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The Quill 26 ~ Kimchi, Coffee and Stable $\infty$-category

Posted on Wednesday, November 5, 2025

1) 'Les avantages du Kimchi sont nombreux et évidents,' writes Grothendieck his essay on his favorite Korean dish, written at Les Aumettes on 15 October 1983, the same year when he began writing Pursuing Stacks. It is, at its essence, a meaningful perspective on fermentation and preservation. The original scan of the recipe was communicated by Johanna Grothendieck and can be found here. Equally important is the significance attributed to the cultural foundation of the food. He also encourages patience during the whole process of cutting the vegetables (and later fruits in what he called Kimchis sucrés), cleaning the pots, environmental factors, waiting during fermentation, careful preservation, and so on. 

An English translation has been done by crowdsourcing and can be found here. 

2) While I have never made or eaten Kimchi, I am, however, fascinated by a different culture, namely that of coffee. On a contrasting level of stooping, I have written a brief essay on the cultural chain of coffee production. It can be found here. There is no recipe for good coffee therein, but I remark on the careful journey of a seed to processing and then to careful and different methods of roasting. I believe there must be more precise and professional accounts of the topics in my essay, but my intention was to provide a commentary on the 'artistic' chain of coffee production.

3) If you do not care about either Kimchi or Coffee, then let there be an $\infty$-category (or quasi-category), and what follows is a (turbulent) definition of stable $\infty$-category. An $\infty$-category is called pointed if it has a zero object. The following is an equivalent definition
  1. The $\infty$-category has an initial object 0.
  2. The $\infty$-category has a final object 1.
  3. $0 \simeq 1$
In a pointed category, a triangle is called a fiber sequence if it is a pullback, and a cofiber sequence if it is a pushout. Now, let there be a pointed $\infty$-category. And let a morphism $f: X \to Y$, then we have the following definition of fiber and cofiber (which are just limits in a category). A fiber of $f$ is a fiber sequence (a pullback square)
so $W= fib(f)$. With its dual language, a cofiber of $g$ is a cofiber sequence (a pushout square)
so $Z=cofib(f)$.

An $\infty$-category $\mathcal{C}$ is called stable if the following is satisfied
  1. It is pointed.
  2. Every morphism in $\mathcal{C}$ admits a fiber and a cofiber.
  3. A triangle in $\mathcal{C}$ is a fiber sequence if and only if it is a cofiber sequence. (Basically, a commutative diagram is a pushout square if and only if it is a pullout square.)
The last point (3) in the above definition, which is the coincidence of fiber sequences and cofiber sequences, contributes to the invertibility of the suspension function $\Sigma$ and loop functor $\Omega$. Any exact triangle can thus be rotated using the invertibility. (Well, these functors are self-equivalences of the homotopy category of the $\infty$-category, which is a triangulated category and hence, exact triangles and $\Sigma$ become the translation functor.)

4) But why do we care about a stable $\infty$-category? We have seen previously in The Quill (see here) that there is no canonical functorial way to define (co)-limits in a triangulated category. This poses a problem of gluing, as mentioned by Lurie. For having this feature, we have to develop the stable $\infty$-category for which the homotopy category carries the triangulated structures where the distinguished triangles are the (co)fiber sequences. A prime example developed by Lurie is the $\infty$-category of Spectra (which is an interesting discussion in its own right). It is interesting to study the motivation behind the development of the theory of stable $\infty$-categories. We will discuss them in due course.

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Anyons and Modular Tensor Categories

Posted on Wednesday, October 29, 2025

This post is jointly written by Vaibhav Kalvakota and Aayush Verma (and note that neither of us is an expert in anyons!). We discuss the anyon statistics and later relate the ideas of (fusion and) modular tensor categories with them.

We begin with a couple of words about anyon statistics. In three dimensions, the exchange of two indistinguishable particle positions $(r,p)$ makes the wavefunction $\psi(r,p)$ acquire a phase, but when we exchange the positions once more, it is equivalent to a trivial loop. Hence, the phase acquired must satisfy some constraints $$\psi(r,p) = e^{2i\theta} \psi(r,p)$$ where $\theta=n\pi$, which are exactly the constraints that define boson exchange statistics (when $n$ is an even integer) or fermion exchange statistics (when $n$ is an odd integer).

However, in two dimensions, a closed loop exchange is not equivalent to a trivial loop. The $\theta$-statistics, which are followed by neither bosons nor fermions, lead to anyon statistics in two dimensions. And since the following is true

From Sumathi Rao's paper https://arxiv.org/pdf/1610.09260
the particles are defined using a braid group (which is the fundamental group of the configuration space) and not a permutation group. Moreover, in anyon exchange, the history of double exchanges is not forgotten, unlike for bosons and fermions, where it is a trivial exchange, as we remarked earlier. For more on anyon statistics, see this paper by Rao.

A fusion category $\mathcal{C}$ is a semisimple tensor category which is rigid, composed of finite isoclasses of simples, and has a simple unit object (refer to this for more). The most important algebraic rules in $\mathcal{C}$ is the fusion rules; given $V_{i}$ and $V_{j}$ simples, their tensor product (fusion) can be decomposed into some other simple $V_k$ as \[ V_{i}\boxtimes V_{j} = \bigoplus _{k\in I} N _{ij}^{k}V_{k}\;, \] where $I$ indexes the objects in $\mathcal{C}$, and $N_{ij}^{k}$ are non-negative integers. Physically, $N _{ij}^{k}$ accounts for the number of distinct topologically invariant ways in which anyons $i, j$ fuse to create anyon $k$. Fusion of two anyons does not give a unique anyon, as we saw that there could be more ways to do the fusion. Fusion rules are associative. These fusion rules create a fusion ring, which is also the Grothendieck group of the category $\mathcal{C}$. When there are more than two anyons, the change of basis in the state space (fusion in a particular order) is given by F-matrices (see this talk and this paper). Fusion of non-abelian anyons is more complicated than that of abelian anyons, see this. 

Anyon exchanges are essentially just braiding across objects like $\mathbf{b}_{ij}: V_{i} \boxtimes V_{j} \rightarrow V_{j} \boxtimes V_i$. Here, the hexagonal identities apply as usual:


For each object, we also attribute a twist $\theta _{i}$, and for anyons, for a full $2\pi $ rotation, we obtain a phase or a topological spin. Not that this only happens in two dimensions, and in ordinary 3d, this would just be the usual particle spin statistics. Since we now have two pieces of data corresponding to the $S$ and $T$ matrices -- for the braiding and twisting respectively, they generate a projective representation of SL$_2(Z)$, from which we also get the charge conjugation matrix and the central charge; the $S$ and $T$ matrices follow: $$(ST)^3=\Lambda C, \\ S^2=C,\\ C^2 = I_n, $$ where $\Lambda $ is \[ \Lambda = \frac{1}{\mathcal{D}} \sum d_{i}^2\theta _{i} = e^{2\pi i c/8}\;. \] Here, $d_{i}$ is the quantum dimension and $\mathcal{D}$ is the global quantum dimension $\sqrt{\sum _{i} d_{i}^2}$. Moreover, the Verlinde formula describes the fusion coefficients in terms of the $S$-matrix \[ N_{ij}^{k} = \sum \frac{S_{ax}S_{bx}S^*_{cx}}{S_{0a}}\;, \] where $S_{0a}$ is just $d_a/\mathcal{D}$ which is an important result in CFT as well as modular tensor categories.

This way, not only do we obtain a fusion category, but something with braiding structure and modular data (i.e., the $S$-matrix is invertible), which leads us to modular tensor categories more broadly. These have important consequences in finding a consistent theory of anyons.

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Revering Musings on String Compactification (but mostly de Sitter)

Posted on Tuesday, April 22, 2025

With Vaibhav Kalvakota, I wrote some notes on string compactification starting from supergravity compactification and ending with whether there exists a de Sitter solution in string theory in non-classical cases. This review was meant to initiate discussions on the de Sitter case.


Abstract. These notes are written on the (realistic) string compactifications and the string de Sitter vacua problem. A lot of unanswered questions remain in these regime which are highlighted using a historical canvas and exposition. We discuss also the KKLT proposal and other recent discussions around if there is a de Sitter vacua? In the exposition, we review the Calabi-Yau manifolds, supergravity compactification, flux compactification, moduli stabilization, and all that. This has been written in the same series where we wrote on the de Sitter quantum gravity and observables, ‘Revering Musings on de Sitter and Holography, 2023’.

The preprint is available at this link now.

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Associated Primes

Posted on Tuesday, December 31, 2024

Support of a module $M$ is  $$   \text{supp}(M) = \{\mathfrak{p} \in Spec R: M_\mathfrak{p} \neq 0\} $$ which is basically the collection of points in the $SpecR$ that is non-trivial under the localization of the module to the point $\mathfrak{p}$.

Now, we move on to defining the associated primes of the module. There are finitely many associated points in $SpecR$.

An associated point is called an embedding point if it is in the closure of some other associated point. This means that if $R$ can be reduced, then there are no embedded points.

We also define that $$ \text{Supp}(M)= \overline{Ass_R M} $$ where $M$ is an $R$-module. This means that the support of $M$ is the closure of the associated points of $M$.

We call an element of $R$ a zero-divisor if it vanishes at an associated point. Now we define annihilator ideal $Ann_R\ m$ for $m \subset R$ of a module $M$ $$Ann_R\ m := \{a \in R: am =0\}.$$

There is the localization at these associated primes. Given that $R$ is a Noetherian ring and a module M over $R$, then there is an injection $$ M \rightarrow \Pi_{\mathfrak{p}\in Ass M} M_\mathfrak{p}$$ with a similar injection in the localization at the $SpecR$. (Support of the module and the associated primes commute with localization.) For any reference, see the frequently mentioned notes on this blog.

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Grothendieck's EGA/SGA

Posted on Wednesday, November 20, 2024

Grothendieck's EGA/SGA notes are quite complicated for a beginner to read. EGA starts by assuming that the reader is familiar with homology theory, commutative algebra, sheaves, functors formalism, and category theory. For those who think EGA/SGA is relevant for them (because some believe there are better 'textbooks' out there on Algebraic Geometry), here are some thoughts from my side.

  • If you are an undergraduate and acquainted with basic algebra, you may want to bridge the concepts I mentioned. Some helpful resources are Bourbaki's Commutative Algebra, CRing Project, the Stacks Project (a little advanced), and Commutative Algebra with a View Toward Algebraic Geometry by Eisenbud. You may also try going through Vakil's note on Foundations of Algebraic Geometry which covers category theory in a bare minimum manner to get along the Scheme theory. He does not discuss Topos theory or topics in SGA but the book serves as a brilliant exposition of the scheme theory (schemes, quasicoherent sheaves, ringed spaces, Riemann-Roch, and geometric properties of schemes).
    CRing project was started with the same vision of providing a collaborating workbook for people who would want to study the EGA/SGA or say theory of schemes.
  • If you are wondering if EGA/SGA is still relevant (in the same spirit the question stands for Serre's Faisceaux Algébriques Cohérents), then I believe they are very relevant even today even if there are numerous textbooks/notes around. You may want to read https://mathoverflow.net/q/14695 which is most of what I feel. If you align with the notion of independent inquiries, then reading notes written about the field when such things were only developing will be beneficial.
  • Given EGA/SGA is a wonderful collection of notes set out during the development of modern algebraic geometry, I read EGA whenever I can, as well as FAC. I am afraid I do not read much of SGA. But now I feel tempted (during these years of studying algebraic geometry), to write some 'prenotes' to the style of EGA/SGA. I am not sure how it would unfold given my other commitments. But look for this blog for these notes when I start writing them. I already have a lengthy set of notes on scheme theory but I wish to cover some preliminaries as well. How it would be different than other existing projects? Maybe it would and I am not sure about it. The thought is still in its infancy. But it would be a major EGA style of writing except it would help me to organize the notes I already have/I will write.

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Are four symmetries enough?

Posted on Friday, November 15, 2024

I wrote this post after a talk by Rajarama Bhat at IITK (with the same title as this post's title). 

The question is for the finite-dimensional spaces. One wants, for a unitary operator, to see the product of how many 'symmetries' result in the unitary operator. The motivation is exactly like any other, such as the prime decomposition of rings. We emphasize the von Neumann algebra. Consider (bounded, in this case) unitary involutions (self-adjoint and it will imply a symmetry) $Q^* Q = QQ^* = \mathbb{1}$ and $Q^2=1$.

Theorem [Halmos-Kakatani, 1958 and Fillmore, 1966] - In an infinite dimensional von Neumann algebra $\mathcal{M}$, every unitary symmetry can be decomposed into four symmetries.

In this, one basically says that a set of unitaries $S^4(\mathcal{M})$ will decompose to product four symmetries in $\mathcal{M}$. The proof can be found in Halmos-Kakatani. But this will assume that we have an infinite dimensional algebra (like type $II_\infty$, type $I_\infty$, type$ III$). For finite-dimensional cases, like type $II_1$ one runs into a determinant of unitaries which are $\pm 1$. For this, Radjavi has a theorem for type $I_n$ with matrices in $M_n(\mathbb{C})$ with determinant $\pm 1$ saying that every unitary can be decomposed into four symmetries. But the determinant here has a catch of a slightly different definition of a central-valued determinant. You may refer to this paper by Bhat and Radjavi. But one safely says that in type $I_n$, we can decompose all the unitaries into finitely many symmetries (also see Broise, 1967).

For type $II_1$, [where we have a maximal entropy state], the story is a little different than type $I_n$ for where $S^4(\mathcal{M})$ is norm-closed and not norm-dense in the set of unitaries.

Theorem [Bhat, 2022] - In type $II_1$ algebra $\mathcal{M}$, every unitary can be decomposed into six symmetries in $\mathcal{M}$.
Theorem [Bhat] - For the type $II_1$, if any unitary operator $Q$ has a finite spectrum, it can be decomposed into four symmetries $\mathcal{M}$.

Now, why not three symmetries? The answer is that the product of the symmetries is not norm-dense in the set of unitaries. This can be verified for any von Neumann algebra.
So, the answer to 'Are four symmetries enough', and I don't know one, is that you don't know and it is somewhere between four and six. It is interesting to see such questions (and many are open questions) in these works.

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And Coherent Sheaves...

Posted on Wednesday, October 9, 2024

In the last post, we discussed the quasi-coherent sheaves for some ring $R$ and scheme $X$. On a scheme $X$, the quasi-coherent sheaves form an abelian category and in fact, this category is a sub-category of the category of $R$-modules. So, simply, as Vakil puts it in his notes, one should better look if the category of $R$-modules is an abelian category and prove that the category of quasi-coherent sheaves (call $Q_{coh}$) is indeed a subcategory of the category of $R-$modules ($Mod_R$), so

$$Q_{coh} \subset Mod_R$$
I will leave this to you to prove this.

Similarly, the coherent sheaves for some ring $R$ also form the abelian category (similar proof) and in fact, quasi-coherent sheaves will not always form an abelian category for any arbitrary ringed space while coherent sheaves will always form an abelian category. Coherent sheaves come with a bit more than quasi-coherent sheaves, both attached very strongly to the sheaf of modules of $R$. The extra condition is of finite presentation and finitely generated modules. A quasi-coherent sheaf is a coherent sheaf if the modules $M$ are finitely generated (hence $R^n \to M$ which is a surjection). For the Noetherian scheme, a finitely generated quasi-coherent sheaf will automatically be a coherent sheaf. But for non-Noetherian schemes, it is not guaranteed. That is why, one should be careful defining coherent sheaf as quasi-coherent sheaf which is finitely generated, which is not true always.

In general, local nature of the ringed space will be described better by the coherent sheaf (category). The discussion on coherent sheaf and finite presentation (that I do not discuss in this) will be done later. Some resources on this subject are this, this by Serre and Vakil's notes.

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Quasi-coherent Sheaves and Modules

Posted on Sunday, September 15, 2024

Quasicoherent sheaves are simple but enriched structures, the ones which are used in this note on Fourier theory, that are used to do sheafification. 

To define a scheme, we glue together (spectrum) of rings. Similarly, to define a quasicoherent sheaf, we glue together the modules over those rings. (A module over Ring is defined for a ring morphism $R \rightarrow M$ where $M$ is a generalization of the vector space, $R\times M \rightarrow M$.) So in a fashion, what rings are to schemes, modules are to quasicoherent sheaves.

This to explain briefly what I have not tried defining in the note on Fourier theory. For references check the note.

So, we take a scheme $X$ and define a sheaf over this as $\theta_X$, then the quasicoherent sheaf $\mathcal{F}$ is the sheaf of $\theta_X$-modules such that is defined on every affine subscheme $\mathcal{U}_i \subset X$ and the restriction gives
$$ \mathcal{F}|_M \cong \tilde{M}$$
where $\tilde{M}$ is sheaf for some $R-$module ${M}$. The scheme $X$ is over this ring $R$. So much is packed into this definition. But let us first check the locality aspects.
Basically, we should be able to restrict this sheaf in some affine scheme subscheme $\mathcal{U}_i$ and get the sheaves associated to the modules of the ring $R$. So, we can glue together these (sheaves) modules of the rings and get the globally a quasicoherent sheaf $\mathcal{F}$. So locally a quasicoherent sheaf $\mathcal{F}$ looks like a sheaf of modules over the ring. This helps us to reduce the problem of studying the quasi-coherent sheaf into a problem of studying the modules over the ring of some subsystem.

Moroever, the morphism between quasi-coherent sheaves are basically the morphisms between $\theta_X$ modules.
I have used these quasicoherent sheaves in sheafification over a module in the note since the sheaf $\mathcal{F}$ is isomorphic to the R-mod $M$.
A coherent sheaf is basically a quasi-coherent sheaf with the finiteness condition.
We will discuss more about that later (if I remember).

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Hitchin Equations and Higgs Bundle

Posted on Sunday, August 11, 2024

Hitchin equations were the first solution to the hyperkahler moduli space of the Higgs bundle (see here). Since then, it has appeared many times in the physics of gauge theory. In fact, the reduction of $\mathcal{N} =4$ SYM to a two-dimensional gauge theory has target space (at low energies) which is just he Hitchin's moduli space of a Higgs bundle $\Phi$. 


Let's see what are the Hitchin equations. Take a Higgs bundle on a smooth Reimann surface $C$, then we have 
  1. A holomorphic vector Bundle $E$
  2. Holomorphic Higgs field $\Phi$. This is but the holomorphic section of the endomorphism bundle $End(E) \otimes K_X$ where $K_X$ is the canonical bundle of $X$.
Higgs bundle $\Phi$ is one-form $C$ which takes value in the adjoint representation (which is the adjoint bundle of G-bundle $E$).

Now, we take the bundle and define the complex connection $\mathcal{A}$ over the $G_\mathbb{C}$ bundle
$$\mathcal{A} = A+i\phi$$
$G_{\mathbb{C}}$ is given by the complexification of $E \to \mathbb{C}$ and the structure group $G \mapsto G_\mathbb{C}$. The complex curvature $\mathcal{F}$ is given by
$$\mathcal{F} = d\mathcal{A} + \mathcal{A} \wedge \mathcal{A}.$$
We can get the real and imaginary parts of this curvature as
$$\text{Re}\mathcal{F} = F-\Phi \wedge \Phi$$
$$\text{Im}\mathcal{F} = D\Phi$$
The Hitchin equations are now
$$\mathcal{F} =0 \\ D \star \Phi=0$$
where $D \star \Phi = D^-_Z\Phi_z+D_z\Phi^-_z$, for $z \in \mathbb{C}$, so
$$D^-_z = \partial^-_z+ [A^-_z, \cdot]\\ D_z=\partial_z + [A_z,\cdot]$$
For a $\mathcal{N} =4$ SYM on a four manifold $\Sigma \times \mathbb{C}$ where $\Sigma$ is a 2-manifold  and is very large than $\mathbb{C}$. $\mathbb{C}$ is just the Riemann surface.  Now, we reduce the four dimensional supersymmetric gauge theory on $\Sigma \times \mathbb{C}$ to $\Sigma$. That should make clear why $\Sigma$ is larger than $\mathbb{C}$ and we get the effective field theory on $\Sigma$. This two dimensional theory will be the SUSY $\sigma$-model. The presence of such sigma model is a fortunate situation. The target space of this sigma model is the space of classical supersymmetric vacua that we get while the compactification (this is pretty known in physics). Evidently, what we get is that the target space is space of Hitchin hyperkahler moduli space of the Higgs bundle $\Phi$ that we just discussed. This moduli space is widely studied in the physics literature and usually represented by $\mathcal{M}_H(G,\mathbb{C})$ where $G$ is the structure group of $E$-bundle. So basically the Higgs bundle is given by the pair of a complex connection and Higgs bundle $(\mathcal{A},\Phi)$. And we are interested in solutions in the map $\Sigma \to \mathcal{M}_H$.

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A (Quick) Note on Fourier Theory

Posted on Sunday, July 21, 2024

I wrote a short note about Fourier analysis (of which the generalization is the Langlands program) of finite groups. Basically, we take the action from $G$ on $V$ to $\mathbb{C} G$ on $V$, where $V$ is the representation. Character theory is discussed as well.


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Some Updates

Some of the recent stuff.

  • I recently had a paper on the Local Aspects of Topological Quantization and Wu-Yang Monopoles available at https://arxiv.org/abs/2406.18799.
  • I recently talked at a seminar about the relevance of mathematics in theoretical physics. The primary concern was to notify the wonders of parallel between stuff from both ends. The slides are available at this URL.
    I wish to formalize these into (more proper) notes sometime later in the near future.
  • I wrote a short note on Fourier Analysis of (finite) Groups which is available at this URL.

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The Canvas of Holography of (A)dS/CFT

Posted on Wednesday, April 3, 2024

With V. Kalvakota, we wrote an essay pointing out the traditional points of holography where we have contrasted the case for AdS and de Sitter. The latter has points that are non-trivial in these traditional senses, so people have looked out for answers in different holographic settings. This paper was written for GRF 2024.


(Some may find this review we wrote last year helpful along with this reading.)

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A Correspondence between Algebra and Geometry

Posted on Wednesday, January 10, 2024

In Vakil's notes, I found a quote attributed to Sophie Germain "L'algèbre n'est qu'une géométrie écrite; la géométrie n'est qu'une algèbre figurée." (Algebra is but written geometry; geometry is but drawn algebra.)


Let us first motivate the affine scheme, which is roughly the isomorphism between the spectrum of a ring $R$ and locally ringed space (topological space with a sheaf of rings), such that the topological space admits covering $U_i$ where every $U_i$ being an affine scheme and a general scheme is just gluing together these affine schemes. Remember that $\mathrm{spec}(R)$ is just all the prime ideals of $R$.

For any ring $R$, we have the spectrum of $R$ dual to the affine line over $R$. This is the correspondence we wish to understand. For $\mathbb{C}[X]$, we have
$$\mathbb{A}^1_{{\mathbb{C}}[X]} = \mathrm {Spec}(\mathbb{C}[X])$$
and since $\mathbb{C}[X]$ is an integral domain, the prime ideal is $0$ and then any $(x-a)$ is a prime when $a \in \mathbb{C}$. Geometrically, $\mathbb{A}^1_{{\mathbb{C}}[X]}$ is just the collection of points $(x-a)$ and $0$ on a one-dimensional affine line, but there is no $0$ on this line. Instead, we call $0$ a generic point on this affine line. Now, for every prime ideal in $\mathbb{C}[X]$, one can find a point on this complex affine line. Similar examples exist for other rings.

Such is the correspondence between algebra (prime ideals) and geometry (affine scheme), and it is beautiful. Thanks to Grothendieck.

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Presheaf is a Contravariant Functor

Posted on Thursday, October 26, 2023

It is a natural exercise to check how a presheaf is a contravariant functor from a category of open sets to an abelian category $\mathcal{C}$, that is 

$$\mathfrak{F} \colon {\mathrm Cat_{Open} (X)} \rightarrow \mathcal{C}$$
where the ${\mathrm Cat_{Open} (X)}$ is the category of open sets $U \subset X$ and $X$ is a topological space. We can easily understand why presheaf would be a 'contravariant functor' for these open sets categories by checking the inclusion and restriction morphisms. 

For $X \subset V$ and $X,V \subset X$, we have the restriction morphism
$$f \colon \mathfrak{F}(V) \rightarrow \mathfrak{F}(U)$$
which is rather very straight from our intuition in the 'practice' of the inclusion of sets and restriction maps from it. Or it may be said that the restriction maps are the morphisms to check in the category of presheaves $Psh(\mathcal{C})$ to see if the functor is contravariant or covariant. (A morphism between presheaves are rather the natural transformation of functors.) Since one also naturally defines, for example, in SGA IV
$$\mathfrak{F} \colon \mathcal{C}^{\mathrm Opp} \rightarrow {\mathrm Cat}$$
where $\mathcal{C}^{\mathrm Opp}$ is the opposite category, so one has the opposite arrow of morphisms, which is, if it helps, equivalent to imagining any 'restriction' like situations. It is easy to apply this thought in some categories; however, it is unsettling for some others.

We also think in terms of pullbacks, which is more natural for espace étalé description.

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On de Sitter Quantum Gravity and Holography

Posted on Friday, October 20, 2023

With Vaibhav Kalvakota, we wrote a note on recent developments in de Sitter quantum gravity and some other stuff, such as a review of entanglement entropy in de Sitter and algebra of observables for states in de Sitter static patch.


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Six Dimensional Gauge Theory and 2-Form

Posted on Friday, August 11, 2023

A 6-dimensional theory defined on $M_6$ does not easily admit a quantum field theory action (since $\int_{M_6} H \wedge \star H =0$). On $M_6$, one identifies a 3-form self-dual $H$ with a Bianchi identity

$$dH=0$$
which is similar to $dF=0$ for a 2-form $F$ in a four-dimensional gauge theory. Basically, $F$ is just a curvature of a $U(1)$ connection, and $H$ is a curvature of a $U(1)$ gerbe connection. We can nonetheless study $M_4$ by compactifying $M_6$ on 2-torus
$$M_6 = M_4 \times T^2 $$
Or alternatively, $M_5$ Cauchy hypersurface in $M_6$ has a symplectic form that can be quantized to give a Hilbert space. This will serve the data on $M_6$. But instead, we look at $T^2$ and take
$$ T^2 = {\mathbb{C}^2}/{\Lambda} $$
where $\mathbb{C}^2$ is a $u-v$ plane and $\Lambda$ is a lattice parametrized by $1, \tau$. Here $\tau$ is a point in the complex plane. Let us now endow $M_6$ with a metric $g$, and we can decompose this metric
$$g(M_6) = g(M_4) + g(T^2)$$
and we will now fix the metric on $T^2$. Now we will add a real scaling factor for $g(M_4)$, so
$$g(M_6) = t^2\ g(M_4) + g(T^2).$$ On $T^2$, the conformal strcuture is determined by a point $\tau$ in the upper-half of $\mathbb{C}^2$, modulo the action of $SL(2,\mathbb{Z})$. If one takes $t \rightarrow \infty$, one reduces $M_6$ self-dual gauge theory to a four-dimensional gauge theory; since it is conformal invariant, $t^2$ can be dropped from now. But the whole reduction depends on the canonical structure of $T^2$. The reduced gauge theory will have a symmetry $SL(2,\mathbb{Z})$. We now pull back $F$ to $M_6 = M_4 \times T^2$ with only non-trivial $SL(2,\mathbb{Z})$ terms
$$H = F \wedge dx + \star F \wedge dy$$
Since $dH=0$, we have $dF =0 $ and $d \star F=0$, which are the equation of motion for a $U(1)$ gauge theory in four dimensions. So a self-dual theory of 3-form $H$ in six dimensions is related to a 2-form $F$ defining gauge theory in four dimensions. One can generalize it by relating a self-dual theory in $4k+2$ dimensions with $2k$ form curvature in $4k$ dimensions. We now find that $F$ has a coupling parameter which is determined by $T^2$, namely $\tau$, modulo the action of $SL(2,\mathbb{Z})$
$$\tau = \frac{\theta}{2\pi} + \frac{4 \pi i}{e^2}$$
Even though six-dimensional gauge theory does not have an action, 4-dimensional gauge theory has an action which, in this case, is just the usual action for $U(1)$ gauge theory. But the presence of $SL(2,\mathbb{Z})$ symmetry implies the electric-magnetic duality, which we find inevitably in the reduction to 4-dimensional gauge theory. In this heuristic argument, we prove that reducing a gauge theory on $M_6$ to $M_4$ would admit a hidden symmetry $SL(2,\mathbb{Z})$, which is determined by the conformal structure of $T^2$.

In literature, taking $t \rightarrow \infty$ is also called the infrared limit. So 4-dimensional gauge theory is an infrared limit of 6-dimensional gauge theory. Relevant papers include this, this, this, this, and this.

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Alexander Grothendieck

Posted on Wednesday, May 31, 2023

This is dedicated to mathematician and activist Alexander Grothendieck (1928-2014). I am not fit to serve any opinions about Grothendieck. With this, I only want to share the legacy of Grothendieck, who shapes (and distorts) my mathematical notions daily. Consider visiting https://www.grothendieckcircle.org/.


Note - I do not necessarily agree with every word in these documents. His attacks and meditations are here for archival purpose only.

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Yoneda Lemma

Posted on Saturday, April 8, 2023

Let us take a look at Yoneda Lemma, which might be the most trivial yet the hardest part of Category theory (and algebraic geometry). I would not be drawing any commutative diagrams.


Take a (small locally presumably poset) category $\mathcal{C}$ and hom-functors $h$ on it to ${\bf Set}^C$. So if we have a set of morphism $mor(A,B)$ ($\pi \colon A \rightarrow B$) for $A,B \in \mathcal{C}$, I can construct a functor to ${\bf Set}^C$ out of set of morphism which I write as $H(A, B)$. One does this for every object inside $\mathcal{C}$; in this way, we get many sets of morphisms to form $H(A, X)$. We now find the normal (representation) isomorphism of this functor
$$\xi \colon F \rightarrow Hom(A,X)$$
and this means that an object $A$ is determined up to isomorphism by the pair $(\xi, F)$. We can also say $F$ is the $Hom(Hom(A,X))$. 

Yoneda Lemma states that any information about the local category is encoded in ${\bf Set}^C$. The set of the morphism becomes the objects for ${\bf Set}^C$, and morphism is given by the natural representation of the functor. So any functor in $\mathcal{C}$ can be sent to its functor category ${\bf Set}^C$, which sends $A$ to $h$. Note that we did not say if $h$ is a covariant or contravariant functor, the result is the same for either.

The philosophy of Yoneda Lemma is also encaptured in this video, essentially meaning why only one view is wrong. Another good exercise is to realize how this is a universal property and why taking maps to and from $A$ is important to understand a category.

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Bochner's Tube Theorem

Posted on Sunday, April 2, 2023

Let us say we have an analytic function $f(z)$ where $z \in \mathbb{C}$ defined in a tube $T$

$$T = \{ z \in \mathbb{C}, z = a+ib, b \in \mathcal{C}, a \in \mathbb{R}^n \}$$
where $\mathcal{C}$ is a convex cone at the origin. Given this, we can prove that some $f'(z)$ analytic continuation of $f(z)$ is defined in a similar tube. For this, we say that there exists a connected domain $G \subset \mathcal{R}^n$, which coincides with the boundary values of  $f(z)$ and $f'(z)$. Then it implies that $f(z)$ and $f'(z)$ are the analytical continuations of each other and are analytic around the domain $G$. This is also known as the edge of the wedge problem.

Now we state the classical tube theorem. We also would make use of Malgrange–Zerner theorem. 

Theorem 1. For every connected domain $G \subset \mathbb{R}^n$, there exists a holomorphic envelope $H(G)$ which contains $G$ as its subdomain.

Theorem 2 (Tube Theorem). For every connected domain $G \subset \mathbb{R}^n$, there is a tube given by 
$$T(G) \{ z \in \mathbb{C}^n, Im(z) \in G \}$$
then the holomorphic envelope of the tube $T(G)$ is given by
$$H(T(G)) = T ( Co\ G)$$
where $Co\ G$ is the convex hull of $G$.

This tube theorem (and generalizations like the double cone theorem and Dyson's theorem) provide more insightful results in QFT. See Borchers1961. Timelike tube theorem can also be seen as a quantum generalization of Holmgren's uniqueness theorem, which also deals with analyticity. For more details on the timelike tube theorem, see Witten and Strohmaier & Witten.

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Quantum Error Correction

Posted on Tuesday, December 27, 2022

This is a three pages note on two papers - Scheme for reducing decoherence in quantum computer memory by Peter W. Shor and Error detecting and error correcting codes by R.W Hamming. The latter is about a parity check, and the former is about a quantum error check.


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