I generalize the deconstruction lattice formulation of Endres and Kaplan to two-dimensional super-QCD with eight supercharges, denoted by (4,4), and bifundamental matter. I specialize to a particularly interesting (4,4) gauge theory, with gauge group _{3}/CFT_{2} correspondence. The construction here preserves two supercharges exactly and has a lattice structure quite similar to that which has previously appeared in the deconstruction approach, that is, site, link, and diagonal fields with both the Bose and Fermi statistics. I remark on possible applications of the lattice theory that would test the AdS_{3}/CFT_{2} correspondence, particularly one that would exploit the recent worldsheet instanton analysis of Chen and Tong.

Supersymmetric large

Many promising lattice formulations of supersymmetric field theories occur in two dimensions (2D). (For an extensive list of references on lattice formulations of supersymmetric field theories, both old and new, see [

Broadly speaking, it is the softer ultraviolet (UV) divergences in 2D that generically make it easier to obtain the desired continuum limit in perturbation theory. Whether or not this property holds nonperturbatively is an open question, which at this point can only be answered empirically. In this regard, it is important to note that 2D field theories are more practical to study numerically; a small computer cluster can obtain reasonably accurate results. For some 2D examples, Monte Carlo simulation results have provided information on nonperturbative renormalization. For example, recent simulations of 2D supersymmetric theories that preserve a nilpotent subalgebra seem entirely consistent with continuum expectations [

A well-known example of the AdS/CFT correspondence occurs in the Type IIB superstring, at the intersection of D1 and D5 branes, with four of the directions of the D5 brane wrapped on, say, a torus

In this paper, the (2,2) supersymmetric formulation of Endres and Kaplan (EK) [

I will now summarize the remainder of this paper.

The purpose of this paper is to give a brief outline of the lattice construction and its potential applications. A more thorough discussion of details associated with the lattice system (superspace description, renormalization, etc.), as well as intensive studies of the possible applications mentioned in Section

The 2D theory is most easily obtained from a dimensional reduction of the 4d theory written in

For the gauge multiplet, we have

The hypermultiplet is written in terms of two chiral multiplets, denoted by

The superpotential is the minimal one, which preserves

The action is given by a Grassmann integral over superspace coordinates

The EK approach [

In the present theory, we begin with a matrix model that is the zero-dimensional (0d) reduction (the 0d reduction is obtained by treating all fields as independent of space-time coordinates) of 4d

The next step is to perform an orbifold projection on the mother theory, in order to reduce it to the

It is at this point that the trick to get a weakly gauged

An alternative picture of this trick is the following. We may regard the factor

From this perspective we see that it is necessary to keep the third dimension small so that we never see the effects of the KK states. That is, we want only the

In the discussion of Section

In

1 | 0 | 0 | −1/2 | +1/2 | −1/2 | +1/2 | +1/2 | −1/2 | +1/2 | −1/2 | 0 | 0 | |

0 | 1 | 0 | +1/2 | −1/2 | −1/2 | +1/2 | −1/2 | +1/2 | +1/2 | −1/2 | 0 | 0 | |

0 | 0 | 1 | −1/2 | −1/2 | +1/2 | +1/2 | −1/2 | −1/2 | +1/2 | +1/2 | 0 | 0 | |

0 | 0 | 0 | −1/2 | −1/2 | −1/2 | −1/2 | +1/2 | +1/2 | +1/2 | +1/2 | −1 | 0 | |

1 | 0 | −1 | 0 | 1 | −1 | 0 | 1 | 0 | 0 | −1 | 0 | 0 | |

0 | 1 | −1 | 1 | 0 | −1 | 0 | 0 | 1 | 0 | −1 | 0 | 0 |

0 | 0 | −1/2 | +1/2 | −1/2 | +1/2 | +1/2 | −1/2 | +1/2 | −1/2 | 0 | 0 | |

0 | 0 | +1/2 | −1/2 | −1/2 | +1/2 | −1/2 | +1/2 | +1/2 | −1/2 | 0 | 0 | |

0 | 0 | +1/2 | +1/2 | −1/2 | −1/2 | +1/2 | +1/2 | −1/2 | −1/2 | 1 | 1 | |

+1/2 | +1/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −1/2 | −1/2 | |

0 | 0 | −1 | 0 | 0 | 1 | 0 | −1 | 1 | 0 | −1 | −1 | |

0 | 0 | 0 | −1 | 0 | 1 | −1 | 0 | 1 | 0 | −1 | −1 |

Denote

Introduce “clock operators” that involve roots of unity

To understand the effect of (

The minimal coupling superpotential of the mother theory has the

The

A problem arises for the construction of CKKU [

In the modification, we want to leave the

Relative to the formulation of CKKU, only the following five fields of the gauge multiplet change their nature:

Referring to Table

Note also that the

Having explained how the gauge action is modified, we next turn to the matter action. The daughter theory is obtained in a simple application of the orbifold procedure to the mother theory (

We have already seen from the discussion of the daughter theory gauge action that there are two supercharges that are neutral with respect to the

Straightforward manipulations yield the daughter theory matter action. One merely writes out the fermion components in (

To “Higgs” the theory, such that only the

In terms of

One then gives an expectation value to the

The condition that the KK states decouple from the

A less aggressive prescription is to take

Here I mention one possible application of the lattice theory. Recently, Chen and Tong have studied the D1/D5 effective worldsheet instanton partition function on the Higgs branch. In the gauge theory one looks at the distribution of instanton size

In a numerical study of this phenomenon, one would build up a histogram in the

It would also be interesting to explore the correspondence at finite temperature, since continuum methods start to break down if the temperature is too far from zero. The recent results of Rey and Hikida for small 't Hooft coupling and finite temperature [

In this paper I have generalized the EK construction to 2D (4,4) gauge theories. I have specialized to a

Work in progress includes a careful study of renormalization in the lattice theory, the number of counterterms that need to be fine-tuned, their exact calculation in perturbation theory (the lattice theory is super-renormalizable since the coupling has positive mass dimension), and a numerical study of the correspondence. Renormalization of the theory, such as has been studied in [

Finally, it is of some interest to work out a superfield description of the daughter theory in this model. This would be useful in a super-Feynman diagram perturbative analysis, as well as for understanding the renormalizations to the tree-level action that are possible.

The conditions that we will impose are the following.

The link bosons

The fermion component

At least one other fermion component in

All fields should have integral values of

It is completely general to write

The action can be expressed as three terms,

This work was supported by the U.S. Department of Energy under Grants No. DE-FG02-94ER-40823 and DE-FG02-08ER-41575.

_{3}and BTZ black hole from weakly interacting hot 2d CFT