Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 126, 48 pages      arXiv:1904.07578      https://doi.org/10.3842/SIGMA.2020.126

Small Gauge Transformations and Universal Geometry in Heterotic Theories

Jock McOrist ^a and Roberto Sisca ^b
a) Department of Mathematics, School of Science and Technology, University of New England, Armidale, 2351, Australia
^b) Department of Mathematics, University of Surrey, UK

Received July 30, 2020, in final form November 04, 2020; Published online December 02, 2020; Misprints fixed December 31, 2020

Abstract
The first part of this paper describes in detail the action of small gauge transformations in heterotic supergravity. We show a convenient gauge fixing is 'holomorphic gauge' together with a condition on the holomorphic top form. This gauge fixing, combined with supersymmetry and the Bianchi identity, allows us to determine a set of non-linear PDEs for the terms in the Hodge decomposition. Although solving these in general is highly non-trivial, we give a prescription for their solution perturbatively in $\alpha^{\backprime}$ and apply this to the moduli space metric. The second part of this paper relates small gauge transformations to a choice of connection on the moduli space. We show holomorphic gauge is related to a choice of holomorphic structure and Lee form on a 'universal bundle'. Connections on the moduli space have field strengths that appear in the second order deformation theory and we point out it is generically the case that higher order deformations do not commute.

Key words: string theory; moduli spaces; differential geometry.

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References

1. Anderson L.B., Gray J., Sharpe E., Algebroids, heterotic moduli spaces and the Strominger system, J. High Energy Phys. 2014 (20104), no. 7, 037, 40 pages, arXiv:1402.1532.
2. Anguelova L., Quigley C., Sethi S., The leading quantum corrections to stringy Kähler potentials, J. High Energy Phys. 2010 (2010), no. 10, 065, 27 pages, arXiv:1007.4793.
3. Ashmore A., de la Ossa X., Minasian R., Strickland-Constable C., Svanes E.E., Finite deformations from a heterotic superpotential: holomorphic Chern-Simons and an $L_\infty$ algebra, J. High Energy Phys. 2018 (2018), no. 10, 179, 59 pages, arXiv:1806.08367.
4. Candelas P., de la Ossa X., McOrist J., A metric for heterotic moduli, Comm. Math. Phys. 356 (2017), 567-612, arXiv:1605.05256.
5. Candelas P., de la Ossa X., McOrist J., Sisca R., The universal geometry of heterotic vacua, J. High Energy Phys. 2019 (2019), no. 2, 038, 46 pages, arXiv:1810.00879.
6. Candelas P., de la Ossa X.C., Moduli space of Calabi-Yau manifolds, Nuclear Phys. B 355 (1991), 455-481.
7. de la Ossa X., Svanes E.E., Connections, field redefinitions and heterotic supergravity, J. High Energy Phys. 2014 (2014), no. 12, 008, 27 pages, arXiv:1409.3347.
8. de la Ossa X., Svanes E.E., Holomorphic bundles and the moduli space of $N=1$ supersymmetric heterotic compactifications, J. High Energy Phys. 2014 (2014), no. 10, 123, 55 pages, arXiv:1402.1725.
9. Donagi R., Guffin J., Katz S., Sharpe E., Physical aspects of quantum sheaf cohomology for deformations of tangent bundles of toric varieties, Adv. Theor. Math. Phys. 17 (2013), 1255-1301, arXiv:1110.3752.
10. Donagi R., Guffin J., Katz S., Sharpe E., A mathematical theory of quantum sheaf cohomology, Asian J. Math. 18 (2014), 387-417, arXiv:1110.3751.
11. Garcia-Fernandez M., Rubio R., Tipler C., Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry, Math. Ann. 369 (2017), 539-595, arXiv:1503.07562.
12. Garcia-Fernandez M., Rubio R., Tipler C., Holomorphic string algebroids, Trans. Amer. Math. Soc. 373 (2020), 7347-7382, arXiv:1807.10329.
13. Gillard J., Papadopoulos G., Tsimpis D., Anomaly, fluxes and $(2,0)$ heterotic-string compactifications, J. High Energy Phys. 2003 (2003), no. 6, 035, 25 pages, arXiv:hep-th/0304126.
14. Itoh M., Geometry of anti-self-dual connections and Kuranishi map, J. Math. Soc. Japan 40 (1988), 9-33.
15. Ivanov S., Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685 (2010), 190-196, arXiv:0908.292.
16. Kobayashi S., Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, Vol. 15, Princeton University Press, Princeton, NJ, 1987.
17. Kodaira K., Complex manifolds and deformation of complex structures, Classics in Mathematics, Springer-Verlag, Berlin, 2005.
18. McOrist J., The revival of $(0,2)$ sigma models, Internat. J. Modern Phys. A 26 (2011), 1-41, arXiv:1010.4667.
19. McOrist J., On the effective field theory of heterotic vacua, Lett. Math. Phys. 108 (2018), 1031-1081, arXiv:1606.05221.
20. McOrist J., Melnikov I.V., Half-twisted correlators from the Coulomb branch, J. High Energy Phys. 2008 (2008), no. 4, 071, 19 pages, arXiv:0712.3272.
21. McOrist J., Melnikov I.V., Summing the instantons in half-twisted linear sigma models, J. High Energy Phys. 2009 (2009), no. 2, 026, 61 pages, arXiv:0810.0012.
22. McOrist J., Melnikov I.V., Old issues and linear sigma models, Adv. Theor. Math. Phys. 16 (2012), 251-288, arXiv:1103.1322.
23. Melnikov I.V., An introduction to two-dimensional quantum field theory with $(0,2)$ supersymmetry, Lecture Notes in Physics, Vol. 951, Springer, Cham, 2019.
24. Melnikov I.V., Sethi S., Sharpe E., Recent developments in $(0,2)$ mirror symmetry, SIGMA 8 (2012), 068, 28 pages, arXiv:1209.1134.
25. Melnikov I.V., Sharpe E., On marginal deformations of $(0,2)$ non-linear sigma models, Phys. Lett. B 705 (2011), 529-534, arXiv:1110.1886.
26. Nakahara M., Geometry, topology and physics, 2nd ed., Graduate Student Series in Physics, Institute of Physics, Bristol, 2003.
27. Strominger A., Special geometry, Comm. Math. Phys. 133 (1990), 163-180.
28. Witten E., New issues in manifolds of ${\rm SU}(3)$ holonomy, Nuclear Phys. B 268 (1986), 79-112.
29. Witten L., Witten E., Large radius expansion of superstring compactifications, Nuclear Phys. B 281 (1987), 109-126.