Closed String CFT on Orbifolds

SUSY Gauge Theories

4.3 Closed String CFT on Orbifolds

An orbifold is a quotient of a smooth manifold M by a discrete isometry group

r.

The quotient is taken in the usual sense of identifying the points on the orbits of the f action, a point x E M being identified with gx E M. If the

r

action has fixed sets, the quotient space will have singularities. The reason is as follows. The action of

r

on M lifts to an action of the tangent bundle of M, denoted by TM. It does so in such a way that

for every x in M and every gin

r.

If, however, x is fixed under g, so that x = gx, then g maps Tx to itself. This map must be a rotation of the vectors in Tx, since

r

acts as an isometry and so its action must be norm-preserving. Because

r

is discrete, the quotient singularities produced by the identifications are conical, deficit angle singularities.

The singular set of M /

r

is a union of spaces S ^g = { x lgx = x} fixed by elements g E

r

^*4• Another fact that will be important to us is that the quotient space is Calabi-Yau. One will recall that M is Calabi-Yau manifold if and only if it is a Kahler manifold with a unique nowhere vanishing holomorphic d-form Dd,o_ Then, M/f is Calabi-Yau as well provided the action of

r

preserves D.

4.3.1 Generalities of "Ordinary" Orbifolds

Field theory on M/f is simply defined in terms of truncation of the theory on M to

r

invariant states. In string theory however quotient theory will have states which do not come from M. Since open and closed string theories behave quite differently in this respect, (although, as we will see shortly the difference is only at a superficial level), we will concentrate on the closed string theory first.

In closed string theory the new states go under the name "twisted" string states.

The name derives from the fact that those states come from quantization of strings

*•It is not necessary that M be smooth, but in this section we will assume so, since we consider quotient singularities only.

which are closed on M only up to the

r

action, and thus are created by fields with twisted boundary conditions

8(0"

+

^{27r) =go} 8(0"), g E f.

Classically, the massless string states are constant configurations on the world-sheet so the above equation implies that twisted strings of zero mass must propagate only on the fixed set 5₉.

Now, the problem of computing the massless spectrum of superstring theory on smooth, compact complex manifolds has a well-known solution - the Ramond- Ramond ground states are determined by topological data only, their number is given by the dimension of H*(X). Since 5₉ is smooth for every g in

r ,

the twisted string states are in one to one correspondence with the generators of H* ( 5₉^{) *5•} More precisely, the correspondence is given by the following formula

The shift in the Hodge numbers comes about because the assignment of the cohomol- ogy groups Hp,q to (R, R) states comes via the (p, q) charge of the Ramond- Ramond fields acting on the vacuum under the left and the right-moving U(l) current on the world sheet. In the orbifold, however, the vacuum itself carries non-zero U(l) charge which is computed in [47] (in fact most of the introduction follows this paper), with the result that g : z⁰ -+ e^27rieo z^{0 ,} where 0 :::; () ₀

<

1, then

( 4.1)

It should be clear that the twisted states too must be projected to those that are

r

invariant, so

*sOne can in fact show that 5₉ is a Kahler submanifold (basically the Kahler form on 5₉ is given by the pullback of the Kahler form on on M, and such is preserved by action of r ^[47]).

4.3.2 A Z

₂

x Z

₂

Example

The simplest Calabi-Yau threefold which has an orbifold singularity is a space which is (locally)

C3

/'ll2 x 'll2. If we take Zen a = 1, 2, 3 to be the choice of complex coordinates on

C3

than

r

⁼ 'll2 x 'll2 acts by:

The fixed set of

r

consists of Sg

=

(z1' 0, 0), sh

=

(0, Z2, 0), and Sgh

=

(0, 0, z3). These are three curves of singularities, in the neighborhood of each of which M

/f

looks like <C x <C² /'ll 2, and which intersect over a point (0, 0, 0). The complex manifold

C3

has a unique holomorphic three form, D^3•0 = dz1 /\ dz2 /\ dz3 which survives the quotient since

r

flips the sign of coordinates pairwise. Now let us consider the string spectrum *^6• The fixed set of g is just a copy of S₉ = <C parametrized by z1 . The total cohomology of <C is generated by

which belong to H⁰•0 , H¹•0

, H⁰•1 and H¹•1 respectively. Since h : z1 ---+ - z1 , only 1 and dz 1 /\ dz1 are invariant under

r,

so the contribution to the stringy cohomology of the orbifold of g-twisted states is h^1•1 = 1 = h²•2

, and zero otherwise. There are two more elements similar to g, so we find that

hl,l

=

3, h2,l

=

0,

on the orbifold in string theory. This is a remarkable result. The point is that

*⁵ For our methods, as outlined above, we really need to consider compact spaces. It suffices to think about C3 /Z2 x '1!,,₂ as a piece of a compact manifold, T⁶ /'1!,,₂ x '1!,,₂.This space has 64 fixed points, the neighborhood of each of which looks like our space. Alternatively, one can consider compactly supported cohomology on 5₉ in order to obtain normalizable ground states. We will be loose about this point.

defining cohomology of a singular space is ordinarily fairly hard, and to get something reasonable, one has to do so using "simplicial" rather than de-Rham cohomology.

What we find here is that string theory anticipates the cohomology of the resolved orbifold since the Hodge numbers the CFT computes correspond precisely to what one would have obtained by blowing up the orbifold. This seems to be a generic behavior of string theory on orbifolds.