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Let $X$ be a manifold with $G_2$ holonomy and $\Phi$ be the fundamental associative 3-form on $X$. Let $*\Phi$ be the dual co-associative 4-form on $X$. Now consider a particular associative 3-cycle $Q\in X$, which is Poincare dual to the co-associative 4-form $*\Phi$, i.e. $*\Phi=\lambda\times PD(Q)$, where $\lambda\in R_{+}$. Is there a standard mathematical notation to describe such an associative 3-cycle? Are there any topological restrictions on the properties of such a 3-cycle, e.g. can it be rigid ($S^3$ or $S^3/Z_N$)?

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The cyclic action stems from the role of the $3$- form of $G_2$ on a seven dimensional space. In general we may consider the manifold $M^4\times M^7$, where $M^4$ may be any spacetime $(M, g)$, which for explicit purposes is $AdS_4$. The “dual system” with $AdS_7$ can follow accordingly. A four dimensional field is constructed which for $N$-Killing spinors is $N$ supersymmetric for a $G_2$ holonomy on $M^7$. The exceptional group $G_2$ is the automorphism group of the octonions $o~=~x_0I~+`x_ae_a$ for the basis elements $e_a$ obeying the algebra $$ e_ae_b~=~-\delta_{ab}~+~\omega_{abc} e^c $$ where the tensor $\Omega_{abc}$ is determined by products of three octonionic elements which are $G_2$ invariant. This is the tensor component of a three-form $\Omega$ which is expanded according to elements on the $M^7$ as $$ \Omega~=~{1\over 3!}\omega^{abc}e_a\wedge e_b\wedge e_c $$ $$ =~e_1\wedge e_2\wedge e_3~+~e_4\wedge e_3\wedge e_5~+~e_5\wedge e_1\wedge e_6~+~e_6\wedge e_2\wedge e_4~+~e_4\wedge e_7\wedge e_1~+~e_5\wedge e_7\wedge e_2~+~e_6\wedge e_7\wedge e_3 $$ $$ =~e_1\wedge e_2\wedge e_3~+~(1/2)e_i\wedge e_m\wedge J_{i mn}e_n $$ so the spin tensor $J_{i mn}$ has the element $i~=~1, 2, 3$ and $m,~n~=~ 4, 5, 6, 7$. The second line of this equation is equivalent to the projective Fano plane, or the multiplication rule for octonions. This is an aspect of the alternativity of the octonions which define triplets of quaternions as seen in the index $i$.. The product of the octonionic elements means the product of spin tensors obeys $$ J_i\cdot J_j~=~-\delta_{ij}~+~\epsilon_{ijk}J_k. $$ By definition the $G_2$ holonomy means the $3$-form is closed $d\Omega~=~0$, and the Hopf fibration $S^3~\hookrightarrow~S^7~\rightarrow~S^4$ induces a symmetry between elements in seven dimensions so $d*\Omega~=~0$. In addition for the spin connection $\sigma^{ab}$, the projection with the tensor is zero $\omega_{abc}\sigma^{ab}~=~0$. This means that $\Omega$ is covariantly constant, which is a condition it being a Killing spinor. This gives a set of first order differential equations for the metric elements, The existence of additional covariantly constant field-form restricts the $G_2$ holonomy so the Killing spinor equation has more than one solution and the $4$ dimensional field theory has extended $N~>~1$ supersymmetry

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I'm not aware of a standard notation. Associative 3-folds can be rigid, but need not be. Although it doesn't address your question in any kind of generality, associative 3-cycles can be understood quite easily for a special class of G2 manifolds of the form $(X \times S^1)/Z_2$ where $X$ is a Calabi-Yau 3-fold and the involution acts as $(\sigma_X,-1)$ where $\sigma_X$ is an anti-holomorphic involutive isometry of $X$. There are then "holomorphic" 3-cycles of the form $(\Sigma_2^- \times S^1)/Z_2$ where $\Sigma_2^-$ is a holomorphic curve mapped to $- \Sigma_2^-$ by $\Sigma_X$. This is rigid if $\Sigma_2^-$ is rigid. There are also "Lagrangian 3-cycles" of the form $\Sigma^+/Z_2$ where $\Sigma^+$ is a cycle mapped to plus itself by $\sigma_X$. An example of a rigid "Lagrangian 3-cycle" in a G2 manifold is given in Appendix C of hep-th/9907026.

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Thank you very much @Jeff Harvey ! At some point a while back I went through some parts of your paper you referred to. It's a really nice paper! My specific question was more about the properties of the associative cycle that is Poincare dual, up to a positive multiple, to $*\Phi$. From the responses I'm getting so far, it's not clear if there would be any general constraints on its properties such as rigid vs non-rigid, intersections with other associative cycles, etc. –  stringpheno Feb 14 '11 at 2:26
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A very sensible (and advanced) question. Yes, it is typical for 3-cycles to have the topology you mentioned. For example, this paper

http://arxiv.org/abs/hep-th/0011190

studies only associative 3-cycles that are either $S^3$ or orbifolds of it, or the hyperbolic 3-space $H^3$ and orbifolds of that. And in some cases, the spherical cycle is rigid. The text implicitly says that even with this topology, it doesn't have to be rigid, however. Also, other topologies than those listed above are almost certainly possible.

I don't think that there's a standardized notation to describe associative cycles of this kind. After all, the set of such cycles is not a linear space or a group or any meaningful algebraic structure like that. It's just a set. So you could only write that something is an element of that, or some union or intersection of this set, or another set is a subset, or vice versa, and these statements are simple enough to be expressed in words. But I don't know the terminology of everyone (and not even most people in that field), however.

A mathematical introduction to associative 3-cycles (aside from basic, the text studies the cycles in $R^7$), using "lemmas" and similar words unfamiliar to a physicist, is e.g.

http://www.math.hmc.edu/seniorthesis/archives/2001/iweiner/iweiner-2001-thesis.pdf
http://www.springerlink.com/content/8451j84w08j28432/

The first PDF is free, the second - more general review of all calibrated cycles - is not.

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Thank you @Luboš Motl ! Let me give you the context of my question by bringing an analogy with the Calabi Yau. For a CY manifold $Y$ there exists a particular type of a 4-cycle (divisor) $D\in Y$, called ample. By Poincare duality, a divisor $D$ that is ample is identified with a cohomology class represented by the Kahler form $J$ on $Y$, i.e. $J=\lambda\times PD(D)$, where $\lambda\in R_{+}$. It turns out that there is some tension between rigidity and ampleness, i.e. a generic ample divisor is not rigid and one has to work a bit harder to find examples where both properties are satisfied. –  stringpheno Feb 13 '11 at 20:00
    
Of course, I realize that the above analogy may not necessarily hold. Thus, my original question was whether an associative cycle $Q\in X$, which is Poincare dual to $*\Phi$, would also be classified/possibly restricted in some non-trivial way? –  stringpheno Feb 13 '11 at 20:09
    
Well, I would kind of agree that the analogy may not necessarily hold. ;-) In fact, there could be a better analogy that would be a little bit closer: the associative 3-cycles on G2 holonomy manifolds are "calibrated cycles", and "calibrated cycles" on Calabi-Yau three-folds include the special Lagrangian 3-cycles. So the more analogous Calabi-Yau object, I think, to the associative 3-cycles on G2 manifolds are the special Lagrangian 3-cycles. And they can be rigid, can't they? Isn't it possible to describe all the deformations from an equation? It won't be linked to a "Kähler class" for G2. –  Luboš Motl Feb 13 '11 at 21:12
    
What I want to say is that for G2 manifolds, the fundamental associative 3-form knows both about the metric as well as the "special structure". But when you talk about "associative cycles", you use one part of the information encoded in the 3-form, and it's the part that's analogous to the holomorphic 3-form $\Omega$ of the Calabi-Yau 3-folds, rather than the Kähler form that knows about the metric. So it's not the assoc. 3-cycles that should have a generalized "ampleness". There's no "complex geometry" in G2 manifolds so the ampleness can't work in the same way, however, anyway. –  Luboš Motl Feb 13 '11 at 21:17
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Thank you again @Luboš Motl ! I agree that the SLAGs in CY are much more analogous to the associative 3-cycles in $G_2$. In $G_2$ there is no complex geometry, holomorphic curves, etc. The only true analogue of the Kahler cone in the $G_2$ case are the positive multiples of the $G_2$ structure one already has. The example of the ample divisor was just to stress the point about the Poincare duality between $Q$ and $*\Phi$, which is sort of like the relation between $D$ and $J$ in the CY case but the proper mathematical term/classification for such a 3-cycle $Q$ is what I'm missing. –  stringpheno Feb 13 '11 at 23:35
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