We calculate couplings of arbitrary order from correlation functions among twisted strings, using conformal field theory. Twisted strings arise in heterotic string compactified on orbifolds yielding matter fields in the low energy limit. We calculate completely the classical and the quantum amplitude including normalization, up to a contribution from Kähler potential. The classical action has saddle points which are interpreted as worldsheet instantons described by metastable untwisted strings, formed by twisted strings distributed at certain fixed points. This understanding generalizes the area rule, in the case that the locations of twisted strings do not form a polygon, and provides a general rule for calculating these kinds of instanton corrections. An interpretation of couplings involving linearly combined states is given, which commonly appear in non-prime order orbifolds. The quantum part of the amplitude is given by ratios of gamma functions with order one arguments.
Bibliographical noteFunding Information:
We are grateful to Steve Abel, Massimo Bianchi, Jihn E. Kim, Hans-Peter Nilles, Saul Ramos-Sanchez, Michael Ratz, Robert Richter and Patrick K.S. Vaudrevange for discussion. K.S.C. thanks to Yukawa Institute for hospitality, where part of the work is done. K.S.C. is supported in part by the European Union 6th framework program MRTN-CT-2004-503069 “Quest for unification”, MRTN-CT-2004-005104 “ForcesUniverse”, MRTN-CT-2006-035863 “UniverseNet” and SFB-Transregio 33 “The Dark Univerese” by Deutsche Forschungsgemeinschaft (DFG). T.K. is supported in part by the Grand-in-Aid for Scientific Research #1754025 and the Grant-in-Aid for the 21st Century COE “The Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
- Heterotic string
- Higher order coupling
- Orbifold compactification
- Yukawa coupling