Hyperkähler quotient
Imagine you have a big toy box filled with different colored blocks. Now, you want to organize these blocks so that they make groups of similar colors. To do this, you decide to split the blocks into different piles, with each pile having only one color of block. This makes it easier to find the block you want!
Now, in math, we sometimes have very complicated objects that are made up of many different pieces. For example, we might have a space that has points in it that are moving around. And just like with our toy box, we might want to organize these points into different piles based on some property they have in common.
This is where the idea of a "hyperkähler quotient" comes in. It's basically a way of organizing certain kinds of mathematical objects based on a symmetry they share.
Symmetry is a bit like when you look in the mirror and see that your left hand and your right hand are the same, but opposite. In math, we call this "reflective symmetry." And just like you might divide your blocks into two piles based on whether they're the same on both sides or not, we can divide our mathematical objects into "left" and "right" piles based on their reflective symmetry.
But sometimes our mathematical objects have more than just one kind of symmetry. They might have symmetry in lots of different directions at once! Just like you might spin around in a circle and see that your hands and feet are all in different directions, our mathematical objects might have lots of different pieces that are all pointing in different directions, but still have symmetry.
When this happens, we talk about a "hyperkähler" space. It's a bit like having a toy box filled with blocks of different shapes and sizes, but all of them have some kind of rotational symmetry.
So when we talk about a "hyperkähler quotient," we're basically saying that we want to organize our hyperkähler space into different piles based on the different kinds of symmetry it has. It's like taking our toy box and having separate piles for different shapes or sizes of blocks.
Overall, a hyperkähler quotient is just a way to organize complicated mathematical objects by their symmetries. It's a bit like organizing your toy box so that you can find your favorite blocks more easily!
Now, in math, we sometimes have very complicated objects that are made up of many different pieces. For example, we might have a space that has points in it that are moving around. And just like with our toy box, we might want to organize these points into different piles based on some property they have in common.
This is where the idea of a "hyperkähler quotient" comes in. It's basically a way of organizing certain kinds of mathematical objects based on a symmetry they share.
Symmetry is a bit like when you look in the mirror and see that your left hand and your right hand are the same, but opposite. In math, we call this "reflective symmetry." And just like you might divide your blocks into two piles based on whether they're the same on both sides or not, we can divide our mathematical objects into "left" and "right" piles based on their reflective symmetry.
But sometimes our mathematical objects have more than just one kind of symmetry. They might have symmetry in lots of different directions at once! Just like you might spin around in a circle and see that your hands and feet are all in different directions, our mathematical objects might have lots of different pieces that are all pointing in different directions, but still have symmetry.
When this happens, we talk about a "hyperkähler" space. It's a bit like having a toy box filled with blocks of different shapes and sizes, but all of them have some kind of rotational symmetry.
So when we talk about a "hyperkähler quotient," we're basically saying that we want to organize our hyperkähler space into different piles based on the different kinds of symmetry it has. It's like taking our toy box and having separate piles for different shapes or sizes of blocks.
Overall, a hyperkähler quotient is just a way to organize complicated mathematical objects by their symmetries. It's a bit like organizing your toy box so that you can find your favorite blocks more easily!