Breaking supersymmetry with pure spinors
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Springer
Received: September 9, 2020 Accepted: October 15, 2020 Published: November 19, 2020
Andrea Legramandi and Alessandro Tomasiello Dipartimento di Fisica, Università di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy INFN — Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
E-mail: [email protected], [email protected] Abstract: For several classes of BPS vacua, we find a procedure to modify the PDEs that imply preserved supersymmetry and the equations of motion so that they still imply the latter but not the former. In each case we trace back this supersymmetry-breaking deformation to a distinct modification of the pure spinor equations that provide a geometrical interpretation of supersymmetry. We give some concrete examples: first we generalize the Imamura class of Mink6 solutions by removing a symmetry requirement, and then derive some local and global solutions both before and after breaking supersymmetry. Keywords: Flux compactifications, Superstring Vacua, Supersymmetry Breaking, Dbranes ArXiv ePrint: 1912.00001
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)098
JHEP11(2020)098
Breaking supersymmetry with pure spinors
Contents 1
2 Supersymmetry 2.1 Pure spinors 2.2 The Imamura class 2.2.1 Massive case 2.2.2 Massless case 2.3 A larger IIA system
4 4 6 7 8 8
3 Supersymmetry breaking 3.1 Breaking supersymmetry in the Imamura class 3.2 Larger IIA system 3.3 General R1,3 × S 2 system 3.4 Comparison with the conformal Calabi-Yau class
9 9 11 13 17
4 Examples 4.1 Separation by sum 4.2 Separation by product 4.3 Inverse hodograph transformation 4.4 Breaking supersymmetry
18 18 20 20 22
1
Introduction
If supersymmetry exists, it appears it is broken at high energy scales. In string theory it does play an important role, but nothing prevents it from being spontaneously broken at the Planck scale. Finding supersymmetric solutions, however, is still a lot easier than finding nonsupersymmetric ones. This is in part because the BPS equations provide a first-order system of partial differential equations (PDEs). Moreover, these equations often have compelling geometrical interpretations. These are revealed for example in the G-structure formalism (starting with [1, 2]), and in more complicated cases by generalized complex geometry methods [3]. Indeed the latter provide a system of “pure spinor equations” [3] that partially reduce finding the most general Minkowski or AdS supersymmetric vacuum solution to a geometrical problem. For Minkowski, for example, a condition that emerges is that the internal space be “generalized complex”, an umbrella concept that contains complex and symplectic manifolds. This is not enough to find a solution, but provides a convenient first step.
–1–
JHEP11(2020)098
1 Introduction
1 43 S + ∂z2 S 2 + c(c − 2∂z S) = 0 . 2
(1.2)
For c = 0 this reduces to (1.1). For c 6= 0, we get a deformation of the Imamura equation which still implies all the EoMs. While not all the
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