ACT A - Why Calabi-Yau Manifolds?
In a previous Quill, we saw that one has to do compactifications when dealing with theories in higher dimensions, examples include supergravity (or M-theory) in $D=11$ or superstrings in $D=10$, or the very classic Kaluza-Klein compactification from $D=5$ to $D=4$. We do these compactifications on a very special kind of manifolds for reasonings that are more apt for physics than mathematics.
Let's say we take $10$-dimensional theory and compactify on K like $M_4 \times K$ where $K$ is really a $(3,1)$ manifold of space-time and $K$ is a six-dimensional (compact) Riemannian manifold. The dimensions of $K$ are very macroscopic and $K$ will serve us for three things: 1) Supersymmetry, 2) effective low-energy limit physics, and 3) zero-curvature (Ricci flat). It is thus that $K$ has to be very special, especially from a physicist's point of view. We want to preserve the $\mathcal{N}=1$ (the bare and the simplest number of SUSY) in $M_4$ and we want these $K$ to also provide massless modes of interest. In particular, the holonomy group of the metric of $K$ must be contained in $u(n)$, and in particular, it is of $SU(3)$ holonomy. (It is important for many reasons, to name one, it is crucial for the unbroken supersymmetry in $M_4$.)
Now, there are some good mathematical consequences of these, which were realized by physicists in an amazing period of the 80s following works in algebraic geometry and differential geometry.
- $K$ has a complex structure and its first Chern class vanishes. The metric then becomes Kähler for this. Also, the first Betti number vanishes, $b_1(K)=0$.
- $K$ is Ricci-Flat (that does not necessarily mean the Riemann curvature is zero) and its Ricci tensor would vanish. This implies that the restricted holonomy of the Ricci-Flat Kähler manifold, which is our $K$, has to be in $SU(n)$ (see this or this).
Our $K$ is called the Calabi-Yau manifold (to honor the two greats Eugenio Calabi and Shing-Tung Yau), which is a Ricci-Flat Kähler manifold. Now, what about $\mathcal{N}=2$? We still have Calabi-Yau manifolds, but they have some deformations.
To see how ML is working with the CY manifolds (which exist in thousands of numbers), see this article by Kalvakota here.
One more thing of interest is to notice that the isometry group of $K$ is $G$ (which is $SU(3)\times SU(2)\times U(1)$ for our SM), but if the second Betti-number is non-zero, $b_2(K) >0$, then there are some extra fields and the full gauge group is $G \times \mathbb{R}^{b_2}$.
One more thing of interest is to notice that the isometry group of $K$ is $G$ (which is $SU(3)\times SU(2)\times U(1)$ for our SM), but if the second Betti-number is non-zero, $b_2(K) >0$, then there are some extra fields and the full gauge group is $G \times \mathbb{R}^{b_2}$.
ACT B - Topological Quantum Mechanics and SUSY Algebra
This one is rather on a more mathematical side.
Quantum mechanics is just one-dimensional quantum field theory. In quantum mechanics, one does a lot with Hamiltonian $H$ and the Hilbert space $\mathcal{H}$. $H$ is contained in $\mathcal{H}$, or more appropriately in the algebra. We can now find the time evolution for a state in $\mathcal{H}$ by $$e^{iHt}$$ and something is time invariant if it commutes with $H$. In topological quantum mechanics, we want to, in a sense, kill the time. This can be done, trivially, by setting $H=0$ which limits the eigenspace to a zero eigenspace (which contains only the ground state). It is, however, not interesting in the sense that we do not have much of a theory.
A good alternative is to kill time in a derived sense. This would employ the Hodge theory (for Riemannian manifolds) and cohomology. We can, instead of taking an infinite-dimensional $\mathcal{H} = L^2(M)$ Hilbert space of $L^2$ functions on a manifold $M$, take a space of differential forms $$L^2(p)$$ where $p$ are differential forms. Or more excitingly, attach a complex chain $$(\Omega^\circ, X)$$ where the differential forms are $U(1)$ graded. Doing this adds extra fields to the theory. We can identify, now this has become a SUSY theory, the supercharges $Q,Q^*$ (for $\mathcal{N}=1$ SUSY). These will be our de Rham differential operator $Q=d$ and $Q^*=d^*$ (if you can guess now then we want to pass into cohomology eventually). As in a SUSY algebra, $$[Q,Q] = [Q^*,Q^*] = 0$$ $$[Q, H] = [Q^*,H]=0$$ but $$[Q,Q^*]= H.$$
Quantum mechanics is just one-dimensional quantum field theory. In quantum mechanics, one does a lot with Hamiltonian $H$ and the Hilbert space $\mathcal{H}$. $H$ is contained in $\mathcal{H}$, or more appropriately in the algebra. We can now find the time evolution for a state in $\mathcal{H}$ by $$e^{iHt}$$ and something is time invariant if it commutes with $H$. In topological quantum mechanics, we want to, in a sense, kill the time. This can be done, trivially, by setting $H=0$ which limits the eigenspace to a zero eigenspace (which contains only the ground state). It is, however, not interesting in the sense that we do not have much of a theory.
A good alternative is to kill time in a derived sense. This would employ the Hodge theory (for Riemannian manifolds) and cohomology. We can, instead of taking an infinite-dimensional $\mathcal{H} = L^2(M)$ Hilbert space of $L^2$ functions on a manifold $M$, take a space of differential forms $$L^2(p)$$ where $p$ are differential forms. Or more excitingly, attach a complex chain $$(\Omega^\circ, X)$$ where the differential forms are $U(1)$ graded. Doing this adds extra fields to the theory. We can identify, now this has become a SUSY theory, the supercharges $Q,Q^*$ (for $\mathcal{N}=1$ SUSY). These will be our de Rham differential operator $Q=d$ and $Q^*=d^*$ (if you can guess now then we want to pass into cohomology eventually). As in a SUSY algebra, $$[Q,Q] = [Q^*,Q^*] = 0$$ $$[Q, H] = [Q^*,H]=0$$ but $$[Q,Q^*]= H.$$
The Hamiltonian acts on the differential forms $p$ as a Laplace operator, $H=\Delta$. Then one writes $H$ in the form of these supercharges. Also, given $[Q,Q] =0$, one has $d^2=0$ (which is quite a hint for our pass to homology, since $d$ becomes a boundary operator.). If one takes the de Rham cohomology (or Q-cohomology) $H^*(M)$, then Hamiltonian $H$ acts as $H=0$ (and thus the zero eigenvalues), and for the complex chain $(\Omega^\circ, M)$, one has $H$ homotopic to zero.
In this sense, we killed the time, but in a derived sense. In the process, we have gained a bigger algebra in the form of $\mathcal{N}=1$ SUSY algebra (some super Lie-group). Similarly, one can find theories with a larger number of SUSY supercharges. For a good reference on this, see Witten's paper and the notes on E-M and Langlands by Ben-Zvi.
