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Franco, Sebastian and Hanany, Amihay and Kre, Daniel and Park, Jaemo and Uranga, Angel and Vegh,
David (2007) 'Dimers and orientifolds.', Journal of high energy physics., 2007 (09). 075.
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http://dx.doi.org/10.1088/1126-6708/2007/09/075
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arXiv:0707.0298v2 [hep-th] 30 Oct 2007
Preprint typeset in JHEP style - HYPER VERSION
CERN-PH-TH/2007-099
IFT-UAM/CSIC-07-34
LMU-ASC 41/07
MIT-CTP 3846
MPP-2007-76
PUPT-2238
Dimers and Orientifolds
Sebasti´an Franco
1
, Amihay Hanany
2
, Daniel Krefl
3,4
, Jaemo Park
5,6
,
Angel M. Uranga
7,8
and David Vegh
9
1
Joseph Henry Laboratories, Princeton University
Princeton NJ 08544, USA
2
Perimeter Institute for Theoretical Physics
31 Caroline Street North, Waterloo Ontario N2L 2Y5 Canada
3
Max-Planck-Institut f¨ur Physik
F¨ohringer Ring 6, 80805 Munich, Germany
4
Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians-Universit¨at, Theresienstr. 37, 80333 Munich, Germany
5
Department of Physics, Postech, Pohang 790-784 Korea
6
Postech Center for Theoretical Physics (PCTP), Pohang 790-784, Korea
7
PH-TH Division, CERN
CH-1211 Geneva, Switzerland
8
On leave from Instituto de F´ısica Te´orica, Facultad de Ciencias, Madrid, Spain
9
Center for Theoretical Physics, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge MA 02139, USA
Abstract: We introduce new techniques based on brane tilings to investigate D3-
branes probing orientifolds of toric Calabi-Yau singularities. With these new tools,
one can write down many orientifold models and derive the resulting low-energy
gauge theories living on the D-branes. Using the set of ideas in this pa per one recovers
essent ia lly all orientifolded theories known so far. Furthermore, new o rientifolds of
non-orbifold toric singularities are obta ined. The possible applications of the tools
presented in this paper are diverse. One particular application is the construction of
models which feature dynamical supersymmetry breaking as well as the computation
of D -instanton induced superpo t ential terms.
Contents
1. Introduction 2
2. Some background on dimers and on orientifolds 3
2.1 Quiver gauge theories and dimer diagra ms 3
2.2 Orientifolds 5
2.3 Orientifolding dimers 6
3. Orientifolds from dimers with fixed points 8
3.1 Orientifold rules 8
3.1.1 Example: Orientifolds of C
3
10
3.1.2 A comment on tadpoles/anomalies and extra flavors 11
3.2 Geometric action 12
3.2.1 On the orientifolds of C
3
13
3.2.2 Example: Orientifolds of the conifold 15
3.2.3 Example: C
2
/Z
2
× C 17
3.3 The global sign rule and Higgsing 20
3.4 The global constraint for the fixed point signs revisited 21
3.5 Further examples 22
3.5.1 Orientifolds of C
3
/Z
3
22
3.5.2 Orientifolds of the Conifold/Z
N
23
3.5.3 Orientifolds of SPP 24
3.5.4 Orientifolds of L
1,5,2
25
4. Orientifolds from dimers with fixed lines 26
4.1 Generalities 26
4.2 Few examples and the geometric action 27
4.2.1 Line orientifolds of C
3
27
4.2.2 C
2
/Z
N
× C, even N 28
4.2.3 C
2
/Z
N
× C, odd N 30
4.2.4 The geometric action 30
4.3 Further examples 31
4.3.1 General L
aba
theories 31
4.3.2 Orbifolds of the conifold 33
– 1 –
5. The mirror perspective 35
5.1 Review of the mirror picture 35
5.2 Orientifolds in the mirror system 40
5.3 Classes of orientifold involutions 43
5.3.1 Involutions mirror to orientifold dimers with fixed points 43
5.3.2 Involutions mirror to orientifold dimers with fixed lines 44
5.4 Tadpole cancellation 47
5.5 Calibration 49
6. Applications 49
6.1 Dynamical supersymmetry breaking 49
6.2 D-brane instantons 53
7. Conclusions 54
A. From quivers to dimers and shivers 55
B. Mnemonics: Orientifolded Harlequin diagrams 59
C. L
aba
theories 62
1. Introduction
The study of D-branes at singularities and the gauge theories on them is interesting
for a variety of reasons. On the one hand, branes at singularities give rise to interest-
ing extensions of the original AdS/CFT correspondence to theories with a reduced
amount of (super)symmetry [1, 2]. This front has witnessed remarkable progress in
recent years: in the conformal case, with the precision matching of geometric prop-
erties of new infinite families of Sasaki-Einstein metrics and their gauge theory duals
[3, 4, 5, 6, 7]; in the presence o f fractional branes, with t he dictionary between geo-
metric properties of the singularity and strong infrared dynamics in the dual gauge
theories [8, 9, 10, 11, 12].
On the other hand, they provide a natura l setup for a bottom-up approach to
string phenomenology, a llowing for local constructions of Standard Model-like gauge
theories [13, 14, 15]. Many features of the resulting models depend only o n the local
structure of the singularity and can be investigated without a detailed knowledge of
the full compactification manifold.
Orientifolds are an int eresting new twist in these constructions, with possibly
novel features. To name a few, they can be used to produce interesting spectra
– 2 –
(with new kinds o f gauge factors and representations), they naturally lead to non-
conformal theories (with orientifold charges arising as 1/N corrections), easily lead
to supersymmetry breaking in the infrared (with or without runaway), and lead to
models with interesting non-perturbative superpotential interactions (since orien-
tifolding eliminates superfluous zero modes on certain D-brane instantons). Thus,
the construction of orientifolds of D-branes at singularities is an interesting direction
worth b eing pursued.
Unfortunately, the techniques to construct such orientifolds are very rudimentary.
So far, only a very limited number o f orientifolds of non-orbifold singularities has been
constructed. Orientifolds of orbifold singularities are in principle amenable to direct
construction using worldsheet techniques, although in practice only a few families of
models have been constructed. For orientifolds of non-orbifold singularities, partial
resolution of orientifolds of orbifolds can be used to derive a few new models, but
the a pproa ch becomes non-practical for singularities beyond the simplest ones. A
simple classification o f orientifolds is also possible for the very restricted subset of
singularities that are T-dual to simple Hanany-Witten (HW) setups [16].
The problem of finding the gauge theory on a set of branes probing an arbitrary
toric CY singularity M was fully solved with t he introduction of dimer model meth-
ods [17, 18].
1
One of the main virtues of dimers is its computational simplicity, in
sharp contrast with pre-existent a lternatives such as partial resolution.
Given the striking success of dimer models in the study of branes at singularities,
it is natural to ask how to expand their range of applicability. A natura l problem is
the classification of orientifolds of toric singularities. This paper extends the success
of dimer models in the study of toric singularities to their orientifolds, providing a
general method and explicit construction of them.
The organization of the paper is as follows. In §2, we review some basics of
dimers, quivers and orientifolds. The main results are presented in §3 and §4, where
we explain how to obtain orientifolds of arbitrary toric singularities corresponding to
involutions with fixed points and fixed lines, respectively. The mirror perspective is
discussed in §5. In §6 we present some applications of our framewo r k and conclude
in §7. We collect additional related material in appendices.
2. Some b ackground on dimers and on orientifolds
2.1 Quiver gauge theories and dimer diagrams
In this section we give several aspects of quiver gauge theories that live on D3-
branes at toric Calabi-Yau singularities and their construction in terms of dimer
diagrams (a.k.a. brane tilings). Brane t ilings give the most efficient construction
1
For a recent review, see [19].
– 3 –
