M-theory and orientifolds
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Springer
Received: February 11, Revised: July 20, Accepted: August 10, Published: September 8,
2020 2020 2020 2020
Andreas P. Braun Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Rd, Durham DH1 3LE, U.K.
E-mail: [email protected] Abstract: We construct the M-Theory lifts of type IIA orientifolds based on K3-fibred Calabi-Yau threefolds with compatible involutions. Such orientifolds are shown to lift to M-Theory on twisted connected sum G2 manifolds. Beautifully, the two building blocks forming the G2 manifold correspond to the open and closed string sectors. As an application, we show how to use such lifts to explicitly study open string moduli. Finally, we use our analysis to construct examples of G2 manifolds with different inequivalent TCS realizations. Keywords: String Duality, Differential and Algebraic Geometry, M-Theory, F-Theory ArXiv ePrint: 1912.06072
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)065
JHEP09(2020)065
M-theory and orientifolds
Contents 1 Introduction
1 3 3 6 8 10 13 13 14 16
3 Open string moduli 3.1 Deforming X+ 3.2 Deformations of X+ as D6 moduli 3.3 An example
17 18 18 20
4 G2 manifolds with multiple TCS decompositions 4.1 Example 1 4.2 Example 2
22 22 24
5 Discussion and future directions
25
A TCS G2 manifolds
27
B Acyl Calabi-Yau manifolds, tops, and anti-holomorphic involutions
29
C Nikulin involutions and Voisin-Borcea threefolds
31
1
Introduction
The lift of type IIB orientifolds with O7− -planes to F-Theory [1, 2] is by now a classic result.1 For a Calabi-Yau threefold X and a holomorphic involution fixing a divisor on X, there is an associated IIB orientifold with locally cancelled Ramond-Ramond seven-brane charge that is lifted to F-Theory on the Calabi-Yau orbifold Y = X × T 2 /Z2 . (1.1) Here, the complex structure of the torus encodes the value of the IIB axiodilation modulo SL(2, Z). Deformations of Y can be studied using standard techniques from algebraic 1
See [3, 4] for the F-Theory lift of O7+ -planes.
–1–
JHEP09(2020)065
2 Lifting IIA orientifolds to TCS G2 manifolds 2.1 Review of IIA orientifolds and their M-theory lifts 2.2 IIA orientifolds which lift to TCS G2 manifolds 2.3 The TCS G2 lift of IIA orientifolds 2.3.1 Resolution of TCS and match of degrees of freedom 2.4 Example 2.4.1 The type IIA model 2.4.2 The TCS M-theory lift 2.5 The weak coupling limit of M-theory on TCS G2 manifolds
M = X × S 1 / Z2 . Again, deformations correspond to displacements of D-branes, but it is much harder to give a general description (see [18] for the state of the art) and map it to open and closed string moduli. While this question and the relation to super-Yang-Mills theory has been studied intensively in non-compact setups [19–29], a concise global description at the level of detail available in F-Theory is still missing. We make progress by showing how to map IIA orientifolds based on K3-fibred CalabiYau manifolds with compatible anti-holomorphic involutions to
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