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I have a basic question: if we use 1d string to replace 0d particle to gain insight of nature in string theory, and advanced to use 2d membranes, can we imagine that using $3$- or $n$-dimensional blocks/objects/branes as basic units in physics theory? Where is the end of this expansion?

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Look at – ungerade Jun 3 '13 at 15:36
Possible duplicate: – Qmechanic Jun 3 '13 at 15:59
it is important to point out that blanket "blocks/objects " will not work. Strings work because the are a multidimensional expansion of the basic harmonic oscillator. So are membranes ( think of drums) . imo any symmetric potential can have the first term a harmonic oscillator. To go to more complicated shapes of potentials, which maybe is a future possibility, still would preclude blocks. – anna v Jun 3 '13 at 16:07
Furthermore, the 2-D case is singled out by the Weyl invariance, a symmetry of the action that only appears if the fundamental object you deal with is 2-Dimensional. – Neuneck Jun 4 '13 at 8:43

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 dimensional (from points D0, up to space filling D9 branes if spacetime is 10D) objects.

One way to see why these higher dimensional objects have to be there, is because T-duality transforms (among other things) the von Neuman boundary condition of a free floating open string, that are not stuck on anything, to the Dirichlet boundery condition which means that the endpoints are fixed. So there has to be something the strings can stick on, these objects are called D-branes which can be higher dimensional. That's the way Lenny Susskind introduced D-branes in this last lecture of his string course.

D-branes can for among other things be used to model the interactions of the standard model. For example QCD can be described by 3 D-branes, one for each color.

enter image description here

The mesons are strings which do not need to have both ends on the same "color" brane, quarks and anti-quarks are distinguished by the orientation of the string. Interactions take place when strings break and leave new end points on the brane and when two end points come together.

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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 couple to a Neveu-Schwarz B-field: $$q\int_{}^{} {{{h^{ab}}}\frac{{\partial {X^\mu }}}{{\partial {\xi ^a}}}\frac{{\partial {X^\nu }}}{{\partial {\xi ^b}}}B_{\mu \nu }\sqrt { - \det {h_{ab}}} {{\text{d}}^2}\xi } $$

Now, the $q$ is the EM-charge.

The worldsheet of a string can couple to graviton field (spacetime metric): $$m\int_{}^{} {{{h^{ab}}}\frac{{\partial {X^\mu }}}{{\partial {\xi ^a}}}\frac{{\partial {X^\nu }}}{{\partial {\xi ^b}}}g_{\mu \nu }\sqrt { - \det {h_{ab}}} {{\text{d}}^2}\xi } $$

You can change the "$m$" to any way you like, in terms of the tension/Regge Slope parameter/string length etc.

For a dilaton (now, I DO mean the massless field in the NS-NS sector which determines the coupling) field, $${q }\ell _P^2\int_{}^{} {\Phi R\sqrt { - \det {h_{\alpha \beta }}} {\text{ }}{{\text{d}}^2}\xi } $$ Forget the conformal invariance for the time being.

But what about Ramond-Ramond POTENTIALS? All is fine with the Ramond-Ramond Fields, but the Ramond-Ramond Potentials $C_k$are associated with the Ramond-Ramond field $A_{k+1}$ and it is intuitive (and quite clear) that they can't couple similarly to the worldsheet. But it can for a worldhhypervolume, as long as the world-hypervolume is not 2-dimensional. It would then be given by: $${q_{{\text{RR}}}}\int_{}^{} {C_{{\mu _1}...{\mu _p}}^{p + 1}\frac{{\partial {x^{{\mu _1}}}}}{{\partial {\xi ^{{a_1}}}}}...\frac{{\partial {x^{{\mu _p}}}}}{{\partial {\xi ^{{a_p}}}}}{h^{{a_0}...{a_p}}}\sqrt { - \det {h^{{a_0}...{a_p}}}} {{\text{d}}^{p + 1}}\xi } $$

Just note its similarity to the other couplings! Now, maybe this isn't so much of a necessity .as T-duality's switching of Newmann and Dirchilets, but it is still very important!

Edit: It works with 2-branes (membranes) too, but there's no point stopping there, and t-duality exchanging boundary condition becomes an issue. While in 10-dimensional string theories, all these branes are consistent, in 11-dimensional M-theory, only 2-branes and 5-branes are.

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