Anti-Quill Part 10: Mathematics, Physics, and "Esoteric"

 This post is a slight detour from the quantization discussions, and is meant to be a note of caution to theoretical physicists about what "mathematics" is. 

Here is the average theoretical physics (especially hep-th) scenario. Average Joe theoretical physicist says let $F_{n}$ be this quantity we want to compute, but not only (1) can they not compute it in any sense other than a toy model in $n\to 0$ or $n\to \infty $, (2) $F_{n}$ is not even well-defined. A very popular example is the path integral, in which case not only do physicists not know how to compute them generally, the $\mathcal{D}\Phi $ or whatever term is not even a well-defined measure. A mathematical physicist, then, looks at $F_{n}$ and says huh, this does not make sense. Let me formalise this. And then uses stuff like cohomology theory, differential graded algebras and other formal tools to make a meaningful interpretation of this, which in the path integrals case would lead to the Balatin-Vilkovisky formalism and supermanifolds. Average Joe then says this doesn't make sense. Why are you doing all this when you don't get anything computable from it? As if he could compute it in the first place. Mathematical physicist then says hm, let me formalize QFTs then, since clearly these path integrals and partition functions are still superficial. And then uses category theory to say that a QFT is a functor from some category $\mathbf{Cat}_{A}$ to $\mathbf{Cat}_{B}$. Average Joe does not know what a category is, and says this is all just esoteric nonsense! The smarter theoretical physicist, on the other hand, asks what these $\mathbf{Cat}_{A}$ and $\mathbf{Cat}_{B}$ are. Mathematical physicist then says these could be categories of globally hyperbolic manifolds and categories of $C^{*}$-algebras, as in the case of locally covariant QFTs, or for $n$-bordisms, like for topological QFTs like Chern-Simons theories. Average Joe does not care. He goes there is still nothing to compute, as if he were able to compute anything. However, what you get from this are: (1) a proper program for gauge theories from BV formalism and (2) a formal notion of, eg, algebras of observables and Hilbert spaces associated to not just free field theories in Minkowski spacetime, but also on curved spacetimes (with TQFTs you get a background independent formalism of course which has its own formulations). Average Joe does not find this amusing. He says, where does this even come from?! This comes from picking a Hadamard 2-point function and a formal $\star$-product and you end up with propagators and correlators as usual. 

Then there is the scenario that Urs Schreiber described in this post, which is an excellent observation that circles back to what I am saying. In that example, the topic of discussion is the AdS/CFT duality, which for Average Joe is just the GKPW dictionary combined with entanglement wedge reconstruction. Interestingly, at the time that post was published (2007), the Lewkowycz-Faulkner-Jafferic-Maldacena-Suh works were not published, and therefore notes on modular flows were not a part of that Joe description. In any case, for a mathematical physicist, it does not have to be this precise statement about the original Maldacena duality. In fact, I wrote an essay with Aayush Verma to this effect back in 2024. However, attempts like algebraic holography by Rehren were dismissed very quickly since they did not reflect the physicist approach. Indeed, at the time, algebras of observables were not very well understood, and it would take over a decade to get the Liu-Leutheusser-Witten papers on such constructions of von Neumann algebras and their automorphism groups. Therefore, modular theory was not incorporated into these criticisms of Rehren duality and I take it that incorporating Tomita-Takesaki theory into the picture resolves some issues. In any case, one of the main criticisms at the time was that not only do you not get the exact AdS/CFT correspondence, you just get duality of states and not a state-correlator correspondence! Which in my opinion is interesting, because back then you could not even make sense of reconstruction in a form other than the GKPW dictionary. This takes virtually nothing away from Rehren's duality, and in fact shows that in the two sets of approaches that Schreiber mentions (that of Atiyah-Segal and that of Haag-Kastler, which is synonymous here to perturbative AQFTs), the pAqfts approach does much to axiomatize a general notion of holography in terms of algebras of observables. Which have worked to an extent already in a somewhat more lenient fashion from Liu-Leutheusser works. Now, if a functorial formalism for this is more meaningful I cannot say (yet, as I am working on this precise issue), but the outcome of this situation will still be the same: a formal notion of holography in terms of categories and operator algebras and deformation theory will be found, while Average Joe will still say but why categories?!

What you should take away from this post is that even though a lot in hep-th can be "computed" in some limit without rigorous formalization, the works that mathematical physicists do is not esoteric and is the work that Average Joe was supposed to have done decades back. While it might seem as if getting a categorification of some theory is too unnecessary or irrelevant to what you are computing, if that axiomatization does not make sense in the first place, your computations have been either a very successful mistake or a very irrelevant computation in the first place. In short: no, mathematical physics is not pedantic.

Anti-Quill Part 9: Quantization-2 | Geometric Quantization

 The aim now is to answer what really defines a quantum mechanical system or a quantum field theory, which in a sense is "field degrees of freedom" (quotes because this is very non-trivial physically) that we obtain from some classical mechanical system or a classical field theory. A classical theory consists of a phase space $T^{*}M$ that is a symplectic manifold $(\mathbb{R}^{2n}, \omega )$ and observables are functions $f\in C^{\infty }(T^{*}M)$ that are elements of a Poisson algebra $\mathcal{P}$. which comes with Poisson brackets $\{\cdot , \cdot \} : C^{\infty }\times C^{\infty }\to C^{\infty }(M)$. Quantum observables are then going to be operators on some $\mathcal{A}(\mathcal{H})$ with a commutator $[\cdot, \cdot ]$ on $\mathcal{H}$. Quantization is then just a function $Q: C^{\infty }(\mathbb{R}^{2n})\to L(\mathcal{H})$, space of linear operators on $\mathcal{H}$. In stronger operator algebraic terms we would actually be more careful about what we do with this (i.e. selecting $\mathcal{B}(\mathcal{H})$), but for now we can be more superficial. So the problem is really just going from this classical phase space to a quantum Hilbert space. Note that I specifically said "quantum Hilbert space", since it is usually easier to obtain a pre-quantum Hilbert space from pre-quantization, where we identify from our Poisson manifold a collection of a hermitian line bundle, a connection on this line bundle and a Hilbert space that is etymologically one but not the actual constrained Hilbert space -- in this abuse of terminology one can consider this pre-quantum Hilbert space the auxiliary Hilbert space, on which polarization is done. I will not get into what polarization actually does for now, but this is typically the actual quantization problem. After polarization, we will end up with the actual quantum Hilbert space, which would be the quantised theory. 

So what we did here is called geometric quantization, which is only one of many sorts of quantization. The aim here is of course to end up with a Hilbert space. In a similar fashion, you could ask what happens with field theories. A classical field theory, after all, is also equipped with Poisson algebras and eventually we want to quantize this. Since QFTs describe algebras of observables, you could define it as really a functor from categories of globally hyperbolic manifolds to those of $C^{*}$-algebras satisfying the Haag-Kastler axioms. However, the obvious issue is that field theories have infinite degrees of freedom and therefore the phase space is infinite dimensional. In such cases you would do the so-called refined algebraic quantization, which will be a different post eventually. This was in fact, touched upon in the Hilbert space of de Sitter quantum gravity paper by Suvrat et al, without realising the full technicalities of what they were doing! On the other hand, a more suitable general scheme is that of deformation quantization where the Poisson brackets are "deformed" into commutators. Then there is stochastic quantization and more interestingly, Berezin-Toeplitz quantization which has to do with compact symplectic manifolds and Kostant–Souriau operators, which are related to the Hamiltonian vector field in a certain condition, which I am not entirely sure about.

Anti-Quill Part 8: Quantization-1

  In physics a very natural construction is that of quantization, where we take a classical system and quantize it. This involves, to be very naive, taking a symplectic pair $(M, \omega )$ and turning the functions in $L^{2}(M)$ into operators, and the Poisson brackets $\{\cdot, \cdot \}$ into commutators $[\cdot, \cdot ]$. But getting to $L^{2}(M)$ is hard. In fact, prequantization is as far as we've gotten to here. The objective at the end of the day is to take this symplectic manifold we have and obtain a Hilbert space. Really, what you would work with would be subspaces and not $L^{2}$ spaces, etc etc. There's many sides to how you do this though. In fact, this is a very crucial part of physics -- or in broad terms mathematical physics, since a lot of the developments in this field come not from physicists but from mathematicians. E.g. deformation quantization was formulated by Maxim Kontsevich (the "t" comes before the "s", this comment is directed at my collaborator), a lot of geometric quantization was done by mathematicians just trying to do symplectic geometry and brane quantization, while initiated by Gukov and Witten and is of interest to string theorists primarily, is also of interest for differential geometry and representation theorists. The case with brane quantization is that there's boundary conditions involving Lagrangian and coisotropic A-branes, and stuff I don't fully understand yet. See Gukov-Witten and Gaiotto-Witten for more about this. Interestingly, the ordinary sort of quantization that physicists do is not fully meaningful either. A good example is the work by Suvrat et al on the Hilbert space of de Sitter quantum gravity, where you take the ordinary Hilbert space (obtained from Dirac quantization) and you have to solve the constraints next, for which you define the rigging map and do group-averaging. This forms the so-called refined algebraic quantization scheme, which is really just Dirac quantization extended. So even though quantization as a term seems trivial from undergrad physics a la canonical quantization, the actual mathematically clear way to do it is very nontrivial and they branch out into several formalisms. In the words of Ludwig Faddeev, "quantization is an art, not a science". More on this soon.

Odyssey Sketchbook | Odysseus demo 1

 Here is the track I had composed for Odysseus. Upcoming tracks will be released in the coming weeks. 

Odyssey Sketchbook | Odysseus

arXiv3r

 I made a Discord app that you can use to link arXiv papers with just the identifiers, subscribe to authors and get BibTeX citations. Use [yymm.nnnn] or [cat/nnnnn] to get paper details, use [bib:yymm.nnnn] to get a BibTeX citation, and use [au:Author Name] to subscribe to new papers by an author. Access help using !00arXiv3r.

• Discord app link: add arXiv3r to server
• Github: github.com/vkalvakotamath
• Huggingface Spaces (live status check): huggingface.co/spaces/vkalvakotamath/arXiv3r

There maybe brief downtimes when the app gets updated. Check with the Huggingface Space to see live status.

Things I Wish to Understand.

 Here is a list of things that I do not understand that I wish to learn about in the coming months.

1. Spinorial formalism in string compactifications.
2. String field theory, I started on the review by Sen and Zweibach.
3. Moduli spaces and Riemann surfaces.
4. Random matrices, JT gravity and topological recursion.
5. Explicit $\mathcal{N}=2$ supergravity constructions.
6. More on Hodge numbers and moduli stabilisation.
7. Computing superpotential contributions from F-theory.
8. Explicit F-theory constructions of any sort lolz.
9. Anything compactifications in heterotic string theory.
10. AdS/CFT in KKLT AdS construction: what happens when you break SUSY?
11. Compute pfaffians.
12. Fully read Ashoke's orientifolds/F-theory limits paper. 
13. Understand Gopakumar-Vafa invariants.
14. BPS states, logarithmic corrections, etc.
15. $\alpha '$ corrections to KKLT/LVS scenarios.

Perhaps it is time to dig into each of these now that it is my summer break.

String compactifications review! (Anti-Quill Part 7)

 A review paper I have been collaborating with Aayush Verma has finally been completed. It is a review on string compactifications and the essentials of it. It covers Kaluza-Klein compactifications, string compactifications, moduli stabilisation, flux compactifications and de Sitter compactifications, mostly centred around the KKLT construction with type IIB SUGRA. It has been a 4-month work in progress that is still not fully complete, but it is a pretty decent paper with some bits of humour. I learnt a lot, and there have been some really fun conversations around such de Sitter problems. Enjoy!

REVERING MUSINGS ON STRING COMPACTIFICATION (BUT MOSTLY DE SITTER)

Blade Runner 2099 Sketchbook | Tears In Snow

 Here is the first track in my Blade Runner 2099 Sketchbook, called Tears In Snow. 

Blade Runner 2099 Sketchbook | Tears In Snow - Vaibhav Kalvakota

Anti-Quill Part 6: Type IIB and F-theory -I

 I genuinely forgot that I had a blog. Anyway, here is a short post on F-theory/M-theory duality. Inherently, type IIB superstring theory comes with a bundle of data like D-branes, O-planes and fluxes. I have mentioned how these play an important role in stabilising the moduli. We are yet to talk on the exact nature of these moduli stabilisations and especially in view of fluxes. But first, a short note on the axiodilaton, which is given by the RR field $C_{0}$ and string coupling $g_{s}$:

\[\tau = C_{0}+ig_{s}^{-1}\;.\]

This axiodilaton can be thought of as the modulus of elliptic fibration when KK-compactifying M-theory into type IIA theory, and performing a T-duality action on this type IIA theory gives you your type IIB theory back. That is, taking M-theory on $\mathbb{R}^{9}\times S^{1}\times S^{1}$ with some $R_{S^{1}}\to 0$, you obtain type IIA theory on $\mathbb{R}^{9}\times S^{1}$. Then, acting on this with T-duality gives us $\mathbb{R}^{9}\times \mathbf{S}^{1}$ type IIB theory, and in $\mathbf{R}_{S^{1}}\to \infty $ we get $\mathbb{R}^{10}$. The following tables are from nlab and the TASI lectures by Weigand respectively, which are helpful. 

I'll type more in a minute. Btw I saw Dune Part 2 yesterday from 1320 to 1620 hrs. After 114 viewings, it is more golden than before.

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. A simple approach is to take the 126 monomials minus 25 degrees of freedom under $\mathbb{GL}(5, \mathbb{C})$ transformations, which gives you 101 dof, times 2 plus 1, which gives you 203 real dimension for the moduli space, which really comes from the calculation of the hodge numbers. 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.