In our recent paper, we systematized an inverse algorithm to obtain quiver gauge theory living on the

Initial works of Bagger-Lambert [

Martelli and Sparks [

An elegant combinatorial approach called dimer tilings [

One of the challenging problems was to determine quiver gauge theories corresponding to 18 toric Fano threefold. A Fano variety in

From the forward algorithm and dimer tiling method, quiver gauge theories corresponding to fourteen of the toric Fano

The next immediate question is to understand the embeddings inside the toric Fano

Alternatively, we could determine embeddings from higgsing approach [

Another approach of higgsing called the algebraic method [

For the

The plan of the paper is as follows: in Section

Our aim is to study partial resolution of toric data corresponding to 3-node parent quiver resulting in a toric data corresponding to a

(1) Theory with 4 bifundamental matter fields

Quiver diagram (a).

(2) Theory with 2 adjoints

Quiver diagram (b).

It is pertinent to spell out the following obvious facts.

We can deduce that the two quiver theories are distinct for

When there is no orbifolding—that is,

(3) Theory with 2 bifundamentals

Quiver diagram (c).

However, we cannot determine the CS levels from the inverse algorithm [

A possible choice of toric data for the theory shown in Figure

The charge matrix

We would like to see these Calabi-Yau 4-fold, particularly Fano

The quiver corresponding to the complex cone over Fano

Quiver diagram for Fano

The projected charge matrix (

The toric data of this theory is [

Suppose that we rescale the levels of this theory as

Equivalently, the toric data

From the tiling approach, higgsing of the quiver diagram in Figure

We attempt the higgsing of Fano

As an example, let us take the

By giving VEV to any other matter fields, we have checked that we get the same trivial

In partial resolution, we try to remove the points from the toric diagram. The resulting toric diagram corresponds to some daughter theory which is embedded in the parent theory. From the toric diagram of

Take the toric data

This reduced toric data (

Alternatively, we can take the charge matrix

We have obtained

The quiver gauge theories corresponding to Fano

It is a theory with 3 nodes, 2 adjoint fields

Quiver diagram for Fano

If we give a VEV to any of the

Here, we do the partial resolution of Fano

It is interesting to see that the method of partial resolution does embed the toric Fano

If we remove points

In the following section, we will briefly present the quiver corresponding to Fano

The quiver Chern-Simons theory corresponding to Fano

Cyclic quiver for Fano

Linear quiver for Fano

In this case, we found that then if we remove the set of points

If we remove the points

Taking a row (

This theory was studied in [

Quiver diagram for Fano

The algebraic higgsing in this case gives

Our main motivation was to determine the

For the quiver corresponding to Fano

The

Algebraic higgsing and unhiggsing of quiver theories corresponding to some Fano 3-fold have been studied recently [

It is not obvious whether we can obtain other Fano

We had chosen a charge matrix

The authors would like to thank A. Hanany for discussions. They are grateful to T. Sarkar and Rak-Kyeong for their valuable inputs on the orbifolding issues.