Tuesday, July 03, 2007

A brief survival guide for the brane forest

First of all a quick clarification about the use of two diferents languages, english and spanish, in these blog. Initially I had the intention of using only english, but my participation of some spanish forums about physics derived in posts which I find could be interesting here (in a more complete form that the original ones in the forums). Also in spanish there is less material available about high end physics and not everybody in Spain has good enought level in english to read easily in thath language.

Well, said these I go with these post. When studiying string theory nowadays you find a lot of branes going around and also aparently diferent meaings for the same type of brane. I think that it could be interesting to have a fast guide where you can have a reference of what is everything. As far as I couldn´t find any I have decided to try to write it. As the subject is very extensive I will not goo too deepd in the math details.

Well, let´s beguin by the most basic one, the p-brane. I´ll give first the most broadly used acception of the term. In string theory you have the Nambu-Goto action (see previous post if you speak spanish) which is a generalization of the relativistic particle action. If you allow one-dimensional objects, why stop there and not do a theory for p-dimensinal objects? Mathematically is easy, you simply need a trivial generalization of the Nambu goto action. fo rexample a two dimensional p-brane would be a parametrized surface an son on. In general the action is S= T. V where T is the tension (energy density) of the brane and V is it´s volume.
In fact there are some subleties and you need a cosmological term (see, for example, the Becker-Becker-Green book).

Well, we have a classical action for the p-brane, but if you try to make a quantum theory of it you run into deep problems. Even if you save them for the noninteracting theory you still would have the "small" problem of introducin interactions betwen branes, it is belived that such thing is not possible.

Ok, we have defined a p-brane. But if you go into the literature you find diferents definitions. For exaple the Michio Kakus book "string theory and M-theory" introduces the same terminology with a diferent meaning (also Bachas in his lectures uses the same terminology). I will explain it and i´ll go from there to another famouse branes, the D-branes.

In the midle of the ninties there was a problem with the type II superstring theories. In their spectrum there were antisymmetric fileds coming from the R-R (Ramond-Ramond) part of the spectrum wich are somewhat analogous to gauge fields. It was known that these fields would be charged and that meaned that it was necesary a source for them. The problem was that an string couldn´t be that source. The reason of it is that if you see that fields like a diferential form of diferential geometry is trivial to understand that it must an extended object of diferent dimension than an string. Concretely an Cp+1 field would couple to an extended object of p dimensions. Well, one could that the p-branes I defined previously could do the job. But as I said there were some problems with that branes so in that days people thought that the sources could be black p-branes, which are higher dimensinal analogous of black holes (more on these later).

Well, in fact, as the atent reader could have deduced, these p-branes couldn´t be exactlly the same ones that I introduced firs. One reason for these is that thes branes are charged and in the prevous ones there was no charge. You can introduce charge into these branes adding to them a term similar to the electromagnetic tensor. These takes as into another aspect, in electromagnetism you have electric charges and for the hodege dual of the electromagnetic field you have magnectic charges, that means that you can have electric and magnetic branes. Another thing to consder is that in an extended object the charge is spared. The total charge of the brane can be calculted using the generalized gauss law for a closed surface sourrounding the p-brane. There are many detaills about these, but I guess they are inapropiate for the purpose of these post.

I am going now to introudce the most famous of all branes, the D-branes. They can be introduced from the previous viewpoint and it can be shown that a p-brane can be made piling together d-branes, but I will follow a diferent way.

In open bosonic string theory you can impose Neuman conditions in the end of the string.But it also is possible to impose Dirittlech ones in some of the coordinates. That means that the string can move freelly in the Neuman coordinates but not in the Diritlech ones. If you have Diritlech conditions in p coordinates you have an string that only can move in an p-hyperplane. That is an extended object of p dimensions, and because it is related to Diritlech conditions it is named a Dp-brane where the p indicates the dimension.

Sometimes bosonic p-branes are introduced from T-duality. When performing T-duality in closed strings you get the winding number of an string around the wraped dimension. If you make the analogous and you take R->0 limit you find that the T-dualized open string is efectively constrained to move in one less dimension that the original one. T-duality interchanges Neumman for Diritlech conditions.

These is the very basic idea of p-branes, but I will explain a bit more about them in order to connect with another aspectos of it. In open strig theroy you can associate representations of field theories to ther extrems throguht chan-paton factors. If you do that some new aspectos for D-branes appear. On one hand the brane where the string end becomes charged under the gauge field which the string carries. Another aspect is that it allows that an open string could have their extrems in two diferent D-branes, the way to prove these requires Wilson lines and I´ll not even try to explain it.

Now that we have charged branes we can make a connection with the previous picture of p-branes as sources of antisymmetric RR dields. The idea is easy, you simply can pile together charged D-branes to fit the charge required for the p-brane. There are a few subleties wih these. For example nothing in the p-brane picture requires them beeing hyperplanes but D-branes appeared as such. The solution to these dilema goes back to a characteristic that I had not considered yet. Superstring theory is suposed to be a theory of gravity and in gravity theories you cant have stricitly rigid objects, that means that somehow D-branes mus become dynamical objects. You can go trought these considerations an obtain an efective lagrangian for the perturbations of the d-branes, the Dirac-Bron-infield one. An interesting aspect of it is that the dynamic of the brane is gobernated by the strings ending on it, but I will not go further with these.

Now I´ll itrouduce another viewpoint for D-branes. Superstring theories can be aproximated by effective actions. An efective action for a theory is a classical lagrangian which takes into acoount quantum effects (are tree level)of the original one. For superstring theories these can be done in many ways, for example finding a point particle theory whose amplitudes reproduce the string amplitudes (calculated throught the Polyakov prescription).

The important thing here is that the efeective lagrangian for superstring theories are supergravity theories. You can search solutions to the supergravity theories with some characteristics. I´ll motivate how d-branes appear in these picture. These will lead me to black holes. The most basic one is an Schwarschild one. A generalization of these is to consider a charged (under some gauge field) black hole, these is the Reissner-Nordtrom black hole. You can also look for black hole solutions in supergravity theories. If you search generalizations of these solution in superior dimensions you have what is called a black p-brane.

One interesting aspect of these black p-branes is related to the number of supersymmetric charges. I will not gohere deep into supersymmetry aspects and i will only give a very baci notions. Supersymmetry relates fermions with it´s supersymmetric partners. In the most basic theories you only have a symmetry, but you can have more, if you have one supersymmetry you have an N=1 supersymmetry theorie snd son on. The infnitesimal generators of the symmetri transformations are related throught commutation relations to the generators of the Lorentz group. That imposes an upper boudn of the number of symmetries that you can have in a ginven dimension, for example in four dimmensionsn you can have a maximun of 4 supersymmetries. In fact supersymmetry is broken in the real universe and there are strong reasons to belive that there is only one broken supersymmetry at low energies.

As I said I will not go far into supersymmetry, but I nedded a few basic notions to be able to introduce an important notion. It can be swhown that the black p-brane solutions have half of the supersymmetry of the theory to whcin belong (it is a common thing that solutions of a theory have less symmetry that the actual theory). In general one could be interested in searching for solutions with half the supersymmetry. That solutions are known as BPS states. The BPS states of the supersymmetric theories associated to a superstring theorie can be whown to have the same properties of the p-branes (d-branes) associated to the RR gauge fields I talked before.These shows the aspect of D-branes as BPS states.

Some puntualizations must be made here. I have introduced a pictorial idea of d-branes for the open bosonic string while all the other viewpoint implied closed superstring theories. These means that the D-branes of superstring theories are a generalization of the ones related to the open string theory. An explicit lagrangian for a super p-brane can be made generalizing the p-brane one to superspace. Superpspace is made adding to usual coordinates "supercoordnates", i.e, grassman type coordinates. For p=1 the p-brane is the Green-Scwhartz action of the superstring which is manifestly target space supersymmetric (not like the RamondNeveu-Schawartz one) but it is very ugly to be used in anypractical calculation. A most obscure point is that in the supersymmetric case we had closed strings. If we must keep the analogie these wouuld imply the existence of an open string sector in Type II theories. I hae seem in some papers stating that these is possible but I have not seen an explicit construction. Recently I have seen that people in string field theorie is triying to annalize these from a diferent viewpoint, but I still don´t know too mucho about these.

Untill now we have seen generic p-branes, black pbranes and D-branes. It is time to expose one common propertie of branes. One could think that is thses objects exist they could be important in string perturbation theorie and thay one would need to care about event in whcih an incoming string goes into outgoing branes an so on. In fact these doesn´t happen. The reason is that the mass (or tension, both are related) of the d-branes goes as 1/g where g is the string coupling. These means that for small coupling, the range in chich perturbation theory works, their mass becomes infinite and don´t appear. In the non-perturbative range both branes and strings have similar importance (In fact there are one dimensional D-branes, known as D-strings).

I have not gone into the properties and utility of D-branes. A quick summary is that you can wrap D-branes so that they get geometries very fr from the hyperplane. Thhey are tranformed trought dualities into other branes. Strings betwen diferent branes have a mass which depends on the separation betwen them. D-Branes parallel don´t interact betwen them. You can use apropiates combinations of wrapped D-branes and strings to construct Reissner-Nordstrom black holes and you can reproduce the Haking entropy of them. But counting microscopic states of excitations of strings betwen branes you can have a microscopic description of the black hole. The calculation of these entropy leads to the former implementation of the ADS/CFT concjeture and many more things. But a correct explanation of these subejects imply an understanding of modern string theorie, and that is something that you couldn´t expect from a simgle blog post ;-).

The ones that I have presented till now are by far the most common used branes but there are more, I´ll trate briefly some others.

I´ll begin by the NS-branes. All oriented strings have a common sector consisting of a graviton, a dilaton and a massles antisymmetric tensor field usually dennoted as Bmn. For similr reasons that ofr the RR fileds you can worry about the source of the charge for these field. For the "electric" charge the source can be shown to be the same string, but for the "magnetic" charge these must be an extended object. It´s dimension can be whown to be 5 and it is known as the NS5-brane. For the shake of completity I will mention that analogously as how you can see that d-branes are related to black pbranes it can be seen that a fundamental sring charged with respect to the Bmn field admit solutions somwhat similar to resissner-nodstrom black holes and these solutions are known as "black strings".

Aparently these would be similar to the d-branes but there are a few diferences. Perhaps the most interesting of them is that the d-branes can be shown not to deformate, at the firs order in perturbative calculations, the space around them (despite the fact they have mass). NS5 branes don´t share these propertie and are less addequate for "brane enginering".

A diferent kind of branes are related to M theory. In the same way that N=2 supersymmetric theroies in 10 dimensions are related to string theries one can answer if there is some fundamental theroy related to N=2 11 dimensional supergravity. A carefull analisis of the fields which appear in eleven dimensional supergravity shows that the source for them need to be extended objects (in fact one cna infere the existence of D-branes for type II strings because the 10 dimesnional supersymmetries have the same RR fields that the corresponding superstring theories to whcih they are related. Concretelly it is necessary the existence of 2 and five dimenional branes. Like they are related to M theory they are named M-branes. M theorie also appears as the S-dual of Type II A superstring (the size of the eleventh dimesnion beeing g.l where g is the string couplina nd l the string length). The M2 brane whould be associated to the fundamental string so there are not fundamental strings in M-theory.

The last type of branes I will speak about are G-strings. It can be shown that the global charges in a D-dimensional theory of gravity consist of a
momentum PM and a dual D − 5 form charge KM1...MD−5 , which is related to the
NUT charge. It is possible to construct p-branes for these charges in a very similar way that it was made for the RR gauge fields and you get a D-5 and a 9 branes which is called G-brane (gravity brane) Here D is 11 if the gravity theory comes from M-theory and 10 if it comes from supersymmetric Type II strings.

Hope that the post would be understable and that I wouldn´t have made some mistake in the exposition. Also to say that there are some other types of branes, but I think that the ones trated here are by far the most frequently found ones.

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