The Quill 27 ~ Tannaka Duality, Affine Group Schemes and Hopf Algebra

Posted on Wednesday, December 10, 2025

As I believe, one of the most significant developments in mathematics has been the study of dualities. There are many standard ones, like Hilbert's Nullstellensatz, Schemes, Gelfand duality, or Tannaka duality (on the dualities note, you may see Lecture 1 of Scholze's Gestalten notes). We will see the Tannaka (Grothendieck–Saavedra–Deligne–Milne) duality for a commutative Hopf algebra, however, in an incomplete way here. Other similar posts include theQuill20 and theQuill21. 


An affine group scheme is defined over a field $k$ to be the representable functor $G: \mathsf{CAlg_k} \to \mathsf{Grp}$. The coordinate ring on $G$, denoted by $\mathcal{O}(G)$, forms a commutative Hopf algebra structure (this can be realized using Yoneda's Lemma). The converse also exists, that given a commutative Hopf algebra $A$, we can find the corresponding affine group scheme, which is the spectra $Spec A(R) = Hom_k(A, R)$. The representation of an affine group scheme is given by a natural transformation of functors $\rho: G \to Aut_V$ where $Aut_V(R) = Aut_R(V \otimes_k R)$ is another functor valued in $\mathsf{Grp}$. If rank $n$ vector space $V$ is free, then $\rho: G \to GL_n$ as $Aut_V \cong GL_n$.  Moreover, the comodules over Hopf Algebra, in this case, we will take $\mathcal{O}(G)$, are defined as usual.

The claim is that there is a canonical equivalence between the category of representations of the affine group scheme $G$ and the category of comodules over the Hopf algebra $\mathcal{O}(G)$
$$ \mathsf{Rep_k(G)} \simeq  \mathsf{Comod_k(\mathcal{O}(G))}.$$ Recall, if there exists a symmetric (tensor), faithful, exact, k-linear fiber functor $\omega: \mathsf{T} \to \mathsf{Vect_k}$, then we call $\mathsf{T}$ a (neutral) Tannakian category $\mathsf{T} \cong \mathsf{Rep_k(G)}$ which is a rigid, k-linear,  tensor category and there exists a fundamental group associated to the Tannakian category which is the affine group $G= Aut^{\otimes}(\omega)$. (One can also realize that the tensor structure on $Rep_k(G)$ is induced by the Hopf algebra.) Overall, the duality between affine group schemes and commutative Hopf algebras is $$ \{\text{affine group schemes over k} \} \leftrightarrow \{ \text{commutative Hopf Algebra over k}\}$$ $$ G \to \mathcal{O}(G)$$ $$ Spec A \leftarrow A$$
Given a fiber functor $\omega: \mathsf{T} \to \mathsf{Vect_k}$, we can get a Hopf algebra $H$ from co-end construction (we will discuss (co)ends shortly on this blog) $$H = \int^{X \in \mathsf{T}} \omega(X)^\vee \otimes_k \omega(X).$$ The matrix coefficients from the coend $H$ can be realized to be associated with the representative functions on $G$ for $\omega$ a forgetful functor. Given a monoidal functor $\omega: \mathsf{T} \to \mathsf{Vect_k}$, one says that $\mathsf{T}$ is equivalent to the category $\mathsf{Comod_k(\mathcal{O}(G))}$ and under this Tannakian hypothesis and ($T \cong \mathsf{Rep_k(G)}$), we have $H \cong \mathcal{O}(G)$. More appropriately, the Hopf algebra structure on $H$ is given by the coend construction, where the (co)multiplication and (co)unit are given by the tensor structure of $\omega$. (I apologize for having rushed this part for this post; please see this, this, and this text for more appropriate definitions and follow-ups.) 

Posted in | Leave a comment Print it.

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.

Posted in | Leave a comment Print it.

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.

Posted in | Leave a comment Print it.

The Quill 25 ~ Auslander-Reiten Translate and (a bit of) AR Sequence

Posted on Saturday, September 27, 2025

This post will be very specific (but basic in nature); for those who are not familiar with Quivers, please take a look here or here (or here, which is slightly advanced). Most (if not all!) of my knowledge about Auslander-Reiten theory of quivers comes from Prof. Amit Kuber.

The Auslander-Reiten (AR) quiver is a picture of a quiver with all of its indecomposable representations. For instance, in the $A_3$ quiver with linear orientation ($1 \to 2 \to 3$), we can draw the following AR quiver

AR graph for $A_3$
We are interested in finding an AR sequence in this quiver that is an exact sequence
$$0 \to A \xrightarrow{f}  B \xrightarrow{g} C \to 0$$
which is called an almost split sequence if
1) It does not split 
2) $f$ is a left almost minimal split and $g$ is a right almost minimal split (a lot is here packed, please see the definitions in AR paper, III)
3) $A$ and $C$ are indecomposable in $rep_k Q$ (category of finite-dimensional representations of quiver Q)

Such AR sequences are the main interest of Auslander-Reiten theory.  There is an AR sequence that can be easily observed from the AR quiver, which is
$$ 0 \to \tau V \to E \to V \to 0$$
where $V$ is some indecomposable and non-projective and $\tau V$ is the AR translate of $V$, which is the main subject of this post. There does not exist an AR translation of a projective representation since we begin with a projective resolution of an indecomposable, and if that is a projective, then it is not very interesting, and we fail to get a minimal projective resolution.

A projective resolution of a representation is helpful because of the equivalence between projectives and injectives using the Nakayama functor (which we may discuss in another post, but you may see this). The general idea is that  $$\nu : proj_k Q \to inj_k Q$$ and similarly, there exists an inverse $\nu^{-1}$. So, for any indecomposable $V$, we write its projective resolution as $$\cdots P_2 \to P_1 \to P_0 \to V \to 0$$ where $P_0 \to V$ is a surjection given that our category has enough projectives and, of course, our category is hereditary. We then have the AR translation of $V$ given by (for a minimal projective resolution)
$$0 \to \tau V  \to \nu P_1 \to \nu P_0$$  where $\nu$ is the Nakayama functor $\nu = DHom_A(-,A)$ which sends a projective to its corresponding injective.

For example, in $A_3$, if we take $V=110$ (where the digits are the dimension vectors of the vector spaces of representation $k \to k \to 0$), then we have the minimal projective resolution as $$ 001 \to 111 \to 110\to 0$$ and the AR translation of $110$ is $$0 \to \tau V = 011 \to 111 \to 100$$ (where $\tau V = ker (\nu (P_1) \to \nu(P_0))$ which, if you observe in the AR quiver, then it is the indecomposable which you reach after one step moving to the left side (if you can). Similarly, we can find the AR translation of all the other non-projective indecomposable. Using $\tau V$, we define the AR sequence $0 \to \tau V \to E \to V \to 0$ which is an almost split sequence (and dually, $0 \to V \to E \to \tau^{-1} V\to 0$). From here, we land into the discussion of extensions (derived functors!) and how to define these extensions using the AR quiver for which Auslander-Reiten wrote their beautiful formula (some other time).

If you could tell, there is a lot of homological algebra (and category theory) that exists between these steps, and Auslander-Reiten theory has interesting consequences in triangulated settings as well (see Happel). You can also refer to the notes here.

Posted in | Leave a comment Print it.

The Quill 24 ~ Failure of Functorial Cones in Triangulated Categories

Posted on Thursday, September 4, 2025

In this post, we will quickly discuss the (basic) definition of triangulated categories and why there exist non-functorial cones in these categories. (The whole motivation, at least pointed in, to pass to some stable homotopy theory, like $\infty$-category, is built somewhat around this failure of triangulated categories.) We will discover more about these in the coming posts.


A triangulated category is an additive category with a shift functor that satisfies some axioms (see below). It is represented by $(\mathcal{T},[1],\Delta)$ where $[1]$ is a shift-functor (an autoequivalence $[1]: \mathcal{T} \to \mathcal{T}$) and $\Delta$ are distinguished triangles. A distinguished triangle is of the form
$$X \to Y \to Z \to X[1]$$
and the triangulated category $\mathcal{T}$ satisfies Verdier's axioms (of which an important one is the octahedral axiom, TR4). A good example of a triangulated category is the derived category of an abelian category. Anyhow, I will stop this definition by mentioning that we can see these distinguished triangles as a replacement of short exact sequences in abelian categories. For more on these triangulated categories, please see.

Our goal here is to see why there is no canonical functorial definition of (co)limits in a triangulated category. When working with some nice definition like $Qcoh(X_i) = A_i-mod$ where we have taken an open cover on $X$, we want the gluing to be defined. This motivates one to develop dg-enhancements, stable $\infty$-category, and so on (for some other day) to recover 'functorial cones'. However, I must mention that there exists a notion of homotopy (co)limits (see this work). In a distinguished triangle diagram, those come at the third vertex ($Z$). I do not know much about this theory, but they face certain restrictions too (like totalization, non-functoriality, and so on). Anyway, let us proceed with our problem. We will be mostly dealing with Stevenson's argument.

The argument is as follows. If an idempotent complete triangulated category $\mathcal{T}$ which  admits a functorial cone, then it is a semisimple abelian category. A functorial cone is a functor defined as $\text{cone}: Mor(\mathcal{T}) \to \mathcal{T}$ where $Mor(\mathcal{T})$ is a category with objects $f: X \to Y$ for $X, Y$ in $\mathcal{T}$ (also called arrow category). For any morphism $f$, we can write a distinguished triangle
$$X \xrightarrow{f} Y \xrightarrow{g} cone(f) \xrightarrow{h} X[1]$$
and this triangle satisfies Verdier's axioms. Any morphism between two morphisms $f, f'$ is a commutative square. And there exists a coherent choice, which means that two cones of the same morphism are equivalent up to canonical isomorphism. For a non-functorial cone, cones are defined up to non-unique isomorphisms. Hence, there does not exist a coherent choice for a cone that is natural for the commutative square of morphisms.

Given a functorial cone in a category, this implies the existence of functorial weak colimits, and if a category also admits split idempotents, there exist actual colimits. Let us say our $\mathcal{T}$ is such a category. The functorial cone provides a functorial weak cokernel. In $\mathcal{T}$ for a functorial cone, there would exist an actual cokernel. Similarly, the category has an actual kernel as well. This makes $\mathcal{T}$ into an abelian category (it was already defined to be an additive category). Now, in the triangulated category $\mathcal{T}$, all monomorphisms and epimorphisms split, which makes $\mathcal{T}$ into a semi-simple abelian category. This is proposition 3.1, Stevenson. Without idempotents, one would still arrive at the same conclusion.

The only problem is that there are only trivial examples of triangulated categories that are semi-simple abelian categories. Hence, non-trivial triangulated categories cannot admit functorial cones. For example, the derived category of bounded coherent sheaves on a variety, the stable homotopy category of spectra, and so on are some examples that are not semi-simple categories. This brings us to conclude that a triangulated category admits non-functorial cones. To restore functoriality, one would need more than a triangulated structure, and such motivations will be discussed in due course.

Posted in | Leave a comment Print it.

The Quill 23 ~ Arithmeticity and Thinness of (Hypergeometric) Monodromy Groups

Posted on Monday, August 25, 2025

I was in a talk given on monodromy groups and the arithmetic (and thinness) property of them by Jitendra Bajpai. This post is a quick write-up on that problem.

For any hypergeometric differential equation of order $n$ on a Riemann sphere with three punctures $\mathbb{P}^1(\mathbb{C}) / \{0,1,\infty \}$, one has linearly $n$ independent solutions which are called hypergeometric functions. For the $\alpha, \beta$ parameters, we write the local monodromies using the (local) space of solutions through the monodromy representation $\rho: \pi_1 \to GL(V)$. The subgroup $\rho(\pi_1)$ of $GL(V)$ is called the monodromy group associated with our hypergeometric equation and parameters $\alpha, \beta$. Accoriding to Levelt's theorem (and later worked by Beukers–Heckman), there exists a (local) basis of solution space of hypergeometric equation such that the monodromy group associated to it (corresponding to $\alpha, \beta$) is generated by the companion matrices $A$ and $B$ of polynomials $f,g$ such that the polynomials are cyclotomic (then one can conjugate the monodromy groups) which means their roots are roots of unity, primitive, self-reciprocal, and thus have no common root. The monodromies are defined locally using these companion matrices. This monodromy group $\Gamma(f,g)$ is a subgroup of $GL_n(\mathbb{C})$.

Now, when we restrict to the polynomials $f,g$ with integer coefficients, then the monodromy group becomes a subgroup of $GL_n(\mathbb{Z})$. We are very much interested in this special representation. However, there is a very large gap between the monodromy group $\Gamma$ and $GL_n(\mathbb{C})$, hence we look for something in between, which is the Zariski closure of $\Gamma$. For the order $n=1$, we see that $\Gamma$ sits in $GL_1(\mathbb{Z})$; however, with order $2$ and higher, we have non-trivial things going on like finite and infinite index, and so on. For order $n=2$, the Zariski closure (here it is the smallest algebraic group that contains $\Gamma$ in $GL_n(\mathbb{Z})$) of $\Gamma$ is $SL_2(\mathbb{Z})$. Then a nice question is to ask if $\Gamma$ is of finite index in its Zariski closure, then it is called arithmetic, and if it infinite index, then it is called thin.

Let us denote for $\Gamma$ its Zariski closure $\Gamma^{Zar}$ as $G$, so $\Gamma \subset G(\mathbb{Z})$. Typically, one has $G=Sp_\Omega(\mathbb{Z})$ where $\Omega$ is a symplectic form on $\mathbb{Z}^n$ or $G=O_Q$ where $Q$ is a quadratic form. The details about when $G$ is symplectic and orthogonal are here. Anyway, the important question is to ask for any given order, $\Gamma(f,g)$ has a finite (arithmetic) or infinite (thin) index in $G(\mathbb{Z})$. For the number theoretic motivation behind the arithmeticity and thinness, see these notes by Sarnak. In order $n=2,3$, there are arithmetic monodromy groups. However, for $n=4$, the situation is more non-trivial and it contains both arithmetic and thin groups. There are $14$ cases in which the monodromy groups are associated with Calabi-Yau 3-folds. The periods of the three-form $\omega$ on the moduli of a CY 3-fold solve a fourth-order differential equation. (I do not understand much connection more than this for these 14 special cases connected to CY right now.) Anyway, again, one is asking how many of them are arithmetic and how many are thin. They were studied in this paper. As it turned out, 7 of the 14 cases are thin and the other 7 are arithmetic, and Kontsevich was also involved in the problem.

Posted in | Leave a comment Print it.

The Quill 22 ~ de Rham Class Field Theory and Fourier Transform on $\mathcal{D}$-modules

Posted on Saturday, August 16, 2025

I realize that there have been many posts in the Quill series about math. Many apologies for that.
Let $G$ be an abelian variety. We define the 1-shifted Cartier duality (following Ben-Zvi's notations) to be

$$(G [1])^V = Hom(G, \mathbb{G}_m)$$ But there is a slight issue in the algebraic geometric side of the usual $\mathcal{B}$ side. Instead, let us do the class field theory in de Rham space. We can define $X_{dR}$ for an abelian and smooth variety $X$ as $$X_{dR} = X/\hat{\Delta}$$ where $\hat{\Delta}$ is the formal neighbourhood of the diagonal. It just imposes a relation $x \sim y$ if $x$ and $y$ are infinitesimally close to the identity. (A nice physics equivalent description is modding out the local gauge redundancy.) Though it is not technically a scheme, it is enough to do algebraic geometry. Actually, at many places, it is defined as a functor of points. So we will consider (quasic-coherent) sheaves on it. $$QC(X_{dR}) = QC(X)^\Delta$$ which is just another way of saying that $QC(X_{dR})$ contains flat-connections given by the identification $\mathcal{F}_x \xrightarrow{\sim} \mathcal{F}_y$ if $\mathcal{F}_x$ and $\mathcal{F}_y$ are $x$ and $y$ are infitemsally close. Actually, the following is true and interesting $$QC(X_{dR}) = \mathcal{D}_X-Mod$$ where $\mathcal{D}_X$ is the category of quasi-coherent sheaves. So the flat connections are exactly $\mathcal{D}$ modules. In physics, one can translate this by saying that the sheaves of $X_{dR}$ give the solutions to the Yang-Mills equation $F=dA + A \wedge A= 0$, so basically flat gauge field configurations.

All right. Let us take an abelian group $G$, then we can define the de Rham space of $G$ as $$G_{dR} = G/\hat{G}$$ where $\hat{G}$ is the formal neighbourhood of identity (same as before). A very good example is a vector space. The original Fourier-Mukai dual of $V$ has to do with the classifying space of the formal completion of the dual (we may discuss this in the following Quill(s)). But for $V_{dR}$ it is more interesting. Firstly, for $V$ (an abelian group) $$V_{dR}= V/ \hat{V}$$ Or equivalently, it is a chain complex (derived algebraic geometry) $V_{dR} = [ \hat{V} \to V]$. Now, the $1-$shifted Cartier duality is $$(V/\hat{V})^V [1]= (V^*/\hat{V}^*)$$ so $V_{dR}$ is canonically self-dual. Which means many things. But the best is that $\mathcal{D}(V) \simeq \mathcal{D}(V^*)$. This is better said as a Fourier transform of $\mathcal{D}$-modules. And it is nothing but the algebraic-geometric incarnation of electric-magnetic duality.

This is one nice symmetric way of doing abelian class field theory in algebraic geometry.
To learn more, see this paper by Ben-Zvi and Nadler.

Posted in | 2 Comments Print it.

The Quill 21 ~ The Meaning of Tannakian Construction and Anveshanā July 2025

Posted on Monday, July 28, 2025

Note that this contains some incomplete accounts. 

First, we will see what a tensor category is. Actually, we are interested in a symmetric monoidal category with a tensor functor. For a category \( \mathcal{C} \), we have a bifunctor
$$ \otimes \colon \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}, $$
and for a unit object \( \mathbf{1} \in \mathcal{C} \), we have natural isomorphisms
$$ \mathbf{1} \otimes - \simeq - \simeq - \otimes \mathbf{1}. $$
Furthermore, the category is symmetric, which means that for \( M, N \in \mathcal{C} \), we have a symmetry isomorphism
$$ M \otimes N \simeq N \otimes M. $$
Anyway, we are interested in the case of schemes over a ring. For a commutative ring \( R \), the scheme is given by \( \mathrm{Spec}\, R \), which is constructed by gluing affines. But the functions on a space \( X \) in algebraic geometry are limited, unlike in differential or geometric topology, where functions like \( C^\infty \) span the whole space. So, instead, we step up to discuss stacks. They are algebraic-geometric spaces—more general than schemes and algebraic spaces—and the origin of the mathematics of stacks is in Grothendieck's work on fibered categories and descent (SGA 1).

In this Tannakian construction, one is interested in seeing whether geometric objects like schemes and stacks can be recovered from linear categories associated with them. See the paper by Lurie, Tannaka Duality for Geometric Stacks, and also Tannaka Duality Revisited by Bhatt and Halpern-Leistner.

Informally, one can say that given a commutative ring \( R \), we may view
$$ X = \varinjlim \mathrm{Spec}\, R, $$
and then we are interested in sheaves
$$ \mathrm{QC}(X) = \varprojlim R\text{-Mod}. $$
One then has to realize that
$$ \mathrm{QC}(X) \colon \text{Stacks} \to (\otimes\text{-categories})^{\mathrm{op}} $$
and that there is a right adjoint
$$ \mathrm{Spec} \colon (\otimes\text{-categories})^{\mathrm{op}} \to \text{Stacks}. $$
Given a category \( \mathcal{C} \), we are interested in defining a geometric object \( \mathrm{Spec}\, \mathcal{C} \), which will be the best way to approximate \( \mathcal{C} \) in algebraic geometry.

Now, the Tannakian reconstruction here is about building a space (a stack or scheme) from tensor categories, i.e., symmetric monoidal categories. From the adjunction map, there is a faithful embedding
$$ X \mapsto \mathrm{Spec}\, \mathrm{QC}(X), $$
where \( X \) is a geometric stack (think of an Artin stack, for instance). The above map is also called the 1-affinization or Tannakanization, and it provides the best affine approximation to \( X \) via its category of quasi-coherent sheaves. For the parts I skipped in the above discussion, please see this lecture by David Ben-Zvi.

Also sharing the July issue of Anveshanā magazine. This features interviews with Aparna Dar (IIT Kanpur), Sumathi Rao (TIFR-ICTS, Bengaluru), and Jyoti Hegde (Veenagram, Sirsi), and a selection of articles, including an essay by C.S. Aravinda on the hyphen in Harish-Chandra.  

Posted in | Leave a comment Print it.

The Quill 20 ~ Tannaka-Krein Duality and Reconstruction

Posted on Saturday, July 12, 2025

Tannaka-Krein duality, in its classical form, says that given a compact group $G$, one can reconstruct the group $G$ from the category of the finite-dimensional complex representations $\Pi(G)$. And $\Pi(G)$ is a tensor category (thus having a monoidal structure). For a locally compact abelian group, it is a much easier case and from Pontryagin duality, one has that the dual group $\hat{G}$ is the space of 1-dimensional representations of G (namely characters), and the dual group determines the group $G$, please see this note from last year. However, for non-abelian cases, the category of representation $\Pi(G)$ contains all finite-dimensional $\mathbb{C}-$linear representations, not just 1-dimensional irreducible representations like abelian case. Actually, as I think, it is helpful to see that Pontryagin duality is an abelian prelude to Tannaka-Krein duality. The latter also generalized a lot of the framework later, which culminated in the Tannakian framework of Grothendieck, quantum groups by Drinfeld, Deligne's tensor categories, and so on.

But one can say that given
$$\Pi(G) \cong \Pi(H)$$
which is an equivalence of symmetric monoidal categories, $G \cong H$ will not always be true unless there are some conditions to observe. This was emphasized by introducing a faithful, exact, forgetful, $\mathbb{C}$-linear fiber functor which preserves the tensor products and forgets the group action $\omega$
$$\omega: \Pi(G) \rightarrow Vect_\mathbb{C}$$
which forgets the group action and remembers the underlying vector spaces. The group $G$ can now be associated with the automorphism of the fiber functor
$$G \cong \underline{Aut}^{\otimes}(\omega).$$
This essentially means that all the information about the group is encoded in $\Pi(G)$, and given this fiber functor, we can reconstruct our group. In more standard words, given any symmetrical monoidal category $\mathcal{C}$, if one has a fiber functor $\omega: \mathcal{C} \rightarrow Vect_\mathbb{C}$, then under the suitable conditions, there exists a compact group $G$ such that $\mathcal{C} \cong \Pi(G)$ and $G  \cong \underline{Aut}^{\otimes}(\omega)$.

The framework of Tannakian categories and fiber functor was introduced by Grothendieck-Deligne in the 1960s, while Tannaka originally used the ring of representative functions.

Posted in | 2 Comments Print it.

The Quill 19 ~ A Quick Sheafification

Posted on Friday, June 27, 2025

In this quick post, I wish to state a simple Lemma.


Let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$ on $X$. Then there exists a sheaf $\mathcal{F}^{\#}$ with values in $\mathcal{C}$ and a morphism $\mathcal{F} \to \mathcal{F}^{\#}$ of presheaves with values in $\mathcal{C}$ with the following properties:
   1. The map $\mathcal{F} \to \mathcal{F}^{\#}$ identifies the underlying sheaf of sets of $\mathcal{F}^{\#}$ with the sheafification of the underlying presheaf of sets of $\mathcal{F}$.
    2. For any morphism $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf with values in $\mathcal{C}$, there exists a unique factorization $\mathcal{F} \to \mathcal{F}^{\#} \to \mathcal{G}$.

Proof: We start with a pre-sheaf $\mathcal{F}$ with values in $\mathcal{C}$ (some category). However, this is a pre-sheaf of abelian groups. We can use a forgetful functor
$$F(\mathcal{F}): Open(X)^{op}  \to Sets $$
Now with the classical sheafification, we get a sheaf of sets $F(\mathcal{F})^{\#}$. So, we have the elements and can be glued.

We wish to lift the algebraic structure for this sheaf of sets. We can assume that $\mathcal{C}$ has filtered colimits. These colimits preserve the algebraic operations. For any $U \in X$, $\mathcal{F}^{\#}$ is a canoncial object in $\mathcal{C}$. 

The second part of the lemma, which is a universal property where $\mathcal{G}$ is a sheaf in $\mathcal{C}$
$$\mathcal{F} \to \mathcal{F}^{\#} \to \mathcal{G}$$
Let us assume a morphism $\mathcal{F} \to \mathcal{G}$, then the forgetful functor gives
$$F(\mathcal{F}) \to F(\mathcal{G})$$
and since $F(\mathcal{G})$ is a sheaf, $F(\mathcal{F})^\#$ (which is a sheafification of $F(\mathcal{F}$) is a sheaf. And as we saw that $F$ creates a colimit, there is a morphism in the category $\mathcal{C}$. There is a morphism lifted $\mathcal{F}^\# \to \mathcal{G}$ and thus one has the factorization $\mathcal{F} \to \mathcal{F}^{\#} \to \mathcal{G}.$
□

Posted in | Leave a comment Print it.

The Quill 18 ~ More on $\mathcal{D}$-Modules

Posted on Wednesday, June 11, 2025

$\mathcal{D}$-modules are very important objects of algebraic analysis and algebraic geometry. We briefly talked about it in the last post. $\mathcal{D}$-modules are modules over the ring of differential operators. In this post, let us briefly revise the Weyl Algebra because it is very important to enter the world of $\mathcal{D}$-modules. Weyl Algebra can be constructed as a $\mathcal{D}$- module.


The best application of $\mathcal{D}$-modules has been to solve the differential equations with Bernard Malgrange applied to the (differential) equations with constant coefficients initially. Then, Kashiwara in 1971 applied the equations with analytic coefficients. And if I am right, this was also the second birth of algebraic analysis after Sato.

And as said, Coutinho, $\mathcal{D}$- modules have two branches which depend on the base variety: algebraic and analytic. Moreover, $\mathcal{D}$- modules are not restricted to the Weyl Algerbas.

Let $K$ be a field of characteristic zero and $K[X]$ be the ring of polynomials. Now, $K[X]$ is an infinite-dimensional vector space over K, and the linear operators are denoted by the $End_K K[X]$. These will be two linear operators $x_i$ and $\partial_i$. (Remember that polynomials in $K[X]$ are defined in $n$ commuting indeterminates over $K$). Let us take a polynomial $m$ in $K[X]$. The operators are defined as
$$x_i (m) = x_i \cdot m$$
$$ \partial_i(m) = \frac{\partial m}{\partial x_i}$$
The Weyl Algebra $A_n$ is defined as the subalgebra of $End_K K[X]$ (and hence a $K$-algebra) generated from $x_i$ ($x_1,x_2,\cdots,x_n$) and $\partial_i$ ($\partial_1,\partial_2 \cdots, \partial_n$).

Moreover, the most important part of Weyl Algebra is that they are not commutative. So one finds,
$$\partial_i\cdot x_i =  x_i \cdot \partial_i +1$$
and hence $[\partial_i, x_i]=1$ (it is very easy to see this).

A Historical Note: Weyl Algebra played a crucial role in the development of quantum mechanics (see this). Also, $A_n$ notation was used by Dixmier to denote the algebra that physicists use to describe systems with $n$ degrees of freedom.

Posted in | Leave a comment Print it.

The Quill 17 ~ D-Modules

Posted on Saturday, May 24, 2025

$\mathcal{D}$-modules are very important objects in algebraic geometry and mathematical physics (for example, in string theory). In a previous post, we saw that Hecke eigensheaves on a $Bun_G$ (a set of isomorphisms of vector bundles) are $\mathcal{D}$-modules. They are a quasi-coherent sheaf of the scheme of differential operators.


As the name suggests, $\mathcal{D}$-modules are just modules over the ring of differential operators on some variety $X$. 
Let $X$ be a smooth variety over a field $k$ (of char 0), then $O_X$ is the structure sheaf over it. Let $\mathcal{F}$ be a quasi-coherent sheaf over $O_X$. Let us denote $D$ as a differential operator and for open affine subset $U\in X$, $\mathcal{D}(U)$ denotes the ring of differential operators on $X$. Then $U \to D(U)$ is a quasi-coherent sheaf of $O_X$-modules. And we call this sheaf of differentials operators $\mathcal{D}$-modules.

$\mathcal{D}$ is sheaf of non-commutative algebra. A very good example is Weyl Algebra $A_n(k)$. Locally, they are generated by the Heisenberg commutation relations here. Globally, we are interested in seeing if a solution exists by gluing them. So, $\mathcal{D}$-modules help us to understand local-global pictures of linear differential equations.

There is a specific class of D-modules that were of interest to Kashiwara - holonomic modules in the Riemann-Hilbert correspondence. Anyway, the central concept to understand in $\mathcal{D}$ is that they are a sheaf of modules over the ring of differential operators and they are quasi-coherent to $O_X$-modules. For physicists, geometric Langlands has been putting the duality between the category of D-modules on $Bun_G$ and the Fukaya category of Lagrangians in its cotangent bundle. Moreover, $\mathcal{D}$-modules give a geometric meaning to the automorphic forms.

Posted in | 1 Comment Print it.

The Quill 16 ~ What is $Bun_n(\mathbb{F}_q)$?

Posted on Thursday, May 1, 2025

We take a finite field $\mathbb{F}_q$ and define a curve (smooth, projective) over it, $X$. Now $X$ can also be taken on a Riemann surface when doing geometric Langlands (that is, switching from the curves defined over the finite fields to curves defined over the Riemann surface). Anyway, we define 

$$Bun_n = \{\text{set of isomorphism classes of $n$ rank (holomorphic) vector bundles on curve } X \}$$
$Bun_n$ is a moduli space and an algebraic stack, so a moduli stack! The set would be countable, as there are finitely many vector bundles over each finite field extension. One is then interested in studying a restricted (cuspoidal) space of functions on $Bun_n$.

Well, the Hecke operators (which correspond to modifications at points) from classical Langlands can be geometrized as well. They can be projected to the $Bun_n$ as well. We can define Hecke eigensheaves on $Bun_n$ which correspond to moduli of the rank $n$ local systems on the curve $X$. This is the geometric Langlands correspondence. There is the Hecke correspondence between the Hecke stack and the moduli stack.
Moreover, these Hecke eigensheaves are D-modules on $Bun_n$ satisfying a certain property set by the vector bundle $E$.

Now, a very short remark on why $Bun_n$ is an algebraic stack but not a scheme. Because the moduli space has non-trivial automorphisms. Moreover, a general scheme can not track the automorphisms, so if you were to construct a scheme $Bun_n$, you would fail to account for the automorphism groups. Moreover, the stack structure on $Bun_n$ is important for Langlands correspondence as well, even for defining D-modules on it or for the Hecke correspondence to work. We will look into the Hecke eigenvsheaves and what comes before all of these geometrizations in some later post.

Posted in | 1 Comment Print it.

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.

Posted in | Leave a comment Print it.

The Quill 15 ~ Properties of Morphisms

Posted on Tuesday, April 1, 2025

This will be a very quick act.
Grothendieck emphasized that properties of objects are better understood than the properties of morphism. But his relative perspective also lighted a switch of perspective. Vakil notes that it is a change of perspective while algebraic geometry is still intact. For the properties of a scheme, like quasi-compact, separated, affine, and so on, we can translate them to the properties of morphisms. For the affine case, we can use the Affine Communication Lemma, so let us take that example here.


Let us say that $P$ is a property of a scheme (so it is going to be an object here) of being affine (that it is isomorphic to Spec $A$). Then the a morphism of schemes
$$\pi: R \to S$$
can be said to have this property if $\pi$ is affine. How do we know that? We can say that for every subscheme $U \in S$, the object $\pi^{-1} (U)$ should have the property $P$ (in this case, being affine). Using the Affine Communication Lemma, we can check it for every subset and verify the properties locally on the base scheme $S$.
Now, while it is similar to the saying that every fiber on $S$ should have property $P$, it is more than this fiber-by-fiber analysis. It behaves well when fiber moves. Such a fiber is more of a scheme-theoretic fiber. Such a change of perspective should be helpful, as I understand, in the moduli spaces (and deformation theory) discussion where there is a morphism of universal family, and its properties should determine the properties of the moduli space. A property will ensure that these fibers behave well when you move on the base moduli space.

Posted in | Leave a comment Print it.

The Quill 14 ~ Volume of Moduli Space and Random Matrix Theory

Posted on Tuesday, March 18, 2025


ACT $\alpha$ - One of the less talked about correspondences between algebraic geometry and quantum gravity is the relation of finding the volume of the moduli space, intersection theory on these moduli, random matrix theory, and JT gravity (a two-dimensional solution of gravity), at least on this website. A bit of history can be traced in this paper by Witten from 91. In recent years, we have seen more of these interplays, especially see the following papers
In this paper, Witten tries to argue for an analogy of the volume of moduli space of Riemann surfacthe es to volume of moduli space of super-Riemann surfaces. But let us first see the example of an ordinary Riemann surface. It is defined over a bundle $PSL(2,\mathbb{R})$. The volume of the moduli space (denoted by $\mathcal{M}$) of these Riemann surfaces is given by
$$V = \int_\mathcal{M} Pf(\omega) = \int_\mathcal{M} e^\omega$$
where moduli space $\mathcal{M}$ is defined for some genus $g>1$ cause it admits a symplectic form called Weil-Petterson form and $\omega$ is that form given by
$$\omega= \frac{1}{4\pi} \int_M Tr \delta A \wedge \delta A$$
where $M$ is a Riemann surface and $A$ is the connection on the flat $PSL(2, \mathbb{R})$ connection. 

In the super-Riemann surfaces, the moduli spaces have a clear analogy of volume too, the moduli spaces of super-Riemann surfaces $\mathfrak{M}$ for genus $g>1$ have a similar symplectic form
$$\hat{w} = \frac{1}{4\pi} \int_M Tr \delta A \wedge \delta A$$
but now the connection $A$ is on the $OSp(1|2)$ connections. The volume of $\mathfrak{M}$ is defined by
$$\hat{V} = \int_\mathfrak{M} \sqrt{Ber\ \hat{w}}$$
where Ber is Berezinian, an analog of determinant here for the supersymmetric case.

Here, in the ordinary moduli space $\mathcal{M}$, volume $V$ is given by the intersection theory on $\mathcal{M}$ (the details of this fact will be discussed soon in some other post or somewhere else). The essence is that the Weil-Petterson form (a tautological class) can be computed if one knows the intersection numbers on $\mathcal{M}$. And for physicists, it is that random matrix theory and intersection theory are related. Also, see this work by Kontsevich. 
Here, the Riemann surface $M$ is defined with geodesic boundary of length and then the intersection numbers and volumes are related to these lengths. Moreover, the two-dimensional gravity theory, namely JT gravity, is dual to a random matrix theory. This crucial result is due to Saad, Shenker, and Stanford which uses recursion relations on JT gravity on a disk. More crucially, they replace the notion of a function of energy levels of a particular Hamiltonian by the average level density of an ensemble of Hamiltonians in a random matrix theory. This was surprisingly related to the moduli space of Riemann surfaces and thus to the volume of moduli space. The result can be explained by these papers by Eynard & Orantin and Mirzakhani. This paper is helpful too.

Anyhow, the analogy for the super-Moduli space is not straightforward. And no trivial counting of intersection numbers exists. However, a hopeful topological recursion relation still exists and was computed by Stanford and Witten. Now instead of a matrix ensemble, one studies a supersymmetric matrix ensemble dual to super-JT gravity. Then mimicking the result of Eynard & Orantin and Mirzakhani, one can find the volume of super-Moduli space $\mathfrak{M}$.

Posted in | 1 Comment Print it.

The Quill 13 ~ The Maps of Graded Rings and Projective Schemes

Posted on Tuesday, March 4, 2025

A Quick ACT: We know that a ring morphism $R \to S$ induces a map in the opposite direction $Spec S \to Spec R$. Well, the scheme morphism $X \to Y$ induces a map of Z-valued points $X(Z) \to Y(Z)$. These Z-valued points of an affine scheme are helpful, for instance, the rational solutions to $x^2 + y^2 = 9$ are the $\mathbb{Q}$-valued points of scheme $Spec \mathbb{Z} [x,y]/(x^2+y^2=0)$ and similarly, the integral solutions are the $\mathbb{Z}$-valued points.


Coming to the main ACT. Z-valued points of a projective scheme are a little bit harder. In fact, dealing with projective schemes is subtler and more troublesome when trying first time. Grothendieck only came up with the right interpretation of how the coordinates should look in the projective space, as they should be in a ring.

Now, as the map of rings induces a map of affine schemes in opposite directions, we can show that the map of graded rings induces a map of projective schemes in opposite directions. But again, it is a little subtle. So for a graded ring map
$$\varphi: R_\circ \to S_\circ$$
we have a map of the projective scheme map
$$(Proj S_\circ)/V(\varphi(S_+)) \to Proj R_\circ$$
and when $V(\varphi(S_+)) = \emptyset$ we have a morphism of projective schemes $Proj S_\circ \to Proj R_\circ$. However, it is very important to know that not every map of projective schemes comes from a map of graded rings in opposite directions. This is only a motivating case.

Moreover, the map $\varphi$ is a map of A-algebra and the induced morphism of projective schemes is a map of A-schemes.

Posted in | Leave a comment Print it.

The Quill 12 ~ Between 't Hooft and Wilson

Posted on Thursday, February 13, 2025


This post will contain only one ACT.
The kind of duality that exists between the electric charge and magnetic charge is very advanced and that extends to some of the wild mathematics. Let's explore some of the mathematics here. 
What mathematicians study as the Fourier transform between the lattice $H^2(M,\mathbb{Z})$ and the dual (complexified) torus $T$ is the same as electric-magnetic duality in $\mathcal{N}=4$ super Yang-Mills theory with gauge group $U(1)$. There is a lot to unpack there.

In a past Quill, we saw a twist that produces $\mathcal{N}=1$ quantum mechanics, and that killed time (so the Hamiltonian is zero in cohomology) in a derived sense. We will now see the $U(1)$ gauge theory. For mathematicians, it is a connection on the $U(1)$ bundle, and for physicists, it is the theory of electromagnetism in three dimensions. Now, we will do a twist of a similar kind here to get a bigger Lie algebra (super Lie algebra). There are two kinds of twists (see Ben-Zvi's lectures on Langlands). But the idea is to include some more operators than the Hamiltonian only and study the cohomology. We already have seen the standard SUSY algebra in this post. 

A-type Twist: We include the operators (like the Q-charge with nilpotency $Q^2=0$) and 16 supercharges. Now, instead of asking for the cohomology on the ordinary Hilbert space of $L^2$ functions, we want to look at the cohomology on the space of connections modulo gauge equivalence. The obtained cohomology is again a de-Rham cohomology with differential forms.

B-type Twist: Now, we include a complex connection corresponding to Higgs fields, to the total connection is $d+A+i\sigma$ where $\sigma$ is one-form on the three-dimensional manifold $M$ (see this). Now, the bundle is not ordinary $U(1)$ but a complex bundle which we will call $\mathbb{C}$, which will also complexify the structure group of the bundle. Anyhow, the cohomology is now studied for the connection on the $\mathbb{C}$ bundle. In this case, the vector space contains the holomorphic functions on $M$. The cohomology that will be of interest, in this vector space, is the Dolbeault cohomology with operators like $\bar{\partial}^2=0$ and so on. 

Now, the A-twist can be associated with the locally constant functions in the zeroth de Rham cohomology, i.e., $H^2(M,\mathbb{Z})$. This $\mathbb{C}(H^2(M,\mathbb{Z}))$ corresponds to the dual lattice of the complexified torus (that is locally how the holomorphic functions would look like in the B-twist). More precisely, $\mathbb{C}(H^2(M,\mathbb{Z}))$ and $\mathbb{C}(T_\mathbb{C}$) are related by the Fourier series. And the lattice $H^2(M,\mathbb{Z})$ and the dual torus $T_\mathbb{C}$ are related by the Fourier transform.

Coming to physics, in A-twist, the 't Hooft operators create magnetic charges and the Wilson operators in the B-twist create electric charges. From the above duality, these operators are also related. So A-twist and B-twist are the same theory with different elementary charges. This is a manifestation of electric-magnetic duality. A-twist is gauge group $G$ and the B-twist is with dual group $^VG$.

Posted in | 1 Comment Print it.

The Quill 11 ~ What's in a Moduli?

Posted on Friday, February 7, 2025

ACT A - What's in a Moduli?

In a previous post, we discussed the Kaluza-Klein (KK) compactifications and extended the KK mechanism to string compactifications. Now, we do not really get a unique solution for these compactifications as one can really scale the metric to obtain some other solution. We can thus make a family of these solutions which are called moduli. We will take them on manifold $\mathcal{M}$ with singularities. The metric, let us call it $g$, that we usually consider are Ricci-flat and they are never unique.  If a Calabi-Yau manifold has a $Ricc(g) = 0$, then there can always be perturbations of the kind that can still make the Ricci curvature vanish (see this). One can always construct a cohomology class (with some restrictions of the perturbation) of the perturbations so that the class of non-trivial metric perturbations can be studied which preserves the Ricci flatness conditions.

A moduli space in algebraic geometry, differential geometry, and algebraic topology is about the classifications of objects which are equivalent and those which are not equivalent. (In topology, it is usually called a classifying space.) In essence, it is a geometric interpretation of the solutions. For example, a moduli space of Riemman surface, a moduli space of Elliptic curves, and so on. 

Coming back to the compactifications, here, the moduli space is of Ricci flat metrics. The moduli of the compactification consist of the scalar fields (the massless modes) which are the solutions of the equations that are written for motion. These scalar fields correspond to an infinite tower of modes, also called Kaluza-Klein modes. One can take a family of Ricci flat metric $\mathcal{G}$ and a perturbation around it (as we discussed above). We can then show that some perturbations $\delta \mathcal{G}$ will correspond to these varying of $\mathcal{G}$ (which is equivalent to saying that the perturbations would not hurt the conditions of $K$ that we started with). These will still solve the equations of motion. And each variation (in $\delta \mathcal{G}$) will give us a massless scalar field in our compactified $4-$manifold. As is remarked in this paper, there is no need for symmetry for these variations (like one needs for the description of Goldstone bosons).

One then is left with different values, which only differ by the varying scalar fields. We can then write a family of the vacuas and call it the moduli space $\mathcal{M}$. Now, this is not very exciting because it predicts a large number (a very big number) of vacua for describing our four-dimensional world. This is why moduli stabilization is considered. Moduli is lifted to make the massless modest pick up some mass. (There are non-supersymmetric and supersymmetric realizations of this though.)

ACT $\alpha$ - A Note on the Cohomology of Quasi-Coherent Sheaves

On this blog, we have mentioned about quasi-coherent sheaves and coherent sheaves in multiple places. I wrote a (very) short note on the cohomology of quasi-coherent sheaves. A subject like this is at the heart of algebraic geometry and stems from the ideas of homological algebra (see Tohuku by Grothendieck) and sheaf theory (coherent sheaves, see FAC by Serre). We discuss, in the note, Serre's vanishing theorem, Čech cohomology, and the vanishing of higher cohomology.

Posted in | Leave a comment Print it.

The Quill 10 ~ A Sundry for Thoughts

Posted on Wednesday, January 29, 2025

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 $G \times \mathbb{R}^{b_2}$.

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 $$[Q,Q^*]= H.$$
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.

Posted in | Leave a comment Print it.