## January 15, 2004

### [Exercise] Canonical analysis of D-string action

#### Posted by Urs Schreiber

I want to talk a little about doing the exercise of canonically analyzing the action of the (super) D-string.

My motivation is to try to find the spacetime interpretation of the classical SCFT that has been discussed in section (2.8) of this entry.

I don’t have much time since the computer room at University of Barcelona that I am currently using will close soon. Therefore I’ll begin with just a few observations concerning the bosonic D-string and hopefully say more in a followup.

In a general massless background the action of the bosonic D-string is

(1)$S=-T\int {e}^{-\Phi }\sqrt{-\mathrm{det}{K}_{\mathrm{ab}}}+T\int \left({C}_{2}+{C}_{0}\left(B+\frac{1}{T}F\right)$

where

(2)$-\mathrm{det}{K}_{\mathrm{ab}}=-\mathrm{det}\left({G}_{\mathrm{ab}}+{B}_{\mathrm{ab}}+\frac{1}{T}{F}_{\mathrm{ab}}\right)=-\frac{1}{2}\left\{{X}^{\mu },{X}^{\nu }\right\}{G}_{\mu {\mu }^{\prime }}{G}_{\nu {\nu }^{\prime }}\left\{{X}^{{\mu }^{\prime }},{X}^{{\nu }^{\prime }}\right\}-{\left({B}_{01}+\frac{1}{T}{F}_{01}\right)}^{2}\phantom{\rule{thinmathspace}{0ex}},$

and indices $a,b$ range over the two worldsheet dimensions (I’ll write equivalently ${\partial }_{0}X=\stackrel{˙}{X}$ and ${\partial }_{1}X={X}^{\prime }$) while indices $\mu ,\nu$ range over the target space dimensions. Furthermore, ${F}_{\mathrm{ab}}=\left(\mathrm{dA}{\right)}_{\mathrm{ab}}$ is of course the gauge field strength on the D1 brane and $\left\{{X}^{\mu },{X}^{\nu }\right\}:={ϵ}^{\mathrm{ab}}{\partial }_{a}{X}^{\mu }{\partial }_{b}{X}^{\nu }$ is the Nambu-bracket on the worldsheet.

The canonical momenta with respect to the embedding coordinates are:

(3)${P}_{\mu }:=\frac{\delta ℒ}{\delta {\stackrel{˙}{X}}^{\mu }}=T\frac{1}{{e}^{\Phi }\sqrt{-\mathrm{det}\left({K}_{\mathrm{ab}}\right)}}\left({X}^{\prime \nu }{G}_{\mu {\mu }^{\prime }}{G}_{\nu ,{\nu }^{\prime }}\left\{{X}^{{\mu }^{\prime }},{X}^{{\nu }^{\prime }}\right\}+{B}_{\mu \nu }{X}^{\prime \nu }\left({B}_{01}+\frac{1}{T}{F}_{01}\right)\right)+T\left({C}_{2}+{C}_{0}B{\right)}_{\mu \nu }{X}^{\prime \nu }\phantom{\rule{thinmathspace}{0ex}}.$

There are furthermore canonical momenta conjugate to the gauge field:

(4)${E}_{0}:=\frac{\delta S}{\delta {\stackrel{˙}{A}}_{0}}=0$
(5)${E}_{1}:=\frac{\delta S}{\delta {\stackrel{˙}{A}}_{1}}=\frac{\delta ℒ}{\delta {\stackrel{˙}{X}}^{\mu }}=T\frac{1}{{e}^{\Phi }\sqrt{-\mathrm{det}\left({K}_{\mathrm{ab}}\right)}}\left({B}_{01}+\frac{1}{T}{F}_{01}\right)+{C}_{0}\phantom{\rule{thinmathspace}{0ex}}.$

Because ${A}_{a}$ takes values in $U\left(1\right)$ it is a periodic (canonical) coordinate and hence its momentum must have discrete eigenvalues $p\in Z$.

(6)${E}_{1}:=p\phantom{\rule{thinmathspace}{0ex}}.$

Now ${B}_{01}+\frac{1}{T}{F}_{01}$ can be removed from all expressions by using

(7)${\left({B}_{01}+\frac{1}{T}{F}_{01}\right)}^{2}=-\mathrm{det}\left({G}_{\mathrm{ab}}\right)\frac{\left({E}_{1}-{C}_{0}{\right)}^{2}{e}^{2\Phi }}{1+\left({E}_{1}-{C}_{0}{\right)}^{2}{e}^{2\Phi }}$

and hence

(8)$-\mathrm{det}\left({G}_{\mathrm{ab}}+{B}_{\mathrm{ab}}+\frac{1}{T}{F}_{\mathrm{ab}}\right)=-\mathrm{det}\left({G}_{\mathrm{ab}}\right)\frac{1}{1+\left({E}_{1}-{C}_{0}{\right)}^{2}{e}^{2\Phi }}$

the canonical momenta ${P}_{\mu }$ read

(9)${P}_{\mu }=\stackrel{˜}{T}{X}^{\prime \nu }{G}_{\mu {\mu }^{\prime }}{G}_{\nu {\nu }^{\prime }}\left\{{X}^{{\mu }^{\prime }},{X}^{{\nu }^{\prime }}\right\}+T\left({C}_{2}+{E}_{1}B\right){}_{\mu \nu }{X}^{\prime \nu }$

where

(10)$\stackrel{˜}{T}:=\frac{\sqrt{{e}^{-2\Phi }+\left({E}_{1}-{C}_{0}{\right)}^{2}}}{\sqrt{-\mathrm{det}G}}$

is the tension of a $\left(p,q=1\right)$ string. (The bound state of 1 D-string with p F-strings.)

By staring at this expression for a while one sees that these canonical momenta obey the following two identities:

(11)$\left(P-T\left({C}_{2}+pB\right)\cdot {X}^{\prime }{\right)}^{2}+{\stackrel{˜}{T}}^{2}{X}^{\prime 2}$
(12)$\left(P-T\left({C}_{2}+pB\right)\cdot {X}^{\prime }=0\phantom{\rule{thinmathspace}{0ex}},$

Using the objects

(13)${𝒫}_{±}:=\left(P-T\left({C}_{2}+pB\right)±\stackrel{˜}{T}{X}^{\prime }$

this can be succinctly rewritten as

(14)${𝒯}_{±}:={𝒫}_{±}\cdot {𝒫}_{±}=0\phantom{\rule{thinmathspace}{0ex}}.$

This way one arrives at the usual Virasoro constraints known from the F-string, except that the tension is rescaled and the coupling to the $B$-field is replaced by $p$-times the coupling to the $B$-filed plus once the coupling to ${C}_{2}$.

This demonstrates the familiar fact that the D-string with $p$-units of ‘electric field’ ${E}_{1}$ turned on is nothing but the bound state of $p$ F-strings with one D-string.

In the remainder of this entry I want to briefly say something about my original motivation for this exercise (and its extensions), which was finding the background field that would give me bosonic ‘currents’ ${𝒫}_{±}$ that read something like

(15)${𝒫}_{±\mu }=\left({G}_{\mu }{}^{\nu }±{C}_{\mu }{}^{\nu }\right){P}_{\mu }+\left(\cdots \right){X}^{\prime }\phantom{\rule{thinmathspace}{0ex}},$

because that’s what appeared in section (2.8) of my discussion of classical deformation of SCFTs.

In this context I was toying around with the terms at hand without subjecting myself too much to constraints of physical viability and thereby noticed the following:

When one uses the DBI action mentioned above, but without the ${F}_{\mathrm{ab}}$ term and with a reversed sign on the ${B}^{2}$-term, i.e.

(16)${S}_{\mathrm{trial}}=-T\int \sqrt{-{G}_{\mathrm{ab}}+i{B}_{\mathrm{ab}}}$

then one finds constraints of the form

(17)$\left({K}^{-1}{\right)}^{\left(\mu \nu \right)}{P}_{\mu }{P}_{\nu }+{T}^{2}{X}^{\prime }\cdot {X}^{\prime }=0\phantom{\rule{thinmathspace}{0ex}},$

where

(18)$K=G+\mathrm{iB}$

and hence formally

(19)${K}^{-1}=\sum _{n}\left(-\mathrm{iB}{\right)}^{n}\phantom{\rule{thinmathspace}{0ex}}.$

If we assume that $B$ is small (in the sense that its eigenvalues are), then it might be reasonable to write to first nontrivial order

(20)$\left({K}^{-1}{\right)}^{\left(\mu \nu \right)}={G}^{\mu \nu }-\left(B\cdot B{\right)}^{\left(\mu \nu \right)}+𝒪\left({B}^{4}\right)=\left(G-B\right)\cdot \left(G+B{\right)}^{\mu \nu }+𝒪\left({B}^{4}\right)\phantom{\rule{thinmathspace}{0ex}}.$

This is exactly what I am looking for, because to this lowest order one can now define the bosonic currents as

(21)${𝒫}_{±}:=\left(G±B\right)\cdot P+\left(\cdots \right){X}^{\prime }\phantom{\rule{thinmathspace}{0ex}},$

and they would, to this order, satisfy the constraint

(22)${𝒫}_{±}\cdot {𝒫}_{±}=0\phantom{\rule{thinmathspace}{0ex}}.$

But of course this is wishful thinking as long as there is no reasonable justification for the ansatz of the action ${S}_{\mathrm{trial}}$ above. Does anyone a way to turn this handwaving into something meaningful? Or does it look totally misguided? If yes, does anyone see from what kind of known action bosonic currents of the form ${𝒫}_{±}=\left(G±B\right)\cdot P+\left(\cdots \right){X}^{\prime }$ would follow?

In the paper M. Aganagic, J. Park, C. Popescu, J. Schwarz, Dual D-Brane Actions

some action of the form that I am looking for actually does appear (see their equation (7)-(9)) by a certain rewriting of the original DBI action. However, Lubos says (I hope that I may say that here) that there might be problems with the respective derivation.

Enough for now, I have to fetch some food before the cafeteria closes, then there will be further lectures here at the RTN winter school on Strings, Supergravity and Gauge Fields, and after that the computer room will be closed, unfortunately.

Posted at January 15, 2004 1:36 PM UTC

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