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mitchell porter
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Last week I was considering a post called "Gordon Kane versus Sundance Bilson-Thompson", in which I was going to compare and contrast the "G2 Minimal Supersymmetric Standard Model" (G2-MSSM), a version of the MSSM, motivated by the study of M-theory on G2 manifolds, which Kane has been promoting, with the categorified version of Bilson-Thompson's braid correspondence by means of which Marni Sheppeard and collaborators hope to recover the standard model.
Let me describe very briefly how the G2-MSSM works. M theory has 11 dimensions, G2 manifolds have 7 dimensions, so M theory on a G2 manifold has 4 large space-time dimensions as required. In the compact 7-dimensional space, we focus on two nonintersecting singular 3-dimensional hypersurfaces. Each such surface has a nonabelian gauge theory living on it; then there are even more singular points on the surface, at which chiral fermions are located. These fermions within a surface interact with each other via the gauge fields inhabiting the surface, and the fields on the two surfaces also interact gravitationally with each other across the 7-dimensional bulk. One 3-surface gives us the visible world - this is where the realization of the MSSM is located - the other 3-surface is a hidden sector in which strong interactions break supersymmetry and stabilize the shape of the 7-manifold.
The G2-MSSM research program does not aspire to predict most of the parameters of the standard model, at least not in the foreseeable future. It is concerned with embedding something that looks like the standard model (the MSSM) into string theory, in a way which, first of all, doesn't just contradict reality, and second, makes some testable predictions for beyond-standard-model physics, such as a light gluino. The reason it doesn't aspire to explain, e.g., the mass of the electron is that all those predictions depend on the exact details of the G2 manifold, and, like Calabi-Yau spaces (only worse), specific G2 manifolds are hard to construct. So much of the substance of the G2 phenomenology program involves deducing generic consequences of a whole class of G2 manifolds.
In contrast, one of the defining features of Sheppeard et al's program is that it works directly with the particle masses and mixing matrices. The starting point there is Carl Brannen's version of the Koide formula, connecting the masses of the charged leptons. Using matrices called circulants, Brannen and Sheppeard have developed parametrizations or other formulations of all the standard model masses and mixing matrices, and then Sheppeard wants to obtain these values from the theory of motives, applied to Bilson-Thompson's braids (considered as category-theoretic morphisms rather than as Wilson-loop-like geometric objects), using Michael Rios's work on Jordan-algebra matrix models. In this picture, space-time geometry is totally emergent from an algebraic structure - the motivic association of quantum numbers with braided morphisms.
What I had wanted to do, in a post like this one, is to build a bridge between the two research programs. I don't necessarily mean that I want to combine them, although their approaches to standard model physics (explain the parameters, explain everything else) are complementary. It was more that I wanted to feel out the space between them, perhaps identify a superclass of models to which they both belong. For example, on the algebraic end, they both employ octonion-related structures.
But I believe the biggest barrier to comparison and communication has always been the role of the Bilson-Thompson braid correspondence. There have been a few loop-quantum-gravity papers trying to embed the braids in a dynamical system, but such efforts are preliminary at best, and hardly begin to meet the criticisms that Lubos Motl made when these ideas first received popular attention: how do you get Lorentz invariance, how do you get gauge invariance, how do you get fermionic statistics, how do you imbue braids with all the features of actual quantum particles? Sheppeard, incidentally, seems to have deferred this issue, by focusing instead on producing a combinatorial formulation of twistor gauge theory, with the hope that the precise roles of the braids will be clarified when gravity is included.
Anyway, yesterday a paper came out which demolishes the idea that you can't associate a functioning field theory with a set of braids: http://arxiv.org/abs/1110.2115" , by Cecotti, Cordova, and Vafa. This is a paper about three- and four-dimensional field theories with N=2 supersymmetry, that can be obtained from M5-branes wrapped on special subsurfaces in a Calabi-Yau space. (Many such setups probably have a dual description in terms of a G2 manifold, by the way.) In particular, they talk a lot about situations where two coincident M5-branes fuse along a knot-shaped path in the large dimensions, so that you actually have a single doubly wrapped M5-brane. You then consider circles inside the volume of the M5-brane, which act as the boundary of a disk-shaped M2-brane stretching through the bulk space outside the hypersurface on which the M5-brane is wrapped. The topologically distinct classes of "1-cycle" (embedded circle) in the M5-brane correspond to different M2-brane states and thus to different particles in the field theory limit.
Cecotti et al then consider situations in which the topology of the M5-brane's self-fusion changes along a spacelike direction. The knot describing the M5-brane's self-contact is different on either side of this transition. The transition is described by a braid, there is a domain wall at the transition zone on which certain excitations get trapped, and their properties can be read off the braid. There is a field theory describing the interaction of those excitations within the domain wall, and that is the field theory canonically associated with the braid.
As you can tell, the constructions are quite intricate, and conceptually rather deep. For example, if you want to know what it is that gets braided, apparently it's the "zeroes of the Seiberg-Witten differential" of the domain-wall field theory. Nonetheless, I think this paper will be of great interest to LQG researchers, because it's full of objects - Pachner moves, tetrahedral decompositions - that they have employed elsewhere, e.g. in spin foams. And when we get to the algebra, it features motivic quantities, like Goncharov's quantum polylogarithms, that have been showing up in the twistor uprising.
This is still not a complete vindication of Bilson-Thompson's approach. That domain-wall field theory is 2+1 dimensional, and the particles in the spectrum aren't individually associated with braids; the details are more complicated. But it's not unimaginable that a 3+1 dimensional extension of these ideas, perhaps one in which the three space dimensions are decomposed into simplices, each of which has a "2-3 wall-crossing" associated with it, could look like Bilson-Thompson's scheme. And certainly this is a new beginning for the overall idea of obtaining real-world physics from topological objects.
Let me describe very briefly how the G2-MSSM works. M theory has 11 dimensions, G2 manifolds have 7 dimensions, so M theory on a G2 manifold has 4 large space-time dimensions as required. In the compact 7-dimensional space, we focus on two nonintersecting singular 3-dimensional hypersurfaces. Each such surface has a nonabelian gauge theory living on it; then there are even more singular points on the surface, at which chiral fermions are located. These fermions within a surface interact with each other via the gauge fields inhabiting the surface, and the fields on the two surfaces also interact gravitationally with each other across the 7-dimensional bulk. One 3-surface gives us the visible world - this is where the realization of the MSSM is located - the other 3-surface is a hidden sector in which strong interactions break supersymmetry and stabilize the shape of the 7-manifold.
The G2-MSSM research program does not aspire to predict most of the parameters of the standard model, at least not in the foreseeable future. It is concerned with embedding something that looks like the standard model (the MSSM) into string theory, in a way which, first of all, doesn't just contradict reality, and second, makes some testable predictions for beyond-standard-model physics, such as a light gluino. The reason it doesn't aspire to explain, e.g., the mass of the electron is that all those predictions depend on the exact details of the G2 manifold, and, like Calabi-Yau spaces (only worse), specific G2 manifolds are hard to construct. So much of the substance of the G2 phenomenology program involves deducing generic consequences of a whole class of G2 manifolds.
In contrast, one of the defining features of Sheppeard et al's program is that it works directly with the particle masses and mixing matrices. The starting point there is Carl Brannen's version of the Koide formula, connecting the masses of the charged leptons. Using matrices called circulants, Brannen and Sheppeard have developed parametrizations or other formulations of all the standard model masses and mixing matrices, and then Sheppeard wants to obtain these values from the theory of motives, applied to Bilson-Thompson's braids (considered as category-theoretic morphisms rather than as Wilson-loop-like geometric objects), using Michael Rios's work on Jordan-algebra matrix models. In this picture, space-time geometry is totally emergent from an algebraic structure - the motivic association of quantum numbers with braided morphisms.
What I had wanted to do, in a post like this one, is to build a bridge between the two research programs. I don't necessarily mean that I want to combine them, although their approaches to standard model physics (explain the parameters, explain everything else) are complementary. It was more that I wanted to feel out the space between them, perhaps identify a superclass of models to which they both belong. For example, on the algebraic end, they both employ octonion-related structures.
But I believe the biggest barrier to comparison and communication has always been the role of the Bilson-Thompson braid correspondence. There have been a few loop-quantum-gravity papers trying to embed the braids in a dynamical system, but such efforts are preliminary at best, and hardly begin to meet the criticisms that Lubos Motl made when these ideas first received popular attention: how do you get Lorentz invariance, how do you get gauge invariance, how do you get fermionic statistics, how do you imbue braids with all the features of actual quantum particles? Sheppeard, incidentally, seems to have deferred this issue, by focusing instead on producing a combinatorial formulation of twistor gauge theory, with the hope that the precise roles of the braids will be clarified when gravity is included.
Anyway, yesterday a paper came out which demolishes the idea that you can't associate a functioning field theory with a set of braids: http://arxiv.org/abs/1110.2115" , by Cecotti, Cordova, and Vafa. This is a paper about three- and four-dimensional field theories with N=2 supersymmetry, that can be obtained from M5-branes wrapped on special subsurfaces in a Calabi-Yau space. (Many such setups probably have a dual description in terms of a G2 manifold, by the way.) In particular, they talk a lot about situations where two coincident M5-branes fuse along a knot-shaped path in the large dimensions, so that you actually have a single doubly wrapped M5-brane. You then consider circles inside the volume of the M5-brane, which act as the boundary of a disk-shaped M2-brane stretching through the bulk space outside the hypersurface on which the M5-brane is wrapped. The topologically distinct classes of "1-cycle" (embedded circle) in the M5-brane correspond to different M2-brane states and thus to different particles in the field theory limit.
Cecotti et al then consider situations in which the topology of the M5-brane's self-fusion changes along a spacelike direction. The knot describing the M5-brane's self-contact is different on either side of this transition. The transition is described by a braid, there is a domain wall at the transition zone on which certain excitations get trapped, and their properties can be read off the braid. There is a field theory describing the interaction of those excitations within the domain wall, and that is the field theory canonically associated with the braid.
As you can tell, the constructions are quite intricate, and conceptually rather deep. For example, if you want to know what it is that gets braided, apparently it's the "zeroes of the Seiberg-Witten differential" of the domain-wall field theory. Nonetheless, I think this paper will be of great interest to LQG researchers, because it's full of objects - Pachner moves, tetrahedral decompositions - that they have employed elsewhere, e.g. in spin foams. And when we get to the algebra, it features motivic quantities, like Goncharov's quantum polylogarithms, that have been showing up in the twistor uprising.
This is still not a complete vindication of Bilson-Thompson's approach. That domain-wall field theory is 2+1 dimensional, and the particles in the spectrum aren't individually associated with braids; the details are more complicated. But it's not unimaginable that a 3+1 dimensional extension of these ideas, perhaps one in which the three space dimensions are decomposed into simplices, each of which has a "2-3 wall-crossing" associated with it, could look like Bilson-Thompson's scheme. And certainly this is a new beginning for the overall idea of obtaining real-world physics from topological objects.
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