## Motivations for SUSY

The Jacobi theta function is defined as $$\theta(z, \tau) = \sum_{n \in \mathbb{Z} } e^{\pi i n^2 \tau + 2 \pi i n z}$$ and it transforms under $\theta(z+1, \tau) = \theta(z, \tau)$ and $\theta(z, -\frac{1}{\tau}) = (-i \tau)^{\frac{1}{2}} e^{\pi i z^2 \frac{1}{\tau}} \theta(z, \tau)$. Because it has a unique root $z = \frac{1}{2} (z+1)$ up to periodicity, using Weierstrass factorization theorem we can write it as: $$\theta(z, \tau) = \prod_{n=1}^{\infty} (1-q^n)(1+wq^{n-\frac{1}{2}})(1+\frac{1}{w} q^{n-\frac{1}{2}} )$$ where $w=e^{2 \pi i z}$ and $q = e^{2 \pi i \tau}$. We often need the asymptotic behavior of $\theta(z, \tau)$ as $q \rightarrow 0$ or as $q \rightarrow 1$. The former can be read immediately from either forms above, but the latter is not manifest in either of them, because an infinite number of terms contribute. It can be obtained from the modular transformation $\tau \rightarrow -\frac{1}{\tau}$, which relates the two limits by $q \rightarrow e^{\frac{2 \pi^2}{lnq}}$.

We can also define theta functions with characteristics as follows:

$$\Theta^{a}_{b}(z, \tau) := e^{\pi i a^2 \tau + 2 \pi i a (z+b)} \theta(z + a\tau + b, \tau)$$ $$= \sum_{n \in \mathbb{Z}} e^{\pi i (n+a)^2 \tau + 2 \pi i (n+a)(z+b)}$$. Then, we further define:

$$\theta_{00}(z, \tau) := \Theta^{0}_{0}(z, \tau) = \sum_{n \in \mathbb{Z}} q^{\frac{n^2}{2}}$$

$$\theta_{01}(z, \tau) := \ \Theta^{0}_{\frac{1}{2}} (z, \tau) = \sum_{n \in \mathbb{Z}} (-1)^n q^{\frac{n^2}{2}} z^n$$

$$\theta_{10}(z, \tau) := \Theta^{\frac{1}{2}}_{0} (z, \tau) = \sum_{n \in \mathbb{Z}} q^{(n-\frac{1}{2})^2} z^{n-\frac{1}{2}}$$

$$\theta_{11}(z, \tau) := \Theta^{\frac{1}{2}}_{\frac{1}{2}} (z, \tau) = -i \sum_{n \in \mathbb{Z}} (-1)^n q^{(n-\frac{1}{2})^2} z^{n-\frac{1}{2}}$$

Then, we have the following infinite product representations, respectively:

$$\theta_{00}(z, \tau) = \prod_{n=1}^{\infty} (1-q^n) (1 + w q^{(n-\frac{1}{2})})(1+\frac{1}{w} q^{n-\frac{1}{2}})$$

$$\theta_{01}(z, \tau) = \prod_{n=1}^{\infty} (1-q^n) (1 – w q^{(n-\frac{1}{2})})(1-\frac{1}{w} q^{n-\frac{1}{2}})$$

$$\theta_{10}(z, \tau) = 2 e^{\frac{1}{4} \pi i \tau} cos(\pi z) \prod_{n=1}^{\infty} (1-q^n) (1 + w q^{n})(1+\frac{1}{w} q^{n})$$

$$\theta_{11}(z, \tau) = -2 e^{\frac{1}{4} \pi i \tau} sin(\pi z) \prod_{n=1}^{\infty} (1-q^n) (1 – w q^{n})(1-\frac{1}{w} q^{n})$$

Now, we have the following identity, which is unique of its type:

Theorem (Jacobi, 1829): $$\theta_{00}(0, \tau)^4 – \theta_{01}(0, \tau)^4 – \theta_{10}(0, \tau)^4 = 0$$

or in other words, that:

$$\prod_{n=1}^{\infty} (1 + q^{2n-1})^8 – \prod_{n=1}^{\infty} (1-q^{2n-1})^8 = 16q \prod_{n=1}^{\infty}(1+q^{2n})^8$$

This identity can be seen as a special case of Riemann-Mumford relations (namely R5) as found in Mumford’s Tata lectures. The mathematical interpretation comes from the triality of $Spin(8)$. Jacobi himself, not realizing any of its immediate implications of this identity he referred to it as “aequatio identica satis abstrusa” (a very obscure equation). Some 150 years later, string theory provided a natural explanation for it, namely that, it is the requirement of spacetime supersymmetry at 1-loop order in string perturbation theory. More specifically, this remarkable, non-trivial mathematical identity implies that the full superstring vacuum amplitude vanishes, which provides very strong evidence in favor of spacetime supersymmetry of string theory in $d=10$. As we will see below, it simply reflects the fact that the spacetime N-S sector bosons and R-sector fermions contribute the same way in equal numbers (but with opposite signs due to spin statistics). Indeed, there is a physical interpretation of it, using characters of $\mathcal{N}=1$ for $D_4=SO(8)$, appearing as the transverse part of oscillators in the $10$-dimensional Lorentz group $SO(9, 1)$ under the RNS formalism and the GSO projection.

Another strong mathematical motivation that fundamental, high-energy physics exhibits supersymmetry is Deligne’s theorem on Tannakian reconstruction of tensor categories, when combined with Wigner’s classification of fundamental particles as irreducible, unitary representations of the Poincaré group. Indeed, this theorem is much more intuitive than the usual theorems that are being usually referred as motivation, namely Coleman-Mandula theorem and Haag-Lopuszanski-Sohnius theorem. Recall that, under Klein geometry (or under Cartan geometry if we talk about global spacetime models) the spacetime symmetry groups should be seen as more fundamental than spacetime itself. It is fun to observe that his “sub-exponential growth condition” is obvious under physical grounds. Deligne’s theorem does not state that spacetime symmetry groups need to have odd-supergraded components (they don’t).

What it does say though, is that the largest possible class of those groups that are sensible as local spacetime symmetry groups is precisely the class of algebraic super-groups. Or in other words, the class of algebraic super-groups precisely exhausts the moduli space of possible consistent local spacetime symmetry groups. This does not prove that fundamentally, local spacetime symmetry is a non-trivial supersymmetry. But it means that it is well motivated, at least mathematically, to expect that it might be one.

## Topological catastrophes

René Thom in the 1960s developed “catastrophe theory”, a branch of bifurcation theory in the field of dynamical systems with connections to singularity theory in geometry. Thom proposed that in 4-dimensions there are 7 different equilibrium surfaces and therefore, 7 possible “elementary catastrophes”: fold, cusp, shallowtail, butterfly, hyperbolic umbilic, elliptic umbilic and parabolic umbilic.

In his 1979 speech, into the prestigious Académie des Beaux-Arts of the Institut de France, Salvador Dalí described Thom’s theory of catastrophes as the most beautiful aesthetic theory in the world. In his first (and only) meeting with René Thom he recalled that Thom told him he was studying tectonic plates at the time.

Indeed, Dalí’s penultimate drawing in 1983 was given the title “The topological abduction of Europe”, with eminent seismic fractures in it, while the algebraic equation for the shallowtail appeared in the lower left corner:

His last painting, in May of 1983 was titled “The shallowtail”:

This provoked Dalí  to question him about the railway station at Perpignan in France (a city in south France near the Pyrénées) which happened to be a place of particular significance for the artist, having declared it to be the “centre cosmique de l’univers” in the 1963s after experiencing a vision of cosmogonic ecstasy there. Unfortunately, the place is mostly known the last decades from the serial killer case that took place between 1995-2001. Many art experts that period collaborated with police giving analytic reports in order to spot any patterns or potential links between the killings and Dali’s paintings.

The following two years draw a painting called “The mystic of the railway station of Perpignan”. Thom is reported to have replied, “I can assure you that Spain pivoted precisely -not just in the area of- but exactly there, where the Railway Station in Perpignan stands today”. Dalí was immediately enthralled by Thom’s statement, influencing his series “Série des catastrophes” above.

Anyway, René Thom in his PhD thesis in 1951 under the supervision of Henri Cartan with title “Espaces fibrés en sphères et carrés de Steenrod“, proved (among other things) that in general, a homology cycle of a nonsingular compactification manifold cannot be represented by a nonsingular submanifold. Indeed, it turns out that dimension $6$ is the lowest such dimension in which a homology class may not be representable by a smooth manifold.
On the physical side of things, we now know that while in general the Freed-Witten anomaly is a necessary condition for the homology class of a $D$-brane to lift to a twisted K-theory class, the condition is also sufficient for all p-branes where $p \neq 6$ when the N-S $3$-form is topologically trivial. A brane wrapping a representable cycle carries a $K$-theory charge if and only if its Freed-Witten anomaly vanishes. Therefore, a brane wrapping a representable cycle carries a $K$-theory charge iff its Freed-Witten anomaly vanishes. But does this mean that $D$-branes can wrap such non-representable cycles? The framework of twisted $K$-theory that classifies R-R fields is indispensable here if we want to understand questions of that nature. Indeed, it turns out that some $K$-theory charges are only carried by branes that wrap non-representable cycles.