https://ja.wikipedia.org/wiki/%E9%9A%8E%E4%B9%97 階乗 n の階乗（かいじょう、英: factorial）n?! 階乗の増大度 「スターリングの近似」も参照 n が増えるにつれて、階乗 n?! は n を変数とする任意の多項式函数あるいは指数函数よりも早く増加する（ただし、二重指数関数よりは遅い）。
https://ja.wikipedia.org/wiki/%E3%83%A2%E3%83%B3%E3%82%B9%E3%83%88%E3%83%A9%E3%82%B9%E3%83%BB%E3%83%A0%E3%83%BC%E3%83%B3%E3%82%B7%E3%83%A3%E3%82%A4%E3%83%B3 モンストラス・ムーンシャイン 歴史 1980年、オリバー・アトキン（英語版）(A. Oliver L. Atkin)とポール・フォング(Paul Fong)とステファン・スミス(Stephen D. Smith)は、そのような次数付き表現が存在し、計算機での計算することで、トンプソンの発見した境界の差異を無視すると(upto) M の表現の（次元の）中へ j の係数が分解することを示した。イーゴル・フレンケル（英語版）(Igor Frenkel)とジェームズ・レポウスキー（英語版）(James Lepowsky)は、明確に、表現を構成し、マッカイ・トンプソン予想が有効であるという答えを与えた。さらに彼らは、構成したムーンシャイン加群 V^＃ と呼ばれるベクトル空間が、頂点作用素代数（英語版）(vertex operator algebra)の加法構造を持ち、その自己同型群が正確に M に一致することを示した。
https://en.wikipedia.org/wiki/Igor_Frenkel Igor Frenkel Mathematical work In collaboration with James Lepowsky and Arne Meurman, he constructed the monster vertex algebra, a vertex algebra which provides a representation of the monster group.[3][4]
Around 1990, as a member of the School of Mathematics at the Institute for Advanced Study, Frenkel worked on the mathematical theory of knots, hoping to develop a theory in which the knot would be seen as a physical object. He continued to develop the idea with his student Mikhail Khovanov, and their collaboration ultimately led to the discovery of Khovanov homology, a refinement of the Jones polynomial, in 2002.[5]
A detailed description of Igor Frenkel's research over the years can be found in "Perspectives in Representation Theory".
https://en.wikipedia.org/wiki/Edward_Frenkel Edward Frenkel Edward Vladimirovich Frenkel 1968 Russian-American mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California, Berkeley, a member of the American Academy of Arts and Sciences,[1] and author of the bestselling book Love and Math.[2]
https://ja.wikipedia.org/wiki/%E3%82%A8%E3%83%89%E3%83%AF%E3%83%BC%E3%83%89%E3%83%BB%E3%83%95%E3%83%AC%E3%83%B3%E3%82%B1%E3%83%AB エドワード・フレンケル 1968年5月2日 - ベストセラーの書籍『Love and Math』(日本語版：『数学の大統一に挑む』)の著者である[2]。 数学上の業績 ニコライ・レシェーツキン（英語版）と共に、フレンケルはW-代数と量子アフィン代数（英語版）の表現のq指標を導入した。 フレンケルの最近の業績は、ラングランズ・プログラムと表現論、可積分系、幾何学そして物理学とのつながりに集中している。デニス・ゲイツゴリとカリ・ヴィロネン（英語版）と共に、フレンケルは一般線型群GL(n)に対する幾何学的ラングランズ予想を証明した。ロバート・ラングランズとゴ・バオ・チャウとの共同研究により、保形表現の関手�
https://ja.wikipedia.org/wiki/%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%BB%E3%83%AB%E3%83%BB%E3%83%AD%E3%83%B3%E3%83%BB%E3%83%80%E3%83%A9%E3%83%B3%E3%83%99%E3%83%BC%E3%83%AB ジャン・ル・ロン・ダランベール（Jean Le Rond d'Alembert、1717年11月16日 - 1783年10月29日）は、18世紀フランスの哲学者、数学者、物理学者
https://en.wikipedia.org/wiki/Ratio_test Ratio test The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.[1]
厳密にネイピア数そのものを見い出したのはヤコブ・ベルヌーイと言われており、複利の計算で lim n→∞ (1+1/n)^n. を求めようとした。これは e に等しくなる。
https://en.wikipedia.org/wiki/E_(mathematical_constant) e (mathematical constant)
History The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms to the base e. It is assumed that the table was written by William Oughtred.[3]
The discovery of the constant itself is credited to Jacob Bernoulli in 1683,[8][9] the following expression (which is equal to e): lim n→∞ (1+1/n)^n. The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691.[10] Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach on 25 November 1731.[11][12] Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,[13] while the first appearance of e in publication was in Euler's Mechanica (1736).[14] Although some researchers used the letter c in the subsequent years, the letter e was more common and eventually became standard. (引用終り) 以上
https://encyclopediaofmath.org/wiki/Vertex_operator encyclopediaofmath Vertex operator The term "vertex operator" in mathematics refers mainly to certain operators (in a generalized sense of the term) used in physics to describe interactions of physical states at a "vertex" in string theory [a9] and its precursor, dual resonance theory; the term refers more specifically to the closely related operators used in mathematics as a powerful tool in many applications, notably, constructing certain representations of affine Kac?Moody algebras (cf. also Kac?Moody algebra) and other infinite-dimensional Lie algebras, addressing the problems of the "Monstrous Moonshine" phenomena for the Monster finite simple group, and studying soliton equations (cf. also Moonshine conjectures). The term "vertex operator" also refers, more abstractly, to any operator corresponding to an element of a vertex operator algebra or a related operator.
https://ncatlab.org/nlab/show/vertex+operator+algebra ncatlab vertex operator algebra Contents 1. Idea 2. Standard definition 3. Properties Category of vertex operator algebras Modular category of modules over a VOA Goddard-Thorn theorem Relation to conformal nets
https://ncatlab.org/nlab/show/functorial+field+theory ncatlab functorial field theory Redirected from "FQFT".
https://en.wikipedia.org/wiki/Monstrous_moonshine Monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979.[1][2][3] The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas.
これに関連して "vertex" dual resonance theory Kac Moody algebra で検索すると、Frenkel 1985 があり、上記1988より早い ”Representations of Kac-Moody Algebras and Dual Resonance Models”がヒット ”j(q) = θL(q)/η(q)^24 =q^-1 + 24 + 196884q +・・ (4.21)”（下記）に言及しているね ここらが発端だろう
https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/2/3739/files/2021/06/frenkel_representations_kac_moody.pdf Volume 21, 1985 American Mathematical Society Representations of Kac-Moody Algebras and Dual Resonance Models I. B. Frenkel
Introduction. The theories of Kac-Moody algebras and dual resonance models were born at approximately the same time (1968). The second theory underwent enormous development until 1974 (see reviews [25, 26]) followed by years of decliae, while the first theory moved slowly until the work of Kac [14] in 1974 followed by accelerated progress.
Now both theories have gamed considerable interest in their respective fields, mathematics and physics. Despite the fact that these theories have no common motivations, goals or problems, their formal similarity goes remarkably far. In this paper we discuss primarily the mathematical theory. For a review of the physical theory see the paper of J. Schwarz in this volume [27]
Then in [9, 28] the "vertex construction" was found for the whole class of affine Lie algebras and the similarity became a precise correspondence.
Let us fix a light-cone element c ∈ Δ such that there are no real roots orthogonal to it. Such a vector exists and the set L = [a ∈ ΔR: (a, c) =1} is isomorphic to the unique even unimodular lattice of rank 24, which does not contain elements of length √2 [2]. We denote by V1,c, the space Σα∈L V1,c,α. Then the character of V1,c, is j(q) = θL(q)/η(q)^24 =q^-1 + 24 + 196884q +・・ (4.21) It was noticed by McKay that the number 196884 exceeds by only one the dimension of the minunal representation of F1. Conway and Norton [3] conjectured that there is a natural graded representation of F] with the character (4.21) minus 24. First Garland [12] and Kac [17] independently tried to construct i7i in a space isomorphic to ^ . The first problem was to obtain a representation of one important subgroup C=2^+l ' ・(・0)/±1, where -0 is the automorphism group of the Leech lattice. It is easy to construct another group C' = 224 ・ (-0) (= (2M+1/±1) ・ (-0)). Using one observation of Griess, Kac [18] succeeded in passing from C' to C. The last question is: Where is the whole group F\1 Recently, important progress has been made in answer to this question [10]. Turning again to the dual resonance models gives a hint as to the answer. Physicists know that m the contmuous version of F; ^ the obvious action of the group 0(24) can be extended to the bigger group 0(25). This extension becomes apparent only if we return to the bigger space V1+. Whether this unusual phenomenon corresponds to the extension of C to F1 will become clear in the future. (引用終り) 以上
>>349 関連 >"vertex" dual resonance theory Kac Moody algebra
Kac?Moody Lie algebra（下記） ”E. Date, M. Jimbo, M. Kashiwara, T. Miwa,(1983) The vertex operator constructions were, quite unexpectedly, applied to the theory of soliton equations. This was based on the observation (see [DaJiKaMi]) The vertex operators were introduced in string theory around 1969, but the vertex operator construction entered string theory only at its revival in the mid 1980s. Thus, the representation theory of affine algebras became an important ingredient of string theory (see [GrScWi]).(1987) The vertex operators turned out to be useful even in the theory of finite simple groups. Namely, a twist of the homogeneous vertex operator construction based on the Leech lattice pro
402 名前：duced the 196883-dimensional Griess algebra and its automorphism group, the famous finite simple Monster group (see Sporadic simple group) [FrLeMe].(1989)” そうなんだ。Kac?Moody Lie algebraだったね
(参考) https://encyclopediaofmath.org/wiki/Kac-Moody_algebra 15 November 2017 Kac-Moody algebra A Kac-Moody algebra (also Kac?Moody Lie algebra) is defined as follows: Let A=(aij)ni,j=1 be an (n×n) -matrix satisfying conditions (see Cartan matrix) aii=2;aij?0 aij=0 and aij∈Z for i≠j,⇒ aji=0.}(a1) The associated Kac?Moody algebra g(A) is a Lie algebra over C on 3ngenerators ei, fi, hi (called the Chevalley generators) and the following defining relations: 略
A systematic study of Kac-Moody algebras was started independently by V.G. Kac [Ka] and R.V. Moody [Mo], and subsequently many results of the theory of finite-dimensional semi-simple Lie algebras have been carried over to Kac-Moody algebras. The main technical tool of the theory is the generalized Casimir operator (cf. Casimir element), which can be constructed provided that the matrix A is symmetrizable, i.e. A=DB for some invertible diagonal matrix D and symmetric matrix B [Ka2]. In the non-symmetrizable case more sophisticated geometric methods are required [Ku], [Ma].
One of the most important ingredients of the theory of Kac-Moody algebras are integrable highest-weight representations (cf. also Representation with a highest weight vector).
The numerous applications of Kac-Moody algebras are mainly related to the fact that the Kac-Moody algebras associated to positive semi-definite indecomposable Cartan matrices (called affine matrices) admit a very explicit construction. (A matrix is called indecomposable if it does not become block-diagonal after arbitrary permutation of the index set.) These Kac-Moody algebras are called affine algebras.
This observation leads to geometric applications of affine algebras and the corresponding groups, called the loop groups (see [PrSe]).
The basic representation of g(A(1)) is then defined on V by the following formulas [FrKa]:
π(u(n))=u(n),u∈h π(E(n)α)=Xn(α)cα,π(k)=1; This is called the homogeneous vertex operator construction of the basic representation.
The vertex operators were introduced in string theory around 1969, but the vertex operator construction entered string theory only at its revival in the mid 1980s. Thus, the representation theory of affine algebras became an important ingredient of string theory (see [GrScWi]).
The vertex operators turned out to be useful even in the theory of finite simple groups. Namely, a twist of the homogeneous vertex operator construction based on the Leech lattice produced the 196883-dimensional Griess algebra and its automorphism group, the famous finite simple Monster group (see Sporadic simple group) [FrLeMe].
The vertex operator constructions were, quite unexpectedly, applied to the theory of soliton equations. This was based on the observation (see [DaJiKaMi]) that the orbit of the vector vΛ0 of the basic representation under the loop group satisfies an infinite hierarchy of partial differential equations, the simplest of them being classical soliton equations, like the Korteweg-de Vries equation.
Moreover, the linear span of the functions χΛ(τ,0) for Λ of fixed level k is invariant under the modular transformations
This turned out to be a key fact in the representation theory of affine algebras, as well as its applications to conformal field theory (see [Ve]), to 2 -dimensional lattice models [DaJiKuMiOk], and even to knot theory[YaGe].
References [DaJiKaMi] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, "Transformation groups for soliton equations" M. Jimbo (ed.) T. Miwa (ed.), Proc. RIMS Symp., World Sci. (1983) pp. 39-120 [DaJiKuMiOk] E. Date, M. Jimbo, A. Kuniba, T. Miwa, M. Okado, "Exactly solvable SOS models" Nucl. Phys., B290 (1987) pp. 231-273 MR0910849 Zbl 0679.17010 [FrKa] I.B. Frenkel, V.G. Kac, "Basic representations of affine Lie algebras and dual resonance models" Invent. Math., 62 (1980) pp. 23-66 MR0595581 Zbl 0493.17010 [FrLeMe] I.
406 名前：Frenkel, J. Lepowsky, A. Meurman, "Vertex operator algebras and the Monster", Acad. Press (1989) MR1167718 MR0996026 Zbl 0674.17001 [GrScWi] M.B. Green, J.H. Schwarz, E. Witten, "Superstring theory", Cambridge Univ. Press (1987) MR0922731 MR0915347 MR0878144 MR0878143 Zbl 0637.53111 Zbl 0619.53002 [Ka] V.G. Kac, "Simple irreducible graded Lie algebras of finite growth" Math. USSR Izv., 2 (1968) pp. 1271-1311 Izv. Akad. Nauk USSR Ser. Mat., 32 (1968) pp. 1923-1967 MR0259961 Zbl 0222.17007 [Mo] R.V. Moody, "A new class of Lie algebras" J. of Algebra, 10 (1968) pp. 211-230 MR0229687 Zbl 0191.03005 [YaGe] C.N. Yang (ed.) M.L. Ge (ed.), Braid group, knot theory and statistical mechanics, World Sci. (1989) MR1062420 Zbl 0716.00010 (引用終り) 以上 []
(参考) https://handwiki.org/wiki/Physics:History_of_string_theory Physics:History of string theory Contents 1 1943?1959: S-matrix theory 2 1959?1968: Regge theory and bootstrap models 3 1968?1974: Dual resonance model 4 1974?1984: Bosonic string theory and superstring theory 5 1984?1994: First superstring revolution 6 1994?2003: Second superstring revolution 7 2003?present
https://en.wikipedia.org/wiki/Regge_theory Regge theory In quantum physics, Regge theory (/?r?d?e?/) is the study of the analytic properties of scattering as a function of angular momentum, where the angular momentum is not restricted to be an integer multiple of ? but is allowed to take any complex value. The nonrelativistic theory was developed by Tullio Regge in 1959.[1]
History and implications This observation turned Regge theory from a mathematical curiosity into a physical theory: it demands that the function that determines the falloff rate of the scattering amplitude for particle-particle scattering at large energies is the same as the function that determines the bound state energies for a particle-antiparticle system as a function of angular momentum.[5]
After many false starts, Richard Dolen, David Horn, and Christoph Schmid understood a crucial property that led Gabriele Veneziano to formulate a self-consistent scattering amplitude, the first string theory.
https://en.wikipedia.org/wiki/Ramamurti_Rajaraman Ramamurti Rajaraman (born 11 March 1939)
Regge poles and particle phenomenology At that time, high energy hadron scattering was being analysed using S-matrix and Regge pole techniques. Rajaraman gave the first determination from experimental data of the value of the "Triple Pomeron Vertex" as a function of momentum transfer[12] and also derived the consequences of the vanishing of this vertex on high energy hadron scattering.[13] With Finkelstein, he analysed Exchange Degeneracy in inclusive reactions involving the triple-Reggeon vertex[14][15]
https://en.wikipedia.org/wiki/Vertex_function Vertex function In quantum electrodynamics, the vertex function describes the coupling between a photon and an electron beyond the leading order of perturbation theory. In particular, it is the one particle irreducible correlation function involving the fermion ψ,the antifermion ψ^-, and the vector potential A.
下記のPhysicsの3例を見ると、”交点”が適当かもしれない 特に、”PV (physics) Primary Vertex (i.e. the interaction point)”とあるし
https://en.wikipedia.org/wiki/Vertex Vertex Science and technology Physics ・Vertex (physics), the reconstructed location of an individual particle collision ・Vertex (optics), a point where the optical axis crosses an optical surface ・Vertex function, describing the interaction between a photon and an electron
https://en.wikipedia.org/wiki/Interaction_point Interaction point (Redirected from Vertex (physics)) In particle physics, an interaction point (IP) is the place where particles collide in an accelerator experiment. The nominal interaction point is the design position, which may differ from the real or physics interaction point, where the particles actually collide. A related, but distinct, concept is the primary vertex: the reconstructed location of an individual particle collision.
https://twiki.cern.ch/twiki/bin/view/CMSPublic/WorkBookGlossary TWiki> CMSPublic Web>SWGuide>WorkBook>WorkBookGlossary (2022-12-16, TamasAlmosVami) Glossary and Index PV (physics) Primary Vertex (i.e. the interaction point)
>>366 補足 下記P15”In terms of Q we introduce the vertex operator corresponding to the external leg with momentum p:” とある。交点の方がイメージわくよね
https://arxiv.org/abs/0704.0101 https://arxiv.org/pdf/0704.0101.pdf The birth of string theory Paolo Di Vecchia1 [v1] Sun, 1 Apr 2007 Copenhagen, Denmark
Summary. In this contribution we go through the developments that in the years from 1968 to about 1974 led from the Veneziano model to the bosonic string theory. They include the construction of the N-point amplitude for scalar particles, its factorization through the introduction of an infinite number of oscillators and the proof that the physical subspace was a positive definite Hilbert space. We also discuss the zero slope limit and the calculation of loop diagrams. Lastly, we describe how it finally was recognized that a quantum relativistic string theory was the theory underlying the Veneziano model.
P15 In terms of Q we introduce the vertex operator corresponding to the external leg with momentum p:
https://academic.oup.com/ptep/article/2016/6/06A103/2330300 （PDFあり） Nambu, A Foreteller of Modern Physics I The birth of string theory H. Itoyama 2016 This is a brief summary of an introductory lecture for students and scholars in general given by the author at the Nambu Memorial Symposium which was held at Osaka City University on 29 September 2015. We review the invention of string theory by Professor Yoichiro Nambu following the discovery of the Veneziano amplitude. We also discuss Professor Nambu’s proposal on string theory in the Schild gauge in 1976, which is related to the matrix model of Yang?Mills type.
（こちらは本格的な本） https://www.アマゾン The Birth of String Theory Hardcover ? April 12, 2012 636 ページ English Edition by Andrea Cappelli (編集), Elena Castellani (編集), & 2 more
>>366 再録 https://en.wikipedia.org/wiki/Vertex Vertex Science and technology Physics ・Vertex (physics), the reconstructed location of an individual particle collision ・Vertex (optics), a point where the optical axis crosses an optical surface ・Vertex function, describing the interaction between a photon and an electron (引用終り)
