The Way Theorized

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

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

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 $\Phi$. 

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.

Some of the recent stuff.

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

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.

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

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

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

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. 

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.

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

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 $\mathbb{Z}[x]$.

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

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

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.)

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 

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.

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

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/.

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.

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

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.

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.

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

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. 

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.

We have five categories of consistent string theory;

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

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.

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

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.)

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

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 $x\log x$.

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.

•   JT gravity as a matrix integral by Phil Saad, Stephen Shenker and Douglas Stanford - https://arxiv.org/abs/1903.11115.
  

•  Nonperturbative Jackiw-Teitelboim gravity by Clifford V. Johnson and his other papers on this subject as well- https://arxiv.org/abs/1912.03637. 
  

•  Lower dimensional gravity by Roman Jackiw- https://doi.org/10.1016/0550-3213(85)90448-1.

•  Gravitation and hamiltonian structure in two spacetime dimensions by Claudio Teitelboim- https://doi.org/10.1016/0370-2693(83)90012-6.
  

• Comments on the Sachdev-Ye-Kitaev model by Juan Maldacena, Douglas Stanford- https://arxiv.org/abs/1604.07818.
  

• JT Gravity and the Ensembles of Random Matrix Theory by Douglas Stanford, Edward Witten- https://arxiv.org/abs/1907.03363.

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.
  

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.