Interstellar on IMAX

 I watched Interstellar on IMAX today from 1315 to 1615 hrs. Wow I just can't express how much I love it. Interstellar and Dune will forever be the only things that I hold close to me more than anything. 

Anti-Quill Part 5: D-Branes and O-planes 1

 In Anti-Quill Part 4, we discussed Calabi-Yau manifolds and the essential idea of compactifications on them. Before going forward, it is important to note that a lot of care has to be taken in ensuring SUSY and structures are preserved. As an example, take a type IIB theory on $T^{6}=T^{2}\times T^{2}\times T^{2}$. We have two fluxes, the NS-NS fluxes $H_{3}$ and the RR flux $F_{3}$. If you define an involution operator $\pi : (z)_{3}\to (-z)_{3}$, the fixed points of $\pi $ would correspond to the O3-planes along with the parity conjugation operator $\mathcal{P}$. This has the effect of taking $p$-forms and turning them into -$p$-forms under $\pi $, after which we act with $\mathcal{P}$. In order for $H_{3}$ and $F_{3}$ to be of the same parity after the action of $\mathcal{P}\cdot \pi $, they must have odd parity under $\pi $, so that they have consistent parity with the O3-planes. Having said this, it is a nice simple exercise to convince yourself why this is important in view of tadpole cancellation. Another interesting exercise from van Riet is to take $T^{6}$ with angles $\theta _{i}$ and $\pi _{O3} : \theta _{i} \to -\theta _{I}$, with 64 O3-planes. Taking a simple $F_{3}$ flux $F_{3}=d\theta _{1} \wedge d\theta _{2}\wedge d\theta _{3}$, for dilaton stabilisation, what would be the corresponding $H_{3}$ flux?

Anti-Quill Part 4: Calabi-Yau 2

 In Anti-Quill Part 3, we recalled some interesting properties of Calabi-Yau manifolds. Since a Calabi-Yau metric $g$ can be perturbed via some deformation $g+\delta g$ continuously to produce another Ricci flat manifold, this gives us a moduli space of CY manifolds. Using hodge diamond and specifically calculating hodge numbers with quintic hypersurfaces $\Sigma $ in $\mathbb{CP}^{4}$, one can find the dimension of the moduli space. In string theory, the actual nature of the moduli space is far more complicated because we want to do EFTs and somehow quantify viable vacua (if they exist!). The swampland distance conjecture tells you that taking two points on the moduli space $M(\text{string theory})$, there is an infinite tower of modes with exponentially vanishing mass: $M \sim m\exp (-\lambda s)$, where $s\to \infty $ is the geodesic distance on $M(\text{string theory})$. We expect usually that the ``landscape" theories lie in a particular part of this moduli space and the UV-complete nature of it, and how EFT corrections contribute in certain limits, like when $g_{\text{string}}$ is weak. There are some subtleties around how you view the converse with the light tower of modes, see van Riet and Zoccarato. However, the full moduli space even geometrically is non-trivial. Addition of fluxes with the split $M=M_{c}\times M_{K}$ from before is done by the Gukov-Vafa-Witten superpotential $V_{GVW}(\Omega _{3}, G_{3})$, where $\Omega _{3}$ is the complex structure moduli and $G_{3}$ is the dilaton.

Anti-Quill Part 3: Calabi-Yau 1

 In Anti-Quill Part 2, I mentioned how we add in D-branes and O-planes into the theory and not just the moduli and the fluxes. The way we gauge whether or not a given theory has viable compactifications is by looking at the effective potential for phenomenological reasons. As an example, the Freund-Rubin effective potential is 

\[ V \sim \frac{1}{R^{4}} \left( \frac{2g-2}{R^{2}}+\frac{N^{2}}{R^{2}} \right)\;, \]

where $N$ are the number of fluxes. However, Freund-Rubin are not complete or realistic compactifications (even if you add in perturbative corrections). One way in which you could see this is to note that $g$ is not zero for compactifications we wish to pursue (Candelas et al). We instead wish to work with Calabi-Yau manifolds, which are Kahler manifolds with $c_{1}(K_{M})=0$, where $K_{M}$ is the complex line bundle. Very quickly, we will notice two interesting features. To start, see that $T^{2}$ can be decomposed into to two $S^{1}$  with radii $(r_{1}, r_{2})$, and the moduli space for this would be a 2-dimensional manifold $M(r_{1}, r_{2})$. Defining $\tau = iT$, where $T=R_{2}/R_{1}$, notice that there is a $T$-duality, which more generally is a mirror symmetry for reasons I will defer from here. In string theory, one usually also adds in a $B$-field, but the point here is that Calabi-Yau manifolds have this $T$-duality. For general moduli spaces $M_{D}$ of Calabi-Yau manifolds, we define a Weil-Petersson metric $g_{WP}$, which is a Kahlerian metric on $\mathrm{Teich}(n, g)$. An interesting associated result is that the Weil-Petersson volume $V_{WP}(M)$ is finite, and that $R(g_{WP})$ for $M(CY3)$ is either positive or negative, unlike for general $M(K3)$ (something I am still trying to understand). It is then of interest how the curvature and the Weil-Petersson metric affect the stringy moduli space, which typically decomposes into $M=M_{c}\times M_{K}$.

Anti-Quill Part 2: Orientifolds

 Like I said in Anti-Quill Part 1, when doing compactifications we also have to consider D-branes and O-planes. These orientifolds are quite mysterious; in a mathematical context, these arise from RR tadpole cancellation in type IIA/B superstring theories and are related to orbifolds. For now, however, we will approach orientifolds from the second and the more characteristic nature of orientation reversal. To see this, recall that strings on the worldsheet in closed string theories decompose into left and right moving modes $X^{\mu }_{L}(t+\sigma )$ and $X^{\mu }_{R}(t-\sigma )$. With periodicity $\pi $, we can introduce a parity conjugating operator $\mathcal{P}$ so that $\mathcal{P} : X^{\mu }_{L}(t+\sigma )\to X^{\mu }_{R}(t-\sigma )$. In the grand scheme of M-theory, you have the duality between type IIA and type IIB superstring theories via the T-duality, and you can relate type I superstring theory to type IIB and conversely type IIA by using a combination of T-duality and the action of $\mathcal{P}$. In this sense, taking an un-oriented version of type I string theory on $S^{1}$ and acting on it with T-duality gives you a type IIA string theory. In this language of O-planes, a type I theory is a type IIB theory with an O9-plane, whereas with type IIA theory it would be an O8-plane. Going from type IIB to type I theory, the O-planes break half the supersymmetry of the full theory. 

Anti-Quill Part 1: Compactifications

This is the $CPT$-conjugated version of Aayush's very good Quill series. My series is conjugated in the sense that I won't be as mathematical or elaborate or excessively (and frankly concerningly) obsessed with algebraic geometry.

In string theory, compactifications reduce to the usual 4D Einsteinian gravity in the low energy limit, with  an ``external" (although usually this is considered the internal space) manifold $K$ with the extra dimensions surpassed into a small scale similar to the $S^{1}$ with small $r$ in Kaluza-Klein compactifications. In our case, 10D string theory would be $M_{4}\times K^{6}$ with Ricci flatness and a more special condition, that the first Chern class $c_{1}=0$. These lead us to Kahler manifolds and more specifically, Calabi-Yau manifolds. In general, what comprise compactifications are the following: moduli of massless scalar fields, a moduli space of geometries admissible under these moduli, addition of fluxes $\mathcal{F}$ from the theory, D-branes contributing with positive $T$ and orientifold planes contributing with negative $T$. The last three are a messy subject in an EFTs sense because of the tadpole cancellation conjecture in the swampland. A very simple example of stringy compactifications (although not very realistic here because we ignore perturbative corrections) is that of Freund-Rubin compactifications, where you can take magnetic fluxes across $S^{2}$ and write the effective potential in terms of the addition of these fluxes. This is usually an order $\sim 1/R^{6}$ (where $R$ is the volume modulus) against the genus $g$, but the negative curvature contributions are cancelled out when the number of fluxes $N$ is large. When you have large fluxes, these do much to ``stabilize" the moduli against the negative curvature contributions and in more complicated theories, these fluxes being used for moduli stabilization tells you a lot about the EFT and vacua. A similar philosophy applies to more realistic stringy scenarios. In type IIA superstring theory, you add in O-planes and D-branes as well, but with caution since you have to worry about tadpole cancellation. This is typically a problem in F-theory where Calabi-Yau 4-folds are stabilised, but with fluxes adding to the D3-brane tadpole.