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In this Seiberg's SUSY lecture, the professor said that the following theory with Kinetic and $W$ potential terms:

$$ K=|\phi|^2 $$ $$ W=m\phi^2+g \phi^3 $$

  1. "It is not a valid theory in $4d$, but it is infrared (IR) free, thus it is not an interacting QFT."

  2. "In $2d$ and $3d$, they are valid interacting QFTs."

Could someone explain the logic?

  • what do free (quadratic lagrangian?) and interacting (higher order non-quadratic lagrangian?) mean respect to UV or respect to IR?

  • I thought the $K=|\phi|^2$ requires derivative to be a kinetic term? Is he incorrect?

  • I thought the $W=m\phi^2+g \phi^3$ are both relevant operators in the IR in 4d. Thus $g \phi^3$ changes the IR dynamics? Should this lead to an interacting QFT in 4d at IR?

  • I thought the $m\phi^2$ is relevant and $g \phi^3$ is a marginal operator in the IR in 3d. Thus $g \phi^3$ again changes the IR dynamics? Should this lead to an interacting QFT in 3d?

  • I thought the $m\phi^2$ is marginal and $g \phi^3$ is an irrelevant operator in the IR in 2d. Should this lead to a free QFT in 2d?

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The terms $K$ and $W$ are not kinetic and potential terms, rather $K$ is a K"ahler potential and $W$ is a superpotential. Neither term enters the Lagrangian directly, but they are used to construct it. For details see, for example, Cyril Closset's lecture notes on supersymmetry.

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  • $\begingroup$ thanks but how about the other parts of puzzles on interacting theory or not? $\endgroup$ – ann marie cœur Apr 14 at 17:46
  • $\begingroup$ To check the properties of the theory (i.e. if and how it interacts) you would first need to compute the Lagrangian generated by these potentials. $\endgroup$ – Arthur Morris Apr 14 at 18:41
  • $\begingroup$ thanks I voted up +1 -- does that lecture note contain all these aspects? $\endgroup$ – ann marie cœur Apr 14 at 19:34
  • $\begingroup$ It does :) As an aside, the standard reference for SUSY (which Seiberg mentions in that lecture) is usually Wess and Bagger. $\endgroup$ – Arthur Morris Apr 14 at 19:50

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