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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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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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33 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 ...
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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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40 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 ...
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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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30 views

Covariant derivative of energy-stress momentum tensor of Polyakov action is trivial.

From the equations of motion of the Polyakov action we know that the energy-stress momentum tensor is zero, $T_{ab}=0$. Isn't it then trivial that $\nabla^a T_{ab} =0$? In page 15, this note on ...
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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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93 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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1answer
92 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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50 views

Mathematics for Barton Zweibach String Theory

I'm interested in knowing what mathematics is needed ONLY for Barton Zweibach's "A first course in String Theory"? Note I'm only asking for what mathematics is necessary for just Barton Zweibach's ...
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116 views

What's the latest on Moonshine conjecture? [closed]

Regarding this article, what is the latest concerning the Moonshine conjecture for Pariah groups. As far as I can tell Moonshine has been shown for the Monster group and some of the subquotient ...
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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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1answer
56 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 $2r$ ...
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1answer
181 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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1answer
780 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 ...
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83 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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47 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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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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131 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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40 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 ...
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1answer
80 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}\...
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$1+2+3+4+5+\dotsb =\frac{-1}{12}$ 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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108 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 ...
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1answer
190 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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155 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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1answer
22 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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1answer
45 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} ...
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2answers
157 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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296 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}^...
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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 ...
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208 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 $n(n - 1)/2$ ...
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How many different subsequence in Thue-Morse sequence

Consider the Thue-Morse string. Suppose it has $n$ elements. My question is: how many different substrings(or subsequence) in this string. Actually I'm interest in bound of this value. Is it smth ...
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351 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 ...
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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-...
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1answer
72 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_{\...
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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....
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114 views

Weyl transformation of geodesic distance

Consider a Riemannian manifold $M$ with a metric $g$. For two points $x,y \in M$ the geodesic distance $d(x,y)$ is defined in the usual way. I would like to know if there is a formula expressing how ...
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83 views

Are $T^2/Z_2$ orbifolds just ironed spheres?

(Note that this question is migrated from the physics SE... I apologize for the imprecise language) The only $Z_2$ symmetries I can think of the torus are reflection on plane, whose quotient should ...
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1answer
102 views

An intutive proof of 'replacing two-caps by a handle'

I am trying to understand a statement given in Polchinski Vol.1 - a torus with cross-cap can be obtained either as (g,b,c) = (0,0,3) or as (1,0,1), trading two cross-caps for a handle. Here, g is ...
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3answers
156 views

Calculating euler number of disk

I'm trying to do exercise 3.1 from Polchinski, which should be a rather easy differential geometry problem. I have to calculate the euler number defined by $$\chi = \frac{1}{4\pi}\int_{M}d^{2}\sigma ...
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1answer
346 views

Image of ring homomorphism $\phi : \mathbb{Z}[t] \to \mathbb{Q}$?

Here is a problem I face practicing the theory of rings: Define $\phi : \mathbb{Z}[t] \to \mathbb{Q}$, a ring homomorphism (it does map $1$ to $1$). I'm trying to show that if $\phi(t)=\frac{u}{v}$ ...
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199 views

(Soft Question) How active an area of research is Non-Commutative Geometry? [closed]

I am currently an undergraduate, but I am considering applying for a phd in algebraic geometry or a related field. I am quite interested in the link between non-commutative geometry and theoretical ...
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377 views

How many bit strings of length 7 exist if the string remains unchanged if it is reversed?

How many bit strings of length 7 exist if the string remains unchanged if it is reversed ? 1 1 1 1 1 1 1 and 1 0 0 1 0 0 1 are an example that is unchanged if reversed. 0 0 0 0 0 0 0 and 0 1 1 0 1 1 ...
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Areas of contemporary Mathematical Physics

I have often heard that some developments in Physics such as Gauge Theory, String Theory, Twistor Theory, Loop Quantum Gravity etc. have had a significant impact on pure mathematics especially ...
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Polchinski 12.3.22 - superspace green's function

Forming the supersymmetric string using superfields and superspace, Polchinski claims that the function $$ G \sim \ln{\left|z_{1} - z_{2} - \theta_{1}\theta_{2}\right|^{2}} $$ satisfies the equation $$...
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1answer
789 views

String probability (with conditional prob and combinations)

I'm having trouble with the questions below, all relating to string probability. I'll write the problem and then provide my work for my (incorrect) answer. Please help me figure out what I did wrong. ...
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1answer
175 views

Why are there cubics in a Calabi-Yau manifold?

I heard from a recent talk that the "number of $n$-degree curves" in a Calabi-Yau manifold is an invariant of the space. But what does that mean? (Specifically I would like to ask the following.) ...
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2answers
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Complex and Kähler-manifolds

I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simpelest N=1 supergravity we'll get these. Now in the course-...
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1answer
172 views

Physical interpretation of categorical structures related to Dirichlet Branes

In Dirichlet Branes and Mirror Symmetry by Aspinwall et al, section 5.9 discusses various questions that remain open. In particular they say: "There are many constructions from homological algebra ...
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Status of a question from Freeman Dyson's 1972 article

In a famous article, Freeman Dyson mentions an interesting relationship between the $\tau$ functions of number theory and the dimensions of finite-dimensional simple Lie algebras (section 2). He ...