Index Concordia

Very basic stuff on strings

Posted in AdS/CFT, Gauge Theory, String Theory by Index Guy on June 5, 2008

I would like to share some of the stuff I learned from

arXiv:0711.4467

about basic notions from string theory and the AdS/CFT correspondence. The review is focused on gauge/gravity dualities and mesons, but it contains a nice overview of some concepts to bring readers up to speed. Some stuff bellow might be verbatim; most comes from the reference mentioned above.

The action for a relativistic string is given by the area of its worldsheet. In the Nambu-Goto form the actions looks like

S = T \displaystyle\int d\tau d\sigma \sqrt{\det{P[G_{ab}]}}.

In this expression T is the string tension and P denotes the “pullback” of the background metric G_{mn}

P[G_{ab}] = G_{mn}\displaystyle\frac{dX^{m}}{d\sigma^{a}}\frac{dX^{n}}{d\sigma^{b}}.

On the 2-dimensional worldsheet we have the coordinates \sigma^{a} = (\tau, \sigma). One can argue that the square root in the action looks ugly. The Polyakov form for the string action includes a worldsheet metric h_{ab} as an auxiliary field:

S = -\displaystyle\frac{T}{2}\int d^2\sigma \sqrt{-h}h^{ab}\partial_{a}X^{m}\partial_{b}X^{n}G_{mn}.

In this form the conformal symmetry is more clear. Because of this symmetry we have the vanishing of the energy momentum tensor. Using the reparameterization invariance and the conformal symmetry one can put the worldsheet metric in Minkowski form, h^{ab} = \eta^{ab}.

The unexcited string is classically massless. Excitations of the oscillations on the string surface form a tower of states with masses in units of \sqrt{T}. The zero-point energy modes will contribute to a negative energy shift. This is what one usually call a tachyon. It turns out that these “particles” can be removed by introducing SUSY.

The conformal invariance found on the worldsheet is anomalous when one considers quantization in arbitrary dimensions. The only case where there is no conformal anomaly is for d = 10.

We introduced SUSY to get rid of those tachyons. The open strings will give rise to massless gauge multiplets, while the closed string loops will have left and right moving modes. The closed string’s spectrum contains the metric tensor, the scalar dilaton and the antisymmetric two-index tensor. The GSO projection [???] acts as a chiral projection on spacetime fermions emerging from left and right modes. If the projections have the same chirality, the theory is called type IIA. The bosonic content of this theory includes a gauge field A_{1} and a three-form C_{3}. On the other hand, if the projections have the opposite chirality, then we talk about type IIB. This has as bosonic content a scalar, a two-form and a four-form. It turns out that both type IIA and IIB posses \mathcal{N}= 2 SUSY.

When one includes open strings into type IIB the SUSY breaks down to \mathcal{N} = 1. The strings trace a surface through spacetime as they propagate. One introduces interactions by allowing the worldsheet to have handles and holes. The dilaton’s action measures changes in the topology and \exp{\Phi} plays the role of coupling in the theory. Type IIB is of central importance to gauge theory. In it’s low-energy limit the strings become point-like and the theory becomes SUGRA.

Let us talk about D-branes now. We can consistently restrict the end-points of open strings to a sub-space of the 10-dimensional space. The resulting hyperplane is called a D-brane. It turns out that in SUGRA we can find solitonic solutions that are naturally sourced by these branes. Type IIA allows branes with even dimension that are electric and magnetic sources for A_{1} and C_{3}. For type IIB we find branes with odd dimensionality that fuel the dilaton and the two- and four-index fields.

What about the action for the Dp-branes? This is given by the Born-Dirac-Infeld expression:

S_{Dp} = -\mu_{p}\displaystyle\int d^{p+1}\xi e^{\Phi}\sqrt{-\det{P[G_{ab} + a B_{ab}] + a F_{ab}}} + \displaystyle\frac{a^2}{2}\mu_{p}\int P[C^{(p+1)}]\wedge F \wedge F.

In this expression we have

\mu_{p} = a^{-p} \alpha^{\prime(p-1)/2} and a = 2 \pi \alpha'.

Again, P represents the pullback and B is some external, antisymmetric two-form.

Let us mention a few words about \mathcal{N} = 4 super Yang-Mills theory AKA “MSYM” AKA the “spherical cow theory”. The field content of MSYM is as follows:

  • One gauge field – singlet of the SU(4) global symmetry
  • Four Weyl fermions – in four-dimensional representation of SU(4) global symmetry
  • Six real scalars – in six-dimensional representation of SU(4) global symmetry

All fields are in the adjoint representation of the gauge group SU(N). MSYM can be found on the surface of a D3-brane in IIB. Open strings generate gauge field in 10-dimensions. By restricting the 10-d space to a (3+1)-d space breaks gauge fields into (3+1)-d guage field and 6 scalar fields. The fermionic superpartners separate and complete the (3+1)-d supermultiplets.

For MSYM the beta-function vanishes to all orders in perturbation theory. Then we can say that MSYM is conformal. In four dimensions, the conformal group is SO(4,2). We also have the global R-symmetry group is SU(4). The complete superconformal group is SU(2,2|4), of which both SO(4,2) and SU(4) are bosonic subgroups.

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  1. […] basic stuff on AdS/CFT We continue our discussion (started here) of very basic notions in the study of the AdS/CFT correspondence. I have followed very closely the […]


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