# What is the definition of moduli space, in math vs in physics?

It is easy to find that there are many questions regarding moduli space on MSE: https://math.stackexchange.com/search?q=what+is+moduli+space

But it seems to me that this phrase, moduli space, may mean many different things.

For example, according to Wikipedia, the moduli space is used in physics to refer specifically to the moduli space of vacuum expectation values of a set of scalar fields, or to the moduli space of possible string backgrounds. Moduli spaces also appear in physics in topological field theory, where one can use Feynman path integrals to compute the intersection numbers of various algebraic moduli spaces.

Moduli space occurs in the context of

• in algebraic geometry, as a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.
• intersection theory of Riemann surface.
• moduli space of Calabi-Yau manifolds.
• Weil-Petersson metric of the moduli space of elliptic curves.

So what is the definition of moduli space, in math vs in physics, after all these?

• The precise definition and conditions one wants varies by the specific area and the specific application, but the general ideal is the same: it's a space parametrizing some other stuff you care about. You will not find one definition to rule them all. Commented Jun 16, 2021 at 6:20

Question: "So what is the definition of moduli space, in math vs in physics, after all these?"

Answer: In algebra and algebraic geometry, mathematical physics etc. you want to classify objects such as algebraic varieties/schemes, vector bundles, vector bundles with a connection, vector bundles with a flat connection etc.

Example: Let $$C$$ be a smooth projective curve over a field $$k$$ and let $$E$$ be a finite rank vector bundle on $$C$$. How large is the "parameter space" of connections on $$E$$? What about the "parameter space of flat connections on $$E$$"? In this case there is an obstruction class $$a(E)$$ which measures when $$E$$ has a connection: $$a(E)=0$$ iff $$E$$ has a connection. There is also a "parameter space"

$$Conn(C,E):=H^0(C, End(E)\otimes \Omega^1_{C/k})$$

parametrizing all connections on $$E$$. The space $$Conn(C,E)$$ is a finite dimensional vector space over $$k$$. There is in many cases an algebraic group $$G$$ acting on $$Conn(C,E)$$, and the "orbits" of this action correspond to isomorphism classes of connections $$(E,\nabla)$$. This is where "moduli spaces" appear: You want to take the "quotient" $$Mod(C,E):=Conn(C,E)/G$$ of the action of $$G$$ and you want the quotient $$Mod(C,E)$$ to "behave" like a scheme. You want to have access to notions such as dimension, non-singularity, irreducibility, vector bundles, algebraic cycles, cohomology/homology etc. This is what "moduli spaces" do - they are objects with properties similar to schemes.

Question: "So what is the definition of moduli space?"

Answer: If $$X$$ is a scheme parametrizing a set of objects, and if $$G$$ is an algebraic group (or groupoid) acting on $$X$$, the associated moduli space is the "quotient" $$X/G$$. Sometimes you write $$[X/G]$$ if you consider the stack quotient.

Example: If $$X:=Spec(A)$$ is an affine scheme and if $$G$$ is a finite group acting on $$A$$, the quotient map $$\pi: X \rightarrow X/G$$ is given by the inclusion $$A^G \subseteq A$$ where $$A^G$$ is the sub ring of $$G$$-invariant elements.

Let $$A:=k[x]$$ and $$S_d$$ the symmetric group on $$d$$ elements with $$B:=k[x]\otimes_k \cdots \otimes_k k[x]$$ and obvious action $$S_d \times B \rightarrow B$$. It follows the invariant-ring $$B^{S_d}=k[s_1,..,s_d]$$ where $$s_i$$ are the elementary symmetric polynomials. The polynomials $$s_i$$ are algebraically independent, hence $$B^{S_d}$$ is a polynomial ring. It follows

$$Spec(B^{S_d}) \cong \mathbb{A}^d_k.$$

By definition $$Sym^d(X):=X^{\times d}/S_d$$, hence $$Sym^d(\mathbb{A}^1_k)\cong \mathbb{A}^d_k$$. If $$k$$ is algebraically closed it follows any $$k$$-rational point $$p\in \mathbb{A}^d_k(k)$$ corresponds to a maximal ideal

$$(t_1-a_1,..,t_d-a_d) \subseteq k[t_1,..,t_d].$$

We may view the point $$p$$ as the polynomial

$$f_p(T):=(T-a_1)\cdots (T-a_d)$$

in the variable $$T$$. When we multiply out $$f_p(T)$$ we get

$$f_p(T)=T^d-s_1(a_i)T^{d-1}+\cdots +(-1)^ds_d(a_i)$$

where $$s_j(a_i)$$ are the symmetric polynomials in the numbers $$a_i$$.

The map

$$\pi_d: (\mathbb{A}^1_k)^{\times d} \rightarrow Sym^d(\mathbb{A}^1_k)$$

does the following:

$$\pi_d(a_1,..,a_d):=(s_1(a_i),..,s_d(a_i))$$

hence it maps the polynomial $$f_p(T):=\prod_i (T-a_i)$$ to the polynomial

$$\pi_d(f_p(T)):=T^d-s_1(a_i)T^{d-1}+\cdots +(-1)^ds_d(a_i).$$

Hence $$Sym^d(\mathbb{A}^1_d)$$ is the "moduli space" parametrizing polynomials by its coefficients.

Note: What is mysterious here is that $$\mathbb{A}^d_k/S_d \cong \mathbb{A}^d_k$$. By the "going up theorem", since the ring extension

$$A:=k[s_i ] \subseteq B:=k[t_i]$$

is integral, it follows for any prime ideal $$\mathfrak{q} \subseteq A$$ there is a prime ideal $$\mathfrak{p} \subseteq B$$ with $$\mathfrak{p}\cap A=\mathfrak{q}$$. In particular if $$\mathfrak{q}:=(s_1-a_1,..,s_d-a_d)$$ is maximal, there is a maximal ideal $$\mathfrak{p}:=(t_1-u_1,..,t_d-u_d) \subseteq B$$ with $$\pi_d(\mathfrak{p})=\mathfrak{q}$$. Hence the map

$$\pi_d: \mathbb{A}^d_k \rightarrow Sym^d(\mathbb{A}^1_k) \cong \mathbb{A}^d_k$$

is surjective at the level of topological spaces with finite fibers. This is a paradox: The map $$(\pi_d)_{t}$$

$$(\pi_d)_t: (\mathbb{A}^d_k)_t \rightarrow (\mathbb{A}^d_k)_t$$

is a surjective "endomorphism" of affine space which is not injective. This paradox is due to the fact that the sub ring ring $$k[s_1..,s_d] \subseteq k[t_1,..,t_d]$$ is isomorphic to $$k[t_1,..,t_d]$$: The map

$$\phi:k[t_j] \rightarrow k[s_j]$$

defined by $$\phi(t_i):=s_i$$ is an isomorphism of rings. Hence $$k[t_j]$$ is isomorphic to a strict subring of itself. When dealing with infinite sets we "must accept" such paradoxes.

Group actions on sets: If you (naively) have a finite set $$S$$ and a finite (non trivial) group $$G$$ acting non-trivially on $$S$$ via

$$\sigma: G \times S \rightarrow S$$

it follows the quotient $$S/G$$ has fewer elements than $$S$$. For schemes you may end up with an isomorphism $$X/G \cong X$$.

Closed subgroups of algebraic groups: If $$H \subseteq G \subseteq GL_k(V)$$, are closed subgroups with $$k$$ a field and $$V$$ a finite dimensional $$k$$-vector space, you may always construct the "quotient"

$$\pi: G \rightarrow G/H,$$

and $$G/H$$ is a smooth quasi projective algebraic variety of finite type over $$k$$. The grassmannian and flag varieties are examples of such quotient varieties. If $$W\cong k^m \subseteq V\cong k^n$$ is a sub vector space and if $$P \subseteq SL(V)$$ is the (closed) subgroup fixing $$W$$ it follows $$SL(V)/P \cong \mathbb{G}(m,V)$$ is the grassmannian variety, parametrizing $$m$$-dimensional subspaces of $$V$$.

There are many paradoxes in this theory, but if you are interested there is a homepage devoted to this (the "stacks homepage" in NY) where a "general theory of moduli spaces" is developed. You will find much material on this page (more than 6000 pages of math).

Here you find a relevant discussion at MO:

https://math.stackexchange.com/questions/tagged/algebraic-stacks

https://mathoverflow.net/questions/tagged/stacks

https://mathoverflow.net/questions/123942/how-many-flat-connections-has-a-line-bundle-in-algebraic-geometry/393770#393770

This phenomenon that you get paradoxical situations when taking quotients appears elsewhere. In this thread I construct a "curve" $$C$$ with a non-trivial equivalence relation $$R$$ such that the "quotient" $$S:=C/R$$ has dimension $$2$$.

"Infinity" in mathematics and an elementary question on dimension.

Ex: Let $$A:=\mathbb{Z}$$ and $$Q:=\mathbb{Z}/(p)\mathbb{Z}$$ with $$p$$ a non-zero prime number. Define the map $$\phi: A \rightarrow A$$ by $$\phi(n):=pn$$. It follows $$\phi \in End(A)$$ is an injective endomorphism of the abelian group $$A$$ giving an exact sequence

$$0 \rightarrow A \rightarrow^{\phi} A \rightarrow Q \rightarrow 0,$$

hence the map $$\phi$$ is an isomorphism between $$A$$ and a strict subgroup $$\phi(A) \subseteq A$$ of $$A$$. Hence this type of construction/set theoretic paradox appears everywhere in mathematics. You should ask a set theorist how to fix this problem - a brilliant set theorist.

As you can see: You may ask similar questions for "stacks" as you ask for schemes:

1. What are the irreducible components of a "stack"?
2. What is its dimension?
3. What is the "etale fundamental group" of a "stack"?
4. Can I view "the theory of stacks" as a "black box" and prove important theorems?