Green-Schwarz Strings in TsT-transformed backgrounds

We consider classical strings propagating in a background generated by a sequence of TsT transformations. We describe a general procedure to derive the Green-Schwarz action for strings. We show that the U(1) isometry variables of the TsT-transformed background are related to the isometry variables of the initial back- ground in a universal way independent of the details of the background. This allows us to prove that strings in the TsT-transformed background are described by the Green-Schwarz action for strings in the initial background subject to twisted bound- ary conditions. Our construction implies that a TsT transformation preserves inte- grability properties of the string sigma model. We discuss in detail type IIB strings propagating in the γi-deformed AdS5×S 5 space-time, find the twisted boundary con- ditions for bosons and fermions, and use them to write down an explicit expression for the monodromy matrix. We also discuss string zero modes whose dynamics is governed by a fermionic generalization of the integrable Neumann model.