Quantum Background Independence of Closed String Field Theory

⋆ E-mail address: [email protected], [email protected] † Permanent address: Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India. ‡ E-mail address: [email protected], [email protected]. Supported in part by D.O.E. contract DE-AC02-76ER03069. We prove local background independence of the complete quantum closed string field theory using the recursion relations for string vertices and the existence of connections in CFT theory space. Indeed, with this data we construct an antibracket preserving map between the state spaces of two nearby conformal theories taking the corresponding string field measures dμe2S/h̄ into each other. A geometrical construction of the map is achieved by introducing a Batalin-Vilkovisky (BV) algebra on spaces of Riemann surfaces, together with a map to the BV algebra of string functionals. The conditions of background independence show that the field independent terms of the master action arise from vacuum vertices Vg,0, and that the overall h̄-independent normalization of the string field measure involves the theory space connection. Our result puts on firm ground the widely believed statement that string theories built from nearby conformal theories are different states of the same theory.