And Coherent Sheaves...

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

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

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

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 Quill 7 ~ Spectral decomposition of Modules

For any commutative ring $R$, we have an $R-$module with the map

$$R \rightarrow End(V)$$
where $V$ is a part of $R-$mod. Now, there is a spectral decomposition of this $V$ as 
$$ \mathcal{V} = V \otimes_R \mathcal{O}(U)$$
where $\mathcal{V}$ is a sheaf defined over the Spec $R$ and $U \in Spec R$. So given a commutative $R$ we have a module over $R$ which sheafifies over Spec $R$. See this paper by Serre for motivation and Lemma 7.1 in Stacks Project.

Now, we take a commutative algebra $A$ and the Spec $A$ is just the linearized version of
$$A \rightarrow \mathcal{O}(X)$$
where $X$ is a space, we choose functions over this space, which satisfies the algebra. (Well, most of the time Spectrum has the same meaning, see this post.) 

We know that in quantum mechanics, the observables belong to an algebra and the Hilbert space is where the algebra works. In quantum mechanics, phase space is changed by Hilbert Space $\mathcal{H}$. (In quantum mechanics, we are mostly interested in a *-algebra and hermitian operators.) Now any observable $\mathcal{O}$ acting on the Hilbert space $\mathcal{H}$, there is a spectral decomposition that we have defined at the beginning where $\mathcal{H}$ sheafifies over $\mathbb{R}$ because we considered a single operator and $Spec A = \mathbb{R}$ in that case. So the Hilbert space decomposes over $\mathbb{R}$. Now any state $\psi \in \mathcal{H}$ is defined as a section of this sheaf $\mathcal{H}$ over Spec $R$ and thus we have an eigenspace decomposition from this section of the state vectors. Support of this section gives us measurements.

Such kind of decomposition relies on the definition of quasi-coherent sheaves as these $\mathcal{V}$ sheaves are quasic-coherent sheaves. And this is very appropriate for the physics as well.

(This will be the last post in The Quill Series as I am suspending it indefinitely as I fail to commit with a decent frequency in this category as originally thought. I will continue to post but not with this category in mind.)

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The Quill 6 ~ Langlands Program and Electric-Mangetic Duality

We start by putting a disclaimer that Langlands program is a massive program and contains many abstract subtleties which we neither address here nor we think of them, at least when the focus is on theoretical physics. Langlands program is the study of the map from automorphic forms and representation theory to Galois theory. If we wish to talk about the Langlands correspondence for $GL_n/K$ where $K$ is a finite field extension of $Q_p$, then the correspondence is a canonical bijection between the $\infty$-dim irreducible $\mathbb{C}$ representation of $GL_n/K$ and $n$-dim complex representation of Galois group $Gal(\bar{K}/n)$. This was a local Langlands correspondence but such a duality is of non-interest to physicists. There exist many manifestations of the Langlands program in mathematics and physics. Instead, we are interested in geometric Langlands correspondence. It is surprising to note that mirror symmetry appears when we talk about geometric Langlands correspondence between gauge theories {Kapustin:2006pk}.

Monopoles were first subjected in a paper by Dirac in 1931 {dirac1931quantised}, in which there is a monopole sitting at the origin. Such a thing is without consequence and thus we find a quantization condition, also called {\it Dirac's quantization condition}. For subsequent developments in physics of it, refer to {preskill1984magnetic}. In this paper, we are concerned (mostly) with mathematical developments. We now fast forward to the paper by Goddard, Nuyts and Olive (GNO) in 1976 {Goddard:1976qe}.

GNO puts forward the idea that for a compact gauge group $G$, the electric charges take values in the weight lattice of the gauge group $G$ while magnetic charges take values in the weight lattice of gauge group $G^L$ where $G^L$ is the Langlands dual of the gauge group $G$. This is a very profound result in mathematical physics. The dual group could also be called GNO dual group. When we move ahead, we have the Monotonen-Olive conjecture {montonen1977magnetic,Osborn:1979tq} which states that the coupling constants of the group $G$ and $G^L$ are equivalent if

$$ \alpha \longleftrightarrow \frac{1}{\alpha'} $$

which might look familiar to people who have seen S-duality in string theory. Well, there are quite many similarities between string theory and such rich mathematics but we would not be able to cover all of them in this note. One interesting, however, is the statement that categories of A-branes defined over the moduli space of Higgs bundle over some Riemann surface $C$ for $G$ and $G^L$ are equivalent. A similar statement can be made for categories of B-branes defined over the moduli space of flat connections that are equivalent for $G_\mathcal{C}$ and $G^L_\mathcal{C}$ {Kapustin:2008pk,Kapustin:2006pk}.

In other words, the irreducible representations of the gauge group $G$ are known to be the Wilson loops, then by GNO duality, there exist irreducible representations of a dual group $G^L$ which are called `t hooft loops. Monotonen-Olive conjectured that group $G$ and $G^L$ are isomorphic if the Yang-Mills coupling constant are inverse to each other. For physicists, it same as S-duality of coupling constants and in the right context, indeed, the geometric Langlands correspondence is S-duality of $\mathcal{N}=4$ super Yang-Mills theory.

$\mathcal{N}=4$ super Yang-Mills theory is a perfect setting for this exposition. It means that we have four copies of the representation of supersymmetry (SUSY) algebra. The minimum degree of SUSY is $\mathcal{N}=1$. Note that $\mathcal{N}=4$ is the maximum number of SUSY we can have in six dimensions. The reason why we start with a six-dimensional manifold $\mathcal{M}_6$ is that it naturally leads to S-duality action when compactified {Kapustin:2006pk,Witten:2009at}. We can compactify $\mathcal{M}_6$ on a 2-torus (product space of $S^1 \times S^1$)

$$ \mathcal{M}_6 = \mathcal{M}_4 \times T^2. $$

Recall that $\mathcal{M}_6$ does not easily admit an action. But $\mathcal{M}_4$ does. However, $\mathcal{M}_6$ admits a 3-form $H$ (similar to 2-form $F$ in four-dimensional Maxwell's equation) where $H$ is a curvature term of some ($U(1)$) gerbe connection. This 3-form is related to the 2-form $F$ over $\mathcal{M}_4$ after compactification. On $T^2$, the conformal structure is provided by a complex parameter $\tau$ in the upper half of $\mathbb{c}^2$ (as the imaginary part of $\tau$ is always positive). One can now write the 3-form $H$ after the dimensional reduction as

$$ H = F \wedge dx + \star F \wedge dy $$

and because of the Bianchi identity $dH=0$, we have $dF=0$ and $d \star F =0$. One can similarly generalize it by relating a self-dual theory in $4k+2$ dimensions with $2k$ form curvature in $4k$ dimensions. In this case, a six-dimensional quantum field theory admits a self-dual curvature, which is also the reason for not having a definite action attached to it. See {Witten:2009at} for a discussion over this.

The parameter $\tau$ is given as

$$ \tau = \frac{\theta}{2\pi} + \frac{4 \pi i}{e^2} $$

and it has two symmetries in QFT, where one is classical 

$$ T \colon \tau \rightarrow \tau +1 $$

and the other one is quantum symmetry

$$ S \colon \tau \rightarrow - \frac{1}{\tau} $$

and together these two generators $S$ and $T$ generates the modular group $SL(2,\mathbb{Z})$ which is a subgroup of $SL(2,\mathbb{R})$. This group is very special in string theory and unsurprisingly also in mathematics, especially in representation theory and Langlands program itself. The symmetry $S$ relates the coupling constants and thus we have a S-duality in the four-dimensional reduction. 

Let us now start with a ten-dimensional manifold $\mathcal{M}_{10}$ and compactify six dimensions, then the action of the four-dimensional theory becomes 

$$ S = \frac{1}{g} \int d^4x {\rm Tr} \frac{1}{2}\sum_{\mu, \nu=0}^3 F_{\mu \nu}F^{\mu \beta}\\ + \sum_{\mu=0}^3 \sum_{i=0}^6 D_\mu \phi_i D^\mu \phi_i + \frac{1}{2} \sum_{i,j=1}^{6}[\phi_i,\phi_j]^2 + \cdots $$

where $\cdots$ represents the fermionic part of the action. $g$ in the action is the Yang-Mills coupling constant. For brevity, the fermionic part can be ignored until the SUSY becomes important. In this action, we add a topological term

$$ S_\theta = -\frac{\theta}{8\pi^2} \int d^4x\ {\rm Tr}(F \wedge F) $$

and thus we have the complex parameter with us

$$ \tau = \frac{\theta}{2 \pi} + \frac{4\pi i}{e^2} $$

and the S-duality action with this.


There will be more posts for the Langlands program in the coming weeks.

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The Quill 5 ~ Solitons and sine-Gordon theory

We will look at the Soliton solutions in the sine-Gordon equation (which also shares correspondence with the (massive) Thirring model in perturbation theory). Let us first see a standard example of soliton in field theory. We take a non-linear scalar field theory $\phi$ with Lagrangian

$$ \mathcal{L} = -\frac{1}{2}\partial_\mu \phi \partial^\mu \phi - U(\phi) $$

where potential $U(\phi)$ is described by

$$ U(\phi)=\lambda\left(\phi^2-m^2/\lambda\right)^2/4 $$

and the dimensionless coupling constant is $g=\lambda/m^2$. Here, $m$ is the mass of the elementary solutions of $\phi$. Then we define the  topological current

$$ j_u = \frac{\sqrt{g}}{2}\epsilon_{\mu \nu}\partial^\nu \phi$$

and the topological charge is then

$$ Q = \int_{-\infty}^{\infty}dx\ j_0$$

integrating it becomes

$$ Q = \frac{\sqrt{g}}{2}\left(\phi(\infty)-\phi(-\infty)\right) $$

where the $\infty$ is for a kink solution and $-\infty$ is for an anti-kink solution. These kinks deserve our attention here. $\phi$ varies from the minimum of $U(\phi)$ at $\phi = \mp 1/\sqrt{g}$ at $x=\infty$ to the minimum of $U(\phi)$ at $\phi = \pm 1/\sqrt{g}$ at $x=-\infty$. We can write a solution to this equation, which follows

$$ \phi^{''}=\frac{\partial U}{\partial \phi} $$

integrating this with $U$ with $\phi'$ vanishing at infinity we get

$$ \frac{1}{2}(\phi^{'})^{2} = U(\phi).$$

Integrating this now over our choice of $U$ will give us the kink (k) and anti-kink (k') solution

$$\phi(x)_{k(k')} = \pm \frac{m}{\sqrt{\lambda}} tanh\left[m(x-x_0)/\sqrt{2}\right].$$

The rest mass for the soliton is given by

$$ E = \int dx \frac{1}{2}\left(\phi^{'}\right)^{2} + U(\phi) = \frac{2\sqrt{2}}{3}\frac{m}{g}$$

which clearly states that kink (rest) mass divided by the $m$ is proportional to $1/g$. This is also an indication that solitons are non-perturbative physics.

Anyway, the previous example was about solutions of just one theory, where kink and elementary solutions shared a relation. The nature of these kinks will be apparent in the next post. Now, what about a duality between two sectors of different theories. For this, we will turn to the massive Thirring model, which shares a correspondence with the sine-Gordon theory, in a next post. 

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The Canvas of Holography of (A)dS/CFT

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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The Quill 4 ~ The Grand Dictionary

Vakil's note on algebraic geometry has a lot of emphasis on, of course, the duality between geometry and algebra. 

The Dictionary

For every ring $A$, we can define the spectrum of $A$, which is but the prime ideals of the ring. So, any prime ideal in the ring corresponds to a point in the affine scheme. Any element $x$ in $A$ can be written as a function in Spec $A$. The radical ideals of $A$ are the closed subsets of Spec $A$. The maximal ideals of $A$ are the closed points of Spec $A$. An affine scheme is a ringed space that is isomorphic to $(Spec A, \Theta_{Spec A})$, which we have written for a ring $A$ where $\Theta_{Spec A}$ is the Zariski topology defined on the set. Read a similar post.
__________

A radical of an ideal $I \subset A$ is defined as a set 
$$\rm{rad}(I) = \{  r \in A | r^n \in I\ \forall n \in Z^+\}$$
and a nilradical would be a radical of a zero ideal
$$\rm{nilrad}(I) = \{  r \in A | r^n=0 \in I\ \forall n \in Z^+\}$$

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The Quill 3 ~ On Generic Point

For a domain (or integral domain), the affine scheme is called an integral affine scheme. We know that the spectrum of the ring Spec $R$ is a collection of all the prime ideals of the ring, which is equivalent to geometric points of the affine space. For instance, for the polynomial ring $\mathbb{Z}[x]$, the affine space is 

$$A^1_{\mathbb{Z}} = Spec(\mathbb{Z}[x]).$$

For an algebraic closed field $K[x_1, x_2, \cdots, x_n]$, the prime ideals are of form $(x_1-a_1, x_2-a_2, \cdots, x_n - a_n )$ where $a_n \in \mathbb{C}$. This is known as weak Nullstellanz. (In a previous case, we saw that for $\mathbb{C}[x]$, the prime ideals were the maximal ideals $(x-a)$ where $a \in \mathbb{C}$ and the zero prime ideal $(0)$.)

We called the prime ideal (0) a `generic point' in that picture. But what does the term mean? Generally, a generic point $x \in X$ where $X$ is a topological space if the closure of $x$ is the whole space. (One can find different definitions of generic points in the presence of different motivations.) Equivalently, we say that the generic point is a point that is `generic' for the whole space. Thus, if some function is valued on the generic point, then the function will value the same everywhere in the space. In general, a generic point is not available in the affine space. So, a generic point is contained in any other point of space. In the example of $\mathbb{C}[x]$, the generic point is unique, which is zero ideal. We know that the affine space points correspond to the ring's prime ideals. But a zero ideal can not be `pointed' in the affine space, meaning that the points of affine space will generally correspond to the maximal ideals of the ring.

The prime ideals of $\mathbb{Z}[x]$ are the principle ideals generated by primes $p$, ideals generated by irreducible polynomials $f(x)$, of form $(p,(f(x))$ which are the maximal ideals and zero ideals. Now, we know that the affine space of $\mathbb{Z}[x]$ is just a space with points corresponding to these prime ideals, which are called the spectrum of $\mathbb{Z}[X]$. Interestingly, Mumford has a picture containing these points in his Red Book of Varieties and Scheme, known as Mumford's Treasure Map.

We see that the map has some points on the intersection of the horizontal and vertical curves, and the curves themselves are a collection of prime ideals. The horizontal curves are the prime ideals generated by some irreducible polynomial of form $f(x)$. In the map, one has polynomial $(x)$, so the ideal is $Z[x]/(x) \simeq \mathbb{Z}$ and similarly we have $(x^2+1)$ and so on. (We can see that the curve of $(x)$ is less thickened than $(x^2+1)$, which is because of the number of elements contained in the ideal.)

The vertical curve has points of the principle ideals the primes generate, for example, $\mathbb{Z}[x]/p$. $\mathbb{Z}[x]/2$ which is just $\mathbb{F}_2[x]$ (since $\mathbb{Z}[x]/p \simeq \mathbb{Z}/(p\mathbb{Z}) [x]$) where $\mathbb{F}_p$ is a finite field. Now, we have the points on the intersection of the curves, which are $(p,(f(x))$, and these are the maximal ideals. So, for $(2,(x+1))$, we have $\mathbb{Z}[x]/(3,(x+2)) = \mathbb{Z}/(3\mathbb{Z})[x]/(x+2) = \mathbb{F}_3$. 

But where are the generic points? Mumford has some doodles in the upper right corner of the map. This is the zero ideal of $\mathbb{Z}[x]$ and is called a generic point. Geometrically, it does not make sense to point out a generic point since it is available everywhere, but it is nicely drawn on the map. The doodle has been pointed in every direction and is contained in every other point of the space. Once again, a generic point is quite harder to make sense of geometrically, but this is a nice way of visualizing them for the case of $\mathbb{Z}[x]$.

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The Quill 2 ~ Defining Gauge Theory

This post will contain some random notes on gauge theory and differential geometry.


It was very surprising when two cohorts, namely mathematicians and physicists, found that the stuff they were doing simultaneously was the same in two different languages. Let us take a vector bundle E with $G$ as its structure group on a manifold $\mathcal{M}$. Then, here, one can do lots of differential geometry. But the interesting thing is that this is equivalent to describing a gauge theory with the gauge group $G$. Let's say the connection on the vector bundle $A$ is 1-forms potential in the gauge theory. The curvature is defined as the exterior derivative of $A$
$$F=dA$$
In the case of electromagnetism, which is U(1) gauge theory, one can say that a vector bundle with $U(1)$ structure group and connection $A$ defines the electromagnetism where $A$ follows a gauge transformation
$$G \colon A \rightarrow A'$$
where $G=U(1)$ in this case. The gauge transformations are taken in the overlap regions which we define while defining the 1-forms potentials (which are not global on the manifold). The curvature here is the 2-form $F$, which is the field for electromagnetism. It follows a Bianchi identity
$$dF=0.$$
Similarly, we can describe gauge theory using this way of seeing for $G=SU(2)$ or $G=SO(3)$. However, every case has its own special features; for example, if one has a compact abelian group (like U(1)), one can shoot for Hodge theory to study the 2-forms.

Now, the gauge group $G$ is the automorphism (the homomorphism from $E$ to $E$) of the bundle, and the algebra associated with $G$ is called gauge algebra. Thus, we observe that the Yang-Mills theory is described by these connections, curvature, and gauge group of a vector bundle.

We will expand later on why the connections are defined only locally on a manifold. For reference, one may check this book or this. There exists a very giant literature (expository too) on this.

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The Quill 1 ~ Dirac Strings

In the Quill series, I will discuss works ranging from quantum gravity to mathematics (especially algebraic geometry). I do not have any specific number of posts to write, so they will come as I see them in this month of February. This has been inspired by This Week's Finds by John Baez. 


The Issue with Dirac Strings

There was a paper last year by Gonuguntla and Singleton (https://arxiv.org/abs/2310.06005) that argued that there was an overlooked field momentum in the case of Dirac string, which makes the model inconsistent with the center of energy theorem if one accepts that they are truly real. 
We start with a simple monopole placed at the origin of ${\mathbb R}^3$ so that the magnetic and electric charges are at rest. The field momentum of the electric field by this monopole has two components: Coulomb's term and Dirac's string term. There is a non-zero mechanical field momentum contribution from the interaction of magnetic charge and electric field due to the inclusion of Dirac's string, which does not vanish at all. See https://arxiv.org/abs/2310.06005 for the discussion on this term. It was suggested in same that there are two takeaways from this non-trivial mechanical field momentum: 1) the first is to say that the center of energy theorem is wrong, which implies that this term is an error, and 2) the second is to believe in the center of energy theorem and accept this term as a real contribution which implies that Dirac's string is real and must be physical even though how infinitesimally thin we believe it to be. However, then it becomes a system in which the electric charges generate a monopole-like magnetic field with a solenoidal magnetic flux.

A comment on that paper appeared (https://arxiv.org/abs/2401.02423v1), which points out that the vector potential taken in the paper of Gonuguntla and Singleton, is taken over all the space is not possible because these potentials, which are 1-forms can not be defined globally but only can be defined in certain overlaps using gauge transformations and the quantization is defined because of the locally constant cocycle condition appearing in those overlaps. For basics, see ( T. T. Wu and C. N. Yang, "Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields," Phys. Rev. D 12, 3845 (1975) doi:10.1103/PhysRevD.12.3845)

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A Correspondence between Algebra and Geometry

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

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

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

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

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

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

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

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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Notes on Black Holes Information Problem

I wrote some discussions on the black information problem. Unfortunately, I could not include the AdS/CFT solution to the information problem while I strongly insist on studying it, do consider Harlow Jerusalem's review paper for that.


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Black Holes and Pondering

Since the 1970s, black holes have emerged as a central area for theorists. Here are five points why.

  • Black holes are unique physics; no classical, semi-classical, or quantum descriptions can describe them completely. It provides a way to understand quantum gravity. Understanding horizons, singularities, information through it, and many more make black holes perfect for learning new physics.
  • Black holes have unique entropy. Bekenstein, in 1971 gave (with arbitrary constants only perfected by Hawking later in 1973) a formula for entropy $S=A/4$ in Planck units. Bekenstein also provided a generalized second law $S = S_{BH} + S_{out}$, which conjectured that generalized second law can never be violated, this went through many tests. Bekenstein also proposed a unique bound on the entropy of falling objects, now known as the Bekenstein bound. (Which on the other hand has been explored a lot by many setting up holographic bounds, covariant bounds, and so on.)
  • Black holes evaporate. Hawking, in 1973, did the field theory calculations and showed that the positive flux of radiation goes to the future infinity, and the negative flux goes inside the horizon. This is radiation. With negative flux, black hole mass is reduced, and thus it is called evaporated. But this is not fitting since the black hole starts with a pure state, and in this scenario, it will end in the mixed state as Hawking radiation. This is called the Information problem.
  • The last mentioned point is why most ponder over black holes. Information preservation is a must, so physicists devised many roundabouts and alternatives to information loss proposed by Hawking. This includes Fuzzballs, Firewalls, Complimentarity, baby universes, etc. None of them exactly solves it; some create further paradoxes. It is unsettling that information could be lost.
  • There are beautiful ideas regarding wormholes through the double-sided eternal AdS black holes. ER=EPR is quite a line that joins the geometry and entanglement. Quantum teleportation, in theory, is also possible due to information theory and the web- traversable and non-traversable,

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Memories of a Theoretical Physicist

Joseph (Joe) Gerard Polchinski Jr. was an exceptional physicist who gave a lot to string theory. He has inspired a generation of physicists. He is personally one of my string heroes. In his later life, he suffered from cancer that made him unable to even read. Despite all the constraints, he wrote his autobiography and posted it on arXiv in 2017; you may read it here. Ahmed Almheiri (Joe's student) recently edited the biography to add more scientific explanations and image plates. There is a forward by Andrew Strominger and afterward by Joe's family. The book has been published by MIT Press. 


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Stages of Black Holes

Recently, I spotted a rough yet beautiful representation of the formation, evaporation, and different stages of black holes in https://arxiv.org/abs/2006.06872, which is an excellent review to read. I am reproducing it here.

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Strings Dualities

We have five categories of consistent string theory;

  1. Type I
  2. Type IIA
  3. Type IIB
  4. Heterotic $E_8 \times E_8$
  5. Heterotic $SO(32)$
Type I has both closed and open strings. Type IIA and Type IIB contain only oriented closed strings. Heterotic strings are hybrid of bosonic string theory and superstring theory. There are two of them and much interesting. There are many striking relations between all of them, which were worked out by Sagnotti, Sen, Witten, Polchinski, and a few others. We call these relations dualities. (A critical, relating gauge/gravity theory, duality came from Maldacena in 1997, where he proved that Type IIB in AdS is equivalent to N=4 SYM on the conformal boundary.)


Type IIB is dual to Type I by T-duality. Type IIA and Type IIB are related by T-duality. T-duality relates theories on different tori. (T-duality is a very interesting duality that comes up when we have a compactification, as in toroidal compactification.) Type I with g coupling and Heterotic SO(32) with 1/g coupling are related by S-duality. Mirror symmetry relates string theories compactified on other Calabi-Yau manifolds. Compactification of Type II on a K3 Calabi-Yau manifold is dual to Heterotic theories on the 4-torus. Taking the limit $g\rightarrow \infty$ of Type IIA gives 11d Supergravity. These were significant results given that string theory was thriving for something like these dualities, which proved that all are merely the same string theory in different limits or on different geometry.

Another relevant figure from https://arxiv.org/pdf/hep-th/9607201.pdf


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A Black Hole in Every Way

After reading Harlow's review notes on the black hole information problem, I am convinced that there could be many ways to explain the black holes information problem and black holes interior. It would be worth waiting and seeing which one is correct and which is less accurate. Here, I want to collect those possible ways of describing BHs.

  • Old Hawking Radiation: This was the idea of Hawking where he used Bekenstein entropy and implied negatively about unitarity, which later Hawking graciously accepted.
  • Complimentarity: This is a set of progressive ideas, mainly based on unitarity, purity of Hawking radiation, Einstein equivalence, and low energy EFTs. This one has exciting works from Hayden, Preskill, and Susskind.
  • AMPS: This one was put forward by Almheiri, Marolf, Polchinski, and Sully in this paper. This says that four things assumed (or used as concrete builds) in complementarity cannot all be true. They propose a firewall that would incinerate information. One can say a firewall is just an extended singularity or horizon. The firewall appears to break the Einstein equivalence principle which states that any observer should not see anything unusual at horizon.
  • Fuzzballs: This is for big charged black holes where one gets fuzzball-like solutions. However, hard to find such fuzzballs in uncharged black holes.
There are a few more (of which I am not now fully aware), including the Raju-Papadodimas theory and Horowitz and Maldacena theory. It is interesting to observe that some ideas clash with others, and some just sound crazy out the well, which should only be taken with a grain of salt unless there are concrete understanding of things.

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Black Holes and Information

(Some notes on black holes information problem)
At the moment, black holes and their information diary are intriguing theoretical physics problems. Starting with Hawking-Bekenstein, it is a very engaging problem, which has taken multiple routes over the course. To name a few- unitarity, holography (or AdS-CFT correspondence), and page theorem. Page theorem (and page curve) was one of the most exciting developments. 

Page curve suggests that the radiation $R$ is still maximally entangled with the remaining black hole $BH$. At the page time, both coarse-grained entropy is equal as $S_R = S_{BH}$, and after page time, as for a pure state, the entropy of the black proceeds to zero. It is very profound if you think hard about it. Page curve is a part of the hotter debate of whether infalling information is conserved in the radiation. 

Page Curve

Recovering the information (of course, this is just theoretical because an actual black hole information experiment is out of technical reach) is an arduous task and should be done quantum mechanically. I encountered Hayden and Preskill's experiment in https://arxiv.org/abs/1409.1231 (which I suggest for taking a broad view of the problem). Hayden and Preskill throw a diary in the black hole, and the diary is entangled to a system, early radiation, and black hole is entangled. After the black hole consumes the diary, in a thermalized sense, the question is how fast the information comes out. The answer, among others, is (http://arxiv.org/abs/0708.4025v2) very rapidly for black holes that have already radiated by half (in other words, black holes which have exceeded the page time). This led them to call old black holes as mirrors.

Arrangement of Hayden and Preskill's. From http://arxiv.org/abs/0708.4025v2

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Notes on Information Theory

I wrote some notes on information theory and its application in quantum mechanics and computing. I acknowledge D. Dhar for his constant support throughout the period.

Notes on Information Theory

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Covariant and Contravariant Functor in Category Theory

Recently, I strolled around an exciting fact about a difference of meaning of covariant and contravariant words in mathematics (category theory) and physics (tensor analysis). Well, that makes it harder for a mathematician and a physicist to talk about these two words without knowing in what sense.


In category theory, we can think of functors as the mapping of objects between categories. We can say that if a functor preserves the direction of morphism, then the functor is a covariant one. If it reverses the direction of the morphism, then it is a contravariant functor. John Baez has briefly mentioned them in his book "Gauge Fields, Knots and Gravity". An identity functor is a covariant functor, and so are tangent vectors. While cotangent vectors and 1-forms are contravariant. (1-form in this case is differential of a function, however, if a differential of a function is to be thought as a vector field then the vector fields are covariant.)


Suppose we have a map $\phi:M \rightarrow N$ from one manifold to another. On $N$, we have real valued functions defined from $\psi:N\rightarrow \mathbb{R}^n$. To get real valued functions on $M$ we have pullback $\psi$ from $N$ to $M$ by $\phi$. 

$$\phi * \psi = \psi \circ \phi$$

We see that real-valued functions on $M$ suffer a change of direction in their morphism. So they are contravariant.

 

In tensor analysis, one can say that $X_\mu$ is covariant and $X^\mu$ is contravariant. (It is important to dodge that $\partial_\mu$ is covariant while its component $v^\mu$ can be contravariant.)

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Heterotic Strings

Here is my 10 pages handwritten (rough) notes on Heterotic string theory. We will work on both $SO(32)$ and $E_8 \times E_8$. For any reference, one can use String Theory Vol 1 and Vol 2 by Green, Schwarz and Witten. 

Heterotic Strings

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A Few Comments on Entropy

Entropy $S(x)$ is the measure of randomness of a variable $x$. It is important in the area of information theory, which, on the other hand, shares similarities with the entropy that we have in thermodynamics. We write entropy as

$$S(x) =\sum -p(x) \log p(x),$$

here $p(x)$ is the probability mass distribution of the variable. In quantum information theory (or quantum Shannon theory), we use discrete matrices in the place of mass distribution. We mostly prefer the logarithms in base 2 and the entropy is measured in bits. 

Suppose that Alice has sent a message which contains either $a$ or $b$. There is half-chance probability occurring of either. In this case, the binary entropy looks like the below figure, where when $p=1/2$ and $(1-p)=1/2$ the entropy becomes $1$ bit,

$$S(x)=-p(x)\log p(x) - (1-p) \log (1-p),$$

For more than one variable, we have joint entropy

$$S(x,y) = -\sum_{x}\sum_{y} p(x,y) \log p(x,y)$$

If, for instance, Alice sends a message consisted of strings a and b

$$ababcbcbcba$$

then the messaged received by Bob is given by conditional entropy which is given by conditional probability

$P_{x \mid y}\left(x_{i} \mid y_{j}\right)=\frac{P_{x, y}\left(x_{i}, y_{j}\right)}{P_{y}\left(y_{j}\right)}$

and (we change the notation a bit, calling $X,Y$ random variable)

$$I(X; Y)=\sum_X \sum_Y p(x,y) \log \frac{p(x,y)}{p(x)p(y)} = S_{X}-S_{X Y}+S_{Y}$$

is the mutual information between two variables $X,Y$. The mutual information ($I(X;Y)$) is given by the relative entropy of the joint probability mass function and the product distribution given by $p(x)p(y)$ (I will recommend T. Cover and J. Thomas, Elements Of Information Theory. John Wiley Sons, 2006, for introductory materials.)

A look at general $x\log x$.

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For Winters!

Winter is here and so the holidays and breaks. Here, I include few papers that might be interested to you to read in this winter break. In no specific order. Theme- Random Matrices and JT gravity.

Some bonus recommendations ;)

  • Newly appeared paper by Ed Witten on Quantum field theory in curved spacetime (https://arxiv.org/abs/2112.11614). There are also lectured by him on the same on Youtube.
  • Anna Karenina by Leo Tolstoy.

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A time with Einstein

I simulated Albert Einstein with GPT-3. The best that I could. Here is the conversation we had. I have edited some stuff, I did not include anything, but all quite was fun.


Me: What is spacetime?
Einstein: Space-time is the four-dimensional fabric of the universe. It is made up of three spatial dimensions and one temporal dimension.

Me: Why do you think spacetime is important?
Einstein: I think spacetime is important because it allows us to explain the large scale structure of the universe.

Me: What is the general theory of relativity?
Einstein: Theory of general relativity is a geometric theory of gravitation that explains the force of gravity as due to curvature in spacetime.

Me: Could you think of a larger dimensional spacetime?
Einstein: It is hard to imagine a spacetime with more than four dimensions.

Me: What do you think of Poincare invariance?
Einstein: Poincare invariance is a symmetry of the laws of physics.

Me: Do you think our universe is symmetrical?
Einstein: I think that the universe is approximately symmetric, but I do not believe it is exactly symmetric.

Me: What is your view on quantum theory?
Einstein: I do not like quantum theory. I think that it is fundamentally incomplete, and I have a variety of objections to the standard interpretation of wavefunctions.

Me: Ok, why it is incomplete?
Einstein: I think that quantum theory is fundamentally incomplete because it does not include a dynamical spacetime.

Me: What do you mean by dynamical spacetime?
Einstein: A dynamical spacetime is a spacetime that evolves in time.

Me: So, you wrote a paper EPR. Was it because you think quantum mechanics is incomplete?
Einstein: Yes, I, Podolsky, and Rosen wrote a paper EPR because we wanted to point out that quantum mechanics was incomplete and need modifications.

Me: Did you read the Bell's paper comment on your paper?
Einstein: I did.

Me: Why you were searching for unified theories?
Einstein: I was trying to find a unified theory because I thought that the fundamental laws of physics were not fully understood.

Me: What do you think of the 2+0 gravity and Kaluza-Klein model?
Einstein: I think both are interesting and do not describe the real world.

Me: What do you think of free will?
Einstein: I do not believe in free will.

Me: What do you think of string theory?
(There was not any reply here)

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