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BFSS model is a theory of super-symmetric matrix quantum mechanics describing $N$ coincident $D0$-branes, defined by the action

$$S=\frac{1}{g^2}\int dt\ \text{Tr}\left\{ \frac{1}{2}(D_t X^I)^2 + \frac{1}{2}\psi_\alpha D_t \psi_\alpha + \frac{1}{4}[X^I,X^J]^2 + \frac{1}{2}i\psi_\alpha \gamma_{\alpha\beta}^I[\psi_\beta,X^I]\right\}.$$

You can read about the background and all the details regarding the indices, the gauge field, the spinor representations in a nice paper by Maldacena, To gauge or not to gauge? See section 2 specifically.

Now this is not a quantum field theory in $d+1$ dimensions. This is simply quantum mechanics in $0+1$ dimensions. In quantum mechanics we can similarly define Feynman diagrams for perturbative calculations, but this is not something known as commonly as Feynman diagram techniques in quantum field theory. See for example this nice review by Abbott, Feynman diagrams in quantum mechanics.

When I did a literature search, I couldn't find any results regarding Feynman rules for the BFSS model. Is this something that has never been considered? Or is it just too hard to find the propagators and vertex factors? How can we go about calculating the free propagator? Can we look at an expansion for the full propagator in the large-$N$ limit?

It feels as if we need to introduce the 1PI diagrams, and construct something like this Equation 7.22 in Peskin & Schroeder: writing the Fourier transform of a two-point function as a series of 1PI diagrams.

Is there an effective action for the BFSS? Maybe that can simplify the computations?

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    $\begingroup$ Did you see Becker, Becker, Schwarz string theory book? I think there are some relevant materials in chapter 12 including propagator and vertex factors of BFSS theory $\endgroup$
    – Arian
    Jun 27 '21 at 13:30
  • $\begingroup$ @Arian Are you serious. I never had the chance to read that book! That's a great comment, thank you! $\endgroup$
    – 123456
    Jun 27 '21 at 14:58
  • $\begingroup$ @Arian If you wish you can form your comment into an answer, so that you can get your well deserved bounty. $\endgroup$
    – 123456
    Jun 27 '21 at 15:06
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    $\begingroup$ Dear gsuer, unfortunately I wasn't involved in studying matrix theory calculations, but Katrin Becker and Melanie Becker papers around 97-98 are very useful for your question, specially this paper arxiv.org/pdf/hep-th/9705091.pdf might be useful for you, which is a detailed extension of what they have used in their book. I hope you find the answer in that paper. $\endgroup$
    – Arian
    Jun 27 '21 at 18:25
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You can read the Feynman rules directly from the action you wrote. In general, the propagators and vertices take the same form for these types of theories. The "spacetime" dependance of these is the same as the usual scalar theories that are used to teach QFT. The non-trivial part of the computation comes from the matrix combinatorics. This problem was first considered by ‘t Hooft in Nucl.Phys.B 72 (1974) 461 for large N QCD. This problem is notoriously difficult and very well studied in many contexts. Some of the details on how these computations are done can be found in the AdS/CFT integrability review.

BFSS is a notoriously difficult model to solve, and it is also plagued by problems such as having flat directions in the moduli space of vacua which make numerical analysis difficult. Nowadays there are other models which capture similar things as BFSS that are either more amenable to systematic analysis (such as $\mathcal{N}$=4 SYM which is a cousin of BFSS, or the BMN matrix model which has nicer vacua), or simpler solvable toy models such as SYK.

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  • $\begingroup$ Great answer. I guess it is easy to read it off from the action, especially the propagators. But I'm a bit confused regarding the vertices. Like if we had a simple $\text{Tr}X^4$ potential, it would be alright, that's what t'Hooft considered. But with the commutator involved I believe there should be 4 different vertices. And similarly for the fermionic vertex. How can I really understand these? And also what is that AdS/CFT review you mentioned. Can you provide a link? $\endgroup$
    – 123456
    Jun 30 '21 at 5:50
  • $\begingroup$ Section 5 of this chapter of the review gives details on how some of these perturbative computations are done arxiv.org/pdf/1012.3983.pdf. The computations in $\mathcal{N}=4$ are similar (a little easier) than BFFS because the theory is conformal, but this should give you the gist. $\endgroup$ Jun 30 '21 at 20:42
  • $\begingroup$ @gsuer You are right that there will be many kinds of vertices, but it's easier to keep track of the whole sum of vertices $\Tr \left(\sum_{i<j}[X_i,X_j]^2 \right)$ and to do the Wick contractions explicitly for the corresponding operators. $\endgroup$ Jun 30 '21 at 20:47
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As @AdolfoHolguin remarked it is quite easy to read off the Feynman rules from the action itself. See Developing local RG: quantum RG and BFSS.

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