Open Strings on Orbifolds

SUSY Gauge Theories

4.6 Open Strings on Orbifolds

In string theory there is a complementary way to study singular spaces, and that is by using D-brane probes. In the language of D-branes, the spacetime itself arises indirectly, as the moduli space of the gauge theory living on the D-brane world volume.

To describe a Calabi-Yau threefold, it is necessary to use D p-brane probes with p :::; 3.

For concretness we will henceforth set p

=

3, the other cases being related to it via dimensional reduction. The world volume theory must have

N

⁼ 1 supersymmetry and the singularities in the moduli spaces are resolved by turning on Flayet-Iliopoulos parameters or by deformations of the superpotential.

For a general X there is no known prescription of how to determine precisely the

world volume gauge theory, with the only requirement really being that the moduli space of a single D-brane on X should in fact be X itself. With this requirement alone we can say very few things. First, N D branes at a smooth point in the Calabi-Yau space will neccesarily be described, at energies E

« :, , R'

^{by U(N)} gauge theory on the world volume, with effective

N

⁼ 4 supersymmetry. When the curvatures are large and substringy there must exist an effective gauge theory description of the

"compactification" manifold, as long as the probe itself is small enough to be a "good"

probe of geometry. Beyond this one must aproach the problem on a case by case basis.

Exceptions to this are orbifolds X ~

C3

/f*^7, where a simplification occurs because the theory on

C3 /f

^is a quotient of the theory on

03*

^8• Studies of CC² /Zm, and

C3

/Zm

showed that in both cases. the stringy constructions provide a physical realization of such concepts as Hyper-Kahler quotients and symplectic quotient constructions respectively. It is then interesting to ask if the same phenomena we have found in the closed string theory on

C3

/Zm x Zn will persist in the open string theory as well.

First, let us briefly review the general construction of D-branes probing orbifolds.

Throughout we will mostly keep the discussion at the level of low-every effective field theory on the world volume. The theory of D 3-branes on

C3 /f

is defined as a truncation of the theory on

C3,

where only

r

invariant configurations are kept in the quotient.

What does

r

invariance mean? Forgetting for the moment the non-Abelian nature of the theory (or more properly, thinking about open string CFT with boundary conditions), we can associate a Chan-Paton factor i to a D-brane at z(i) E

C3,

then the

r

action on

C3

translates into

go z(i)

=

z'(J(g)i), Vg E f.

*7ln general c3 can be replaced with some other Ricci-fiat three-dimensional manifold M admit- ting a symmetry r which is useful if the probe theory on M is known. Recently, this was done for the case when M is a conifold, [49].

*s It is true in fact that the knowledge of the c3

/r

theory allows one to describe all other sin- gualrities which are toric[50]. The unsolved problem is what to do for Calabi-Yau hypersurface singularities.

A

r

invariant configuration of D branes must then consist of orbits of

r

action on

<C3,

a generic orbit in this case consisting of

If I

points on

<C3.

Thus, to describe N D 3-branes on

<C3

/f we must start with a d

=

4,

N =

4 supersymmetric U(Nlfl) gauge theory, as an effective open string theory of Nlfl branes on the covering space. The action of

r

on the open string CFT induces the action on the effective bosonic degrees of freedom of the form

g A ---+ 1(g)A1(g)-1,

zi ---+ i(g) (go zi) i(gt1,

( 4.4)

(4.5)

where zi are the scalar fields whose diagonal pieces parametrize the position of the D-branes on the covering space, and 1(f) is an embedding of the orbifold group

/: f---+ U(Nlfl).

At the end of the day, if

r

is a subgroup of SU(3)*⁹ as in the case we are interested in, the quotient theory will have N

=

1 supersymmetry in d

=

4.

Let's now take

r

⁼ Zm x Zn, so that

r

^is generated by two elements g, and h, satisfying gm = 1, hn = 1, gh = hg.

<C3

/f is then a quotient of

<C3

by

g : (z,z^{1 2} ,z)-+(emz^{3 2rri 1} ,z,e mz) ^{2 - 2rri 3}

h:

(z,z,z)-+(z,enz,e nz). ^{1 2 3 1 2rri 2 - 2rri 3}

Orbits of

r

are generated by g, h, so a D-brane at a generic point in

<C3

/Zm x Zn must have mn preimages on

<C3.

It is convenient to label D-branes with a biindex (i, a) where i, a naturally label points on the orbits generated by g, h respectively, so that i E {O, ... ,m-1} and a E {O ... ,n-1} *¹⁰•

*⁹f is the holonomy group of the orbifold. If r E SU(3) upon resolution of singularities in the orbifold the holonomy becomes (at most) SU(3).

*¹⁰There are also N-valued indices labeling distinct physical branes, but since we will consider here only the generic orbits of r, whatever goes through for a single brane holds for any number of them.

We will thus set N

=

1 for clarity of the text.

The orbifold fixes the action of

r

on the single open string states, which is inherited from

r

action on

C3,

but we must still pick an action on the Chan-Paton factors.

There is a natural "geometric" choice of action which can be described as follows.

Since a generic orbit of

r

is just a copy of f*^11, action of

r

on its orbit is an action of

r

on itself which gives, by definition, the regular representation, 1(g )icx,jf3 = gioi,joa,f3, and 1(h)icx,jf3 = oi,jhcxocx,f3· With this definition of the

r

projection, N D branes on the orbifold are described by a quiver

It,cx

U(N)i,cx gauge theory with chiral matter in bifundamental representations, and a superpotential which is a reduction of the

N

⁼ 4 superpotential W = Tr Z¹ [ Z², Z³].