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Misha
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In order to form this quotient, you can simply take the slice $\{z_1=1\}$. Then $M_N$ is the $N-1$-fold product of the circle $S^1$. You are interested in the subset $M_N(t)$ of $M_N$ consisting of configurations $(z_1,...,z_N)$ satisfying the inequality $$ |z_1+z_2+...+z_N|\le t, $$$$ \frac{1}{N}|z_1+z_2+...+z_N|\le t, $$ $0\le t<1$. Such spaces are actually well-studied from various viewpoints. One way to think of this is as the moduli space of $N+1$-gons in $\mathbb C$ with first $N$ sides of the unit length and the remaining side-length $\le t$$\le tN$. (The points $z_k, k=1,...,N$ are directions of the edges of the polygon and the barycenter zero condition corresponds to the fact that we have a closed polygonal chain, i.e. a polygon. Considering the "moduli space" means that we consider polygons up rigid Euclidean motions: translations and rotations.) If $t=0$ then this is the moduli space of unit length $N$-gons in the plane. This space is a smooth compact connected manifold if $N$ is odd; if $N$ is even, then this space is "mostly" a manifold, but it has finitely many isolated quadratic singularities. Dimension of $M_N(t)$ is $N-3$. If $t>0$ then $M_N(t)$ is a smooth compact connected manifold with nonempty boundary; the manifold has dimension $N-2$.

Varying $t$ changes the space but does not alter its topology (as long as $0<t<1$$0<t<1/N$), it is a consequence of some Morse-theoretic considerations. The space $M_N(t)$ deformation-retracts to $M_N(0)$. (I do not suppose you know what it means, but all basic algebraic topology invariants are the same for $M_N(0)$ and $M_N(t)$.) Here is how you can get configurations $(z_1,...,z_N)\in M_N(t)$:

In order to form this quotient, you can simply take the slice $\{z_1=1\}$. Then $M_N$ is the $N-1$-fold product of the circle $S^1$. You are interested in the subset $M_N(t)$ of $M_N$ consisting of configurations $(z_1,...,z_N)$ satisfying the inequality $$ |z_1+z_2+...+z_N|\le t, $$ $0\le t<1$. Such spaces are actually well-studied from various viewpoints. One way to think of this is as the moduli space of $N+1$-gons in $\mathbb C$ with first $N$ sides of the unit length and the remaining side-length $\le t$. (The points $z_k, k=1,...,N$ are directions of the edges of the polygon and the barycenter zero condition corresponds to the fact that we have a closed polygonal chain, i.e. a polygon. Considering the "moduli space" means that we consider polygons up rigid Euclidean motions: translations and rotations.) If $t=0$ then this is the moduli space of unit length $N$-gons in the plane. This space is a smooth compact connected manifold if $N$ is odd; if $N$ is even, then this space is "mostly" a manifold, but it has finitely many isolated quadratic singularities. Dimension of $M_N(t)$ is $N-3$. If $t>0$ then $M_N(t)$ is a smooth compact connected manifold with nonempty boundary; the manifold has dimension $N-2$.

Varying $t$ changes the space but does not alter its topology (as long as $0<t<1$), it is a consequence of some Morse-theoretic considerations. The space $M_N(t)$ deformation-retracts to $M_N(0)$. (I do not suppose you know what it means, but all basic algebraic topology invariants are the same for $M_N(0)$ and $M_N(t)$.) Here is how you can get configurations $(z_1,...,z_N)\in M_N(t)$:

In order to form this quotient, you can simply take the slice $\{z_1=1\}$. Then $M_N$ is the $N-1$-fold product of the circle $S^1$. You are interested in the subset $M_N(t)$ of $M_N$ consisting of configurations $(z_1,...,z_N)$ satisfying the inequality $$ \frac{1}{N}|z_1+z_2+...+z_N|\le t, $$ $0\le t<1$. Such spaces are actually well-studied from various viewpoints. One way to think of this is as the moduli space of $N+1$-gons in $\mathbb C$ with first $N$ sides of the unit length and the remaining side-length $\le tN$. (The points $z_k, k=1,...,N$ are directions of the edges of the polygon and the barycenter zero condition corresponds to the fact that we have a closed polygonal chain, i.e. a polygon. Considering the "moduli space" means that we consider polygons up rigid Euclidean motions: translations and rotations.) If $t=0$ then this is the moduli space of unit length $N$-gons in the plane. This space is a smooth compact connected manifold if $N$ is odd; if $N$ is even, then this space is "mostly" a manifold, but it has finitely many isolated quadratic singularities. Dimension of $M_N(t)$ is $N-3$. If $t>0$ then $M_N(t)$ is a smooth compact connected manifold with nonempty boundary; the manifold has dimension $N-2$.

Varying $t$ changes the space but does not alter its topology (as long as $0<t<1/N$), it is a consequence of some Morse-theoretic considerations. The space $M_N(t)$ deformation-retracts to $M_N(0)$. (I do not suppose you know what it means, but all basic algebraic topology invariants are the same for $M_N(0)$ and $M_N(t)$.) Here is how you can get configurations $(z_1,...,z_N)\in M_N(t)$:

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Misha
Misha
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Let me try to point you in the right direction. Let $M_N$ denote the quotient of the $N$-fold product $$ S^1\times ... \times S^1 $$ by the action of the group of rotation $SO(2)=S^1$: $$ (z_1,...,z_N)\mapsto (z z_1,..., z z_N). $$ Here and below, $S^1=\{z\in \mathbb C: |z|=1\}$.

In order to form this quotient, you can simply take the slice $\{z_1=1\}$. Then $M_N$ is the $N-1$-fold product of the circle $S^1$. You are interested in the subset $M_N(t)$ of $M_N$ consisting of configurations $(z_1,...,z_N)$ satisfying the inequality $$ |z_1+z_2+...+z_N|\le t, $$ $0\le t<1$. Such spaces are actually well-studied from various viewpoints. One way to think of this is as the moduli space of $N+1$-gons in $\mathbb C$ with first $N$ sides of the unit length and the remaining side-length $\le t$. (The points $z_k, k=1,...,N$ are directions of the edges of the polygon and the barycenter zero condition corresponds to the fact that we have a closed polygonal chain, i.e. a polygon. Considering the "moduli space" means that we consider polygons up rigid Euclidean motions: translations and rotations.) If $t=0$ then this is the moduli space of unit length $N$-gons in the plane. This space is a smooth compact connected manifold if $N$ is odd; if $N$ is even, then this space is "mostly" a manifold, but it has finitely many isolated quadratic singularities. Dimension of $M_N(t)$ is $N-3$. If $t>0$ then $M_N(t)$ is a smooth compact connected manifold with nonempty boundary; the manifold has dimension $N-2$.

Varying $t$ changes the space but does not alter its topology (as long as $0<t<1$), it is a consequence of some Morse-theoretic considerations. The space $M_N(t)$ deformation-retracts to $M_N(0)$. (I do not suppose you know what it means, but all basic algebraic topology invariants are the same for $M_N(0)$ and $M_N(t)$.) Here is how you can get configurations $(z_1,...,z_N)\in M_N(t)$:

Start with any $N$-tuple of points $z_1,...,z_N\in S^1$, such that at most $\lfloor N/2 \rfloor-1$ of these points are allowed to be the same. (For instance, choose $N$ distinct points in the unit circle.) Then there is a Moebius transformation $g$ of the unit disk in the complex plane, $$ g(z)= e^{i\theta} \frac{z-a}{1-\bar{a}z}, |a|<1 $$ such that the tuple $$ (w_1,...,w_N)= (g(z_1),...,g(z_N)) $$ satisfies the property that $w_1=1$ and $$ \sum_{k=1}^N w_k=0. $$ In other words, $(w_1,...,w_N)\in M_N(0)$. The way to find this $g$ is to take the conformal barycenter $a=C(\mu)\in \mathbb C$ of the measure $$ \mu=\sum_{k=1}^N \delta_{z_k}. $$ This conformal barycenter satisfies the inequality $|a|<1$. Then take $g$ as above. (You choose $\theta$ so that $w_1=1$ if it is important to you, otherwise, take $\theta=0$.) This is all pretty much contained in my papers with John Millson, [2] and [3].

There are ways to approximate the value of $a$ given the points $z_1,...,z_N$, see [1]. If you prefer to have a configuration $(w_1,...,w_N)\in M_N(t)$ which is not in $M_N(0)$, this is also easy to accomplish, just use $$ g(z)= e^{i\theta} \frac{z-b}{1-\bar{b}z}, $$ where $|b|<1$, $b$ is close to $a$ but is different from $a$. You can read more on this for instance in the references below. But the literature on this subject is vast, including papers of in topology, geometry, engineering, physics, etc. I no longer can keep track of this literature.

[1] Cantarella, Jason; Schumacher, Henrik, Computing the conformal barycenter, SIAM J. Appl. Algebra Geom. 6, No. 3, 503-530 (2022). ZBL1515.65051.

[2] Kapovich, Michael; Millson, John, On the moduli space of polygons in the Euclidean plane, J. Differ. Geom. 42, No. 1, 133-164 (1995). ZBL0847.51026.

[3] Kapovich, Michael; Millson, John J., The symplectic geometry of polygons in Euclidean space, J. Differ. Geom. 44, No. 3, 479-513 (1996). ZBL0889.58017.