Quiver path algebra and F-term relations in melting crystals

[Monocles]

EDIT: fixed TeX issues

Hi, I'm learning about the correspondence in string theory between the geometry of Calabi-Yau manifolds and melting crystals. I care more about the math and know almost nothing about string theory, so navigating the literature littered with so much string theory jargon has been difficult.

Given a brane tiling $F$, we associate a quiver $Q$. We then consider the path algebra $\mathbb{C}Q$ associated with $Q$. Then, for a reason that I do not understand yet, we consider $\mathbb{C}Q$ modulo some equivalence relations called the F-term relations. I understand the geometric interpretation - given any path $p$ in $\mathbb{C}Q$ between two nodes $i,j$, modulo F-term relations we can write $p$ as $p_{i,j}\omega^n$, where $p_{i,j}$ is a shortest path between $i$ and $j$, and $\omega$ is a loop around a face located at $j$.

Thus far, though, I have been having a difficult time extracting the mathematics of what the equivalence relation precisely is from the references I've been looking at - there is too much string theory jargon. Am I worrying about details too much? Is the fact that I already know how to write down a path modulo F-term relations (even if I don't know how to compute the n in $\omega^n$) fine?

I am brand new to this game so I apologize if there is a well-known reference that I'm unaware of or something like that.

 

[George Jones]

Replace every tex by itex. Only use tex for mathematics that you want to appear on a separate line. As an example, I changed the tex and /tex around the first F to itex and /itex.

 

[Monocles]

Ah, thanks!

 

[mitchell porter]

I assume we are talking about pages 21-22 of http://arxiv.org/abs/1002.1709.

It looks like n is just the number of loops in the path p that you start with. The factor of $\omega^n$ in the equivalent path just means that all n loops are now on top of each other.

As for your other question: A field theory is usually specified by a Lagrangian density. Sometimes, along with the physical fields, you have "auxiliary fields" appearing whose dynamics is specified by an extra algebraic condition. In supersymmetric field theories you have http://en.wikipedia.org/wiki/F-term" .

I will admit that I am still more than a little fuzzy on how all this hangs together. But in section 1.5.1 (second-last link above) it is remarked that F-term equations can define an underconstrained system. I assume this has to do with the "F-term equivalence" appearing in brane tilings - that the formal equivalence of paths in the path algebra corresponds to something like a "gauge equivalence" of corresponding field configurations, and/or an equivalence of ordered field-operator products.

 

[suprised]

I guess some brief physical explanation may help. F-term equations are
nothing but the equations of motion of a superpotential in the theory;
the vanishing of these equations encodes the minimum of the potential
(and unbroken SUSY; a similar story holds for the D-terms, the main difference
being that F-terms are holomorphic, the D-terms not).

Thus the equations F_i=0 describe the ground states, or vacuum manifold,
or moduli space, of the theory (together with Di=0).

The relation to quivers and path algebras is as follows. The nodes
of the quiver corresponds to branes, and the links to open strings
mapping between them. Closed paths on a quiver essentally correspond
to possible terms in the superpotential; each link represents a
chiral superfield in the superpotential. Note that the ordering
of the fields is important, which reflects the non-commutative
nature of open-string interactions.

Thus, quivers together with F-term relations encode the vacuum
manifold, or moduli space, of certain gauge theories with
superpotentials, which arise from intersecting brane configurations.
This is a quite general mathematical construction, without an
intrinsic relation to melting crystals (which are related to a
special class of non-compact, toric brane geometries).

 

[mitchell porter]

Thanks for stepping in :-)

I was going to say: but what about quivers in AdS/CFT? But it finally dawned on me that there's no difference. The AdS side is some configuration of branes, and the boundary theory is the field theory on the branes, so quivers work in exactly the same way there, too.

 

[Monocles]

This helps a lot, thanks! I should have mentioned that I've almost completed my physics BS and have learned a little string theory from Zwiebach's book, so I did find these explanations very helpful.