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Questions tagged [compactification]

Compactification entails changing a theory with respect to one of its space-time dimensions. Instead of this dimension ranging to infinity, the theory is changed so that this dimension has a finite range, and may be periodic. In the limit where the size of the compact dimension goes to zero, no fields depend on this extra dimension, and the theory is *dimensionally reduced*. Further use for dimensional reduction.

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Extra Dimensions (in String Theory) - What does it mean?

I have been reading a lot about string theory and the necessity of extra dimensions (particularly as visualized in Calabi-Yau spaces), as "curling-ups" in our apparently 3-dimensional (or 4-...
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Particle spectrum in dimensional reduction

First of all, sorry if this question is a bit stupid, but my knowledge of certain aspects of particle physics and group theory is a bit limited. I am compactifying the heterotic $E_{8}\times E_{8}$ ...
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Compactification of space in Hamiltonian formulation of Yang-Mills theory

I am reading David Tong's lecture notes on Gauge Theory where he talks about Hilbert space interpretation of Yang-Mills theories in Section 2.2 of Chapter 2. When discussing the gauge dependence of ...
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Relationship between compactification moduli and generations in standard model

The situation I am describing is a $10D$ heterotic string theory which is compactified on a Calabi-Yau to get a $N=1$, $4D$ effective theory. It is mentioned in Ashoke Sen's notes on string ...
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Are the “extra dimensions” in string theory universal?

Are the extra (compactified) dimensions from string theory universal, in that any particle/field with a sufficiently small enough wavelength will be able to propagate through them? The reason I want ...
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Root weights and states in orbifold compactifications

I have the following question regarding orbifold compactifications of the heterotic string: What is the relation between a certain representation and the weights of the root lattice? I mean: take ...
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Are particles of fields that arise from compactification and strings treated differently in string theory?

I am aware that particles in string theory are different vibrating modes of strings. I am also aware that compactification leads to emergent fields from the parts of the metric of the compactified ...
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How do we assign length to dimensions in string theory?

My current understanding of the variations of string theory includes the sentence "string theory needs (at least) 6 extra spatial dimensions to work, but because our observable universe consists of 3 ...
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Questions about the landscape in string theory

If I understand correctly, the string theory landscape is the totality of possible Calabi-Yau manifolds to make up the compact factor of space in string theory, in which there are of the order of $10^{...
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3 generations as extra unconventional dimensions?

Kaluza-Klein resonances arise in theories with extra dimensions like superstrings or old KK models. Usually, extra dimensions and KK states provide regular spacing in the spectrum. Question: can some ...
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What is meant by this variant of the euclidean plane: $\mathbb{R}^2_{\hbar}$?

I am reading some papers in mathematical physics (https://arxiv.org/abs/1006.0977) 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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Kaluza-Klein and Fourier expansion

In every book/reference on Kaluza-Klein (KK) dimensional reduction, one uses that fluctuations $\delta\Phi(x,y)$ can be expanded as follows $$\delta\Phi(x,y)= \sum_n\delta\Phi_n(x)\,h_n(y)$$ where $\{...
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Compactification from non-orthogonal symmetries

For many theories like string theory, one extends the dimensions to a higher number D and then requires the space-time rotational symmetry group to be O(D-1,1). Then one compactifies the excess ...
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What generates the curvature which is necessary for curling-up extra dimensions?

To say it right away, I am not an expert in string theory, but I know well General Relativity. So I wonder how the curling up of extra-dimensions which is assumed in many "Kaluza-Klein" like theories (...
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Understanding a joke in Zwiebach - “A first course in string theory” [closed]

So fairly early on Zwiebach discusses the quantum mechanics of a one-dimensional square well. He then goes on to add an extra dimension which is compact to demonstrate how one can understand ...
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Spectrum of 11d supergravity on a CY 3-fold

I'm trying to understand how various fields arise in the spectrum of 11d supergravity compactified on a Calabi-Yau 3-fold, as described in arXiv:hep-th/9506144. Specifically, my understanding is ...
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How seriously should one take the energy eigenvalues of an electron's wavefunction in a compactified space?

In chapter 2 (section 2.10) of Zwiebach's string theory book, there is a neat derivation of the energy eigenvalues for the two dimensional square well in which one dimension is compactified to a ...
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Effective action of IIB Calabi-Yau compactification

I'm currently reading K. Becker, M. Becker and John. H. Schwartz book on string theory. I have a question about Calabi-Yau compactifications of IIB string theory. In chapter 9 page 403, Why do we don'...
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Compactification of Bosonic Closed Strings on $T^2$ and $T^3$

I am looking for a text to explain compactification of bosonic closed strings on $T^2$ and $T^3$ by focusing on its gauge groups enhancement. In fact, I want to know in each case ($T^2$ and $T^3$) how ...
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What is really M-theory? (non-pertubatively)

I don´t really understand what M-theory is supposed to be. Going beyond the dualities relating different string theories (for example the common $11-D$ limit of IIA and $E_{8}\times E_{8}$) I don't ...
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Algebraic geometry and topology for string theory [duplicate]

I am looking for a comprehensive book or notes in algebraic geometry and topology techniques used in string theory compactifications covering topics like orientifolds, orbiolds, Calabi Yau manifolds ...
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The role of the Hopf Fibration in Kaluza-Klein theory

I have been learning recently about the Hopf Fibration and its relation to physics. My professor has told me that it is one of the simplest methods of dimensional reduction in Kaluza-Klein theory. ...
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Can branes obstruct complex structure deformations in string theory?

In compactifications of string theory, to preserve supersymmetry often branes need to wrap subspaces that are specified by holomorphic equations in the compactification space $X$. Additionally there ...
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For making gravity supersymmetric is a modification of Einstein's vacuum field equations necessary ( apart from adding the R-S Lagrangian)?

Upon building up supersymmetric Supergravity in 4D, is it necessary to modify the Einstein's vacuum field equations (apart from adding the Rarita-Schwinger Lagrangian for the gravitino) in order to ...
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Where are the infinite-dimensional spaces of Quantum Mechanics?

There are two questions here: One of the confusing points about String Theory is the existence of extra dimensions. These are explained by saying that the these extra dimensions are compactified. ...
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The space-time metric in string theory

I always wondered about the metric to be used in string theory. As 10-4 (11-4) or 26-4 dimensions are supposed to be curled up or being a Calabi-Yau manifold, in this part of the space-time the ...
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Can superstring theories predict our 4 dimensions? [duplicate]

Popular science literature is replete with the necessity for superstring theories to live in at least 10 dimensions, requiring at least 6 of them to be compactified so that they are not observable. ...
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Fundamental string theory questions

Can anyone answer some basic string theory questions for me? The Veneziano Amplitude is celebrated for predicting the scattering amplitude of mesons and for practically giving birth to string theory....
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Geometric interpretation of dimensional reduction of $E_8 \times E_8$ string

Following on from this question. I'm trying to get a geometric picture of what happens under dimensional reduction. Let me focus on a single $E_8$ factor, in 10 dimensions - the Lagrangian will ...
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Does String Theory predict a particle with twice the mass of the electron?

As far as I know, the spectrum of any (?) String Theory is of the form $$ M^2\propto N $$ where $N$ is the number operator. The lightest known particle being the electron, I am led to think that we ...
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How does one actually apply the M-theory/heterotic duality “fiberwise”?

It seems to be generally accepted ([1], [2]) that one can apply the duality between a $T^3$ compactification of heterotic string theory and a $\mathrm{K3}$ compactification of M-theory "fiberwise" to ...
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Boundary conditions on the Euclidean Schwarzschild black hole

This question is based on page 71 of Thomas Hartman's notes on Quantum Gravity and Black Holes. The Euclidean Schwarzschild black hole $$ds^{2} = \left(1-\frac{2M}{r}\right)d\tau^{2} + \frac{dr^{2}}{...
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Dimensional reduction of Rozansky-Witten theory

Rozansky-Witten theory is a 3d topological sigma model which is used to study topological invariants of 3-manifolds. In what follows, $X$ will denote its target space. In a question posted here - ...
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Instability of higher dimensional universes

According to superstring theory, there are at least 10 dimensions in the universe (M-theory actually suggests that there are 11 dimensions to spacetime; bosonic string theories suggest 26 dimensions). ...
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In string theory why are the “extra” dimensions super-compact?

Is the only reason the "extra" dimensions of string theory are considered to be super-compact, so we can avoid dealing with the question "why can't we experience them?"?
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ADE Gauge Theory and String Theory

I should preface this my saying that I'm an algebraic geometer, and am not terribly knowledgeable about physics, nor the physics literature. I want to consider the singular surfaces $\mathbb{C}^{2}/...
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Compact vs non-compact scalars in dimensional reduction

Starting from the vector multiplet in (5+1) dimensions with 8 supercharges (whose field content is a gauge field $A_\mu$ and a spinor field), if we successively dimensionally reduce to (2+1) ...
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Does the universe have some geometry and fundamental constants are nothing but the result of this geometry?

The Physical Constant that appears every where might be possible that related to geometry of the universe that still needed to be uncovered. For example, if a person is confined inside a room which in ...
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Compactifications of 6d (2,0) SCFT

It is conjectured that 6d (2,0) SCFT has no known description in terms of the action or the Lagrangian. However, it has many interesting compactifications for example 3d-3d correspondence which ...
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The determinant of the Dirac operator in Euclidean signature

Suppose the Dirac operator determinant in Euclidean space-time with manifold $\mathbb R^{4}$: $$ d = \text{det}(iD), \quad iD = i\gamma^\mu (\partial_\mu +A_{\mu}) $$ The Dirac operator is elliptical, ...
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Why is full M-theory needed for compactification on singular 7-folds and what does that even mean?

In "M-theory on manifolds of $G_2$ holonomy: the first twenty years" by Duff, it is claimed (e.g. in section 8) that for compactification on singular 7-folds to be possible, we need to consider not ...
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Massless particles in a universe with compact extra-dimensions

One common idea behind many extensions to the Standard Model (such as String Theory or Kaluza-Klein Theory) are small or hidden "Extra-Dimensions", that are compactified. According to my ...
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Can compactified dimensions enlarge and large dimensions compactify in string theory?

In string theory, can six or more dimensions rolled up into high-dimensional compact manifold become large and our 3+1 large dimensions collapse into compact manifold?
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Why spacetime has 3+1 large dimensions and remaining higher dimensions underwent compactification? Why spacetime hasn't got eg. 5+4 or 3+5 dimensions?

Yesterday I wrote very, very big thread which proved to be too broad, so according to your instructions I divided it. Content is unchanged. I read that it might be related to the phenomenon of ...
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What is the stringy interpretation of the cohomology classes arising from the Kähler class?

In superstring theory, one usually considers compactifications on Calabi-Yau 3-manifolds. These manifolds are in particular compact Kähler, hence possess a Kähler class which gives rise to nontrivial ...
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What is meant by “smooth” instantons, and is there a constraint on the winding number of $E_8$ instantons?

In the paper, ``Comments on String Dynamics in Six Dimensions" (arXiv:hep-th/9603003) by Seiberg and Witten, there is a sentence on page 8 (section 3) which reads classically, smooth $E_8$ ...
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AdS/CFT, QCD & String Phenomenology

The AdS/CFT correspondence in its prototypical case states that $\mathcal{N}=4$ Super Yang-Mills is dual to string theory in an $AdS_5 \times S^5$ background. Since this duality is of the type strong/...
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Kaluza-Klein in superstring theory

In superstring theory, it says that they wrap 16 dimensions on a torus given by $\mathbb{R}^{16}$ divided by a SO(32) or $E_8 \times E_8$ lattice and this gives a gauge group of the same name. But in ...
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How does superstring theory explain the inverse square gravity law, given that it requires 9 spatial dimension?

In superstring theory, the spacetime dimension is either 10, one of them is time, the rest are spatial dimensions. But based on geometrical argument, we can say that $F\propto r^{1-D}$, where $D$ is ...
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$\mathcal{N}=8$ Gauged supergravity from $d=11$ supergravity

In this lecture video on maximal supergravity, H. Nicolai mentions that dimensional reduction of $d=11$ supergravity on $T^7$ gives us $\mathcal{N}=8, d=4$ (ungauged) supergravity found by Cremer-...