{"metadata":{"kernelspec":{"language":"python","display_name":"Python 3","name":"python3"},"language_info":{"pygments_lexer":"ipython3","nbconvert_exporter":"python","version":"3.6.4","file_extension":".py","codemirror_mode":{"name":"ipython","version":3},"name":"python","mimetype":"text/x-python"}},"nbformat_minor":4,"nbformat":4,"cells":[{"cell_type":"markdown","source":"This is an essay about  artificial intelligence and machine learning (AI/ML), but before we can delve into the AI/ML part, we must take a sizable detour to present the context of interest in which the techniques of AI/ML are deployed. This will take us on a lightning tour of contemporary theoretical physics together with some description of some specific problems that can be tackled with AI/ML in the past few years. This is not an exhaustive account of all theoretical physics problems that are amenable to the techniques of AI/ML but rather an in-depth description of a few selected problems that the author has some familiarity with.","metadata":{}},{"cell_type":"markdown","source":"<h1 id=\"sec:phys1\">1. The struggle of contemporary theoretical physics</h1>\n\nThe field of physics concerns itself with the understanding of natural phenomena around us and the unlocking of the natural laws that govern everything that exists: from the largest to the tiniest of scale, ranging from the entire universe with its infinitely many galaxies to the subatomic particles that make up all known matter in the universe. On the one hand, we rely on  the theory of quantum mechanics to understand the behaviors of these constituents of matter; on the other hand, we rely on the theories of special and general relativity to understand the behavior of space and time. The synthesis of quantum mechanics with special relativity led to the successful construction of many quantum field theories such as quantum electrodynamics, electroweak theory and quantum chromodynamics that form the basis of the *Standard Model of particle physics* - A model that has explained and predicted all observed physical interactions of the three fundamental forces of nature - electromagnetism, strong and weak nuclear forces - to a stunning degree of precision [[1]](#1). However, as successful as the Standard Model (SM) is currently, it cannot be the final model that explains everything, as gravity - the fourth fundamental force of nature - is not part of its framework. This is due to the incompatibility between quantum mechanics and general relativity that has prevented the synthesis of these two theories into a *final quantum gravity theory*. Constructing this final theory is the ultimate goal of contemporary theoretical physics - this is an endeavor that has been going on since at least the 1980s with currently no end in sight. Several worthy candidates for the theory of quantum gravity have been constructed, perhaps most notable among them are the two theories known as loop quantum gravity and *string theory*. In this essay, we will only focus on the latter and study the various ways in which AI/ML methods have been applied to explore and tackle some difficult problems in string theory. ","metadata":{}},{"cell_type":"markdown","source":"<h1 id=\"sec:string\">2. The struggle of the potential \"theory of everything\"</h1>\n\nString theory is more than just a contender for the final theory of quantum gravity - it is arguably also the most successful candidate for the position of \"theory of everything\" in contemporary physics [[2]](#2). Not only is the theory celebrated  for its mathematical beauty and consistency, the many results that came out of string theory have proved transformational in many other areas of physics and pure mathematics - perhaps the most important among these is the AdS/CFT holographic duality conjecture that states that quantum gravity in a negatively curved universe (Anti-de-Sitter space, or AdS for short) is dual to a conformal field theory (CFT) in a universe without gravity in a spacetime with one dimension less [[3]](#3). However, string theory is notorious for requiring many exotic physical structures that have yet to be observable with our current technology. These structures are required to exist at very high energies - higher than the level currently accessible by our most powerful particle collider - The Large Hadron Collider. As such, their existence is unverifiable currently (nor can they be invalidated). Among these are:\n\n - **Supersymmetry** - a type of hypothetical symmetry between bosons and fermions (between  'force' particles and 'matter' particles). \n\n - **Extra dimensions** - spacetimes in string theory are not limited to exist in just four dimensions like our observable universe. \n\nIn this essay, for the first part, we will focus on the second point above, namely the existence of extra spacetime dimensions. This idea is not unique to string theory, since long before its conception, extra dimensions were proposed as a way to unify gravity with electromagnetism in the work of Kaluza and Klein in the 1920s [[4]](#4),[[5]](#5). While it might seem physically ludicrous to the uninitiated, there are compelling mathematical arguments/reasons for not ruling out the potential existence and validity of theories incorporating extra dimensions in their framework. It would be too much of a detour to get into these arguments in the scope of this essay, which focuses instead on the AI/ML aspects - namely, how to leverage the power of AI/ML to study *string theory problems involving spacetimes with more dimensions than four*. \n\nIn the second part of this essay, we will describe another problem that has recently been tackled using AI, namely *solving conformal field theories using the bootstrap approach*. Conformal field theories (CFTs) occupy a very important place in the construction of string theory and appear very often in various string theory contexts. Two of the most notable are: \n - Two-dimensional ($D=2$) CFTs describe the worldsheet theory of strings' propagation - these are the theories that explain how the various modes of vibration of strings correspond to all the fundamental particles known in our world (such as spin-0 scalars, spin-1 photons (or gauge bosons, more generally), spin-1/2 fermions (quarks and leptons), spin-2 gravitons) [[2]](#2).\n \n - $D$-dimensional CFTs (with $2\\leq D\\leq 6$) are one of the two ingredients, or one side, of the AdS/CFT holographic duality. The other ingredient is $(D+1)$-dimensional AdS spacetimes. Because of the (exact) correspondence between the observables of AdS and CFT, quantum gravity in AdS spaces can be understood by studying CFTs, which, in some cases, are far more tractable. \n \nGiven these, it is perhaps not too much of an exaggeration to state that understanding CFTs is synonymous with understanding string theory. ","metadata":{}},{"cell_type":"markdown","source":"<h1 id=\"sec:ai_string\">3. Using AI to find string theory compactifications</h1>\n\nIf our starting point is a spacetime geometry in $n$ dimensions (with $n>4$), the immediate problem is to find ways to compactify the $(n-4)$ extra dimensions such that there are only 4 macroscopic, observable dimensions. The idea of compactification means that these extra dimensions are hidden from plain view at low energies (where life exists) and only detectable at very high energies. Using compactification, the original $n$-dimensional spacetime can be effectively expressed as a product of two geometries: an inner, compactified geometry (or manifold) with no time dimension, and an external, observable four-dimensional universe with both space and time dimensions. The key point here is the fact that the geometry or symmetries of the internal manifold play a determining role in the observable particle spectrum of the external universe [[6]](#6). For our 4D universe, this particle spectrum is documented in the Standard Model mentioned earlier. So, loosely speaking, if there exists a higher dimensional spacetime model that admits a compactified internal space with the right symmetries that can give rise to the correctly observed 4D particle spectrum, we cannot rule out the possibility that our 4D universe is part of this higher dimensional world, and alongside our 4D world, there is an unseen internal space accessible only at very high energies. \n\nThe problem is, in string theory, there is a great number of possibilities for such higher-dimensional spacetimes, together with an even greater number of possibilities for the compactified internal space. In fact, the number is so large (initially conservatively estimated to be about $10^{500}$ [[7]](#7), with the latest estimate being $10^{272000}$ [[27]](#))  that it is not humanly possible to fully explore the space of all possibilities. This is where AI/ML techniques have been put to use effectively for the last couple of years, yielding useful results. The gist of the problem involves finding and classifying geometries with the correct symmetries that correspond to the 4D SM particle spectrum. \n\nSince we are dealing with string theory, supersymmetry is a prerequisite - As such, AI has been deployed to identify and classify geometries with the right symmetries that can give rise to the minimally supersymmetric standard model (MSSM) particle spectrum. This spectrum is a doubling of the SM spectrum because, for every particle that exists (and is documented in the SM model), there is a supersymmetric partner. Working with a specific type of string theory known as heterotic theory, there is a special type of manifolds that can be used as the internal space in string theory compactifications, known as Calabi-Yau (CY) manifolds,  with the right geometric properties that can guarantee the existence of the  \"correct\" MSSM particle spectrum in the observed four-dimensional external world [[7]](#7),[[8]](#8), [[9]](#9) (A concise explanation of this mathematical fact can be found in Chapter 1, section 1.2.1 \"Triadophilia\" of the book [[7]](#7)). Since CY manifolds are \"complex\" (in the sense of \"complex\" numbers), the terminology goes as follows: a CY $n$-fold refers to a CY with $2n$ real dimensions, e.g. a 3-fold means a CY with 6 real dimensions, a 4-fold means a CY with 8 real dimensions, etc. The mathematical setting of the string compactification problem involving CY manifolds is typically of the following cases:  \n\n- A 10-dimensional theory (heterotic string theory) compactified on an internal CY 3-fold. This is the setting that we will discuss further below. \n      \n- A 12-dimensional theory (F-theory) compactified on an internal CY 4-fold. (Although this case is not discussed, we will point out some works on a similar setting after discussing the 10D heterotic case above.)\n\nFurthermore, there exists a particular construction of CY manifolds as the vanishing loci of a number of homogeneous polynomial equations in an ambient space that is the product of all the individual spaces in which each polynomial equation is defined [[7]](#7), [[8]](#8). This type of CY manifolds is known as \"complete intersection CY\" (CICY), and it is this specific type of CY that is used in those compactification problems involving AI/ML techniques. \n\nThere are two ways AI techniques can be applied to the study of CY manifolds and their associated use as compactification manifolds:\n\n - **Supervised learning**: AI/ML can be used to learn certain geometric properties of the CICY 3-folds and 4-folds. The dataset for the former consists of 7890 3-folds, while the dataset for the latter contains about $10^6$ 4-folds. This learning problem is a supervised one, in which a neural network is used to learn and predict certain geometric or topological quantities of interest. \n \n - **Reinforcement learning (RL) ** is used to find consistent string compactification models.  \n\nWe will describe each of the two cases above in more detail below.\n\n","metadata":{}},{"cell_type":"markdown","source":"<h2 id=\"ai_sup\">3.1. Supervised learning of geometric/topological properties </h2>\n\nFor this problem, we will first focus on the CICY 3-fold dataset of 7890 manifolds. Each data point in this dataset is a matrix of different size, so for practicality, one takes the dimensions of the largest matrix and uses padding on the rest to create a uniform dataset of 7890 $13\\times 15$  matrices  - all of which have positive integer entries denoting the degrees of the polynomials defining the CY manifolds. Geometric and topological properties of each manifold are also included in each data sample in the dataset. Hence, the problem of learning certain properties of interest is a conventional supervised learning problem involving a neural network.  In particular, the most often studied quantities for this dataset are the Hodge numbers $h^{1,1}$ and $h^{2,1}$ (very roughly speaking, the higher these numbers are, the more physically complex the CY manifold is and the more geometric structures it has). Different machine learning techniques such as linear regression, support vector machines, decision trees and neural networks were used to predict $h^{1,1}$ and $h^{2,1}$ [[11]](#11). For the neural network option, different convolutional neural network architectures were used and experimented with to get increasingly better results, with the most successful one involving inception blocks (with nearly a 100 percent accuracy for learning $h^{1,1}$) [[10]](#10).\n\nFor the case of CICY 4-folds, the dataset consists of almost $10^6$ samples (the actual number is closer to $9\\times 10^5$) - Each sample consists of a configuration matrix describing the CICY 4-fold and a list of topological invariants associated with it. This data structure is identical to the 3-fold case, but each data sample contains more information - this is necessarily the case because a CICY 4-fold is characterized by more topological invariants than its 3-fold counterpart. The largest size of the configuration matrix is $16\\times 20$  so padding is again applied to all matrices in the dataset to obtain a uniform list. The topological properties of interest are again the Hodge numbers, of which there are now four: $h^{1,1}$, $h^{2,1}$, $h^{2,2}$, and $h^{3,1}$. In [[26]](#26), the authors applied supervised learning using dense neural networks (classifier and regressor networks) to predict two of the Hodge numbers - $h^{1,1}$ and $h^{3,1}$. While they achieved almost 100 percent accuracy on the $h^{1,1}$ learning problem, far poorer results were obtained for the $h^{3,1}$ case due to the much larger spread of the $h^{3,1}$ values using the same classifier/regressor network architectures. By restricting the dataset to only those configuration matrices with a small size of $4\\times 4$ and employing a two-branch network architecture (a dense network on one branch and a single dense layer on the other branch), better results (around 98 percent accuracy) for $h^{3,1}$ were obtained. \nIn a later work [[27]](#27), a specialized network, named CICYMiner and built using multiple inception blocks, was proposed to simultaneously predict all 4 Hodge numbers. This architecture presents a great improvement on the problem of learning Hodge number for 4-fold CICYs with the achieved accuracy of 100 percent for both $h^{1,1}$ and $h^{2,1}$, 96 percent for $h^{3,1}$, and 83 percent for $h^{2,2}$. \n\nIn a broader context not necessarily related to the string theory compactification problems, supervised learning involving conventional machine learning techniques (regression, support vector machines, decision trees, etc.) and those involving the use of deep neural networks have been successfully applied to the field of computational algebraic geometry - A list of the typical problems can be found in Chapter 4 (\"Machine Learning the Landscape\")  of the book [[7]](#7). ","metadata":{}},{"cell_type":"markdown","source":"<h2 id=\"ai_reinforce\">3.2. Reinforcement learning in compactification model building</h2>\n\nFor this problem, in addition to the CY 3-folds that act as the internal compactified space (in the heterotic string context), another essential ingredient is the vector bundle that can be constructed from these CY manifolds. The presence of said vector bundle is required in order to facilitate the gauge symmetry breaking of the higher-dimensional model from a single large symmetry group to an intermediate gauge symmetry group that unifies the smaller, observed gauge symmetry group of the four-dimensional world. The unification gauge group can be either $E_6$, $SO(10)$, or $SU(5)$, which are the groups capable of unifying the Standard Model gauge group $SU(3)\\times SU(2)\\times U(1)$ - the 4D gauge group characterizing the three fundamental forces of nature (electromagnetism, strong and weak nuclear forces). \n\nTo build a compactification model, one starts by selecting a CICY 3-fold of interest and then proceeds to construct the vector bundle associated with this CY 3-fold. Depending on whether the choice of the 4D unification group is $E_6$, $SO(10)$ or $SU(5)$, the rank of the vector bundle is 3, 4, or 5, respectively. Crucially important is the fact that the constructed vector bundle must satisfy a list of *topological constraints* which translate directly to properties of the 4D particle spectrum [[12]](#12). Among the desired physical properties for the 4D particle content are the following (the list below only highlights the most important features and is not exhaustive): \n\n- The number of 4D fermionic matter generations must equal three (as in reality, quarks and leptons come in three generations)\n- There must not be any exotic matter (such as those associated with the unification gauge symmetry of either $E_6$, $SO(10)$ or $SU(5)$) charged under the 4D symmetries $SU(3)\\times SU(2)\\times U(1)$ of the Standard Model. This means that if there are \"exotic\" particles, they must behave like scalars (or singlets) under the SM gauge group (to specify a particle as a scalar/singlet under a certain symmetry is to mean that the particle remains invariant when acted on by those transformations generated by this symmetry. A macroscopic example that can be readily understood is the case of a rotationally invariant object (such as a sphere) looking the same no matter how it is rotated - in this example, the object is said to behave as a \"scalar/singlet\" under rotational symmetry).\n \nEach configuration of the vector bundle that satisfies the constraint list constitutes a semi-realistic string compactification model. Two types of vector bundles were considered in the majority of the works found in the literature: monad bundles and line bundles. Reinforcement learning was applied in problems involving both types of bundles.\n\n For the purposes of further technical description, we take the case of line bundles and explain in detail how RL is performed. The line bundle can be represented as a matrix $V$ of size $rk(V)\\times h^{1,1}$ where $h^{1,1}$ is one of the Hodge numbers, and $rk(V)= 3,4,5$ is the rank of the bundle which is directly tied to the unification gauge group of $E_6$, $SO(10)$ and $SU(5)$ as mentioned above.  The starting point of the RL problem is the matrix $V$ with randomly initialized integer entries $k_a^i$  ($i = 1, \\ldots, h^{1,1}$ and $a = 1, \\ldots, n$) chosen to be within a certain range. The entries $k_{a}^i$ are then changed in steps of +1 or -1 by the agent performing the learning task. Each time a change is effected, the topological properties of vector bundles are altered (these are directly calculated from $k_{a}^i$, for example, the condition of having three generations of matter is $\\frac{1}{6} d_{ijl} \\sum_{a=1}^n k_a^i k_a^j k_a^l = -3|\\Gamma|$, where $d_{ijl}$ and $|\\Gamma|$ are geometric quantities associated with the CICY 3-fold of choice). Depending on whether the altered topological quantities satisfy the list of constraints, the agent is rewarded or punished accordingly. A new line bundle model is found when the agent successfully finds a configuration of the $V$ matrix satisfying all topological constraints. This RL problem was implemented in the work of [[13]](#13) using the Asynchronous Advantage Actor Critic (A3C) method for the case of $n=5$ ($SU(5)$ models) and $h^{1,1} = 5$ and $h^{1,1} = 6$, corresponding to the CICY 3-folds named \"$M_{5256}$\" and \"$M_{5302}$\", respectively. For each of these chosen CICY 3-folds, thousands of new $SU(5)$ line bundle models have been found, demonstrating the success of the approach. \n\nFor the monad bundle case (which requires two matrices $B$ and $C$ to represent the bundle instead of a single matrix $V$, but  remains qualitatively similar to the line bundle case), RL implemented in Mathematica (either with the REINFORCE or actor-critic package) was performed in the work of [[14]](#14), (see also [[15]](#15) for a comparison of the results obtained by RL  and genetic algorithms) to identify hundreds of new $SO(10)$ monad models that can give rise to the correct 4D phenomenology. A nice summary of these results (as well as discussions on other aspects related to CICY string compactifications), by one of the authors of [[14]](#14)-[[15]](#15), can be found in [[16]](#16). Incidentally, [[16]](#16) was written as a chapter contribution to the book \"Machine-learning in Theoretical Physics and Pure Mathematics\" [[17]](#17), which contains a comprehensive collection of AI/ML techniques that have been applied to the problems in string theory compactification and algebraic geometry that were briefly described/mentioned in this section of the essay, together with some other topics such as AI/ML in number theory and statistical inference. \n\nBefore moving on, we note that for a different kind of compactification problems concerning a different type of string theory (namely 10D type IIB theory) that are closely related to (or, in some cases, even equivalent to) the 12D F-theory setting mentioned above, RL involving A3C and double deep-Q learning algorithms was used (also compared against and found to outperform other algorithms such as random walk or metropolis) in the work of [[18]](#18) to successfully identify many new low-energy effective field theory models arising from the type IIB theory compactified on either a 6-dimensional torus or a 6D conifold (both are special kinds of CY 3-folds). A later work [[19]](#19) builds on these results and uses both A3C RL and genetic algorithms to generate new models with the desired phenomenology. \n\nCollectively, the promising results of these works demonstrate the utility and flexibility of RL in this type of string compactification problems.","metadata":{}},{"cell_type":"markdown","source":"<h1 id=\"sec:ai_cft\">4. Using AI in conformal bootstrap approach</h1>\nTo understand the context of this problem, several fundamental facts about CFTs must be enumerated without delving too deep into the technical details: \n\n- These theories are quantum field theories with additional symmetries  (conformal symmetries) which place strong restrictions on the observables of the theories. Intuitively, the more symmetries a physical system has, the more restrictive and predictable its properties become. \n\n- The observables of a general CFT are its $n$-point correlation functions of the form $\\langle \\phi_1 \\phi_2\\rangle$, $\\langle \\phi_1 \\phi_2 \\phi_3\\rangle$, $\\langle \\phi_1\\phi_2\\phi_3 \\phi_4\\rangle$, etc., where $\\phi_i$ (with $i=1,2,\\ldots,N$) constitute the field content of the theory. \n\n- Conformal symmetries determine the form of the 2-point and 3-point functions\n$\\langle \\phi_1 \\phi_2\\rangle$, $\\langle \\phi_1 \\phi_2 \\phi_3\\rangle$ exactly, while leaving the 4-point (and $n$-point with $n>4$) functions constrained but not exactly determined.\n\n- Associated with each primary operator $\\phi_i$ of a CFT is a quantum number known as the \"scaling dimension\" $\\Delta_i$ (This is a simplified statement of the more general fact that all operators present in a CFT must form representations under conformal symmetries. These representations are labeled by a set of quantum numbers that are the eigenvalues of the Cartan generators of the symmetries - $\\Delta_i$ is only one of such quantum numbers).\n\n- Any two primary operators in a CFT (with $N$ primary operators) can be expanded into a sum of the remaining operators of the form $\\phi_i(x_i) \\phi_j(x_j) = \\sum_{\\phi_k} C_{ijk} f(x_i-x_j, \\partial_k)\\phi_k(x_k)$ under an operation called Operator Product Expansion (OPE). The coefficients $C_{ijk}$ are the OPE coefficients, and $f$ is a differential operator acting on $\\phi_k$. \n\n- A CFT is completely solved if its data (consisting of the  OPE expansion coefficients $C_{ijk}$ and the operator scaling dimensions $\\Delta_i$) are determined. From here on, we refer to this data simply as ($C, \\Delta)$. \n\nThe goal of the conformal bootstrap program is to solve for the CFTs data $(C, \\Delta)$ by solving the so-called \"crossing symmetry equation\". This equation is derived based on the fact that 4-point correlation functions can be reduced to a sum of 2-point functions using OPE, and the order of reduction should not matter, i.e. given $\\langle \\phi_1\\phi_2\\phi_3\\phi_4\\rangle$, the reduction of this into either $\\langle \\phi_1\\phi_2 \\rangle\\langle \\phi_3\\phi_4\\rangle$ or into $\\langle \\phi_1\\phi_4\\rangle\\langle \\phi_2\\phi_3\\rangle$ (or $\\langle \\phi_1 \\phi_3\\rangle \\langle \\phi_2 \\phi_4\\rangle$) should yield an identical result. Unfortunately, the fact that this equation actually is an infinite set of equations with an infinite set of unknowns that are the CFT data ($C, \\Delta)$  hampered the progress on solving it for decades. Recently, this problem has been studied from the opposite perspective - by assuming a minimal set of CFT data, one can check whether the consistency conditions required by the crossing symmetry equation are satisfied. This approach paved the way for the development of many advanced numerical techniques to study the crossing equations for CFTs of different dimensions [[28]](#28). \n\nA novel implementation of this same approach using reinforcement learning with the Soft Actor Critic algorithm was introduced in the work of [[20]](#20) for the case of 2D CFTs. The crossing symmetry equation is truncated to a finite set of $N_z$ equations for each point $z_i$ ($i= 1, \\ldots, N_z)$ in space. For each of the $i^{th}$ equation in this set, the sum over the infinite number of intermediate OPE coefficients and scaling dimensions is truncated to a finite set of $(C_I, \\Delta_I)$, where $I = 1, \\ldots, N_{u}$, by defining a spin partition function that specifies a fixed number of operators per spin that can appear in the set of intermediate operators. The space of parameter $(C_I, \\Delta_I)$ is  $2N_{u}$-dimensional, and it serves as the environment for the RL problem. The agent can operate in three different modes, given the inputs being the spin partition function, the lower bounds and search windows for $C$ and $\\Delta$. The agent is rewarded or punished based on whether the crossing symmetry equation is satisfied (within a desired level of accuracy) with the chosen value of $C$ or $\\Delta$.  In [[20]](#20), the authors applied  this RL algorithm to the 2D Ising, tricritical Ising, and compactified free boson models, obtaining results close to the analytic ones to a high degree of accuracy. In a more recent work [[21]](#21), the authors applied a similar RL strategy to solve the crossing symmetry equations (in the special case of scalar primaries in the energy-momentum tensor multiplet) for the six-dimensional supersymmetric CFT (which is much more challenging than the 2D case), and obtained results in agreement with those obtained from previous numerical studies. ","metadata":{}},{"cell_type":"markdown","source":"<h1 id=\"sec:concl\">5. Conclusion</h1>\n\nIn this essay, we have discussed two theoretical physics problems in the context of string theory that have recently been tackled with machine learning and artificial intelligence techniques. There are many outstanding challenges in connecting the framework of string theory to the observable four-dimensional physics due to the profoundly complex nature of string theory and the enormous number of possibilities for a viable 4D phenomenology that are allowed  by the theory (for the latest status on string theory, see [[22]](#22)), but the use of AI in the problems discussed in this essay has demonstrated their utility and versatility in producing new and promising results. This serves as compelling proof that AI can be employed effectively to expand the frontier of knowledge in this domain. In fact, it was not long ago - only in the middle of the year 2017 - that AI/ML techniques were introduced into string theory, starting with the seminal works [[23]](#23), [[24]](#24), [[25]](#25) among a few others. Since then, the list of string theory problems amenable to AI/ML techniques has been steadily growing as theoretical physicists incorporate ML/AI into their approach and find new ways to apply ML/AI to physics problems of increasingly diverse natures. ","metadata":{}},{"cell_type":"markdown","source":"<h1 id=\"sec:reference\">References</h1>\n\n<a id=\"1\">[1]</a> M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Avalon Publishing, 1995 \n\n<a id=\"2\">[2]</a> K. Becker, M. Becker, J. H. Schwarz, String theory and M-theory - A Modern Introduction,\nCambridge University Press, 2012 \n\n<a id=\"3\">[3]</a> J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv.\nTheor. Math. Phys. 2 (1998) 231-252, hep-th/9711200.\n\n<a id=\"4\">[4]</a> Theodor Kaluza, Zum Unitätsproblem in der Physik, Sitzungsber. Preuss. Akad. Wiss. Berlin.\n(Math. Phys.) (1921) \n\n<a id=\"5\">[5]</a> Oskar Klein, Quantentheorie und fünfdimensionale Relativitätstheorie\". Zeitschrift für Physik\nA (in German). 37 (12): 895–906, 1926 \n\n<a id=\"6\">[6]</a> M. Green, J. Schwarz, and E. Witten, Superstring Theory Vol. 2: Loop Amplitudes, Anomalies\nand Phenomenology, Cambridge University Press (2012). See Chapter 14 \"Compactification of\nhigher dimensions\" \n\n<a id=\"7\">[7]</a> Y.-H. He, The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning, Lecture Notes in Mathematics, Springer Cham (2021), arXiv:1812.02893v2 (https://arxiv.org/abs/1812.02893)\n\n<a id=\"8\">[8]</a> L. E. Ibanez and A. M. Uranga, String theory and Particle Physics: An Introduction to String\nPhenomenology, Cambridge University Press (2012). 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