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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 ...


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

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

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 ...


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 $1$, $\gamma_\mu$, ...


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 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

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 ...


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 ...


2

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, ...


2

There is. They're all included in the name "string theory". While it originally began as a theory of 1-dimensional strings, today it describe a quantum theory of many other $p$-dimensional (D$p$-branes, membranes, etc.). We just didn't bother finding another name for the theory. The key difference is that while a fundamental string is perturbative, the ...


2

Your diagram looks like an illustration of the Ekpyrotic universe. In this model the extra dimensions are not compactified (i.e. curled up) so there is no uncurling of them. The reason we don't see the extra dimensions is because our universe is confined to a 3D brane, not because the extra dimensions are curled up. One well known theory for what determines ...


2

In some extra-dimensional models, such as brane cosmology, the fields (except gravity) are indeed confined to a lower-dimensional surface, which is sort of like "sharing almost the same coordinates in the extra dimensions". In Kaluza-Klein theory with compact extra dimensions, the fields are basically spread evenly across the entire size of the extra ...


2

Yes, if all the dimensions are compact, well, we really mean that all spatial dimensions are compactified on a torus $T^9$, then (multiple) T-duality may map any simple D$p$-brane aligned with some dimensions to a D0-brane. Under T-duality, D$p$-brane is mapped either to a D$(p+1)$-brane or a D$(p-1)$-brane, so its dimension either increases or decreases. ...


2

Where a string carves out a $2$-dimensional world-sheet and a point particle carves out a $1$-dimensional world-line of spacetime, the instanton carves out a $0$-dimensional world-point. Counting only spatial dimensions, a string is $1$-dimensional and a point particle is $0$-dimensional. By logical extension, an instanton has dimension $-1$, if we only ...


2

I would guess that Kaku is referring to the brane world scenario, though he has had to simplify it for popular consumption to the point where it is barely recognisable. String theory is most naturally formulated in ten spacetime dimensions, one time dimension and nine space dimensions, so the question is why don't we see all these dimensions. The brane ...


2

When you try writing a quantu, theory of strings, you get supergravity in the classical limit. Branes are just solitonic solutions to those supergravity theories. You can classify string/brane theories based on the type of SUGRA theory you get in the classical limit. That gives you four kinds in (9+1)dim and M-theory in (10+1)dim, all related by various ...


2

AdS black holes exist in various dimensions, $p=3$ is not the only choice. The parameter can take on values above or below $3$. One famous example is the three-dimensional BTZ black hole, and higher dimensional ones are also frequently used in the correspondence. Furthermore, I think there is a misunderstanding on the concept of a "gravity dual". The metric ...


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

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 ...


2

From what we understand today, p-branes are honest degrees of freedom, on equal footing with strings. Shop you have a good question. But I don't think anyone had so far managed to consistently quantize a p-brane. Loosely, a brane has much more degrees of frerdom than a string and it's difficult to get them under control. So quantizing it is a technical ...


2

This is not a constraint which appears at the level of perturbative open string worldsheets! Where does this additional constraint come from? Your problem has absolutely nothing to do with any special features of string theory; the very fact that you introduced string theory into this discussion only has one effect, namely to completely mask the actual ...



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