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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A very sensible (and advanced) question. Yes, it is typical for 3-cycles to have the topology you mentioned. For example, this paper 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.
The first PDF is free, the second - more general review of all calibrated cycles - is not. |
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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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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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