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While reading a paper by Gaiotto et al., I stumbled upon the concept of momentum maps where they are used in section 2.1 to characterise the vaccum structure of some theory.

I tried reading the literature about it and I couldn't find a satisfactory answer which wasn't too mathematical, diving into the details of the geometry of the moduli space and such. I saw that there is a similar question on the site but neither it nor the linked one received any answers.

So I would like to understand

  1. What are momentum maps and why they come up in supersymmetry (here one can be as mathematical as one likes, but a bit less than these lectures if possible :D )
  2. How do, explicitly, Gaiotto et al. arrive at the result quoted in eq (2.2) and why are they relevant for the specific breaking. Is it really relevant to gauge the $\text{U}(1)$ factor?
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    $\begingroup$ A good reference is arxiv.org/abs/1312.2684. Also, (2.2) are not momentum maps but moment maps which might help you find more information. In theories with many scalars there is a current rotating between them and many ways to give them mass (bilinears). In SUSY theories, one can choose special scalars $Q$ (hypermultiplets) such that these masses and currents are related by an action of supersymmetry. This is when people refer to the mass as a moment map. $\endgroup$ Jul 9, 2023 at 21:10
  • $\begingroup$ I understood that moment map = momentum map, the former being a mistranslation of the French. $\endgroup$ Jul 13, 2023 at 10:31

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I think this question is a little too broad for a good answer which stands alone, so I will start with some references. I concentrate on your first question (1) which is of wider interest. You don't really need to know much about (1) in order to answer (2), but I think if you have a definition of moment functions in SUSY (as in the lecture notes below), your (2) will not be so difficult. There are people more expert in supersymmetric theories than I am so perhaps they can join in.

First, a momentum map (perhaps more commonly known - incorrectly - as a moment map) is a standard idea in symplectic mechanics where some Lie group $G$ acts on a phase space $M$. It is a map which in some sense generates the action of $G$ on $M$. A good reference is chapter 2 of the book Geometric Quantization by Woodhouse', although many other good lecture notes are available.

Second, in SUSY, what are referred to as `momentum maps' are the $\mu = \bar{\varphi} T^a \varphi$ D-terms that occur in the scalar potential, which are essential to studying the vacuum structure of the theory. See these notes in particular. Typically one solves $\mu = 0$, but note that adding an FI term shifts the vacuum solutions to $\mu = \xi$. This may be all you need to know depending on what you are doing, i.e. you may not need to know why they have 'momentum' in their name. On the other hand, it doesn't take long to learn this basic symplectic mechanics.

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Very quickly and schematically, here is a symplectic mechanics primer. We work over some symplectic phase space with symplectic form $\omega$. In these situations, $\omega$ can be used to flatten or sharpen tangent vectors into cotangent vectors and vice versa, using the interior product. A function $h$ on $M$ can in this way generate a vector field $X_h$ defined by $\iota_{X_h} \omega = dh$. In the reverse situation when one is given $X_h$, existence of $h$ is not guaranteed, when it does $X_h$ is called Hamiltonian.

Now let a Lie group $G$ (not necessarily gauged) act on $M$. Then every $e^{it A} \in G$ 'moves' each point $p \in M$ infinitesimally, giving rise to a collection of vector fields $V_A$. This gives a map $\mathfrak{g} \rightarrow TM$ which you can check to be compatible with the Lie bracket and hence a morphism.

To visualise, try either translations $(x,p) \mapsto (x+t,p)$ with symmetry group $\mathbb{R}$ (and $\omega = dx \wedge dp$), or complex $U(1)$ rotations $e^{i \alpha}$ acting on $\mathbb{C}$ (with $\omega = dz \wedge d\bar{z}$). In the first case, the Lie algebra elements induce vector fields $\partial/\partial x$, whilst in the second case they induce vector field $iz \partial_z - i \bar{z} \partial_{\bar{z}}$ ($=\partial_\theta$).

We now assume that this action is `integrable' in the sense that some functions $h_A$ exist which generate the $V_A$. This is just a technical restriction to 'nice' actions. The map $A \rightarrow h_A$ is called a Hamiltonian (one typically requires it to be a Lie algebra morphism, which is a further restriction). This is just a generalisation of standard physics Hamiltonians which generate time translations. Just as often, one gives the dual notion $\mu: M \rightarrow \mathfrak{g}^*$ by $\mu(m)(A) = h_A(m)$. This is the momentum map.

You can check in the first example above one has $\mu(x,p)(A) \equiv h_A(x,p) = p$ (the RHS is the projection onto $p$) whilst in the second example we have $h_A(z,\bar{z}) = iz \bar{z}$ [where $A$ is just the generator of $\mathbb{R}$ or $U(1)$]. If I have my signs right. The first example is where the name momentum comes from.

The next standard and revealing exercise is to show that a matrix Lie group $G$ acting on $\mathbb{C}P^n$ has momentum map $\mu(z)(X) = \bar{z} X^R z$, where $X^R$ is the representation of $X \in \mathfrak{g}$.

It's should be clear now why the D-terms are sometimes called moment(um) functions.

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  • $\begingroup$ This was perfect, clear and concise. Thank you very much! $\endgroup$ Jul 15, 2023 at 8:56

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