## June 13, 2004

### Pohlmeyer invariants and states of the IIB matrix model

#### Posted by urs I have been jabbering about a possible relation between Pohlmeyer invariants and the IIB matrix model for a while now, without being able to give any systematic evidence for this intuition. Now that I have learned a bit about boundary state techniques I think the relation is pretty obvious:

[Update 16/06/04: Details can be found in section 3.6.1 ‘Boundary DDF/Pohlmeyer invariants’ of these notes.]

Consider a stack of $N$ flat $\mathrm{Dp}$ branes without any gauge fields or other stuff turned on. As is very well known (and as is sort of reviewed in the last entry) such can conveniently be characterized by the closed string source term $\mid {\alpha }_{p}〉$ that comes with it, which, equivalently, is the ‘mirror’ state of a closed string that yields the desired open string boundary condition at the brane. More precisely, when we write ${X}^{i}\left(\sigma \right)$ and ${X}^{\mu }\left(\sigma \right)$ for the string’s worldsheet coordinate fields transverse and longitudinal to the branes, respectively, and ${P}_{i}\left(\sigma \right)$, ${P}_{\mu }\left(\sigma \right)$ for the respective canonical momenta, then the defining condition on $\mid {\alpha }_{p}〉$ is

(1)${X}^{i}\left(\sigma \right)\mid {\alpha }_{p}〉=0={P}_{\mu }\left(\sigma \right)\mid {\alpha }_{p}〉\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\forall \sigma \in \left(0,2\pi \right),\phantom{\rule{thinmathspace}{0ex}}\mu \in \left\{0,1,\cdots ,p\right\},\phantom{\rule{thinmathspace}{0ex}}i\in \left\{p+1,\cdots 9\right\}\phantom{\rule{thinmathspace}{0ex}}.$

This can be derived in full generality by some CFT gymnastics but is also pretty obvious since ${X}^{i}\left(\sigma \right)\mid {\alpha }_{p}〉=0$ just says that the $\mathrm{Dp}$-branes are sitting at ${X}^{i}=0$ and ${P}_{\mu }\left(\sigma \right)\mid {\alpha }_{p}〉=0$ is just the T-dual (${X}^{\prime }↔P$) assertion (up to the 0-mode, which I just ignore here).

By similar reasoning one gets the boundary states which describe fluctuations of the D-branes as well as gauge field excitations:

Let ${\varphi }^{i}\left({X}^{\mu }\right)$ be the scalar field which describes the embedding of the brane in the transverse space, then the above condition obviously generalizes to

(2)$\left({X}^{i}\left(\sigma \right)-{\varphi }^{i}\left({X}^{\mu }\left(\sigma \right)\right)\right)\mid {\alpha }_{p}\left(\varphi \right)〉=0$

which is simply solved by acting with the ordinary translation operator on the boundary state:

(3)$\mid {\alpha }_{p}\left(\varphi \right)〉=\mathrm{exp}\left(i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{\varphi }^{i}\left({X}^{\mu }\left(\sigma \right)\right){P}_{i}\left(\sigma \right)\right)\mid {\alpha }_{p}〉\phantom{\rule{thinmathspace}{0ex}}.$

By applying T-duality $P↔{X}^{\prime }$ again this tells us that the boundary state $\mid {\alpha }_{p}\left(A\right)〉$ which describes the flat branes with an abelian gauge field ${A}_{\mu }$ turned on is

(4)$\mid {\alpha }_{p}\left(A\right)〉=\mathrm{exp}\left(i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{A}_{\mu }\left({X}^{\mu }\left(\sigma \right)\right){X}^{\prime \mu }\left(\sigma \right)\right)\mid {\alpha }_{p}〉\phantom{\rule{thinmathspace}{0ex}}.$

Since this is just the Wilson line of $A$ along the string it is clear that for general $A$ we should take the trace over the $\sigma$-ordered exponential:

(5)$\mid {\alpha }_{p}\left(A\right)〉=\mathrm{Tr}\phantom{\rule{thinmathspace}{0ex}}\mathrm{P}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left(i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{A}_{\mu }\left({X}^{\mu }\left(\sigma \right)\right){X}^{\prime \mu }\left(\sigma \right)\right)\mid {\alpha }_{p}〉\phantom{\rule{thinmathspace}{0ex}}.$

By T-dualizing back once again this tells us that displacing the $N$ $\mathrm{Dp}$ branes by different amounts, i.e. by matrix valued coordinates ${\stackrel{̂}{\varphi }}^{i}$ amounts to setting

(6)$\mid {\alpha }_{p}\left(\stackrel{̂}{\varphi }\right)〉=\mathrm{Tr}\phantom{\rule{thinmathspace}{0ex}}\mathrm{P}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left(i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{\stackrel{̂}{\varphi }}^{i}\left({X}^{\mu }\left(\sigma \right)\right){P}_{i}\left(\sigma \right)\right)\mid {\alpha }_{p}〉\phantom{\rule{thinmathspace}{0ex}}.$

Now that I have bored everyone who knows about boundary states to death let me come to the Pohlmeyer invariants:

As one may recall, these are nothing but the objects

(7)$Z\left[A\right]:=\mathrm{Tr}\phantom{\rule{thinmathspace}{0ex}}\mathrm{P}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left(\frac{i}{\sqrt{2}}\oint d\sigma {A}_{\mu }\left({P}^{\mu }\left(\sigma \right)±{\mathrm{iX}}^{\prime \mu }\left(\sigma \right)\right)\right)$

for constant matrices $A$. In principle, one could also use $A=A\left(X\left(\sigma \right)\right)$ in the above definition of the Pohlmeyer invariants - and they would still be invariants (classically or, up to some extra conditions, also quantumly), but there is a certain beauty to the fact that by just choosing (arbitrarily) large constant matrices $A$ in the above expression one still obtains a ‘complete’ set of invariants - and this is how they are defined.

It is clear that these Pohlmeyer invariants are very similar to the unitary operators that were used above to take ordinary boundary states to those describing gauge fields and brane fluctuations. All one has to note to see this in full detail is the following simlple fact:

When the exponential in the Pohlmeyer invariant is Taylor expanded one gets terms of the form

(8)${\int }_{\circ }{d}^{N}\sigma \phantom{\rule{thinmathspace}{0ex}}\left(P+i{X}^{\prime }{\right)}^{{\mu }_{1}}\left({\sigma }^{1}\right)\cdots \left(P+i{X}^{\prime }{\right)}^{{\mu }_{1}}\left({\sigma }^{N}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{Tr}\left[{A}_{{\mu }_{1}}\cdots {A}_{{\mu }_{N}}\right]$

where the ordered periodic integral ${\int }_{\circ }{d}^{N}\sigma$ may be rewritten as

(9)${\int }_{\circ }{d}^{N}\sigma ={\int }_{0}^{2\pi }d{\sigma }^{1}\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\sigma }^{1}}^{{\sigma }^{1}+2\pi }d{\sigma }^{2}\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\sigma }^{2}}^{{\sigma }^{1}+2\pi }d{\sigma }^{3}\phantom{\rule{thinmathspace}{0ex}}\cdots {\int }_{{\sigma }^{N-1}}^{{\sigma }^{1}+2\pi }d{\sigma }^{N}\phantom{\rule{thinmathspace}{0ex}}$

(cf. equation (2.17) of hep-th/0403260). It must be noted that ${\sigma }^{1}$ here seems to play a preferred role, but in fact it does not and we may choose any of the $\sigma$s as ${\sigma }^{1}$ as long as we preserve the correct periodic order of the $\sigma$s. The point is that this implies that even at the quantum level we may re-order the $\left(P+{\mathrm{iX}}^{\prime }\right)\left({\sigma }^{i}\right)$ terms in the above expression. That’s because their commutator is a total ${\sigma }^{i}$-derivative which vanishes under the total ${\int }_{0}^{2\pi }d{\sigma }^{i}$-integral.

This somewhat technical consideration has the sole purpose of showing that when applied to a state which is annihilated either by $P$ or by ${X}^{\prime }$, we may simply cancel the respective operator from the exponential of the Pohlmeyer invariant.

More precisely, let $\mid {\alpha }_{9}〉$ be the boundary state of a space-filling (stack of) brane(s), annihilated by all the ${P}_{0,1,\cdots ,9}$, then the application of the Pohlmeyer operator $Z\left[A\right]$ to the respective boundary state

(10)$Z\left[A\right]\mid {\alpha }_{p}〉=\mathrm{Tr}\phantom{\rule{thinmathspace}{0ex}}\mathrm{P}\mathrm{exp}\left(\frac{i}{\sqrt{2}}{A}_{\mu }{X}^{\prime \mu }\left(\sigma \right)\right)\mid {\alpha }_{9}〉=\mid {\alpha }_{9}\left(A\right)〉$

is nothing but that stack of branes with the constant gauge field $A$ turned on, because, by the above considerations, we can move all the $P\left(\sigma \right)$ that enter the Pohlmeyer operator to the right, where they annihilate the boundary state.

The nice thing is the following: It is known from

T. Maeda & T. Nakutsu & T. Oonishi: Non-linear Field Equation from Boundary State formalism (2004)

(taken together with hep-th/9909027) that applying the above exponential is well defined (has no divergencies from nearby quantum fields that appear in the expression) precisely (to lowest non-trivial order) only when the classical equations of motion hold, i.e. when the matrices $A$ satisfy

(11)$\left[{A}^{\mu },\left[{A}_{\mu },{A}_{\nu }\right]\right]=0\phantom{\rule{thinmathspace}{0ex}},$

which is nothing but an operator version of the familiar vanishing of the $\beta$-functionals of the string’s $\sigma$-model.

Taking this result and T-dualizing all directions (including the time-like one) one finally sees that applying the Pohlmeyer invariant to $N$ coincident $D\left(-1\right)$-branes (instantons) yields the configuration of $N$ such instantons distributed according to the non-commutative coordinates ${\varphi }^{\mu }={A}_{\mu }$ iff (to lowest non-trivial order), the classical equations of motion of the IIB matrix model hold.

And this, after all pretty trivial, observation is what I wanted to get at. It shows that the Pohlmeyer invariants are related to coherent states of closed strings which constitute a distribution of space-time ‘events’ (D(-1)-branes) that is a solution of the classical equations of motion of the IIB matrix model and can be argued to describe a discrete/noncommutative approximation to 1+9D spacetime.

It just remains to note that by using the relation to DDF invariants it is easy to generalize the Pohlmeyer invariants to the superstring (section 2.3.3 of ), which should carry all of the above disucssion over to its supersymmetric extension.

Posted at June 13, 2004 8:53 PM UTC

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