Questions tagged [string-theory]
For questions about string theory, which is a research framework in theoretical physics and mathematical physics that attempts to unify quantum theories and general relativity.
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Is it possible to construct a vertex algebra from the Witt algebra?
I am trying to understand the importance of the central charge in the construction of the Virasoro vertex algebra, since I don't yet understand the role it plays in a vertex operator algebra. (I ...
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Bijection between composition algebras over R and classical superstring theories
In the page for superstring theory, Wikipedia states:
Another approach to the number of superstring theories refers to the mathematical structure called composition algebra. In the findings of ...
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path integral of scalar field action on Riemann surface with boundary
In their paper 'Sewing Polyakov Amplitudes I: Sewing at a Fixed Conformal Structure' Carlip et al. start by considering the path integral $$Z_{\Sigma'}[\tilde{X}] = \int [dX]e^{-S[X]} $$ for the ...
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Section of the projection map [closed]
I have a CY four-fold as a hypersurface of degree $(4,3)$ in $P^3\times P^2$ and I have the projection map from this hypersurface say $X$ to $P^3$ as $\pi:X \rightarrow P^3$. Does this admit a section?...
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Origins of conformal symmetries and generators and similar things. Solving and making sense of an equation.
Let me try to ask. Notation might be sloppy. You can just skip to the question somewhere below. I am basically asking about the meaning of solutions of a differential equation in the context of ...
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Compactification in String Theory [closed]
It seems to me that the idea of compactification in String theory is related to the concepts of Compactification and Quotient topology in topology theory. How compactification in String theory can be ...
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What does $Y(1,z)$ = id for vertex algebras mean?
I'm adding an update to this post here with my current understanding of the situation for context. I read some Wikipedia articles and two texts. I am having some trouble so I figure I would attempt to ...
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Period of Cartan 3-form
A Cartan 3-form in a Lie group $G$ is defined as:
$$\omega = Tr(g^{-1}dg\wedge g^{-1}dg\wedge g^{-1}dg)$$
For $g$ being maps $g:B\longrightarrow G$, with $B$ a given manifold.
The so-called periods of ...
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Exceptional Lie groups and algebras in maths and physics
By the beautiful classification theorem of complex semisimple Lie algebras, we know that there are exactly $5$ types of exceptional Lie algebras, say type $E_6,E_7,E_8,F_4$ and $G_2$. We have a ...
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Violin String PDE Modeling
I have this exercise in my differential equations book...
If you pluck a violin string, and then finger the string, fixing it
precisely in the middle, the tone increases by one octave. In
...
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Geometric intuition for $\mathcal{L}^{-m} \oplus \mathcal{L}^{m} \rightarrow T^{2}$ Calabi-Yau threefolds
I'm having some problems to understand the geometry involved in the section 3.1 of the paper Two Dimensional Yang-Mills, Black Holes and Topological Strings.
Concretely, I was wondering to know if it ...
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Is algebraic geometry actually used in string theory?
Many times I have heard string theorists say that string theory has a lot of algebraic geometry, but physicists seem to have identified complex differential geometry with algebraic geometry and ...
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How do we count the number of ways a loop can go around a torus?
As Cumrun Vafa explains in this video, he was able to calculate the number of micro-states for a black hole by counting the number of ways a loop can go around a torus. Obviously, there are infinite ...
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String Concatenation Comparision
Let $A$ and $B$ be strings consisting of small latin alphabets.
We will say $A<B$ iff $AB$ is lexicography smaller than $BA$. ($AB$ is string concatenation of $A$ and then $B$. For example, $A="...
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$yB$ is not a prime ideal in $B.$
Let $B = \mathbb R[x,y]$ where $x^2 + y^2 = 1$ which is called the coordinate ring of the unit circle.
I am trying to prove that $yB$ is not a prime ideal in $B.$
I have the following information ...
user889267
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Topologically, what is a 'string' from string theory?
To begin: I am not a crank. I am not sure how well-founded my titular question is, but it was interesting enough that I decided to bring it to MSE.
For context: I am an undergraduate mathematics ...
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What this parameter in the Riemann surface metric has to do with its complex structure?
As far as I'm aware, a complex manifold $M$ is a topological manifold together with an atlas ${\cal A}$ of charts $(U_i,\varphi_i)\in{\cal A}$ such that the open sets $U_i$ cover $M$, the maps $\...
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$\begin{align}&(x\! -\! a,f_1(x),\ldots,f_r(x))\\ =\ &(x\! -\! a,f_1(a),\ldots,f_r(a))\end{align}\ $ [Ideal evaluation = mod reduction]
$\begin{align}&(x\! -\! a,f_1(x),\ldots,f_r(x))\\[.1em] =\ &(x\! -\! a,f_1(a),\ldots,f_r(a))\end{align}\ $ [Ideal evaluation = mod reduction]
I am solving Aluffi chapter 0. I am completely ...
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What is a Gauge symmetry, intuitively (string theory)?
I'm writing an essay for a popular (but mathematically mature) audience on the history of mathematical physics, wherein I have a section devoted to string theory. Unfortunately, neither I (nor my ...
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Riemannian compact six-dimensional manifolds Ricci-flat
Are there Real compact six-dimensional manifolds Real Ricci-flat?
It is known that Calabi-Yau manifolds exist, that is, Complex compact three-dimensional Ricci-flat, but I don't know if Real compact ...
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Levenshtein Distance vs Damerau Levenstein vs Optimal String Alignment Distance
Could anyone explain the differences between Levenshtein Distance vs Damerau Levenstein vs Optimal String Alignment Distance?
And the Math behind calculating distance between the strings
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Find a dual lattice basis for lattice $e_1 = \sqrt 2 \hat e_x, \ e_2 = -\frac{1}{\sqrt 2} \hat e_x + \sqrt{\frac{3}{2}}\hat e_y$
Given the basis vector of a lattice $L$:
$$e_1 = \sqrt 2 \hat e_x, \ e_2 = -\frac{1}{\sqrt 2} \hat e_x + \sqrt{\frac{3}{2}}\hat e_y$$
I want to find a set of basis vector for the dual lattice $L^*$.
...
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Origin and source of this quote pertaining to the Monster group
I asked an almost exact question on History of Science and Mathematics SE. I vaguely remember reading a quote/listening to a statement by (I think) John Conway which I can paraphrase as follows:
"I ...
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Limit of a string without recurence
Let the string $(x_n)_{n\geq 0}$ such that : $$\frac{1}{1}, \frac{1}{2}, \frac{1}{2}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\cdots$$
Find $$\lim_{n\...
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Prerequisites for Quantum Fields and Strings: A Course for Mathematicians
I'm interested in learning more about the mathematical structure underneath of quantum field theory and string theory. I've taken a few courses on quantum field theory before, so am getting more ...
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The function is considered: $f: (0,\infty) \rightarrow \mathbb{R}, f(x)= \frac{\ln x}{x}$, how to prove that $(b_n)_{n\geq 1}$ has no limit? [closed]
The function is considered:
$$f: (0,\infty) \rightarrow \mathbb{R}, f(x)= \frac{\ln x}{x}$$
to be determined a string $(a_n)_{n \geq 1}$ so the string $(b_n)_{n \geq 1}, b_n= \frac{f^{(n)}(1)}{a_n}$ ...
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What are the mathematical prerequisites for the string theory
If one were to start self-studying string theory with a background only in calculus and some undergraduate linear algebra and ODE, what would the path and resources to cover to reach the string theory ...
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Turning an integral into a Gamma Function
I have a problem with a step in a physics text book [1]. It claims that
\begin{equation}
I(a,b) = \int_0^\infty dt\int_0^\infty du\, \frac{t^{a-1}u^{b-1} }{t+u}\exp \left(-\frac{tu }{t+u}\right)
\end{...
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How can num2str function cut off some digits in Matlab?
I am reading the book " MATLAB: A practical introductoin programming and problem solving", written by Stormy Attaway and getting curious with the function "num2str".
On page 256, the author used the ...
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How can I prove that if a domain $R$ is an injective module, then $R$ is a field? [duplicate]
What are the similarities between a field and a projective module that can support my question?
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Index of a subgroup of the Modular Group
The subgroup of $SL(2,\mathbb{Z})$ generated by
$\begin{pmatrix}1&0\\1&1\end{pmatrix}$ and $\begin{pmatrix}1&5\\0&1\end{pmatrix}$
has come up in a research question in string ...
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Confusing spinor contraction with gamma matrices in group theory
I'm recently confused with spinor contracting with gamma matrices. Here is an example in String Theory on SO(8):
$$\psi^{AB} \underbrace{|A\rangle\otimes|B\rangle}_{8_s\otimes 8_s}=[\underbrace{A(x)(...
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Unitarity of a representation in the physics literature (in particular in CFT)
In many places in the physics literature, and specifically in conformal field theory, a representation of the Virasoro algebra is defined to be unitary when $L_n^\dagger = L_{-n}$.
For example in the ...
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Non-trivial fibration of $SU(3)$ over $S^1$?
In String Theory it is well known that a string can propagate on backgrounds such as a $T^2$ fibred over a circle. This fibration can be non-trivial in the following sense:
Given $T^2$ generators $J^...
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Clarification on Kac Moody algebras and the different meanings in mathematics and physics
I am confused by the way that mathematicians and physicists use the words "Kac Moody algebra", and "loop algebra", and how exactly these concepts relate to one another. I will write down what I ...
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Two equivalent definitions of fibration structure on toric CY $3$-fold
I've been trying to reconcile two seemingly different definitions of what a toric space is, specifically a toric Calabi-Yau 3-fold. The first definition is from the paper Branes and Toric Geometry, by ...
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Grasping the idea of Virasoro Algebras in 2D Conformal field theory
I have been trying to understand the connection between Virasoro algebras and CFT. After a course in string theory, I was under the impression that the Virasoro algebra was simply the Lie algebra of ...
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Understanding Moonshine and Heterotic E8xE8
Recently I have become familiar with the conjectured relationship of monstrous moonshine and pure $(2+1)$-dimensional quantum gravity in AdS with maximally negative cosmological constant and, it’s ...
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Notation for (skew) Young Tableaux in Hubsch
I'm looking at the notation given for Young tableau in Hubsch's book "Calabi-Yau Manifolds: A Bestiary for Physicists" - see Chapter 3, p. 96.
A Young tableau (for a $U(n)$ representation) is denoted ...
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How can I justify that the compact manifold in string theory must be orientable in one dimension
I am preparing a presentation on Calabi-Yau manifolds where I do not want to use advanced mathematics, but I would like to show that under some minimal assumptions in one (complex) dimension, the ...
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Size and shape moduli of Calabi-Yau manifolds
Let's say that Calabi-Yau manifolds are compact Kähler manifolds with a trivial canonical bundle. We know that $H^2(X,\mathcal T_X) = H^1(X,\Omega^2_X)$ is the tangent space to the deformation functor ...
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What is meant by the symbol $\mathbb{R}^2_{\hbar}$?
I am reading some papers in mathematical physics (https://arxiv.org/pdf/1006.0977.pdf) and I came across the following symbol $\mathbb{R}^2_{\hbar}$ I don't recognize nor could I find any background ...
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Are there branches of mathematics that are unique to String Theory?
I have read that String Theory has developed “new” math but the sources do not specifically identify the math. My question: are there branches of mathematics that are unique to String Theory? By ...
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On the Numbers of Representations of a Number as a Sum of $2r$ Squares, Where $2r$ Does not Exceed Eighteen
I am reading the article "Mathematicians Chase Moonshine’s Shadow" [1], and want to follow up on one of its sources "On the Numbers of Representations of a Number as a Sum of $2r$ Squares, Where $2r$ ...
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Algebraic geometry and algebraic topology used in string theory
I am looking for a comprehensive book or notes in algebraic geometry and algebraic topology techniques used in string theory compactifications covering topics like orientifolds, orbiolds, Calabi-Yau ...
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Question in String Theory / Mass of States / Number Operator
The problem statement, all variables and given/known data
I have the following definition of the space-time coordinates
Relevant equations
Working in a certain gauge we can also do:
From which we ...
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Question about the relation between the Weierstrass equation and weighted projective space
I am reading a review on Toric Geometry for string theorists by Harald Skarke (arXiv:hep-th/9806059). In section 3, the author says
A standard way of describing an elliptic curve is by embedding it ...
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What is the relation between BRST quantization and gauge fixing quantization
To quantize gauge field, one usually use gauge-fixing procedure and then plus ghost field, my question is what the relation between BRST quantization and gauge fixing quantization is? Because the ...
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Consider the continuous map $ f: \Sigma_{2}^{+} \rightarrow \Sigma_{2}^{+} $ , which is shift map also by
Define $ \Sigma_{2}^{+}=\{x=\{x_n\}: x_n=0 \ or \ 1 \} $ is infinite string of $ \ 0 's \ \ and \ \ 1 's $. Consider the continuous map $ f: \Sigma_{2}^{+} \rightarrow \Sigma_{2}^{+} $ , which is ...
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Gauged Wess–Zumino–Witten (WZW) term and its anomaly
First of all, I am confused about the WZW model itself. Let us consider the following WZW-NLSM model on 2-D manifold $X_2=\partial(X_3)$ with level-1 $SO(4)$ WZW term:
\begin{eqnarray}
S=\int_{X_2}\...
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