Constructing symplectomorphisms between symplectic torus quotients

Abstract

We identify a family of torus representations such that the corresponding singular symplectic quotients at the 0-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may have graded isomorphic algebras of real regular functions and graded Poisson isomorphic complex symplectic quotients yet not be graded regularly diffeomorphic nor graded regularly symplectomorphic.

Constructive approach to Theorem 1

We first obtained a proof of Theorem 1 for Type $\hbox {I}_k$ matrices by determining an explicit description of the symplectic quotient $M_0$ and algebra $\mathbb {R}[M_0]$ of regular functions. This description may be of independent interest and illustrates the structure of these spaces, so we include it here. The proofs of these results are cumbersome computations and hence only summarized.

Proposition 2

Let $A = \big (D, \overbrace{\varvec{n},\ldots , \varvec{n}}^k\big )\in \mathbb {Z}^{\ell \times (\ell +k)}$ be a type $\hbox {I}_k$ weight matrix such that $V_A$ is a faithful $\mathbb {T}^\ell$-module. Then a generating set for the algebra $\mathbb {R}[V_A]^{\mathbb {T}^\ell }$ of invariants is given by

• 1. the $\ell$ quadratic monomials $r_i := z_i \overline{z_i}$ for $i = 1,\ldots ,\ell$,

• 2. the $k^2$ quadratic monomials $p_{i,j} := z_{\ell +i}\overline{z_{\ell +j}}$ for $1\le i,j\le k$,

• 3. the ${\alpha +k-1 \atopwithdelims ()k-1}$ degree $\eta$ monomials $q_{\varvec{s}} := \prod _{i=1}^\ell z_i^{m_i} \prod _{i=1}^{k} z_{\ell +i}^{s_i}$ where $\varvec{s} = (s_1,\ldots ,s_k)$ and the $s_i$ are any choice of nonnegative integers such that $\sum _{i=1}^{k} s_i = \alpha$, and

• 4. the ${\alpha +k-1\, choose\, k-1}$ degree $\eta$ monomials $\overline{q_{\varvec{s}}}$ for each choice of $\varvec{s}$.

For a generating set for $\mathbb {R}[M_0(A)]$, the generators in (1) can be omitted using the on-shell relations.

Proof

A simple computation demonstrates that each of the monomials listed in Proposition 2 is invariant. To prove the proposition, one first establishes the result when $k=1$ by induction on $\ell$; the base case is simple, and the inductive step is accomplished by comparing the invariants of A to those corresponding to submatrices formed by removing a single row and the resulting column of zeros. For general k, consider the map $\phi : \mathbb {R}[z_1,\ldots ,z_{\ell +k},\overline{z_1},\ldots ,\overline{z_{\ell +k}}] \rightarrow \mathbb {R}[w_1,\ldots , w_{\ell +1},\overline{w_1},\ldots , \overline{w_{\ell +1}}]$ that maps $z_i\mapsto w_i$ and $\overline{z_i}\mapsto \overline{w_i}$ for $i \le \ell$, $z_{\ell +i}\mapsto w_{\ell +1}$, and $\overline{z_{\ell +i}}\mapsto \overline{w_{\ell +1}}$. It is easy to see that $\phi$ maps A-invariants onto $(D, {\mathbf {n}})$-invariants, and then the proof is completed by considering the preimages of the $(D, {\mathbf {n}})$-invariants, a case with $k=1$. $\square$

Proposition 3

Let $A = \big (D, \overbrace{\varvec{n},\ldots , \varvec{n}}^k\big )\in \mathbb {Z}^{\ell \times (\ell +k)}$ be a type $\hbox {I}_k$ weight matrix such that $V_A$ is a faithful $\mathbb {T}^\ell$-module. The (off-shell) relations among the $r_i$, $p_{i,j}$, $q_{s}$, and $\overline{q_{s}}$ are generated by the following.

• 1. $p_{g,h}p_{i,j} - p_{g,j}p_{i,h}$ for $1\le g,h,i,j \le k$ with $g\ne i$ and $h\ne j$.

• 2. $p_{g,h} q_{\varvec{s}} - p_{i,h} q_{\varvec{s^\prime }}$ where $s_g^\prime = s_g + 1$, $s_i^\prime = s_i - 1$, and $s_j^\prime = s_j$ for $j \ne g,i$. Note that we must have $s_i \ge 1$.

• 3. $p_{g,h} \overline{q_{\varvec{s}}} - p_{g,i} \overline{q_{\varvec{s^\prime }}}$ where $s_g^\prime = s_g + 1$, $s_i^\prime = s_i - 1$, and $s_j^\prime = s_j$ for $j \ne g,i$. Note that we must have $s_i \ge 1$.

• 4. $q_{\varvec{s}} q_{\varvec{s^\prime }} - q_{\varvec{t}} q_{\varvec{t^\prime }}$ where $\varvec{s} + \varvec{s^\prime } = \varvec{t} + \varvec{t^\prime }$ and $\varvec{s}\ne \varvec{t}$.

• 5. $\overline{q_{\varvec{s}}}\, \overline{q_{\varvec{s^\prime }}} - \overline{q_{\varvec{t}}}\, \overline{q_{\varvec{t^\prime }}}$ where $\varvec{s} + \varvec{s^\prime } = \varvec{t} + \varvec{t^\prime }$ and $\varvec{s}\ne \varvec{t}$.

• 6. $\prod _{i=1}^\ell r_i^{m_i} \prod _{j=1}^\alpha p_{g_j,h_j} - q_{\varvec{s}} \overline{q_{\varvec{s^\prime }}}$ where the vector $(g_1, \ldots , g_\alpha )$ contains each value g exactly $s_g$ times and the vector $(h_1,\ldots , h_\alpha )$ contains each value h exactly $s_h^\prime$ times.

On-shell, the monomials additionally satisfy the defining relations of the moment map, $-a_i r_i + n_i \sum _{j=1}^k p_{j,j}$ for $i=1,\ldots ,\ell$.

Proof

One verifies that each of these relations holds by direct computation using the definitions of the monomials given in Proposition 2. The proof that all relations are generated by these is by induction on k. For the case $k = 1$, there is only one nontrivial relation, $p_{1,1}^\alpha \prod _{i=1}^\ell r_i^{m_i} - q_{(\alpha )} \overline{q_{(\alpha )}}$; a simple yet tedious consideration of cases demonstrates that this generates all relations. The induction step is demonstrated by considering the preimages of invariants under the map $\mathbb {C}[z_1,\ldots ,z_{\ell +k+1}]\rightarrow \mathbb {C}[z_1,\ldots ,z_{\ell +k}]$ given by $(z_1,\ldots ,z_{\ell +k+1})\mapsto (z_1,\ldots ,z_{\ell +k} + z_{\ell +k+1})$. $\square$

One then verifies the following by direct computation.

Proposition 4

Let $A = \big (D, \overbrace{\varvec{n},\ldots , \varvec{n}}^k\big )\in \mathbb {Z}^{\ell \times (\ell +k)}$ be a type $\hbox {I}_k$ weight matrix such that $V_A$ is a faithful $\mathbb {T}^\ell$-module. The Poisson brackets of the Hilbert basis elements given in Proposition 2 are as follows. Note that the indices g, h, i, j need not be distinct unless otherwise noted.

• $\{ r_g, r_h\} = \{ r_g, p_{h,i}\} = \{ q_{\varvec{s}}, q_{\varvec{s^\prime }} \} = \{ \overline{q_{\varvec{s}}}, \overline{q_{\varvec{s^\prime }}} \} = 0$.

• $\{ r_i, q_{\varvec{s}} \} = - \frac{2}{\sqrt{-1}} m_i q_{\varvec{s}}$.

• $\{ r_i, \overline{q_{\varvec{s}}} \} = \frac{2}{\sqrt{-1}} m_i \overline{q_{\varvec{s}}}$.

• $\{ p_{g,h}, p_{i,j}\} = {\left\{ \begin{array}{ll} \frac{2}{\sqrt{-1}} p_{i,h}, &{} g = j \text{ and } h\ne i,\\ - \frac{2}{\sqrt{-1}} p_{g,j}, &{} g \ne j \text{ and } h = i, \\ \frac{2}{\sqrt{-1}}(p_{h,h} - p_{g,g}) &{} g = j \text{ and } h = i, \text{ and } g\ne h \\ 0, &{} g \ne j \text{ and } h\ne i \text{ or } g = j = h = i. \end{array}\right. }$.

• $\{ p_{g,h}, q_{\varvec{s}} \} = {\left\{ \begin{array}{ll} -\frac{2}{\sqrt{-1}} s_g q_{\varvec{s^\prime }}, &{} s_g > 0, \\ 0, &{} s_g = 0,\end{array}\right. }$

  where $s_g^\prime = s_g - 1$, $s_h^\prime = s_h + 1$, and $s_i^\prime = s_i$ for $i\ne g, h$.

• $\{ p_{g,h}, \overline{q_{\varvec{s}}} \} = {\left\{ \begin{array}{ll} \frac{2}{\sqrt{-1}} s_g \overline{q_{\varvec{s^\prime }}}, &{} s_g > 0, \\ 0, &{} s_g = 0,\end{array}\right. }$

  where $s_g^\prime = s_g - 1$, $s_h^\prime = s_h + 1$, and $s_i^\prime = s_i$ for $i\ne g, h$.

• $\{ q_{\varvec{s}}, \overline{q_{\varvec{s^\prime }}} \} = \frac{2}{\sqrt{-1}} q_{\varvec{s}} \overline{q_{\varvec{s^\prime }}} \left( \sum _{i=1}^\ell \frac{m_i^2}{r_i} + \sum _{j=1}^k \frac{s_j s_j^\prime }{p_{j,j}}\right)$, which we note is polynomial as the $r_i$ and $p_{j,j}$ divide $q_{\varvec{s}} q_{\varvec{s^\prime }}$.

The above results give an explicit description of the Poisson algebra of regular functions. It remains only to determine the semialgebraic description of the symplectic quotient.

Proposition 5

Let $A = \big (D, \overbrace{\varvec{n},\ldots , \varvec{n}}^k\big )\in \mathbb {Z}^{\ell \times (\ell +k)}$ be a type $\hbox {I}_k$ weight matrix associated such that $V_A$ is a faithful $\mathbb {T}^\ell$-module. Using the real Hilbert basis given by the real and imaginary parts of the monomials listed in Proposition 2, the image of the Hilbert embedding is described by the relations given in Proposition 3 as well as the inequalities $r_i \ge 0$ for $i = 1,\ldots , \ell$ and $p_{j,j} \ge 0$ for $j = 1, \ldots , k$.

Proof

From the definition of the monomials, it is easy to see that these inequalities are satisfied. For the converse, choose values of the $r_i$, $p_{i,j}$, and $q_{\varvec{s}}$ such that each $r_i \ge 0$, each $p_{i,i} \ge 0$, and the remaining values are arbitrary elements of $\mathbb {C}$ such that the each $p_{i,j} = \overline{p_{j,i}}$ and relations in Proposition 3 are satisfied. It is then easy to see that the values $|r_i|$, $|p_{i,j}|$ for $i\ne j$, and $|q_{\varvec{s}}|$ are determined by the $p_{i,i}$. Specifically, using the relations of Proposition 3(1), we have

$$\begin{aligned} |p_{i,j}| = \sqrt{p_{i,i} p_{j,j}}, \end{aligned}$$

using the moment map, we have

$$\begin{aligned} |r_i| = \frac{n_i}{a_i} \sum \limits _{j=1}^k p_{j,j} \end{aligned}$$

and using the relations of Proposition 3(6), we have

$$\begin{aligned} q_{\varvec{s}} = \sqrt{\prod \limits _{i=1}^\ell \left( \frac{n_i}{a_i}\right) ^{m_i} \left( \sum \limits _{j=1}^k p_{i,i}\right) ^{\sum _{i=1}^\ell m_i} } \left( \prod \limits _{j=1}^k p_{i,i}^{s_i}\right) ^{\alpha /2}. \end{aligned}$$

Similarly, using the relations of Proposition 3(3), one checks that the arguments of the $q_{\varvec{s}}$ where $\varvec{s}$ has only one nonzero coordinate (which must be equal to $\alpha$) determine the arguments of the $p_{i,j}$ and the other $q_{\varvec{s^\prime }}$. It follows that one can find a point $(z_1,\ldots ,z_n)$ mapped via the Hilbert embedding to these values of $r_i$, $p_{i,j}$, and $q_{\varvec{s}}$ by choosing the modulus of each $z_{\ell +i}$ to be $\sqrt{p_{i,i}}$, the modulus of each $z_i$ for $i\le \ell$ to be determined by the moment map, the argument of each $z_i$ for $i\le \ell$ to be 0, and the argument of each $z_{\ell +i}$ to be the argument of $q_{(0,\ldots ,0,\alpha ,0,\ldots ,0)}$ where $\alpha$ occurs in the ith position. $\square$

With this explicit description of $M_0(A)$ and $\mathbb {R}[M_0(A)]$ the following can be verified by direct computation.

Theorem 6

Let $A\in \mathbb {Z}^{\ell \times (\ell +k)}$ be a Type $\hbox {I}_k$ matrix such that $V_A$ is a faithful $\mathbb {T}^\ell$-module, and let $B = \big (-\alpha (A), \beta (A),\ldots , \beta (A)\big ) \in \mathbb {Z}^{1\times (k+1)}$. Using coordinates $(w_1,\ldots ,w_{k+1})$ for $V_B$, define the map $\varPhi : \mathbb {C}[V_A]^{\mathbb {T}^\ell } \rightarrow \mathbb {C}[V_B]$ by

$$\begin{aligned} r_i&\longmapsto \frac{m_i(A)}{\beta (A)} w_1 \overline{w_1}, \quad \quad \quad 1\le i \le \ell ,\\ p_{ij}&\longmapsto w_{i+1}\overline{w_{i+1}}, \quad \quad \quad 1\le i,j \le k,\\ q_{\varvec{s}}&\longmapsto \sqrt{\beta (A)^{-\beta (A)} \prod \limits _{j=1}^\ell m_j(A)^{m_j(A)}} \quad w_1^{\beta (A)} \prod \limits _{j=1}^k w_{j+1}^{s_j},\\ \overline{q_{\varvec{s}}}&\longmapsto \sqrt{\beta (A)^{-\beta (A)} \prod \limits _{j=1}^\ell m_j(A)^{m_j(A)}} \quad \overline{w_1}^{\beta (A)} \prod \limits _{j=1}^k \overline{w_{j+1}}^{s_j}. \end{aligned}$$

Then $\varPhi$ is a well-defined homomorphism $\varPhi : \mathbb {C}[V_A]^{\mathbb {T}^\ell } \rightarrow \mathbb {C}[V_B]^{\mathbb {T}^1}$ inducing an isomorphism $\mathbb {R}[M_0(A)]\rightarrow \mathbb {R}[M_0(B)]$ and a graded regular symplectomorphism between $M_0(A)$ and $M_0(B)$.