Understanding Moduli Spaces: O(d, d,R) & Beyond

[wam_mi]

Hi there,

Could someone explain to me what a moduli space is.

(i) What does the generating group O(d, d,R) mean?
(ii) Why is it then the modular space is written as O(d,d,R)/O(d,R)×O(d,R)
What role does the division play here?

Thanks a lot!

 

[bapowell]

(ii) Why is it then the modular space is written as O(d,d,R)/O(d,R)×O(d,R)
What role does the division play here?

Not sure what that group is, but the 'O' stands for Orthogonal and indicates that orthogonal matrices make up the group. The 'd' is most probably the dimension of the group, and the R could indicate that the entries of the matrices are taken from the reals.

As for the division, that's how you form a quotient group: by 'dividing' one group by another. It's really a modulus: the group A/B is the group A modulo the action of B. What this means is pretty simple. You take an element of A and act on it with each element of B. This forms a set of numbers, $$\{ab|\forall b \in B\}$$, called a coset. Now, do this for each $$a \in A$$. Then you get a group of cosets. Each coset is now an element of the group A/B. You've modded out by the group B in that you've 'removed' its action. You've done this by taking elements that differ by B's action and calling them one element -- the coset.

To offer a nice example, consider how you construct the group $$Z_2$$, the group of two integers, 0 and 1 mod2. Start with the additive group of integers, $$Z$$. Next take the additive group of all even integers, $$2Z$$. Now, just form the quotient group: $$Z/2Z$$. You've modded out by the action of adding an even number each element of $$Z$$. Even numbers remain even numbers when you add an even number, and odd remain odd. And so there are only 2 elements of the group $$Z_2$$: an even element (really the coset of all even numbers, represented by just a single element), 0, and an odd element, 1.

 

[xepma]

The moduli space is, generally speaking, the "space of all solutions" to some equations. You can think of the Dirac equation, Maxwell's equations or the Yang-Mills equations as a mapping from one space to another. In the presence of sources such equations take on the form

$$D\Psi(x,t) = J(x,t)$$

I've simply wrote down the equations of motion here. $D$ is the (differential) operator corresponding to the equations of motion you are considering (e.g the Dirac operator for the Dirac Equation). J is some source distribution and [itex]\Psi[/tex] is a scalar field, a vector field, or a gauge field etc. If the field $\Psi$ is a solution to this equation, then it is said to live in the moduli space of the operator D. In the absence of sources it is also lives in the kernel of D (the right hand side becomes zero). So the moduli space is nothing but your space of solutions to this equation, also called the parameter space.

It has a special name, because this moduli space can have all sorts of nice algebraic, geometric or topological features which you can study. Identifying the moduli space can tell you a great deal about the properties of the solutions, without the need to explicitly solve the differential equations.

As I said before the moduli space itself is a space on its own, frequently identifyable as a type of manifold. The manifold itself can be endowed with a group structure turning it into a Lie group. The group multiplication acting on the moduli space takes you from one solution to another -- think of it as performing a Lorentz transformation. This changes the solution is written down, but it's still a solution if the theory is Lorentz invariant.

Another class of symmetries are, ofcourse, the gauge symmetries. A gauge transformation also maps one solution into another. However, physically we do not distinguish two solutions which are gauge equivalent. Hence we want two gauge equivalent solutions to be represented by the _same_ point in the moduli space. This is accomplished by "modding out" the gauge group from your moduli space. The resulting space is a coset space, described by bapowell above. It basically means that a gauge transformation induces a transformation in the way you write your solution, but not in your solution \emph{space}. This also shows some advantages of moduli spaces: the gauge redundancy is eliminated, so any topological/geometrical property of your moduli space is automatically physical.

 

[diazona]

Not sure what that group is, but the 'O' stands for Orthogonal and indicates that orthogonal matrices make up the group. The 'd' is most probably the dimension of the group, and the R could indicate that the entries of the matrices are taken from the reals.

As for the division, that's how you form a quotient group: by 'dividing' one group by another. It's really a modulus: the group A/B is the group A modulo the action of B. What this means is pretty simple. You take an element of A and act on it with each element of B. This forms a set of numbers, $$\{ab|\forall b \in B\}$$, called a coset. Now, do this for each $$a \in A$$. Then you get a group of cosets. Each coset is now an element of the group A/B. You've modded out by the group B in that you've 'removed' its action. You've done this by taking elements that differ by B's action and calling them one element -- the coset.

To offer a nice example, consider how you construct the group $$Z_2$$, the group of two integers, 0 and 1 mod2. Start with the additive group of integers, $$Z$$. Next take the additive group of all even integers, $$2Z$$. Now, just form the quotient group: $$Z/2Z$$. You've modded out by the action of adding an even number each element of $$Z$$. Even numbers remain even numbers when you add an even number, and odd remain odd. And so there are only 2 elements of the group $$Z_2$$: an even element (really the coset of all even numbers, represented by just a single element), 0, and an odd element, 1.

I think this may be the first clear explanation I've seen of what group "division" means - thanks!