A generalised Garfinkle–Vachaspati transform

  • Deepali Mishra
  • Yogesh K. Srivastava
  • Amitabh VirmaniEmail author
Research Article
Part of the following topical collections:
  1. The Fuzzball Paradigm


The Garfinkle–Vachaspati transform is a deformation of a metric in terms of a null, hypersurface orthogonal, Killing vector \(k^\mu \). We explore a generalisation of this deformation in type IIB supergravity taking motivation from certain studies of the D1–D5 system. We consider solutions of minimal six-dimensional supergravity admitting null Killing vector \(k^\mu \) trivially lifted to type IIB supergravity by the addition of four-torus directions. The torus directions provide covariantly constant spacelike vectors \(l^\mu \). We show that the original solution can be deformed as \( g_{\mu \nu } \rightarrow g_{\mu \nu } + 2 \, \Phi k_{(\mu }l_{\nu )}, \ C_{\mu \nu } \rightarrow C_{\mu \nu } - 2 \, \Phi k_{[\mu }l_{\nu ]}, \) provided the two-form supporting the original spacetime satisfies \(i_k (dC) = - d k\), and where \(\Phi \) satisfies the equation of a minimal massless scalar field on the original spacetime. We show that the condition \(i_k (dC) = - d k\) is satisfied by all supersymmetric solutions admitting null Killing vector. Hence all supersymmetric solutions of minimal six-dimensional supergravity can be deformed via this method. As an example of our approach, we work out the deformation on a class of D1–D5–P geometries with orbifolds. We show that the deformed spacetimes are smooth and identify their CFT description. Using Bena–Warner formalism, we also express the deformed solutions in other duality frames.


Garfinkle–Vachaspati transform D1–D5 system Fuzzball paradigm 



We thank Swayamsidha Mishra, Ashoke Sen, David Turton, and especially Oleg Lunin for discussions. AV thanks NISER Bhubaneswar, AEI Potsdam, and ICTP Trieste for warm hospitality towards the final stages of this project. The work of AV is supported in part by the DST-Max Planck Partner Group “Quantum Black Holes” between CMI Chennai and AEI Potsdam.


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Authors and Affiliations

  1. 1.National Institute of Science Education and Research (NISER), BhubaneswarKhurdaIndia
  2. 2.Chennai Mathematical InstituteKelambakkamIndia
  3. 3.Institute of PhysicsBhubaneswarIndia
  4. 4.Homi Bhabha National InstituteMumbaiIndia

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