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.

Filter by
Sorted by
Tagged with
0
votes
0answers
10 views

Equivalence of the Dirac representation of the Lorentz algebra and its conjugate in even dimensions

In Polchinski's String Theory, Appendix B.1 we look at the smallest irreps of the Clifford algebra in even-dimensional spacetimes $d=2k+2$. In (B.1.16) he defines two operators from the gamma matrices ...
1
vote
0answers
8 views

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
0
votes
2answers
21 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 $L^*$. ...
1
vote
0answers
19 views

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 ...
-2
votes
1answer
40 views

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 $$\...
2
votes
0answers
71 views

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 ...
0
votes
1answer
30 views

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 $(b_n)_{n \geq 1}, b_n= \frac{f^{(n)}(1)}{a_n}$ ...
0
votes
0answers
38 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 ...
2
votes
1answer
71 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{...
0
votes
0answers
36 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 ...
0
votes
0answers
40 views

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?
6
votes
1answer
84 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 ...
1
vote
0answers
19 views

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)(...
0
votes
0answers
19 views

How Quintic 3-fold is a Calabi–Yau manifold and has non vanishing Ricci scalar?

It’s well known that quintic 3-fold is a Calabi-Yau manifold in the complex projective space $\mathbb{CP}^{n+1}$ , see for instance: https://en.m.wikipedia.org/wiki/Quintic_threefold Now the main ...
3
votes
1answer
56 views

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 ...
3
votes
1answer
96 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^...
3
votes
0answers
70 views

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 ...
0
votes
0answers
15 views

Creating a CFG given a language specification

L = { 0^p 1^q 0^r 1^s: p,q,r,s in N and p+q = r+s}. Design and create a CFG that generates L1. I'm confused on how to even approach this problem. I tried to break it into a simpler problem, Let L = {...
1
vote
0answers
37 views

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 ...
0
votes
0answers
44 views

What “dimensions” means in “Superstring theory has 10 dimensions”.

I read about Clifford Algebras and different Geometries and its relation to Lie Algebras and to String Theory. You have heard that a version of String Theory has 10 dimensions. Stuff like this make no ...
2
votes
0answers
102 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 ...
1
vote
0answers
58 views

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 ...
0
votes
0answers
33 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 ...
1
vote
0answers
42 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 ...
3
votes
1answer
162 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 ...
2
votes
1answer
97 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 ...
4
votes
0answers
132 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 ...
1
vote
1answer
69 views

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 $2r$ ...
5
votes
1answer
230 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 ...
1
vote
1answer
1k views

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 ...
3
votes
0answers
124 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 ...
2
votes
0answers
60 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 ...
0
votes
2answers
45 views

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 ...
1
vote
0answers
174 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}\...
1
vote
0answers
57 views

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 ...
0
votes
1answer
101 views

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}\...
1
vote
0answers
68 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 ...
4
votes
0answers
129 views

Sphere fibrations over Riemann surfaces

In a physics application, there's a 5-dimensional space which consists of a fibration of $S^3$ over a genus-$g$ Riemann surface $\Sigma_g$. In fact this nontrivial fibration is part of a bigger where ...
6
votes
1answer
261 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 ...
4
votes
0answers
179 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{...
1
vote
1answer
23 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 ...
0
votes
1answer
55 views

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} ...
0
votes
2answers
180 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 ...
4
votes
2answers
335 views

Why does a Gorenstein isolated three-fold singularity have a canonical (3, 0) form on the singularity?

I'm trying to read a physics paper, and when talking about rational, graded, Gorenstein, isolated three-fold singularities they say: "Here graded means that the singularity should have a $\mathbb{C}^...
1
vote
0answers
59 views

A basis with rational entries for the Niemeier lattices

The Niemeier lattices are defined to be the 24 even, self-dual lattices in 24 dimensions. Can anyone provide me with explicit bases for them, such that all the entries of the bases are rational ...
2
votes
2answers
321 views

Trace identities for $\text{SO}(n)$

The Green-Schwarz mechanism in Type I string theory involves certain identities relating traces in the vector and adjoint representations of $\text{SO}(n)$ of dimension $n$ and $n(n - 1)/2$ ...
0
votes
0answers
437 views

What is spectral flow symmetry?

I can't find much about this, and am looking into this to satisfy personal curiosity. I will like to know what spectral flow is, and what spectral flow symmetry is. I tried looking for this on ...
1
vote
0answers
80 views

Semi-infinite forms?

I am reading Vafa's paper 'Topological Mirros and Quantum Strings'(arXiv:hep-th/9111017). In this paper, the author says the Hilbert Space of a fermionic string theory corresponds to the space of semi-...
3
votes
1answer
87 views

Polyakov action in complex coordinates

Let $\Sigma$ be a compact $2$-manifold with riemannian metric $g$ and $f:\Sigma \to \mathbf{R}^n$ given locally by $f_1(x_1,x_2),\dots,f_n(x_1,x_2)$. Define $$ S(f,g) = -\frac{1}{2\pi\alpha'}\int_{\...
10
votes
0answers
396 views

Open questions in Topological K-Theory

I am interested in knowing about current research in the Topological K-Theory, especially its interactions with String Theory. About one and a half decade back, there were some papers by Physicists (e....