/2 flavours, not 32.

 

[mitchell porter]

With the SO(32) theories, to get chiral fermions you quotient the background space. e.g. http://arxiv.org/abs/hep-th/0410232 http://arxiv.org/abs/hep-th/0603086 for orbifolds of heterotic SO(32), and numerous papers for orientifolds of the type I string - where the D-branes are crucial.

 

[lpetrich]

I've read that paper, and it's difficult for me to tell from it where the SO(10) spinors come from.

It's possible to get them from the adjoint rep of E8*E8:
E8 -> E6*SU(3)
: 248 -> (78,1) + (1,8) + (27,3) + (27*,3*)
E6 -> SO(10)*U(1)
: 27 -> (16,1/3) + (10,-2/3) + (1,4/3)
: 27* -> (16*,-1/3) + (10,2/3) + (1,-4/3)
: 78 -> (45,0) + (1,0) + (16,-1) + (16*,1)

But can one do anything similar for the adjoint rep of SO(32)?

One can get SO(10) from SO(32) by extension splitting, but one requires a SO(32) spinor to get a SO(10) one, and a SO(32) spinor is huge: 32768

However, I've discovered how to get a SO(10) spinor from a SO(32) vector.

Demote one of the fork-end roots (15 or 16):
SO(32) -> SU(16)*U(1)

Vector:
32 -> (16,1/2) + (16*,-1/2)
Traceless symmetric square of vector:
527 -> (255,0) + (136,1) + (136*,-1)
Adjoint: antisymmetric square of vector):
496 -> (255,0) + (1,0) + (120,1) + (120*,-1)


Map the fundamental onto the spinor:
SU(16) -> SO(10)

Fundamental:
16 -> 16 (spinor)
Fundamental conjugate:
16* -> 16* (spinor conjugate)
Symmetric square of fundamental and its:
136 -> 10 (vector) + 126 (self-dual antisymmetric 5th power of vector)
136* -> 10 (vector) + 126* (anti-self-dual antisymmetric 5th power of vector)
Antisymmetric square of fundamental and its conjugate:
120 -> 120 (antisymmetric cube of vector)
120* -> 120 (same)
Adjoint: fundamental * (fund conj) - (scalar)
255 -> 45 (adjoint) + 210 (antisymmetric 4th power of vector)
Adjoint of SO(10) = antisymmetric square of vector


Thus, this SO(32) -> SO(16)*U(1) reduction does:
32 -> (16,1/2) + (16*,-1/2)
527 -> (45,0) + (210,0) + (10,1) + (126,1) + (10,-1) + (126*,-1)
496 -> (45,0) + (210,0) + (1,0) + (120,1) + (120,-1)

Vector -> spinors -- elementary fermions
Symmetric traceless 2-tensor -> vectors -- Higgs particles
Adjoint -> adjoint -- gauge particles

Unlike the E8*E8 case, one does not get the SO(10) vectors or spinors from the adjoint.

 

[fzero]

There isn't any way to generate the 16 of SO(10) from open string perturbation theory. There aren't many details given, but http://arxiv.org/abs/arXiv:1001.0577 confirms this on pages 5 and 12. There are ways to generate spinor representations on D-branes in type II theories that are T-dual to type I, c.f. http://xxx.lanl.gov/abs/hep-th/9711098, but I think these are related to the enhanced symmetry points where you actually have an $E_6$. Heckman mentions the F-theory version of this on page 31 of that review.

 

[lpetrich]

I don't understand branes very well. :(

How might they work? Each point in them being given by x in the full space and y in the brane space, with x being a function of y?

So does field Lagrangian L_full(x) become L_full(x(y)) ~ L_brane(y) on them? Do branes get additional Lagrangian terms?

 

[mitchell porter]

D-branes are hypersurfaces where open strings can end. So if e.g. you get a U(N) gauge theory from a stack of N coincident branes, the gauge boson A_ij is a spin-1 state of an open string connecting a point on brane i with a point on brane j. (You can get other groups if the branes lie on the fixed point of an orbifold or an orientifold, since in those spaces the string spectrum is different from that on a smooth manifold.) Bear in mind that the endpoints can move, so the string can spin; they just can't move off the branes.

There is also a brane-centric perspective in which the strings are understood as quantum excitations of the branes. For example, displacement of a brane in the other directions of the full space creates scalar degrees of freedom. But I can't tell you how this works.

It occurs to me that an important string construction for you to consider might be the nonsupersymmetric SO(16) x SO(16) string. On this forum, both smoit and suprised have expressed an interest in this string vacuum. Keith Dienes has written a few papers promoting it. It seems to lie in a part of the "landscape" intermediate between the two SO(32) theories, and one might find non-susy SO(10) GUTs nearby.

 

[lpetrich]

There is also a brane-centric perspective in which the strings are understood as quantum excitations of the branes. For example, displacement of a brane in the other directions of the full space creates scalar degrees of freedom. But I can't tell you how this works.

Here's what may be going on. Let's construct a brane Lagrangian from a volume element, much like the Nambu-Goto action for strings. The brane's independent variables are $\xi$ (n of them), and its position variables are $x(\xi)$ (N of them). Construct $X_{ij} = \frac{\partial x}{\partial \xi^i}\cdot\frac{\partial x}{\partial \xi^j}$. The action is

$I = T\int \sqrt{\det{X}} d^n\xi$

with tension T. Now set n of the x's to $\xi$'s, and let the N - n remaining x's be small: y's, like the light-cone gauge for strings. The action becomes

$I = I_0 + \frac{T}{2}\int \frac{\partial y}{\partial \xi}.\frac{\partial y}{\partial \xi} d^n\xi + O(y^4)$

Doing Calculus of Variations gives for each component of y the equation $\nabla^2 y = 0$, the massless Klein-Gordon equation. Thus, the scalar degrees of freedom that you had mentioned.

But is there any action for branes that's comparable to the Polyakov action for strings?

 

[mitchell porter]

There's a huge literature on brane actions. It gets complicated because of supersymmetry, the diverse background spaces, and the noncommutativity which appears when you have more than one brane. This MSc thesis might be a place to start, and it begins with a Polyakov action for membranes. Clifford Johnson's "D-brane Primer" might be useful later on, but it's a lot more advanced, and presupposes a lot of string "lore".

I won't be much help, but if you're interested, I encourage you to pursue this direction, maybe with the interim objective of understanding intersecting braneworld models such as in Barton Zwiebach's textbook.

 

[arivero]

Suppose you are able to find the SM there in the HO theory. What do you expect to happen under T-duality to HE? And then, what under M-theoretical duality?

 

[lpetrich]

Thanx for that paper. It looks like I got it right. The Nambu-Goto action, the Polyakov action, and the light-cone gauge all carry over into membranes.


The Polyakov action features an induced metric, and one can construct the Riemann tensor from it and put it in the action.

For strings, one can add an integral over the Ricci scalar to the Polyakov action. It is a function of the topology of the string surface:
4*pi*(Euler characteristic) = 8*pi*(1-g)

where g is the genus of the surface (0 for sphere, 1 for torus, -1 for two spheres, etc.). It's easy to visualize the genus: it's the number of cuts necessary to give a surface the topology of a sphere. Negative genus means that one has to do the inverse of cutting: joining.


For an even number of dimensions, one can construct a version of the Euler-characteristic integral using a power of (number of dimensions)/2 of the Riemann tensor, something that one cannot do for an odd number of dimensions. For 4 and above, one can construct a different version of that power of the Riemann tensor that does not integrate into a topological invariant. How might such extra Riemann terms affect membrane dynamics?

 

[mitchell porter]

Suppose you are able to find the SM there in the HO theory. What do you expect to happen under T-duality to HE? And then, what under M-theoretical duality?

Stringy dualities translate to field-theoretic global symmetries, in the case of the maximal supergravities. And the eleventh dimension appears at strong coupling in IIA and HE. But I don't know how this would translate to an SM vacuum from HO strings. It might not even have a T-dual.

How might such extra Riemann terms affect membrane dynamics?

It sounds like a great question but I don't know the answer. Everyone focuses on finding the field-theory limits of various brane configurations. There's no perturbative theory of brane scattering, comparable to the topological expansion in string theory, that I can remember seeing. In fact M-theory doesn't have an expansion parameter. The stringy expansion parameter has some other origin in M-theory, see http://arxiv.org/abs/hep-th/0601141. Also see http://arxiv.org/abs/0707.1317 for how M2-M5 brane interactions are studied. The Basu-Harvey equation is a construction of basic importance here.