ER-EPR correspondence and bipartite closed strings

This text is just an experiment. I want to know the number of viewers of a blog entry on a related topic and the composition and character of comments if there are any. The insight below is not necessary correct; and it is not necessarily fundamental. ;-)

The Maldacena-Susskind ER-EPR correspondence invites one to think that (non-traversable) wormholes are natural, sort of inevitable – because spacetimes with this topology are physically equivalent to ordinary spacetimes with entangled degrees of freedom in two regions.

One of the things I was thinking about was whether there are other dual descriptions of such spacetimes. Consider, for example, two faraway Strominger-Vafa black holes in type IIB stringy vacua with 5 large dimensions and connect them with the ER bridge. Now, reduce the coupling \(g_s\) to a very low value. What will you get?




The single Strominger-Vafa black hole's microstates are mapped to particular excited states of D1-branes, D5-branes, and momenta. All these excitations of collections of D-branes may be expressed as degrees of freedom carried by open fundamental strings.




Open strings have Neumann boundary conditions for some spacetime coordinates, e.g.\[

\partial_\sigma(X^0)|_{\sigma=0,\pi} = 0

\] and, as Joe Polchinski in particular has taught us, Dirichlet boundary conditions for other coordinates, e.g.\[

X^1_{\sigma=0,\pi} = X^1_\text{D-brane location}.

\] It seems that if you want to entangle the two Strominger-Vafa D-brane stacks, you have to create the same open string to the vicinity of both stacks. You literally want to "tensor square" the information in the microstates, so the state should contain each open string in pairs of copies. Just to be sure, this is different from having a single open string in a permutation-symmetric state.

These open strings end on the two stacks of D-branes and they correspond to objects connected by a bridge. So it seems natural to imagine that there is another object, a closed string that is split to two open strings, that has\[

X^1_{\sigma=0^\pm} = (X^1)_{\pm,\text{D-brane location}}

\] This was an example of what happens with a Dirichlet boundary condition; the two branes were labeled by the symbols \(\pm\). So the closed strings are allowed to have a discontinuity \(\Delta X^1\) at \(\sigma=0\) and \(-\Delta X^1\) at \(\sigma=\pi\) but only if these points of the string are sitting at the right position in space (locations of the D-brane stacks etc.).

Assuming that the dynamics away from \(\sigma=0\) and \(\sigma=\pi\) is ordinary, the only way to guarantee that these points of the closed string remain at the right place is to make the function \(X^1(\sigma)\) odd. If we ask what it means for the excited strings, it means that we only allow the excitations by\[

\alpha^1_{-n}\tilde \alpha^1_{-n}

\] i.e. by products (not sums – that would be a way to produce unorientable closed string states whose Hilbert space is "less reduced") of left-moving and right-moving oscillators. Needless to say, such a Hilbert space of "very special" closed string states is isomorphic to the Hilbert space of open string states (but the basis vectors are squared). One may see that the \(L_0=\tilde L_0\) level-matching condition isn't challenged because the left-moving and right-moving excitations are paired.

This picture generalizes the old Susskind's idea of an open string as a closed string whose half was stuck beneath the event horizon. Now, "deeply" beneath the event horizon, there's the other black hole, so an open string should really be one-half of a closed string whose other half is an open string on the opposite side of the wormhole or ER bridge.

Note that the individual D-branes represent a "topological defect" which allows the existence of open strings with the corresponding boundary conditions. The pairs of perfectly entangled D-brane stacks also changes the state of the spacetime. This topological defect allows the existence of the closed strings with the discontinuity that may only exist at points where the closed string hits a throat of the wormhole.

The closed string states considered above – in which the excitations are only allowed in the \(\alpha^1_{-n}\tilde \alpha^1_{-n}\) pairs – create extremely special states of the closed string and it is not allowed to "permanently" impose non-local identifications on the paths taken by closed strings. At any rate, if you allow this closed string looking like a "doubled open string" to interact with other, more ordinary strings, those that ignore the bridge, you will generically create closed strings that don't respect the \(\ZZ_2\) symmetry and that consequently deviate from the right value of \(X^1(\sigma)\) for \(\sigma=0,\pi\) at a later time.

I am confused what it could mean. The discontinuity allowing the closed string to jump from one throat to another is only allowed at the right location of the D-brane (stack) but if the closed string gets kicked, it will deviate from the place where the discontinuity is allowed. Does it mean that the bipartite strings aren't allowed at all? Is the picture inconsistent?

Quite generally, the Hilbert space of the connected pair should have the form\[

\HH_\text{one BH}\otimes \HH_\text{one BH}

\] but the closed-string visualization of this Hilbert space could give us a new natural basis optimized for the bridge, i.e. for the heavily entangled states.

How many readers who aren't shy are there who can give sensible answers or observations about similar questions? String theorists who are in e-mail contact with me may send me an e-mail message, too, of course. ;-)