All Questions
Tagged with quantum-anomalies string-theory
58
questions
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How many dimensions are in string theory? [duplicate]
How many dimensions are in string theroy? I heard that there are 11 but to my understanding, there is an infinite, also can strings be on a 2D plane?
3
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1
answer
124
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How is the dimensionful renornalization scale $\mu$ related to break of scale invariance in String Theory?
In the $7.1.1$ of David Tong's String Theory notes it is said the following about regularization of Polyakov action in a curved target manifold:
$$\tag{7.3} S= \frac{1}{4\pi \alpha'} \int d^2\sigma \ ...
- 1,290
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0
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60
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Unitarity of Effective String Theory away from critical dimesions ($D=26$) , in the static gauge
Starting from compete UV description of QCD (in the confined phase), if we integrate out the quarks and Glueballs, in principle, we will get an effective theory of strings (QCD flux tube and not ...
- 11
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132
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Normalization of zero point energy in string theory
Following Joe Polchinski’s Little Book of String, page 12, he use the sum $$1+2+3+...=-1/12$$ to find the zero point energy of the bosonic string (and later used the result to argue that we must have ...
- 1,734
2
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1
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Light-cone quantization of open string as derived in Polchinski
Polchinski uses the following gauge conditions, but I don't follow this procedure of gauge fixing and quantization:
\begin{align}
X^+ = \tau, \tag{1.3.8a} \\
\partial_\sigma \gamma_{\sigma \sigma} = 0,...
- 433
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32
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Why M-theory has eleven dimensions? [duplicate]
Why M-theory has exactly 10+1 dimensions?
Some combinatorics with tensor indices will do.
- 11
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1
answer
150
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Weyl Anomaly for Old Covariant Quantization in String Theory?
In the context of quantization in string theory, the modern approach is the path integral/modern covariant quantization approach. As known from QFT, we fix our gauge and represent the arising Fadeev-...
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How did the two copies of the Witt algebra become two copies of the Virasoro algebra in the CFT?
The Virasoro algebra
\begin{equation}
[L_m,L_n]=(m-n) L_{m+n} +\frac{c}{12} (m^3-m) \delta_{m+n,0}
\end{equation}
of the stress energy tensor $T$ was said to follow from the witt algebra of the local ...
- 3,256
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65
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How does one arrive at the relation of commutator $\left[M^{-i}, M^{-j}\right]$ of Lorentz generators $M^i$ in terms of the string modes $\alpha_n^i$?
I am reading the book "String theory demystified" by David McMahon.
On page 149, the author discusses the "critical dimension" for superstrings.
the number of spacetime dimensions ...
- 251
2
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130
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How do I understand this conformal transformation?
I am learning conformal transformation, and this is by far the most confusing transformation for me.
For the 2D bc system
$$S=\frac{1}{2\pi}\int d^2 z b\overline{\partial}c,$$
we have the ghost ...
- 216
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answer
456
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Polchinski's first derivation of the Weyl anomaly
So, i've been reading volume 1 of Polchinski's String Theory text book and have a doubt.
His first derivation of the Weyl anomaly goes as follows:
From dimensional analysis, we know that:
$$\begin{...
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0
answers
50
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Existence of Weyl invariant regulator for bosonic string theory
In sec $(3.4)$ Polchinksi says
It is easy to preserve the diff- and Poincare invariances in the quantum theory. For example, one may define the gauge fixed path integral using a Pauli-Villars ...
- 3,043
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Why are there only two 496-dim. gauge groups $E_8\times E_8$ and $SO(32)$ allowed in string theory? Why not $E_8\times U(1)^{248}$ or $U(1)^{496}$?
While constructing anomaly-free string theories with $\mathcal N=1$ supersymmetry (16 supercharges constituting a Majorana-Weyl spinor), we learn that the gauge group must be 496-dimensional in order ...
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1
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Is Weyl transformation part of diffeomorphism? Does a gravitational anomaly capture also the anomaly due to Weyl transformation? [duplicate]
Weyl transformation is a local rescaling of the metric tensor
$$
g_{ab}\rightarrow e^{-2\omega(x)}g_{ab}
$$
Diffeomorphism maps to a theory under arbitrary differentiable coordinate transformations
(...
- 4,336
4
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1
answer
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Critical dimension of ${\cal N}=2$ strings
In "A tour through ${\cal N}=2$ strings" by Neil Marcus (https://arxiv.org/abs/hep-th/9211059) the following problem - among others - is noted:
The critical dimension of the ${\cal N}=2$ ...
- 232
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298
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Holomorphic instantons in target torus
For computing instantons contributions from worldsheet torus to target torus, one can evaluate zero modes contribution of genus 1 partition function given by following expression:
$$Tr(-1)^FF_LF_Rq^{...
- 662
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2
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759
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Inconsistency in the normal ordered Virasoro algebra
I seem to have found a basic contradiction when it comes to the commutation relations of the Virasoro algebra with normal ordered operators and I am not sure what the resolution is.
If we have a ...
- 11.6k
5
votes
1
answer
167
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Holomorphic anomaly at genus 1
Partition function on torus can be defined using a generalized Witten like index as given below:
$$F_1=\int_\mathbb{T}\frac{d^2\tau}{\tau_2} Tr(-1)^F F_LF_R \;q^{L_0} \bar{q}^{\bar{L_0}},$$
where $\...
- 662
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1
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243
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Anomalies in the self-dual Yang-Mills theory and $\mathcal{N}=2$ open-string theory
I am reading a paper, written by G. Chalmers and W. Siegel - https://arxiv.org/abs/hep-th/9606061, where they discuss the action of self-dual Yang-Mills theory, which in light-cone formalism is ...
- 4,883
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1
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123
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How do we know there doesn't exist an anomaly that implies that there is no good choice of dimension for the bosonic string?
By considering $\langle T^\alpha_\alpha\rangle$, the Weyl anomaly, we can show that the critical dimension, $D=26$ is the only possible choice of dimension for the bosonic string.
However, how do we ...
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264
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Weyl Anomaly Derivation in Polchinski Eq (3.4.21)
In Polchinski's longer derivation of the Weyl anomaly, he arrives at the result (equation 3.4.19):
$$ \ln{\frac{Z[g]}{Z[\delta]}} = \frac{a_1}{8\pi} \int d^2\sigma \int d^2\sigma' g^{1/2} R(\sigma) G(\...
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Are there versions of String Theory formulated in $D$ spacetime dimensions or even in infinitely many dimensions?
There are a lot of different versions of string theory, and almost all of them differ in the number of dimensions. The most famous ones are formulated in 10, 11 or 26 dimensions.
But are there any ...
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339
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Polchinski Weyl Anomaly from perturbing the flat background. Eq (3.4.22)
In deriving the Weyl anomaly for the bosonic string using a perturbation around a flat background, Polchinksi uses Eq. (3.4.22), i.e.
$$
\ln \frac{ Z[\delta+h] }{Z[\delta]} \approx\, \frac{1}{8\pi^2}\...
- 3,126
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1
answer
515
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OPE of stress tensor in CFT
I come aross an OPE between stress tensor components in CFT which is
\begin{equation}
T(z)\bar{T}(\bar{w})\sim -\frac{\pi c}{12}\partial_{z}\partial_{\bar{w}}\delta^{(2)}(z-w)+...
\end{equation}
I am ...
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260
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Casimir Force and bosonic String Theory dimensions
I was reading the lecture notes on Quantum field theory by David Tong. In the section on Casimir force he derived the force of attraction felt by the plates due to the field vacuum energy in $1+1$ ...
- 1,490
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66
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Critical dimension from the symmetries of the string action
(Related: This post and this post.)
In this thesis it is said (on page 13) that just by assuming that we have some general action with the same symmetries as the Polyakov action (Poincare invariance, ...
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295
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Is there a way to make this simple "derivation" of the Trace Anomaly correct?
I think I came up with a simple yet sketchy almost-proof of the trace anomaly (A.K.A. Weyl anomaly) in 2D CFT, but it has the wrong prefactor. I was wondering if anyone could assess whether this "...
- 11.6k
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1
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279
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Gauge anomaly in Polyakov string and Faddeev-Popov method
I am currently trying to gain a better understanding of the gauge fixing procedure used in chapter 5 of David Tong's notes.
Since the central charge of the Polyakov action for, say, the bosonic ...
- 261
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1
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256
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Is string theory self-consistent? (Conformal anomaly)
Recently I attended a very short course on string theory. We went through the standard presentation in light-cone gauge for brevity. We ‘derived’ the Einstein field equation in the following manner. ...
- 516
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300
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How does the Weyl anomaly imply $\langle T^{\mu}_{\mu} \rangle \neq 0$?
I want to consider the case of euclidean field theory in 2 dimensions with the action
$$S[\phi]=\int \! d^2\!x \sqrt{\det(g)}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$$
which leads to a partition ...
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