# Casimir Effect and parallel $D$-Branes

In the well-known setup for the calculation of the Casimir effect, we take 2 perfectly reflecting plates, impose the appropriate boundary conditions on the relevant fields (scalar, vector, etc.) and calculate the energy of this configuration.

So the most natural analog of plates (that is objects that impose boundary conditions on fields) in String Theory are $$D$$-Branes. This got me wondering whether a similar scenario can be realized in String Theory where we consider the energy of an open or closed string field theory interacting with 2 parallel $$D$$-Branes.

The scenario I had in mind is loosely related to figure1. in Scattering of Strings from $$D$$-branes. I know that $$D$$-branes have open strings stretched between them, but that scenario would not be analogous to the Casimir effect setting because there is no propagation.

• Out of curiosity, why string field theory? Mar 18, 2021 at 13:13
• @NiharKarve Isn't string field theory the natural habitat for such a question?
– user242231
Mar 18, 2021 at 15:12
• I mean, string theory has D-branes and spacetime fields as well. Restricting to string field theory unnecessarily reduces the potential answerer pool for this question by a huge amount. Mar 18, 2021 at 15:16
• @NiharKarve OK, now I understand your point. Excuse my ignorance, but how does introducing another tag reduce the answerer pool? Shouldn't it increase the number, if the tag is related that is.
– user242231
Mar 18, 2021 at 15:19
• The string field theory formalism of string theory is a lot more difficult and much less well-known than the "standard" formulation of string theory that is presented first. Remember that "string field theory" $\ne$ "fields in string theory" (which is what I assume you were going for there) Mar 18, 2021 at 15:35

No, there is no analogue of Casimir effect in between two parallel D-branes. The reason in supersymmetry.

Two D-branes interact with each other by means of open strings stretched among them. The one-loop amplitude for open strings streched between two D-branes is exactly computable (see equation (48) in TASI Lectures on D-Branes) and shown to be exactly zero.

This shouldn't be so surprising, since parallel D-branes break only half of supersymmetry, therefore the "no-force condition" between two BPS states is satisfied.

• OK $D$-branes have open strings stretched between them, but what about string interacting with $D$-branes. Consider a scenario where a closed string is scatterred of from a $D$-brane.
– user242231
Mar 18, 2021 at 15:44
• @hep-py That scenario is exactly equivalent to the one I described ( see open-closed string duality).Also notice that you can't possibly circumvent the argument because is based on supersymmetry. It doesn't matter how do you prove the system of parallel D-branes. The invariant fact is that both preserve half of the supersymmetries and as a consequence there is a no-force condition between them. Mar 18, 2021 at 15:53
• @hep-py Personally, I work on matrix models and I can't recommend nothing better than Review of Matrix Theory. The standard reference TASI Lectures on Matrix Theory is not pedagogical at all, but it's a great roadmap to the literature. Mar 18, 2021 at 16:56
• @hep-py For a conceptual understanding of how string theory achieves a successful microscopic description of the black hole thermodynamics see [Conceptual Analysis of Black Hole Entropy in String Theory ](arxiv.org/abs/1904.03232), [Emergence and Correspondence for String Theory Black Holes ](arxiv.org/abs/1904.03234) and [TASI lectures on the Holographic Principle ](arxiv.org/abs/hep-th/0002044). Technical reviews are The quantum structure of black holes Mar 18, 2021 at 16:57
• @hep-py ... and and Black Holes in String Theory. For excellent reviews of the recent developments on the information paradox at the undergraduate level see Lessons from the Information Paradox and The entropy of Hawking radiation. Mar 18, 2021 at 16:57