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.

76 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
20 votes
1 answer
527 views

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 ...
ACuriousMind's user avatar
  • 125k
18 votes
0 answers
532 views

Compactifying on a circle and the exchange of R and NS sectors

I've noticed a general phenomenon in compactifying on a circle where if you start with, say, an NS field, then the KK fields with an index along the circle will be in the R sector, and those without ...
Ryan Thorngren's user avatar
14 votes
0 answers
310 views

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 ...
Danu's user avatar
  • 16.3k
10 votes
0 answers
391 views

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&...
ACuriousMind's user avatar
  • 125k
7 votes
0 answers
152 views

Is classical Kaluza Klein theory stable or not?

Set Up In the original classical Kaluza Klein theory, you have a $d+1$ dimensional manifold where one space dimension is a circle $S^1$. In the "low energy limit," none of the metric ...
user1379857's user avatar
  • 11.4k
7 votes
0 answers
318 views

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, ...
Name YYY's user avatar
  • 8,808
6 votes
1 answer
813 views

String theory and trace anomaly in semiclassical gravity?

what does string theory have to say about the trace anomaly in the expectation value of the stress-energy tensor of massless quantum fields on a curved background and its interpretation as the ...
Curious George's user avatar
5 votes
0 answers
168 views

$T$-duality in effective gauge theories of a $D(p+1)$-brane

I am considering a $D(p+1)$-brane in a space $\mathbb R^{1,p}\times S_R^1$ where $S_R^1$ is the circle of radius $R$. I am assuming low energies $ER\ll1$, so that only the massless spectrum of the ...
Adrià Berrocal's user avatar
5 votes
0 answers
148 views

Calabi Yau compactification based on U(1) charges

In Green-Schwarz-Witten Volume 2, chapter 15, it is argued (roughly) that we need 6-dimensional manifolds of $SU(3)$ holonomy in order to receive 1 covariantly constant spinor field. And it turns out ...
Frank Müpe's user avatar
4 votes
0 answers
149 views

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 ...
Pranay's user avatar
  • 329
4 votes
0 answers
131 views

Type I string theory on $K3 \times \mathbb T^2/\mathbb Z_2$ and the K3 orbifold limit

Consider Type IIB string theory with 4 O7-planes and 32 D7-branes on $K3 \times \mathbb T^2/\mathbb Z_2$. The K3 induces D3-charge on their world-volumes which can be cancelled by the introduction of ...
faddeev's user avatar
  • 221
4 votes
0 answers
152 views

Projector and delta function on a cycle $\Sigma$ of a manifold $\mathcal{M}_6$

In the paper ``Hierarchies from Fluxes in String Compactifications'' by Giddings, Kachru and Polchinski, the following example is considered for a localized source that may have negative tension (my ...
leastaction's user avatar
  • 2,095
4 votes
0 answers
136 views

A question on the Bousso-Polchinski paper

In this famous paper by Bousso and Polchinski, Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant an example in M-theory compactification is given in section ...
mastrok's user avatar
  • 387
3 votes
0 answers
100 views

Why open strings must all end on same D brane?

Consider 10-d open string theory with a D9 brane (i.e. an open string), and $X^9$ compactified on a circle. T-dualising, we find a D8 brane. Why is it that the endpoints of all open strings in this ...
user984949's user avatar
3 votes
0 answers
68 views

Number of unbroken supersymmetries in compactifications

In type II compactifications, we take a 10/11-d spinor $\epsilon$ to decompose into internal $\eta$ and external $\zeta$ pieces, $$\epsilon^1=\zeta^1\otimes\eta^1\ \ (+c.c.)$$ $$\epsilon^2=\zeta^2\...
user56517852's user avatar
3 votes
0 answers
113 views

The M2 brane of M theory creates the Type IIA string and D2; the M5 brane creates the D4 and NS5. What are the other objects grouped with the D0?

Type IIA string theory is related to M theory with the 10th spatial dimension compactified on a circle.  The origin of the F1 string, D2 brane, D4 brane and NS5 branes is simple: they come from the M2 ...
Felix Wotter's user avatar
3 votes
0 answers
78 views

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 ...
Graphite's user avatar
  • 125
3 votes
0 answers
94 views

Heterotic/F-theory duality beyond spectral covers

In string theory models there is known to be a duality between heterotic string theory and F-theory. In particular, a heterotic model (on an elliptically fibered Calabi-Yau three-fold) can have an F-...
diracula's user avatar
  • 510
3 votes
0 answers
306 views

Articles discussing examples of Kaluza-Klein Reduction

The notes for my class on Kaluza-Klein reduction are a bit all over the place and at times it's difficult to follow what's going on. (I plan on asking a specific question about an example later). For ...
3 votes
0 answers
233 views

Relationship between lightlike and spatial compactification

The compactification of a spatial dimension, say $x^1$ given by the identification $x \sim x^1 + 2\pi R$ is said to be related to the lightlike compactification by a Lorentz boost : $$ \left( \...
Dilaton's user avatar
  • 9,483
3 votes
0 answers
136 views

Examples of manifolds and fluxes coming from generalized complex geometry

The paramount object in generalized gomplex geometry is the Courant algebroid $TM\oplus T^\star M$, where the manifold $M$ is called background geometry. More generally this Courant algebroid can be ...
Benjamin's user avatar
  • 101
2 votes
0 answers
72 views

The string spectrum with several compactified dimensions

In his String Theory Vol. I book, Polchinski wants to compute the string spectrum when $k$ of the 26 dimensions are compactified $$X^m \sim X^m + 2 \pi R, \quad 26 - k \leq m \leq 25 \, . \tag{8.4.1}$...
Adrien Martina's user avatar
2 votes
0 answers
87 views

Do the "extra dimensions" in string theory have equivalent physical value to regular dimensions?

I've seen hyperspace dimensions being discussed in models for superstring theory, where there are 6-7 hyperspace dimensions iirc. But the explanation as to why we don't perceive these extra dimensions ...
Adithya's user avatar
  • 77
2 votes
0 answers
114 views

Kaluza-Klein compactifications with ${\cal N}=1$ supersymmetry

I'm trying to understand some of the important properties of KK-compactifications of 10-dim heterotic string supergravity on 6-dim Calabi-Yau ($CY_3$) manifolds to a 4-dim theory with ${\cal N}=1$ ...
tomdodd4598's user avatar
2 votes
0 answers
47 views

Does compactification of a Nambu-Goto string in one direction break Diff invariance?

Assume we have a Nambu-Goto action, in phase space, for a closed string. If I compactify one coordinate of the target space, do I reduce the diff invariance of the system. We have $$S=\int d^2\sigma \...
physshyp's user avatar
  • 1,359
2 votes
0 answers
43 views

Doubt for open string toroidal compactification

Let $(\tau,\sigma)$ be our coordinate system on a local path of worldsheet. For open string $\sigma \in [0,\pi]$ and the end points are different. Now if we do compactification on this string in $26$...
aitfel's user avatar
  • 2,973
2 votes
1 answer
104 views

Coordinates on a compactified dimension in bosonic string theory

In the simple case of compactification on the circle of radius $R$, $S^1_R$, most sources on string theory, e.g. here (Kevin Wray, An Introduction to String Theory, page 197), it is stated that the ...
Bedge's user avatar
  • 319
2 votes
0 answers
150 views

Why does one need to compactify space in a Euclidean CFT?

It is often claimed [1] that in Euclidean CFT's on $\mathbb{R}^d$ one needs to first compactify space to the sphere $S^d$, as $\mathbb{R}^d$ is not invariant under conformal transformations. I cannot ...
Luke's user avatar
  • 2,240
2 votes
0 answers
81 views

How many different vacua really are there in the string theory landscape?

How many different vacua are there in the string theory landscape? Different sources give different estimates: some sources talk about the number $10^{500}$, others $10^{272\ 000}$, still others say ...
Арман Гаспарян's user avatar
2 votes
0 answers
126 views

Basic confusion about interpretation of the 5th dimension in Kaluza-Klein theory

As has been mentioned in other posts, Kaluza originally didn't require the 5th dimension to be curled up/compactified. So how exactly would our 4D world emerge from a non-compactified 5D manifold? I ...
Adam Herbst's user avatar
  • 2,423
2 votes
0 answers
66 views

Toroidal compactifications of type IIB string Theory and $SO(5,5)/(SO(5)\times SO(5))$ invariant 6D sugra action

It is usually stated that the compactification of (the bosonic part of the) type IIB ($D=10$, ${\cal N}=(2,0)$) supergravity on $\mathbb{T}^4$ gives a six-dimensional ${\cal N}=(4,4)$ supergravity ...
ifidamas's user avatar
  • 123
2 votes
0 answers
176 views

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 $\{...
Roberto Sisca's user avatar
2 votes
0 answers
60 views

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 ...
diracula's user avatar
  • 510
2 votes
0 answers
54 views

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 ...
nonreligious's user avatar
2 votes
0 answers
133 views

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 - ...
Mtheorist's user avatar
  • 1,171
2 votes
0 answers
303 views

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). ...
pooja somani's user avatar
2 votes
0 answers
108 views

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?
NET_BOT's user avatar
  • 95
2 votes
0 answers
182 views

Can spin angular momentum be understood as orbital angular momentum in extra dimensions?

It is to my understanding that in Kaluza-Klein theories the mass of particles can be understood as linear momentum in the extra dimensions. Let's consider in $\mathbb{R}^{1,3}\times{}B$ space-time a ...
Yossarian's user avatar
  • 6,017
2 votes
0 answers
111 views

In KK theory, is proper time defined using the 5 dimensional or the 4 dimensional line element?

Let's consider five dimensional KK theory. This is Klein's metric $\hat{g}_{AB}= \begin{pmatrix} g_{00}+A_{0}A_{0}&g_{01}+A_{0}A_{1}&g_{02}+A_{0}A_{2}&g_{03}+A_{0}A_{3}&A_ 0\\ g_{10}+...
Yossarian's user avatar
  • 6,017
2 votes
0 answers
159 views

Laplacian in 4 spatial dimensions; 4th dimension warped

How can I prove the form of the Laplacian in four spatial dimensions, using the identification $y = y + 2\pi R$ for the fourth dimension and assuming the others as the usual Cartesian ones? I want to ...
user48847's user avatar
2 votes
0 answers
52 views

Compact manifold taken as an Einstein Manifold

In Kaluza-Klein theories I often see that the compact space is assumed to be an Einstein manifold, that is, its Ricci tensor is proportional to its metric. So, why is this done?
Yossarian's user avatar
  • 6,017
2 votes
0 answers
186 views

Can decompactification explain the inflation of the early universe?

I've just reread chapter 11 of this book where it is explained among other things, that our four dimensional universe could be unstable concerning a decompactification transition, since potential ...
Dilaton's user avatar
  • 9,483
1 vote
0 answers
34 views

Five-form flux in Giddings-Kachru-Polchinski (GKP)

I'm studying the work of Giddings-Kachru-Polchinski (GKP) for hierarchies in string theory and I came across the five-form flux defined in eq. 2.9. Now, if one calculates the Ricci tensor for the ...
Fredrigo6's user avatar
1 vote
1 answer
179 views

Examples of the Kaluza-Klein dimensional reduction method

I am looking for references or articles that apply or explain how to apply the dimensional reduction method to known metrics such as Minkowski, Schwarzschild, Kerr, etc. The references I found ...
1 vote
0 answers
49 views

What is the most compact arrangement of superstrings?

I was thinking about crystals and how they might apply to really dense arrangements of matter near the Big Bang or in collapsing stars. It seems that the bosonic strings and superstrings can form ...
user avatar
1 vote
0 answers
71 views

Visualizing the conformal compactification diagram of $G$

I asked a question a year and 3 months ago on mathstackexchange but after 3 bounties and still no answer I've decided to try here. Here's the link: conformal compactification. Construct a conformal ...
geocalc33's user avatar
1 vote
0 answers
59 views

Quantized momentum in compactified direction

I'm reading Szabo's book. Here is exercise 6.1. : A relativistic particle of mass $m$ and charge $q$ in $d$ Euclidean spacetime dimensions propagates in a background electromagnetic vector potential $...
xpsf's user avatar
  • 1,042
1 vote
0 answers
39 views

Meaning of $A^{25}$ in toroidal compactification of open strings

In ch-$8$ sec-$8.6$ (String theory vol $1$) Polchinski starts with “a constant background” U$(1)$ gauge field when doing toroidal compactification of an open string and mention this gauge field $$A_{...
aitfel's user avatar
  • 2,973
1 vote
0 answers
40 views

Using $c$ function to calculate $H$ spectrum in bosonic string theory

Ch-$8$ sec-$8.4$ Polchinski (String theory vol I) states In the canonical approach, focus on the zero-mode contribution to the world-sheet action. Inserting $$\color{red}{X^m(\sigma)=x^m(\sigma^2)+w^...
aitfel's user avatar
  • 2,973
1 vote
0 answers
33 views

Identical OPE behaviour of CFT of T-dual $X’^{\mu}$ of $X^{\mu}$

In chapter $8$ (String theory vol $\mathrm{I}$) sec $8.3$ Polchinski states that The field $X’^{25}$ has same OPE and energy momentum tensor as $X^{25}$ the minus signs always entering in pair $$X^{...
aitfel's user avatar
  • 2,973