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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D-brane gauge theories

Reading Yang-Hui He's review on quiver gauge theories, I read that D-brane gauge theories manifest as a natural description of symplectic quotients and their resolutions in geometric invariant theory....
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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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Equivalence between the equations of motion derived from the two variants of Nambu-Gotto action...

I am trying to show two equations of motion derived from 2 variants of Nambu-Goto action are equivalent... We have the Nambu-Goto action in terms of the induced matrix as $$S=-T\int d^2 \sigma\sqrt{-\...
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3 votes
1 answer
110 views

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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1 vote
0 answers
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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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1 vote
1 answer
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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, $A="...
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1 vote
1 answer
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$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 ...
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10 votes
1 answer
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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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1 vote
1 answer
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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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5 votes
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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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2 answers
71 views

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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-2 votes
1 answer
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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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155 views

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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2 votes
1 answer
109 views

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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115 views

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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6 votes
1 answer
154 views

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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1 vote
0 answers
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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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4 votes
1 answer
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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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5 votes
1 answer
131 views

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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3 votes
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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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2 votes
0 answers
174 views

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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1 vote
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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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36 views

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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1 vote
0 answers
58 views

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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3 votes
1 answer
244 views

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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2 votes
1 answer
106 views

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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5 votes
0 answers
214 views

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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1 vote
1 answer
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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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5 votes
1 answer
274 views

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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1 vote
1 answer
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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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3 votes
0 answers
198 views

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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3 votes
0 answers
77 views

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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2 answers
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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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1 vote
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236 views

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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1 vote
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Connection of irreducible representations of holonomy and differential forms

Say I have some specific irreps of e.g. $SU(3)$ in the case of a Calabi-Yau 3-fold. Now assume you have some section transforming in a specific representation of the holonomy group. Is it possible to ...
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0 votes
1 answer
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String Theory-Virasoro Generators -- show commutator relation

The problem statement, all variables and given/known data (I have dropped the hats on the $\alpha_{n}^{u}$ operators and $L_{m}$) $[\alpha_{n}^u, \alpha_m^v]=n\delta_{n+m}\eta^{uv}$ $L_m=\frac{1}{2}\...
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1 vote
0 answers
70 views

$1+2+3+4+5+\dotsb =\frac{-1}{12}$ Is this true in real domain? Can you tell where this series is used? [duplicate]

In our sequence and series class, my teacher wrote this series and asked if it makes any sense. We all were saying "this makes no sense at all." But when I Googled it, the whole world knows it. So ...
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6 votes
1 answer
369 views

Riemann zeta-function regularization in string theory

First of all let me say that I am a physicist and therefore it is sometimes hard for me to understand some mathematical steps... Now, I've been trying to obtain the well known result for the zeta ...
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4 votes
0 answers
233 views

Instanton Moduli Space on ALE Spaces?

I've recently been learning about moduli spaces of instantons on $\mathbb{C}^{2}=\mathbb{R}^{4}$. From what I can gather, one can consider the framed moduli space of torsion-free sheaves on $\mathbb{...
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1 vote
1 answer
27 views

The group $O(n,n,\mathbb{Z})$ and its properties.

While reading "String Theory" by Becker$^2$, Schwarz I encountered a group called $O(n,m,\mathbb{Z})$, which is a group of symmetries of the bosonic/heterotic strings compactified on a torus. I would ...
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0 votes
1 answer
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Show that for any n ≥ 1, there is an error-detecting set of strings of length n, using the digits 0, 1, and 2, that has 3n−1 strings?

Can anyone please show me a proof by induction for this? A set of error-detecting strings is a set of strings that differ by more than one character. For example: for {0,1,2} for n = 2: {00,11,22} ...
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2 answers
233 views

Explain these concepts of String Theory in easy words for mathematicians, from a popular point of view

In the Wikipedia's article for the number 496 in the section Physics is related the condition found by Green and Schwarz about this perfect number in their string theory. Question. Can you tell us ...
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