{"version":"1.0","provider_name":"SOUL OF MATHEMATICS","provider_url":"https:\/\/soulofmathematics.com","author_name":"Rajarshi Dey","author_url":"https:\/\/soulofmathematics.com\/index.php\/author\/rajarshidey1729gmail-com\/","title":"MIRROR SYMMETRY - SOUL OF MATHEMATICS","type":"rich","width":600,"height":338,"html":"<blockquote class=\"wp-embedded-content\" data-secret=\"kjBKrjAN8N\"><a href=\"https:\/\/soulofmathematics.com\/index.php\/mirror-symmetry\/\">MIRROR SYMMETRY<\/a><\/blockquote><iframe sandbox=\"allow-scripts\" security=\"restricted\" src=\"https:\/\/soulofmathematics.com\/index.php\/mirror-symmetry\/embed\/#?secret=kjBKrjAN8N\" width=\"600\" height=\"338\" title=\"&#8220;MIRROR SYMMETRY&#8221; &#8212; SOUL OF MATHEMATICS\" data-secret=\"kjBKrjAN8N\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\" class=\"wp-embedded-content\"><\/iframe><script type=\"text\/javascript\">\n\/* <![CDATA[ *\/\n\/*! This file is auto-generated *\/\n!function(d,l){\"use strict\";l.querySelector&&d.addEventListener&&\"undefined\"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!\/[^a-zA-Z0-9]\/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret=\"'+t.secret+'\"]'),o=l.querySelectorAll('blockquote[data-secret=\"'+t.secret+'\"]'),c=new RegExp(\"^https?:$\",\"i\"),i=0;i<o.length;i++)o[i].style.display=\"none\";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute(\"style\"),\"height\"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):\"link\"===t.message&&(r=new URL(s.getAttribute(\"src\")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener(\"message\",d.wp.receiveEmbedMessage,!1),l.addEventListener(\"DOMContentLoaded\",function(){for(var e,t,s=l.querySelectorAll(\"iframe.wp-embedded-content\"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute(\"data-secret\"))||(t=Math.random().toString(36).substring(2,12),e.src+=\"#?secret=\"+t,e.setAttribute(\"data-secret\",t)),e.contentWindow.postMessage({message:\"ready\",secret:t},\"*\")},!1)))}(window,document);\n\/* ]]> *\/\n<\/script>\n","thumbnail_url":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2021\/05\/b43a7a50872995-5608972ba15af.gif?fit=600%2C338&ssl=1","thumbnail_width":600,"thumbnail_height":338,"description":"With the appreciation of the importance of Yang\u2013Mills gauge theories in describing the physics of particle interactions, and with the appreciation of its importance in the mathematics of vector bundles, renewed interaction between the two fields began to take place. For example, the importance of instantons and monopoles came to be appreciated from both the physical and mathematical points of view. With the discovery of supersymmetry and its logical completion to superstring theory, a vast arena of interaction opened up between physics and mathematics and continues today at a very deep level for both fields. Mathematicians and physicists often find it difficult to discuss their work and interact with each other. The reason for this appears to be twofold. First, the languages used in the two fields are rather different. This problem is gradually being resolved as we recognize the need to become \u201cbilingual.\u201d The second and more serious problem is that the established scientific methods in the two fields do not converge. Whereas mathematics places emphasis on rigorous foundations and the interplay of various structures, to a physicist the relevant aspects are physical clarity and physical interconnection of ideas, even if they come at the cost of some mathematical rigor. This can lead to friction between mathematicians and physicists. Strings and Compactification In physics,&nbsp;string theory&nbsp;is a&nbsp;theoretical framework&nbsp;in which the&nbsp;point-like particles&nbsp;of&nbsp;particle physics&nbsp;are replaced by one-dimensional objects called&nbsp;strings. These strings look like small segments or loops of ordinary string. String theory describes how strings propagate through space and interact with each other. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its&nbsp;mass,&nbsp;charge, and other properties determined by the vibrational state of the string. There are notable differences between the world described by string theory and the everyday world. In everyday life, there are three familiar dimensions of space (up\/down, left\/right, and forward\/backward), and there is one dimension of time (later\/earlier). Thus, in the language of modern physics, one says that&nbsp;spacetime&nbsp;is four-dimensional.&nbsp;One of the peculiar features of string theory is that it requires&nbsp;extra dimensions&nbsp;of spacetime for its mathematical consistency. In&nbsp;superstring theory, the version of the theory that incorporates a theoretical idea called&nbsp;supersymmetry, there are six extra dimensions of spacetime in addition to the four that are familiar from everyday experience. Lets Talk About&#8230; 1. Manifolds We describe a manifold by coordinate charts. Let {U\u03b1} be an open covering of the topological space M. We endow M with the structure of an n-dimensional manifold with the following information. Let \u03d5\u03b1 : U\u03b1 \u27a5 Rn be a coordinate chart (one may think of coordinates x\u03b1 = (x\u03b1)i , i = 1, \u2026, n as representing the points themselves, i.e., their pre-images under \u03d5\u03b1). On U\u03b1 \u2229 U\u03b2, we can relate coordinates (x\u03b1) to coordinates (x\u03b2) by x\u03b1 = \u03d5\u03b1 \u25e6 \u03d5\u03b2-1(x\u03b2). The map g\u03b1\u03b2 = \u03d5\u03b1 \u25e6 \u03d5\u03b2\u22121 is a transition function. Note that g\u03b1\u03b2 = g\u03b2\u03b1\u22121 and g\u03b1\u03b2 g\u03b2\u03b3 g\u03b3\u03b1 = 1. As an alternative to this structure, we could form a manifold based solely on the data of patches and transition functions satisfying the above relations. A manifold is called \u201cdifferentiable\u201d if its transition functions are differentiable, and \u201csmooth\u201d if the transition functions are smooth (C\u221e). If n = 2k and one can (and does) choose \u03d5\u03b1 : U\u03b1 \u27a5 Ck with holomorphic transition functions, the manifold is called \u201ccomplex.\u201d Note that this extra structure is restrictive. Two complex manifolds may be diffeomorphic as real manifolds (meaning there are invertible, onto, differentiable maps between the two), but there may be no complex analytic mapping between them (we then say they have different complex structures). Likewise, two homeomorphic manifolds may have different structures as differentiable manifolds. Differentiability depends on the coordinate chart maps \u03d5\u03b1. 2. Vector Bundles Vector bundles are constructed similarly, only now every point carries an additional structure of a vector space (\u201cfiber\u201d) over it. Clearly, by retaining the information of the point but forgetting the information of the vector space, we get a map to the underlying manifold. From the description above, it is clear that the simplest vector bundle, E, will be a product space E = M \u00d7 V, where M is a manifold and V is an r-dimensional vector space. E is said to be a rank r vector bundle. E is equipped with the map \u03c0 : E \u2192 M, namely \u03c0((m, v)) = m. Such a vector bundle is called \u201ctrivial.\u201d Locally, all vector bundles are trivial and look like products. So a rank r vector bundle E is a smooth manifold with a map \u03c0 : E \u2192 M to a base manifold, M, such that every point x \u2208 M has a neighborhood x \u2208 Ux with \u03c0\u22121(Ux) \u223c= Ux \u00d7 Rr. Algebraic Geometry Algebraic geometry deals with curves or surfaces (or more abstract generalizations of these) which can be viewed both as geometric objects and as solutions of algebraic (specifically, polynomial) equations. Algebraic geometry sets out to answer these questions by applying the techniques of&nbsp;abstract algebra&nbsp;to the set of polynomials that define the curves (which are then called \u201calgebraic varieties\u201d).&nbsp;The mathematics involved is inevitably quite hard, although it is covered in degree-level courses. Other common questions in algebraic geometry concern points of special interest such as&nbsp;singularities,&nbsp;inflection points&nbsp;and&nbsp;points at infinity&nbsp;\u2013 we shall see these throughout the catalogue. More advanced questions in algebraic geometry concern relations between curves given by different equations and the topology of curves, and many other topics. CLICK THE LINK TO KNOW MORE ABOUT Algebraic Geometry. Calabi\u2013Yau Manifolds Compactification can be used to construct models in which spacetime is effectively four-dimensional. However, not every way of compactifying the extra dimensions produces a model with the right properties to describe nature. In a viable model of particle physics, the compact extra dimensions must be shaped like a&nbsp;Calabi\u2013Yau manifold.&nbsp;A Calabi Yau manifold is a special&nbsp;space&nbsp;which is typically taken to be six-dimensional in applications to string theory. It is named after mathematicians&nbsp;Eugenio Calabi&nbsp;and&nbsp;Shing-Tung Yau. After Calabi\u2013Yau manifolds had entered physics as a way to compactify extra dimensions, many physicists began studying these manifolds. Lance Dixon, Wolfgang Lerche,&nbsp;Cumrun Vafa, and Nick Warner noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi\u2013Yau manifold.&nbsp;Instead, two different versions of string theory called&nbsp;type IIA string theory&nbsp;and&nbsp;type IIB&nbsp;can be compactified on completely different Calabi\u2013Yau manifolds giving rise to the same physics.&nbsp;In this situation, the manifolds are called mirror manifolds, and the relationship between the two physical theories is called mirror symmetry. Homological Mirror Symmetry In string theory and related theories in physics, a\u00a0brane\u00a0is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. The word brane comes from the word &#8220;membrane&#8221; which refers to a two-dimensional brane. In string theory, a string may be open (forming a segment with two endpoints) or closed (forming a closed loop).\u00a0D-branes\u00a0are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter &#8220;D&#8221; in D-brane refers to a condition that it satisfies, the\u00a0Dirichlet boundary condition. Mathematically, branes can be described using the notion of a\u00a0category.\u00a0This is a mathematical structure consisting of\u00a0objects, and for any pair of objects, a set of\u00a0morphisms\u00a0between them. In most examples, the objects are mathematical structures (such as\u00a0sets,\u00a0vector spaces, or\u00a0topological spaces) and the morphisms are\u00a0functions\u00a0between these structures.\u00a0One can also consider categories where the objects are D-branes and the morphisms between two branes\u00a0\u03b1\u00a0and\u00a0\u03b2\u00a0are\u00a0states\u00a0of open strings stretched between \u03b1\u00a0and\u00a0\u03b2. In the B-model of topological string theory, the D-branes are\u00a0complex submanifolds\u00a0of a Calabi\u2013Yau together with additional data that arise physically from having charges at the endpoints of strings.\u00a0Intuitively, one can think of a submanifold as a surface embedded inside the Calabi\u2013Yau, although submanifolds can also exist in dimensions different from two.\u00a0In mathematical language, the category having these branes as its objects is known as the derived category of coherent sheaves on the Calabi\u2013Yau.\u00a0In the A-model, the D-branes can again be viewed as submanifolds of a Calabi\u2013Yau manifold. Roughly speaking, they are what mathematicians call\u00a0special Lagrangian submanifolds.\u00a0This means among other things that they have half the dimension of the space in which they sit, and they are length-, area-, or volume-minimizing.\u00a0The category having these branes as its objects is called the Fukaya category. SYZ conjecture The\u00a0SYZ conjecture\u00a0is an attempt to understand the\u00a0mirror symmetry\u00a0conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by\u00a0Strominger,\u00a0Yau, and\u00a0Zaslow, entitled &#8220;Mirror Symmetry is\u00a0T-duality&#8221;. Along with the\u00a0homological mirror symmetry conjecture, it is one of the most explored tools applied to understand mirror symmetry in mathematical terms. While the homological mirror symmetry is based on\u00a0homological algebra, the SYZ conjecture is a geometrical realization of mirror symmetry. The initial proposal of the SYZ conjecture by Strominger, Yau, and Zaslow, was not given as a precise mathematical statement.\u00a0One part of the mathematical resolution of the SYZ conjecture is to, in some sense, correctly formulate the statement of the conjecture itself. There is no agreed upon precise statement of the conjecture within the mathematical literature, but there is a general statement that is expected to be close to the correct formulation of the conjecture, which is presented here.\u00a0This statement emphasizes the topological picture of mirror symmetry, but does not precisely characterize the relationship between the complex and symplectic structures of the mirror pairs, or make reference to the associated\u00a0Riemannian metrics\u00a0involved."}