nLab conifold transition

Idea

In 1987, Miles Reid conjectured that a combination of contracting (resp. introducing) some isolated curves and smoothing (resp. singularizing) deformation connects all Calabi-Yau 3-folds to each other, as well as to a (non-Kähler, $b_2=0$) connected sum of copies of $S^3\times S^3$, called a rakshasa, thus connecting their individual moduli spaces into an irreducible object: The moduli space of 3-folds with $K = 0$ may nevertheless be irreducible, Math. Ann. 278 no. 1-4, (1987) 329–334 (doi:10.1007/BF01458074). This implies a corresponding connection among all string compactifications using them.

• This process was shown to connect all Calabi-Yau 3-folds realized as:

  • complete intersections in products of projective spaces: P. S. Green and T. Hübsch, Connecting moduli spaces of Calabi–Yau threefolds, Comm. Math. Phys. 119 (1988) 431–441. (doi:10.1007/BF01218081)
  • hypersurfaces in toric varieties: A. C. Avram, M. Kreuzer, M. Mandelberg, and H. Skarke, The web of Calabi–Yau hypersurfaces in toric varieties, Nucl. Phys. B505 (1997) 625–640 (doi:10.1016/S0550-3213(97)00582-8), (arXiv:hep-th/9703003)
  • at least some global finite quotients of the above: R. Davies, Hyperconifold transitions, mirror symmetry, and string theory, Nucl. Phys. B850 (2011) 214–231 (doi:10.1016/j.nuclphysb.2011.04.010), (arXiv:1102.1428)

• The Weil-Petersson-Zamolodchikov distances between topologically distinct Calabi-Yau 3-folds in this web are finite: P. Candelas, P. S. Green, and T. Hübsch, Finite distances between distinct Calabi–Yau manifolds, Phys. Rev. Lett. 62 (1989) 1956–1959 (doi:10.1103/PhysRevLett.62.1956) and Rolling among Calabi–Yau vacua, Nucl. Phys. B330 (1990) 49–102 (doi:10.1016/0550-3213(90)90302-T)

Related concepts

• conical singularity

• flop transition

• ADE singularity

• Sasakian manifold

References

Discussion in the context of gauged supergravity includes

• J. D. Edelstein, A. Paredes, A. V. Ramallo, Singularity Resolution in Gauged Supergravity and Conifold Unification, Phys. Lett. B554 (2003) 197-206 (arXiv:hep-th/0212139)

Discussion in M-theory

• Bobby Acharya, On Realising $N=1$ Super Yang-Mills in M theory (arXiv:hep-th/0011089)

• Michael Atiyah, Juan Maldacena, Cumrun Vafa, An M-theory Flop as a Large N Duality, J. Math. Phys. 42:3209-3220, 2001 (arXiv:hep-th/0011256)

• Edward Witten, Anomaly Cancellation On Manifolds Of $G_2$ Holonomy (arXiv:hep-th/0108165)

• Sergei Gukov, James Sparks, David Tong, Conifold Transitions and Five-Brane Condensation in M-Theory on $Spin(7)$ Manifolds, Class. Quant. Grav. 20 (2003) 665-706 [arXiv:hep-th/0207244]

• Kenneth Intriligator, Hans Jockers, Peter Mayr, David Morrison, M. Ronen Plesser, Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes, Adv. Theor. Math. Phys. 17 (2013) 601-699 (arXiv:1203.6662)

Review includes

• Bobby Acharya, Sergei Gukov, M theory and Singularities of Exceptional Holonomy Manifolds, Phys. Rept. 392:121-189, 2004 (arXiv:hep-th/0409191)