Heterotic phenomenology

I have talked quite often in this blog about F-theory. This is partially due to "historical" reasons, that is, the F-theory GUT revolution happened recently, while this blog growth. Also the influence of a friend of mine to let learn algebraic geometry was a plus because F-theory relies a lot in that area of maths.
Of course another reason is that they are very good developed framework.

But that doesn't mean that there is not development in other areas of string theory. In particular from the eighties heterotic strings where the best candidate for a phenomenological model. Even today most books in string theory (such as the Becker- Becker-Swhartz one) use the heterotic to teach the math of compactification.

Today it has appeared an interesting paper in heterotic phenomenology so I will say a few things about the subject. Heterotic string/m-Theory models are mainly build by the compactification mechanism. In that aspect they differ from many advances in string theory phenomenology. The fact that a group of parallel N D-branes automatically give an U(n) gauge theory was an star point for building local models in which gravitational degrees of freedom can be ignored for many purposes. That derived in a lot of development of D-brane models in Type II A and type II B string models, and, later, the non-perturbative counterpart of type II-B, F theory, where in addition to D-branes one has (P,q) branes. In fact something is done about non local F-Theory models and some is made for M-theory from the duality among some F-theory and M-theory set ups. But here I am going to talk about the most conventional approach, full compactifications. I am not sure about it, but I think that the reason why not too much development in local M-theory models is because not too much is known for certain (despite the Bagger-Lambert minirevolution of two years ago) about the M-theory branes, although possibly that would apply better to type II A M-theory that to heterotic M-theory.

I am going now with some references. The article of today roots in his model of the year 2006: The Exact MSSM Spectrum from String Theory. I let here the abstract of the paper:

We show the existence of realistic vacua in string theory whose observable sector has exactly the matter content of the MSSM. This is achieved by compactifying the E_8 x E_8 heterotic superstring on a smooth Calabi-Yau threefold with an SU(4) gauge instanton and a Z_3 x Z_3 Wilson line. Specifically, the observable sector is N=1 supersymmetric with gauge group SU(3)_C x SU(2)_L x U(1)_Y x U(1)_{B-L}, three families of quarks and leptons, each family with a right-handed neutrino, and one Higgs-Higgs conjugate pair. Importantly, there are no extra vector-like pairs and no exotic matter in the zero mode spectrum. There are, in addition, 6 geometric moduli and 13 gauge instanton moduli in the observable sector. The holomorphic SU(4) vector bundle of the observable sector is slope-stable.

The observable sector of the theory has an SU(3)C × SU(2)L × U(1)Y × U(1)B−L gauge group. The B-L additional group is beyond the MSSM, but that is not as bad as it seems as they discuss in the paper of today. Additionally they have:

Matter spectrum:
– 3 families of quarks and leptons, each with a right-handed neutrino
– 1 Higgs–Higgs conjugate pair
– No exotic matter fields
– No vector-like pairs (apart from the one Higgs pair)

3 complex structure, 3 K¨ahler, and 13 vector bundle moduli

This sector is obtained by by two steps. First a Spin(10) group can arise from the
spontaneous breaking of the observable sector E8 group by an SU(4) gauge instanton
on an internal Calabi-Yau threefold. Later The Spin(10) group is then broken by discrete Wilson lines to a gauge group containing SU(3)C × SU(2)L × U(1)Y as a factor.

The structure of the hidden sector depends on the choice of a stable, holomorphic
vector bundle V ′. The topology of V ′, that is, its second Chern class, is constrained by two conditions: first, the anomaly cancellation equation:

$$ c2(V´) = c2(TX) - c2(V) - [W] $$

Here c2 means the second Chern class of the vector bundle V and [W] is a possible effective five-brane class. Ok, 'll stop writing the details that can be read in the paper. The important part is that they don't obtain in detail the aspects of the hidden sector (the sector of the other E8 group of the E8xE8 heterotic string). They simply assume it's existence.

Since 2006 that model has been further developed and has lead to this paper today: The Mass Spectra, Hierarchy and Cosmology of B-L MSSM Heterotic Compactifications

The two papers even share one co-author, Burt A. Ovrut. The abstract reads:

The matter spectrum of the MSSM, including three right-handed neutrino supermultiplets and one pair of Higgs-Higgs conjugate superfields, can be obtained by compactifying the E_{8} x E_{8} heterotic string and M-theory on Calabi-Yau manifolds with specific SU(4) vector bundles. These theories have the standard model gauge group augmented by an additional gauged U(1)_{B-L}. Their minimal content requires that the B-L gauge symmetry be spontaneously broken by a vacuum expectation value of at least one right-handed sneutrino. In previous papers, we presented the results of a quasi-analytic renormalization group analysis showing that B-L gauge symmetry is indeed radiatively broken with an appropriate B-L/electroweak hierarchy. In this paper, we extend these results by 1) enlarging the initial parameter space and 2) explicitly calculating all renormalization group equations numerically, without approximation. The regions of the initial parameter space leading to realistic vacua are presented and the B-L/electroweak hierarchy computed over these regimes. At representative points, the mass spectrum for all sparticles and Higgs fields is calculated and shown to be consistent with present experimental bounds. Some fundamental phenomenological signatures of a non-zero right-handed sneutrino expectation value are discussed, particularly the cosmology and proton lifetime arising from induced lepton and baryon number violating interactions.

Since the 2006 paper math sophistication has grown and in the way of the theory the have used things such as monads, spectral covers or cohomological methods to calculate the texture of Yukawa couplings and other parameters. The key ingredient is still a Calabi-Yau manifolds with Z3xZ3 homotopy and a vector bundle with SU(4) structure group. The observable matter spectrum is basically the same of the previous paper. As I said before they state that: The existence of the extra U(1)B􀀀L gauge factor, far from being being extraneous or problematical, is precisely what is required to make a heterotic vacuum with SU(4) structure group phenomenologically viable. The reason is the following. As is well-known, four-dimensional N = 1 supersymmetric theories generically contain two lepton number violating and one baryon number violating dimension four operators in the superpotential. The former,
if too large, can create serious cosmological di culties, such as in baryogenesis
and primordial nucleosynthesis , as well as coming into conflict with direct measurements of lepton violating decays.

Well, the details can be read in the paper. The important thing is that the model is mature enough to allow explicit and accurate renormalization group analysis of the effective field theory and do precise predictions of some aspects.

This is not the only line of investigation in heterotic string theory. As I have stated this gives mainly an MSSM but no unification group scheme is followed. But there are such kind of constructions. As early as in 2005 there is a paper doing such a thing from heterotic M-theory: An SU(5) Heterotic Standard Model-

The authors are Vincent Bouchard, Ron Donagi. and the abstract says:

We introduce a new heterotic Standard Model which has precisely the spectrum of the Minimal Supersymmetric Standard Model (MSSM), with no exotic matter. The observable sector has gauge group SU(3) x SU(2) x U(1). Our model is obtained from a compactification of heterotic strings on a Calabi-Yau threefold with Z_2 fundamental group, coupled with an invariant SU(5) bundle. Depending on the region of moduli space in which the model lies, we obtain a spectrum consisting of the three generations of the Standard Model, augmented by 0, 1 or 2 Higgs doublet conjugate pairs. In particular, we get the first compactification involving a heterotic string vacuum (i.e. a {\it stable} bundle) yielding precisely the MSSM with a single pair of Higgs.

If one reads the paper one can see that it cites the papers in heterotic string that are the basic of the other models. This can look a bit surprising since one article is about heterotic string (which has 10 dimensions)and the other about heterotic M-theory (which has 11 dimensions). Nut they actually work with compactifications in a calaby-Yau threfold. That is because the eleventh dimension of the heterotic M-theory has an special character and no compactification of is is done. On the contrary the type II A M-theory has a more conventional eleventh dimensions and it requires compactification on G2 holonomy bundles, which are harder to work. Well, I am far from being an expert in the heterotic phenomenology, but I thought that the today paper was a good occasion to say some things about it.

Besides this paper today there has been many other interesting papers. Fortunately Lubos has written an entry doing a brief comments on them and I prefer link you to that entry to get the info: A generous hep-th Tuesday

Update: In the Lubos entry (where very gently this post is linked, thanks Lubos ;)) the papers about heterotic phenomenology have been discussed and someone posted two papers about G2 heterotic compactifications, concretelly : http://arxiv.org/abs/0810.3285 and http://arxiv.org/abs/0905.1968.

It has also been discussed an issue about the prediction of the paper considered here saying that the mass of the Higgs boson was around 101 - 106 GeV. while the LEP had excluded with a 95% confidence level a Higgs mass minor than 114 GeV. Lubos argues that 95% is not enought to exclude the value f there are goo theoretical reasons to do s. I would add that Jester (resonance) wrote a post stating that the LEP exclusion ny worked for conventional Higgs. I don't remember the details so I can't say if that is relevant, but I recommend the readers of this blog to make a search in resonances.

Uber-naturalness

The most interesting paper today n arxiv hep_th is very probably this: Uber-naturalness: unexpectedly light scalars from supersymmetric extra dimensions

This is the abstract:

Standard lore asserts that quantum effects generically forbid the occurrence of light (non-pseudo-Goldstone) scalars having masses smaller than the Kaluza Klein scale, M_KK, in extra-dimensional models, or the gravitino mass, M_3/2, in supersymmetric situations. We argue that a hidden assumption underlies this lore: that the scale of gravitational physics, M_g, (e.g. the string scale, M_s, in string theory) is of order the Planck mass, M_p = 10^18 GeV. We explore sensitivity to this assumption using the spectrum of masses arising within the specific framework of large-volume string compactifications, for which the ultraviolet completion at the gravity scale is explicitly known to be a Type IIB string theory. In such models the separation between M_g and M_p is parameterized by the (large) size of the extra dimensional volume, V (in string units), according to M_p: M_g: M_KK: M_3/2 = 1: V^{-1/2}: V^{-2/3}: V^{-1}. We find that the generic size of quantum corrections to masses is of the order of M_KK M_3/2 / M_p ~ M_p / V^{5/3}. The mass of the lighest modulus (corresponding to the extra-dimensional volume) which at the classical level is M_V ~ M_p/V^{3/2} << M_3/2 << M_KK is thus stable against quantum corrections. This is possible because the couplings of this modulus to other forms of matter in the low-energy theory are generically weaker than gravitational strength (something that is also usually thought not to occur according to standard lore). We discuss some phenomenological and cosmological implications of this observation

One of the good things of this kind of theories is that a combination of supersymmetry and large extra dimensions allow that the gravitino wouldn't be the LSP in the kind of theories where it would usually be it. That is good if the controversial results claiming the existence of dark matter of 10 GeV are confirmed. As far as I know the gavitino couldn't be the particle responsible for that kind of DM. That's bad because the gravitino is the LSP in the Vafa F-theory GUTs. But if the idea of this paper can be translated to F-theory (which after all shares some of the characteristics of the LV theories disused in this paper) it could give them the flexibility to not be immediately ruled out if the DM experiments confirm the discovering.

Obviously the interest of the paper goes beyond that particular purpose of my own concern. It is a theoretic paper of wide interest and not a phenomenologistic paper devoted to a narrow particular subject. Anyway, it is clear that if someone wants to properly appreciate this paper beyond string theory he needs a good familiarity with supersymmetry, as explained in, for example, the book of the previous entry and cosmological issues of dark matter and inflation.

Update: A litle of meditation shows an obvious obstacle to use plainly this constructions inf F-theory, The idea of this paper relies in a variant of gravity mediated supersymmetry breaking. In F-theory supersymmetry is broken bye a variant of the guidice-massiero mechanism that belongs to gauge mediated supersymmetry breaking. Worse, the very idea of F-theory GUTS relies in the gravity decoupling so it seems hard to incorporate this idea in an straightforward way. Ok, I never have claimed to be at all an expert in F-theory or phenomenology. But I fell s if my understanding of this topics is growing fast, maybe in a few centuries I could publish something worthfull ;).

A watch at the string landscape

Like many physicist I am a reader of science fiction. String theory is not a topic which is too broadly covered in SF, and, anyway, it is not covered too properly. For example, it could be that the author limits to cite the words "calaby-yau" as some kind of manra. Even thought there is one particular novel, writen in the eiguthies, where there was a fine usage of string theory. There an alien spae-craft arrived to earth and they tripulants beguined a discusion with relevant human figures in art, politics an scince. In paarticular, in the sicence area, they tolked with string theorists and discused with them many mathematical aspects and conceptula developments that they found terribly exciting. Despite that no concrete experimental evidence was provided. While doing that the aliens had throught a black hole inside the earth which growed slowly, but fast enought to eat the whole earth a few mounths later, toward the end of the novel. Fortunately anonther space-craft had appeared, tripulated by a diferent alien specie, and saved some selected humans. I guess that any informed reader will be able to see the possible funny possible analogies with the actual situation :-).

The purpose of this introduction was to sign the fact that string theory has grown a lot in many directions since the eighties, and it is somewhat discouragint to try to get a prcise idea of the many lines (some of them alsmost death) of development followed in the while. But if I wuld be one of the "bad aliens" that would try to give some guidance to an eighties string theoretic maybe I could use this post as a begining, or at least that is my intention.

The great chalenge in string theory is to get a proer way to get a decent way to go from 10 to 4 dimensions. In the eighties the most pomising way was to look for compactifications of heterrotic string theory in calaby-yau mamifolds, or maybe in orbifolds. Soon it as realized that it was interesting to study not one, but families of calaby-yaus. One went from one to other by variiying some moduli. Another easy way to compatify were orbifolds, tori acted by some discrete group. The fixeed points of that action were singular, and the studie of that singularities revealed to be very interesting. It was necesary to go troguht a revolution, the discovering of the importance of branes, to give more fuel to the compactifications. One could use branes to solve the singularities of the orbifold fixed points. And it was found that that pints could do transitions among calaaby-ayus with diferent topologies. Also the Calabi-Yau moduli space revealed to hae singular points, called conifold points. Coriosly the own moduli space of a C-Y could be, in some sense, characterized a calaby yau of an special type, one with conifold points (i.e., a point similar to the edge of a cone, that is, a continuous but not diferentiable point). If one suits a D-Brane at that point one can "blow-up" the singularitie. But, anyway, the thing is that conifolds can also give transitions betwen vacua of diferent topology. In fact the scenarie is worst. Ther ecan be transitions to phases where the vacua doesnt´admit an obvious gemoetric description and one must use CFT/non linear sigma models, to describe the theory. In fact Witten argued that in M-theory, an aditional development of string theoyr corresponding to strongly coupled type II A strings, only geometric phases were allowed.

In adition to compactification "braane worlds" were considrd. The idea was that the observable world would be some kind of brane. Precise realizations of that idea were purchased form many viepoints (I guess that the most recent try use the idea of intersecting D6-Branes).

In the mean time it was discovred that the universe is accelerating. And there are som kind of consensun that at a constant rate. That means that "phantm energy" scenaries seen to be ruled out and we must look for a de sitter universe emerging from string theory. The firs realization of this was the KKLT theory. In that scenarie there were required vacuums where supersymmetry was broken in a way that it gived some cosmological constant. It was argued that the univrse could be populated by manu diferne vacua. Each vacuawith a diferent value of the cosmological constant would expand at diferents speeds so we would live in some buble of a particular vacua. That lead to the counting of vacua that shrd some properties, and to do an analisys of the statistical distribution of other properties. For example, in some kind of vacua compatible with aa certain value of the cosmological constant there were more solutions with large extra dimensions. But another kinds of such vacua ere in the opposite direction. By the way, vaua with cosmological constants are not tru vacuums, they are metastable states whose decay rate is graater that the actual ge of the universe, oh yeah ;-).

Some interesting remarks about this models are that they give a potential for the scalar fields that describe the moduli of the vacuas. Taht is, they re, in a certain sense, properly defined theories with all the measurable values fixed. This had proved to be a very difcould task. The Dine-Seiberg conjecture stated that a proper determination of the value of the modulis required to go to non-perturbative range of string theory. But the hope was that once one had a theory with aall that values fixed one would have a unique, of almost unique, theory. In fact one has, in some scenaries, around 10^500 theory (i.e. vacua). whose average cosmologicla constant is the observed one (the counting was made by first time by Bousso and Polchinsky for some particular kind of models). Another point is that there is not a natural way to make statistic mechanic for that diferent vacua. I.E, Ine can´t make a proper statistichal ensmble out of them because that vacua should be separated into diferent sectors with superselection rules not allowing going from one to another. I recmend to llok at the blog of Dimityr (non equibrium net) to get a mch better discusion of this topic.

By the way. Most of this studies were made for type II theories. What was of the hetrotic string?. Well, infact there is an heterotic landscape also. It is courious. Another development of string theory was to prove a counting of the benckenstein entropy of a black holes (or at least a paarticular kind of them). for that puropose Type II theories, and their D-Branes, were used. But later it was seen that one also could use heterotici strings to describe black holes. It seems like if heterotic string theory always has aa delay in the achievement of the resoults. But, in the positive point, heterotic strings still seem to be promising. FOr example teh heterotic landscape contians many fewer vacua.

In the eighties ther was a hope that string field theory could provide some kind of dynamics which could indicate how these compactifications could be achieed. Unfortunately string field theory had not succes and has proved to be very dificoult anyway. I fact one would have an string field theory fo rany of the diferen string theories.

With all that I have exposed it looks like if there are too many things going on. In fact it is so. I think taht what I would like to see is a way to see how topologicla transitions could be used to connect diferent vacua. In fact the vacua of the landscape aare, as I said, not aall of them supersymmetric vacua. That would mena to consider a more generic king of compactifications, and studie the possible topologicla transitions betewem them. One way to consider taht could be the use of instanton/euclidean wommholes. And also to see how to describe this in some kind of SFT. Also it one consider that the diferent string theories are related by dualities, meaning that in some sense they re a single one, one could study wormholes, ot whatever, connecting them. A way to beguin this program could be to try to describe some kind of wormhle like solution connecting diferent compactifications (or a noncapctifed space to a compactified one).

I muist advertize that like this post contain many, many, topics, I have not pretended to be very exact in the descriptions. My idea was just to give a broad perspective. I hope to wite in a near future more detailed posts on more concrete topics, but I guess it was too much tiem since the last posts and that It was a good idea not to bee too lazy and write smething ;-).

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 C^p+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 B_mn. 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 B_mn 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.

One string to rule them all...

En este journal se ha hablado mucho sobre la teoría de cuerdas, pero, sin embargo, no se ha hecho nínguna exposicion formal de la misma, Bien, es tiempo ya de ser un poco mas precisos respecto a la teoria de cuerdas,

Empezamos por lo más sencillo, explicar que es una cuerda dentro de esta teoria. Bien, en realidad es la cosa mas sencilla del mundo, una cuerda (bosónica), matemáticamente, es una curva (real) que evoluciona en el tiempo. ¿Por que alguien se preocupó de trabajar en una cuerda cómo un objeto fundamental en vez de hacerlo con partículas puntuales? La respuesta, curiosamente, es "nadie". La primera motivación para ocuparse de una teoria de cuerdas proviene de la cromodinámica cuántica, o más bien al estatus de la físca de hadrones antes de aparecer la cromodinámica cuántica. Sin entrar en muchos detalles señalar que se sabe que el neutrón y el protón, las partículas que forman el núcleo atómico no son partículas elementales, estan formadas por (3) quarks. Esos quarks se describen por una teoria gauge, la SU(3). Lo curioso es que si los quarks, y las partículas que median su interacción, los gluones, deben formar estados ligados (protones, neutrones, y en realidad todas las partículas hadrónicas) debe haber algo que impida que haya quarks libres, que nunca se han observado. Eso llevó a que en un momento dado se propusiera un modelo fenomenológico bastante descriptivo. Los quarks estaban unidos por algún tipo de cuerda, es decir, existían cuerdas que tenían un quark en cada uno de sus extremos, el confinamiento (ausencia de quarks libres) se debería a que si se estiraba demasiado esa cuerda se rompía en dos nuevas cuerdas cada una con su pareja de quarks, en realidad un quark y un antiquark, en sus extremos (para el protón o neutron era necesario tres cuerdas unidas por un extrem oentre sí y con un quark en los otros extremos). Hacia falta ponerle mates a esa idea, y es lo que se hizo allá por el 75. El problema es que esa teoria tenía un "inconveniente", en su espectro aparecía una partícula de spin 2 que claramente no encajaba en el modelo de quarks, más adelante se reinterpreta la teoria de cuerdas cómo una teoria fundamental y esa partícula de spin 2 pasa a ser el gravitón. He hblado que en el espectro de una teoria de cuerdas hay partículas, bien, esto significa, hablando de manera simplificada, que las cuerdas vibran y que cada modo de vibración se identifica con algún tipo de partícula. Según esto cada partícula conocida sería un modo de vibración de una cuerda. Como ese rango de partículas incluye los fermiones (por así decirlo las partículas que forman la "materia") y los bosones (las partículas que median las interacciones entre la materia) tenemos que la teoria de cuerdas sería una teoria que explicaría toda la física conocida, serí auna teoria unificada. Y además sólo tiene un parámetro libre, la tensión de la cuerda, así pués con la media de un sólo parámetro se tendria el valor de todos los demás parámetros de la fisica pués sería deducibles matemáticamente a partir de esa tension. Tras este previo sobre fenomenológia, no especialmente riguroso, vamos con algo de mates.


En matemáticas, geometría diferencial básica (sin usar formalismo de variedades), una curva es un lugar del espacio de dimensión uno que puede, en un sistema de coordenadas, describirse mediante una parametrizacion.



Aquí σ es el parámetro que describe la curva y las X^μ son las coordenadas. El índice μ varia desde 0 hasta D-1, dónde D es la dimension del espacio-tiempo donde se situa la cuerda. Bien, esto es una curva, una cuerda es una curva que se deja evolucionar enel tiempo, es decir, que aparte de la dependencia en σ hará una dependencia en τ (tiempo propio).




Bien, esto es la "cinemática" de la cuerda, nada particularmente complicado, pasemos a la dinámica. Cómo se ha discutido por aquí, y es bien sabido, en física la dinámica suele inferirse a través de una función lagrangiana, ¿que lagrangiana debe describir la cuerda? Bien, hay dos posibles, la más sencilla, conocida cómo la de Nambu-Goto surge de generalizar el lagrangiano de una partícula libre en relatividad especial, que recordemos es:



dónde, cómo es habitual en física, el punto sobre la coordenada denota derivacion respecto al tiempo. Esta acción representa la longitud de la línea de universo de la partícula relativista, es decir, una partícula puntual, matemáticamente un punto, al evolucionar en el espacio-tiempo describe una rayectoria, parametrizada por el tiempo τ. La acción es la longitud (en la métrica de Minkowsky) de esa curva. Pués bien, una particula al evolucinar en el tiempo describe una curva. Una curva al evolucionar en el tiempo describe una superficie, ergo la acción de Nambu-Goto de la cuerda va a ser el área (minkowskiana) de esa superficie:



Bien, esta acción es sencilla de entender, mera genralización de la acción de la partícula clásica. El problema es que aparece una raiz cuadrada, y eso, cuando se quiere proceder a tareas de cuantización, es algo muy molesto. Así pués se prefiere usar otra accion, la de Polyakov. El truco es expresar el área mediante una métrica intrínseca de la superficie, denotada por h, en concreto tenemos:



dónde la fórma concreta para h es:



Bien, esta es la forma de la acción. En mecánica clásica una vez que tenemos la acción normalmente lo siguiente que hacemos es calcular las ecuaciones de movimiento asociadas a ella (ecuaciones de Euler-Lagrange). Pero antes de hacer eso hace falta señalar unos aspectos importantes. Esta acción, cómo muchas otras que aparecen en teoria cuántica de campos, tiene simetrias, es decir, existe un grupo de transformaciones de los campos que dejan invariante la acción. La accion de Polyakov tiene tres simetrías:

(i) Invariancia Poincaré , (ii) invariancia bajo difeomorfismos de la Worldsheet , y (iii)invariancia Weyl (invariancia de escala).

Estas invarianzas se expresan matematicamente en términos del tensor energía-momento, análogo al de la relatividad general, cuya expresión es:





La invarianza bajo difeomorfismos implica que este tensor (que nos da cuenta de la energía y el momento de la cuerda) debe conservarse, es decir:



La invarianza Weyl se traduce en: .

Bien, esto concluye la breve por ahora el análisis de las simetrías, vamos a poner la ecuación de movimiento:

*

Una vez se tiene la ecuación de movimiento se debe proceder a resolverla.

Habíamos dicho que teníamos siemtrías. La invariancia de la acción bajo esas simetría se traduce en que hay grupos de soluciones equivalentes. Necesitamos un modo de deshacernos de las soluciones redundantes, eso esta relacionado con las ligaduras de las que hablé en los post de LQG. No obstante sin necesidad de saber los detalles de la teoria de ligaduras de dirac podemos entender bastantes cosas, sigamos.

Cuando queremos resolver ecuaciones diferenciales (en este caso en derivadas parciales) se imponen condiciones de contorno. En este caso estas condiciones tiene interpretación cómo condiciones en los extremos de las cuerdas, tenemos cuerdas abiertas (condiciones de Neuman) y cerradas (Diritlech).

En realidad más adelante se comprobó que había mas detalles a tener en cuenta en esto en relacion conla teoria de branas, pero no merece la pena ocuparse de ello en esta introducción.

Tenemos las condicones de contorno, vamos a proceder a encontrar soluciones a la ecuación de movimiento (*). Para hacerlo hay primero que fijar un gauge, elegimos el conocido como gague conforme ahí la ecuacion de movimiento se reduce a la ecuación de Laplace y la solución nos queda para la cuerda cerrada:





y para la cuerda abierta:





dónde son la posición y el momento del centro de masas de la cuerda.

Bien, hasta aquí lo básico, la parte clásica. Enla cuantización, que no trataré en este post, los α de las dos últimas ecuaciones se convertirán en operadores de creación/aniquilación que se corresponderían con las partículas observadas en la fisica del modelo standard. Habrá que imponer la anulacion de la derivada del tensor de energía momento lo quedará lugar a la famosa álgebra de Virasoro. Y además habrá que comprobar que las simetriás de la teoria clásica se respetan, esto no es algo precisamente trivial, todo lo contrario, esas simetrías sólo se respetan si la dimensión (conocida como dimension crítica) en que se propaga la cuerdas es distinta de 4. Aquí he estado explicando la cuerda mas sencilla posible, la cuerda bosónica; para esta cuerda la dimension crítica es 26 (25+1). En realidad la cuerda bosónica no es realista, para empezar, cómo su nombre indica, no tiene nada mas que bosnoes en su espectro. Cuerdas realistas requieren fermiones, eso implica introducir supersimetría y así entramos en el reino de las supercuerdas, par estas la dimension crítica es 10 (9+1). Desde luego hay muchísimo más que decir sobre la teoria de cuerdas, no en vano un libro de 750 páginas tiene algunos capítulos que más que un libro de texto parece un rápido review de resultados, pero creo que lo expuesto puede servir de orientación de a que nos estamos enfrentando al hablar de teoria de cuerdas.