I'm a mathematician doing some work which is related to $\mathcal N=2$ supersymmetric quantum field theory in $d=4$ and am a little confused about the physical notion of moduli space in this context. I apologize in advance if this question is too basic, or too vague, or just kind of stupid. References to appropriate literature would be most welcome.

From skimming articles, there are at least three things people refer to as a moduli space:

  1. the moduli space of supersymmetric vacua,
  2. the collection of all of the parameters, like masses, coupling constants, etc.,
  3. the moduli space of complexified Kähler structures (A-model) or the moduli space of complex structures (B-model), both usually considered on a fixed Kähler or Calabi-Yau manifold.
Question. What is the relationship between these various moduli spaces? Are they the same in some appropriate sense? If they are different which is the one with a "Coulomb branch"?

The impression I get is that the appropriate moduli space has some singular points, at which the corresponding physical theory is special (e.g, conformal?).

Question. What sort of nice things happen with physical theories corresponding to singular points of the moduli space? Are these the theories we're actually interested in and the generic points correspond to more calculable approximations of these theories?

This is going to be a severely incomplete answer, more importantly, I am also new to the subject, so this is a personal understanding, not a definitive answer, I apologize for that. A couple of good references on 4d N=2 theories that I am aware of are [1,2].

Some upfront answers:

Q1) Moduli spaces of type 2 and 3 can be cast in the same light, type 1 is somewhat different. Coulomb branch belongs to type 1.

Q2) You're right I think. Roughly speaking, the points in the moduli spaces correspond to theories of a certain type, with a certain type of field contents, as we move along the moduli space, generically, only some continuous parameters of these theories vary, but at the singular points discontinuous changes occur, such as:

  • the number of massless fields jumps
  • non-conformal theories can become conformal
  • in cases where we can assign some cohomology groups to the theories, the groups don't generically change under movement along the moduli spaces, but they can change at singular points.

Certainly, the singular points are more colorful than a generic point, what we are more interested in can, I think, vary from problem to problem.

Comments about moduli spaces of type 2 and 3 in your list: Any field theory is defined with a set of "static" data, the idea is that given a fixed "type" of theory there's a space of static data such that each point of that space defines a theory of the same type.

Example 1: suppose we are interested in conformal field theories (CFTs) in some dimension with some fields. The "type" in this case is defined by a representation of the conformal group (reducible, consisting of infinitely many irreducible representations) with the structure of an associative algebra (called an operator product expansion (OPE) algebra, example includes vertex operator algebras). If we want CFT with a certain supersymmetry we must consider representations of the appropriate superconformal algebra. Given such a representation, the static data necessary to pinpoint a particular CFT is a collection of numbers, one for each highest weight state of a certain conformal dimension (eigenvalue of the dilatation operator in the conformal algebra). These numbers are called marginal couplings and for any arbitrary choice of these numbers we have a CFT (not true in general, numbers corresponding to a subset of the set of highest weights may need to be fixed to zero for reasons not visible at the static level). So the space of static data in this case, or equivalently the space of CFTs, is going to be (locally homeomorphic to) $\mathbb{R}^n$ or $\mathbb{C}^n$ where $n$ is the number of highest weight states of the particular conformal dimension dimension (minus the number such states whose corresponding coupling constants must be set to zero) and the choice of real or complex numbers depends on the theory. [3]

Example 2: Suppose we are interested in the so called "A-twisted sigma model in 2 dimension". This is a type of theories with a certain supersymmetry algebra. The static data in this case is a choice of coupling constants that happens to be parametrized by the space of complexified Kähler structures of some Kähler manifold. [4]

Example 3: Replace "A-twisted" with "B-twisted", and "Kähler structure of some Kähler manifold" with "complex structure of some Calabi-Yau manifold" in example 2. [4]

The summary here is that, given a symmetry algebra, there may be a family of theories with that symmetry algebra, the family is parametrized by a moduli space. Singular points are special, they can have enlarged symmetry.

Comments about moduli space of type 1 in your list: Apart from the static data, and considerably harder to determine, are the "dynamic" data, these are the things we can find out if we can observe the evolution of the system being described (an analogy in compute science would be the difference between the code of a program (static) and its output (dynamic)). A ubiquitous and particularly important example of such data is the moduli space of vacua. In situations where we can associate a Hilbert space with our field theory the space of vacua is simply the space of states with the minimum possible energy. An alternate and more general definition of this space is in terms of the "effective potential". All the dynamical data of a theory is contained in an entity called the "effective action" which is a functional of fields. The effective action is essentially the classical action modified by quantum corrections (and in general impossible to compute exactly, exceptions are therefore highly interesting). Just like the classical action the effective action has two parts, a "kinetic" part and a "potential" part. The potential part is the classical potential with quantum correction. If we denote the potential part as $V(\phi)$ then the field configurations that minimize the potential constitute the space of vacua. In supersymmetric theories it can be shown that the minima of the potential is zero, i.e., the space of vacua is the space of solution to the equation $V(\phi) = 0$. In perturbative field theories, we expand the fields in our classical action around the vacuum solutions. So if $\phi$ is a field in our theory and $\phi_0$ is a solution to the vacuum equation then we will define $\widetilde\phi := \phi - \phi_0$ and consider $\widetilde\phi$ as the dynamical field of our theory. (Note that in terms of $\widetilde\phi$, the vacuum is given by $\widetilde\phi=0$). The hope is that at low enough energy the choice of vacua is the dominating feature governing the dynamics, everything else can be treated as perturbation. If there's a family of solutions to the vacuum equation then depending on which vacuum we choose to expand around, the low energy dynamics can look somewhat different. In this sense the space of vacua is the moduli space of the low energy description of the theory. Coulomb branch is the name of a certain subspace of the space of supersymmetric vacua in certain supersymmetric theories (the name Coulomb branch refers to the fact that if we expand around a generic point of this subspace, the low energy theory we find is an abelian gauge theory, like the Maxwell's theory of the "Coulomb force"). A final note is that, usually we get a CFT only at the origin of the Coulomb branch, the conformal symmetry is broken everywhere else on the Coulomb branch. [1,2]

[1] https://arxiv.org/pdf/1312.2684.pdf

[2] http://member.ipmu.jp/yuji.tachikawa/tmp/review-rebooted3.pdf

[3] http://www.sciencedirect.com/science/article/pii/0370269389900282

[4] https://projecteuclid.org/euclid.cmp/1104162092

*I would love to be corrected for anything I might have said wrong.


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