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14

A classical1 theory of (relativistic) $p$-dimensional membranes exists for any non-negative integer $p$. Such classical membrane-like objects appears in the full non-perturbative formulation of string theory. The problem arises if one tries to quantize a membrane (in a first quantized sense) using standard perturbative quantization methods, which have ...


9

BPS objects are stable because they're the lightest objects with given values of certain conserved charges. So there exists no potential final state that would be lighter and that the BPS state could decay into, by conserving the energy. The excess energy may be invested to the kinetic energy of the final energy but a deficit energy means that the decay is ...


8

The BFSS matrix model is a quantum mechanical model – i.e. quantum field theory in 0+1 dimensions - that describes uncompactified M-theory in 11 dimensions assuming that we study the large $N$ limit of the model with the $U(N)$ symmetry. As myself and later Susskind determined, one may also directly interpret the finite $N$ BFSS matrix model as describing ...


8

Extra-dimensional scenarios may be described as "inspired" by string theory but they are independent hypotheses and they may be true even if string theory is not. However, one has to reduce the ambitions and standards of consistency. Sociologically, it's surely true that the research of models with extra dimensions has been adopted and pursued by many ...


7

In quantum field theory and its extensions including string theory, the electric charge is a generator of a $U(1)$ symmetry which should be promoted to a local symmetry i.e. gauge symmetry. In string theory, the $U(1)$ symmetry and the gauge field often appear as parts of the low-energy effective action. This could be enough to answer the question: we ...


7

If you're asking whether there is any chance we'll be able to do this in the forseeable future then answer is no. If, however, you're asking whether general relativity allows constructions like this then the answer is yes. Your question refers to a brane which would exist inside a bulk of our own design, and I don't know how literally you meant this but it ...


6

The type of branes a theory has in constrained by the p-form fields it contains. There is only one supersymmetric gravity theory in 11d, and it has a single $p$-form with $p=4$, call it $G_4$. This means that the theory can possess brane solutions that either couple electrically to $G_4$ (M2 branes) or magnetically to $G_4$ (M5 branes). To understand the ...


6

In theories with unbroken supersymmetry, the energy can be written as $$E=\sum_i c_i Q_i Q_i $$ where $Q_i$ are some Hermitian supercharges and the coefficients are positive. This is a sum of squares of Hermitian operators which is why it's positively semidefinite. It can't be negative. Tachyons obey $E^2-p^2=m^2$ for a negative $m^2$ so the 4-momentum (or ...


5

There exists such a model and it is described in a wikipedia article, the Ekpyrotic Universe. The ekpyrotic model came out of work by Neil Turok and Paul Steinhardt and maintains that the universe did not start in a singularity, but came about from the collision of two branes. This collision avoids the primordial singularity and superluminal expansion ...


5

Matrix string theory http://arxiv.org/abs/hep-th/9701025 http://arxiv.org/abs/hep-th/9702187 http://arxiv.org/abs/hep-th/9703030 is indeed an exact description of fundamental type IIA strings (and similarly $E_8\times E_8$ heterotic strings) at any (e.g. weak) coupling where you can explicitly see the off-diagonal degrees of freedom. You could say ...


5

There is no 2-D metrics here, because we are working with the boundary $\partial M$ You could imagine a standard action $S_0 = \int_{\partial M} d\tau A_a \frac{d X^a}{d \tau}$, where $A_a$ is constant. With a small perturbation, we will have : $A_a(X) = A_a + \epsilon_a(X)$, and we have an action $S = \int_{\partial M} d\tau A_a(X) \frac{d X^a}{d \tau}$ ...


5

This formula is actually pretty simple to understand. First, the $2^8$ is the number of possible $D4$ states. Then for each (indistinguishable) $D0$, they can be in either a fermionic or bosonic state, of which there are $8$ each. Next, the coefficient of $q^n$ in $(1+q)^8$ is the number of ways for $n$ independent $D0$ branes to fit in $8$ fermionic ...


5

First, the tensor product of two Dirac spinors is the direct sum of differential $p$-forms for all values of $p$. It's because this tensor product is the same thing as the space of all matrices acting on the Dirac spinor space. But all matrices (e.g. $4\times 4$ Dirac-like matrices in 4 dimensions) may be written as linear combinations of $1$, $\gamma_\mu$, ...


5

I) In this answer we will consider the standard Nambu-Goto (NG) string and show that the Hessian has co-rank 2. The target space metric has $(-,+,\ldots,+)$ sign convention, and $c=1=\hbar$. The NG Lagrangian density is $${\cal L}_{NG}~:=~-T_0\sqrt{{\cal L}_{(1)}}, $$ $$ {\cal L}_{(1)}~:=~-\det\left(\partial_{\alpha} X\cdot \partial_{\beta} ...


4

I) In this alternative answer we resolve the singular Hessian $H_{\mu\nu}$ of the Nambu-Goto string action by introducing two auxiliary variables from the onset, thereby indirectly showing that the Hessian $H_{\mu\nu}$ must have co-rank 2. The target space metric has $(-,+,\ldots,+)$ sign convention, and $c=1=\hbar$. Consider the extended Nambu-Goto ...


4

The higher-dimensional objects are the "branes", which were first found as extended black holes in supergravity. In M-theory, the fundamental objects are the 2-brane and the 5-brane, and strings arise from compactifying a brane, e.g. a 2-brane is wrapped around the eleventh dimension, and shows up as a string in the 10-dimensional limit. The classical ...


4

D-branes are not restricted to planar geometries. They can take on many different forms, and you often encounter branes wrapped around spherical manifolds, like $S^1$ or $S^4$. To determine whether a given configuration is stable, you have to evaluate the action of the D-brane configuration, which is given by the Dirac-Born-Infeld action. For a $Dp$-brane, ...


4

Yes, there must be branes filling the ordinary large spacetime dimensions in order for gauge fields to propagate there. Even the Type I string theory containing gauge fields (open strings) in 10 dimensions can be understood as having space-filling D9 branes on which the gauge fields live (as well as an orientifold O9 plane). The exception is heterotic ...


3

There are, actually. Dilaton (I don't mean the massless field in the NS-NS sector that determines the coupling constant, nor the $g_{55}$ component of the Kaluza-Klein Spacetime metric tensor) already covered the reason through T-duality, so I will discuss the requirement of $p$-branes imposed by Ramond-Ramond potentials. The worldsheet of a string can ...


3

There is really no complication in arriving at equation (5) given equation (5). We have: $$ \frac{d}{d\rho}\left[\frac{\rho^3}{\sqrt{1+\left(\frac{dy_6}{d\rho}\right)^2}}\frac{dy_6}{d\rho}\right]=0. $$ We solve this differential equation. $$ \frac{\rho^3}{\sqrt{1+\left(\frac{dy_6}{d\rho}\right)^2}}\frac{dy_6}{d\rho}=\tilde{c} $$ $\tilde{c}$ being a ...


3

Yes, whenever the momentum is conserved and T-duality holds, T-duality must map a conserved quantity such as this momentum to another conserved quantity, i.e. the string winding number in this case, and this fact is independent of the carrier of the momentum or the winding charge. In the general nonperturbative case, you shouldn't think about the charges as ...


3

There can not only, there have to be heavy higher dimensional objects (as for example D-branes) in string theory, as Joseph Polchinski discovered. So it is strictly speaking no longer appropriate to talk about "string theory", since M-theory is now known to relate all the different string theories known before by dualities and which contains these higher ...


3

It depends on the context whether M-theory refers to the full UV completion or just the 11D supergravity limit. For example when people say "type IIA string theory is related to M-theory by T-duality" and go on to calculate something in "M-theory", they really mean the supergravity limit. However, when you hear vague statements like "M-theory is the ...


2

Type II string theory also contains open strings. The statement on Wikipedia is misleading. The GSO projection removes part of the open string spectrum. Most notably, it removes tachyonic modes from the theory, making it stable. You can read this up in detail in Polchinski, Volume II, Chapter 10.


2

A brane is more than just a manifold, it is a physical object. You can think of it as a higher dimensional version of a particle. It can carry a charge, it can couple to gauge fields, it can decay, etc. We tend to study a brane in the same way we study a string. We study strings in terms of their worldsheet, we study branes in terms of their worldvolume. The ...


2

The branes of string theory are all reflected already in the underlying low-energy supergravity theories as "black branes" and particularly as the extremal black branes, the BPS solutions to the sugra equations of motion. A much-cited lecture note on this is Kellogg Stelle, BPS Branes in Supergravity (arXiv:hep-th/9803116) where for instance figure 6 on ...


2

Probe brane means that the brane is not backreacting on the geometry. It's very similar to the idea of a test particle in GR. A test particle follows geodesics of the spacetime. In reality, of course the particle will backreact on the geometry, but for small objects it's a good approximation to neglect this backreaction. In the context of AdS/CFT, this ...


2

Analyzing the spectrum of the strings, one finds that it contains $N^2$ massless vector states, which is precisely the number of gauge fields corresponding to a $U(N)$ group. Note that this is only true for massless oriented open strings; the unoriented case yields $SO(N)$ or $Sp(N)$. As is described in the same chapter of the book, open string states can ...


2

There are models where the extra dimensions don't need to be curled up. The main issue with extra dimensions is, 'why don't the particles/fields we interact with travel in those directions?' We have extremely good limits on standard model particles (electrons, photons) travelling in extra dimensions. However, it is possible to imagine a string inspired ...


2

Maybe it should be remembered that the late 90s not only saw M(atrix) speculation, but also the observation that most, if not all, higher branes in string/M-theory, at least as far as their "worldvolume theory" is concerned, have a quantum description in terms of AdS-CFT duality. Notably the quantum M2-brane (which is the super-membrane in 11-dimensional ...



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