JHEP10(2019)049 Springer July 30, 2019 superstring, : 4 October 4, 2019 -deformations August 13, 2019 q : T : -deformation can × η 3 S Received × 3 Accepted Published ˆ G, the × ˆ G Published for SISSA by https://doi.org/10.1007/JHEP10(2019)049 2) and construct two supergravity back- | 1 , ˆ G = PSU(1 superstring 4 . T 3 × 1907.05430 The Authors. 3 c Integrable Field Theories, Sigma Models, Superstrings and Heterotic Strings S

For supercosets with isometry group of the form , [email protected] × 3 Institut f¨urTheoretische Physik, Eidgen¨ossischeTechnische Hochschule Z¨urich, Wolfgang-Pauli-Strasse 27, Z¨urich8093, Switzerland E-mail: Open Access Article funded by SCOAP ArXiv ePrint: as well as exploreweaker their generalised supergravity limits. equations of We motion also and construct compare backgrounds themKeywords: to that the are literature. solutions of the Abstract: be generalised to a two-parameter integrableof deformation the with two copies. independent We studyRamond its fluxes. kappa-symmetry and We write down then agrounds focus formula for for on the the Ramond- two-parameter integrable deformation of the AdS Fiona K. Seibold AdS Two-parameter integrable deformations of the JHEP10(2019)049 21 23 superstring, ]. In special 3 2 , 1 -dimensional Anti- d offers a particularly 3 4 1 dimensions [ 20 − d 24 9 16 superstring 4 – 1 – 19 T 2 2 × ) ) 3 4 # 11 S 13 − ⊗ 30 29 × backgrounds are dual to two-dimensional CFTs which are 2) 2) # | 3 | 3 1 1 − ⊗ − , , 12 # ⊗ − (1 (1 28 ] and admits integrable deformations. AdS psu psu superstring 4 , 3 4 T 1 28 26 × 24 3 S × 3 A.3 Two copies of A.1 Gamma matrices A.2 Generators of deformation of the AdS 4.1 Dynkin diagram4.2 ( Dynkin diagram4.3 ( Limits 4.4 Different Dynkin diagrams in the two copies 3.2 Supergravity backgrounds 3.3 Limits of3.4 the supergravity backgrounds Mirror model and mirror duality 2.1 The action,2.2 equations of motion Extracting and the kappa-symmetry Ramond-Ramond fluxes AdS 3.1 Choice of R-matrix better-understood AdS/CFT instances, whichadmit are deformations often that the preserve most their (super-)symmetricwhich solvability. ones, is Here we integrable consider [ thegood AdS setting, providing more controlFirstly, than string in theories other on dimensional AdS cases for two main reasons. 1 Introduction The AdS/CFT correspondence is ade-Sitter duality space between and string Conformal theoriescases, Field in Theories it living is in superymmetry, possible conformal to field theory, investigate and the integrability. AdS/CFT It is setup natural thanks to to ask whether techniques the based on 5 Conclusions A Conventions 4 Examples of generalised supergravity backgrounds for the two-parameter 3 Supergravity backgrounds for the two-parameter deformation of the Contents 1 Introduction 2 Two-parameter deformation JHEP10(2019)049 ] 11 and – (1.1) 5 6 ] for a T 16 × 2 S × 2 ]. This S-matrix 38 1 limit can also be reached 0 gives the S-matrix of the → q case. Moreover, specific points -deformed AdS ]. The shape of the deformation -deformation or inhomogeneous , → 5 η η 2 21 -deformations of the principal chi- and then sending the deformation κ η η η -deformations, but a number remain, 2 T ) the positive (respectively negative) − η i 2 ]. 1 κ is the string tension. A conjecture for 33 = T → T ], which are defined through their action on ]) or by CFT (Wess-Zumino-Witten model) ]. However, superalgebras admit inequivalent 2) invariant S-matrix, which gives a quantum theory. But the | 24 12 – 2 – 5 – 26 (2 , κ S q 22 × 5 κ/T (respectively + psu ]. There it was shown that in order to obtain a super- − i e ⊕ 32 − ]. Progress has been made in trying to give a good string = ]. This generalises the 2) | q 28 worldsheet S-matrix, has been given in [ , 18 (2 ones. , 5 q 27 S 3 17 string theories also naturally emerge when studying black-hole ]. An elegant formula for the target-space supergeometry of the × psu 3 31 5 – ], with parameter ]. Under the deformation the superisometry algebra of the model gets 29 37 , 36 ]. AdS – ] and the symmetric space sigma model [ 21 15 34 , ]. Rather, they satisfy a set of generalised supergravity equations of motion [ – 20 , 18 13 26 superstrings were finally presented in [ , 19 5 is the strengh of the deformation and 25 S η × This solved one of the main puzzles in the field of In this paper we will focus on one particular type of integrable deformation of the type 5 -deformed model was derived in [ -deformed [ deformation of the AdS admits three interestingundeformed limits. light-cone gauge fixed AdS Firstin of another all, way, by sending firstparameter rescaling to infinity. the In tension this maximal deformation limit one gets the S-matrix of the mirror q where the (centrally extended) Dynkin diagrams, and supergravityAdS backgrounds for the including the behaviour inmirror the theory maximal deformation limit and its link to the undeformed of remaining issues [ η gravity background the R-matrix hassupertrace to of satisfy the the so-called structurenot unimodularity constants condition: the built case the out for of the the R-matrix R-bracket chosen should in vanish. [ This was roots. Generically, the deformedmotion [ backgrounds do notthat solve follow the from supergravity imposing equationsbut kappa-symmetry of not or its equivalently Weyl invariance scale [ theory invariance interpretation of to the these generalised model, supergravity backgrounds but there are a number is governed by an R-matrixon solving the the superisometry non-split algebra modifiedto of classical R-matrices Yang-Baxter of the equation Drinfel’d undeformeda Jimbo background. Cartan-Weyl type basis [ We will ofCartan restrict elements the and ourselves superisometry multiply by algebra. More precisely, they annihilate the review). Consequently, it is of interestas to construct deformations new, of exactly solvable the string AdS backgrounds II Green-Schwarz superstring known inYang-Baxter deformation the [ literature as ral model [ NS-NS fluxes, offering a richerallow landscape for than particularly in simple the solutions(for AdS which a can be review analysedtechniques and either [ further by integrability references [ configurations, see such [ as those arising in the celebrated D1–D5 system (see e.g. [ often exactly solvable. Secondly, the background can be supported by a mixture of R-R and JHEP10(2019)049 ] 26 and (1.2) L η ] to semi- 44 , the results are 20 4 -deformed S-matrix q ˆ G copy. The limiting -deformed S-matrix is q ], where the metric of 45 and explore their limits. In . Our conventions for gamma -deformation. We extend the 5 3 . we review the construction of η superstring was embedded into superstring with explicit mirror 2 5 4 S T × × 5 3 . S A × 3 -deformed S-matrix with deformation paramter – 3 – q ˆ G and thus allows for a two-parameter integrable ] reduce to the undeformed mirror theory in the × ˆ G 33 superstring in section -deformed AdS 4 η case. Studying their limits, we will observe that they T R × is again the ⊗ − ⊗ − ⊗ ⊗ − ⊗ − ⊗ η 3 q S = × -deformed S-matrix thus interpolates between the S-matrices of L superstring. The novelty in this case is that the superisometry q η 3 ]. However, neither the generalised supergravity background of [ 4 ]. It is an interesting and open question whether these latter two T superstring and present two supergravity backgrounds corresponding 33 reduces to the usual one-parameter 42 – 4 × η T 40 3 ]. The latter generalises the bi-Yang-Baxter sigma model of [ S = × 3 43 × R ] to the two-parameter deformation and write down an explicit formula for the S η theory. The 3 32 × 5 = 3 S L × ]. This behaviour is referred to as mirror duality. Another interesting case is when η The outline of this paper is as follows. In section To shed more light on this, as yet, unresolved problem we will consider deformations 5 controlling the strength of the deformation in the left and right 39 [ is pure phase, which, in a particular scaling limit, gives the S-matrix of the Pohlmeyer R 0 deformation of the AdS particular we will show how mirroralso duality arises compared in to this backgrounds context. thatof In are motion. section solutions Finally to we the endmatrices generalised with supergravity and some equations generators conclusions are in given section in appendix supergravity backgrounds for the duality. the two-parameter deformation, studyfor its the kappa-symmetry R-R and fluxes. present We a then closed extract formula the supergravity backgrounds for the two-parameter the two-parameter deformationtype of II the AdS supergravitydistinct in in two the different limiting have ways. different Pohlmeyer The and maximal twoof deformation the supergravity limits. background backgrounds Interestingly, exhibits we remain insights mirror find into how duality, that to while one recover the the other mirror background does in not. other cases This and may construct give integrable new to two different unimodularDynkin Drinfel’d diagram Jimbo R-matrices associated to the fully fermionic The two backgrounds are similar to the ones constructed in [ η case results of [ Ramond-Ramond (R-R) fluxes in thisthe more general AdS setting. We then apply these results to of the AdS algebra has a group-product structure deformation [ symmetric space sigma models. There are now two real deformation parameters reduced theory [ limits are also realisedfixed at worldsheet the S-matrix. level Forhave of the been the Pohlmeyer obtained background, limit, [ as encouragingnor well results the as in supergravity in this background themaximal direction of deformation light-cone limit. [ gauge the undeformed theory and itsthat mirror. it An is interesting covariant under feature the ofwith mirror the deformation transformation: parameters the mirrorq of the q AdS JHEP10(2019)049 f ) ˆ α (2) ∈ f ) 1; (2.4) (2.2) (2.3) (2.1) , R and ,X L (0) of grades 1 f X α ˆ G. We denote 2 ˆ ]. Let us also G = D(2 ˆ F respectively. , which project R × Q F ˆ , G X P and , . , = diag( ! and 4 ) ˆ G and , ! ˆ can then be split into T F = α R ! and X ) 1 ) f ˆ is related to the relative R ! × mod 4) R B Q ) iX ] + STr[ 3 P α R X iX S L j + . The subspaces X + − X i L × 0 − ( 3 f L + L X ( L X ⊂ X F ( ( ] X B ) F grading is defined through ( j iP ( P ] is reviewed. The kappa-symmetry , B f 4 − iP ] = STr[ ] 3 , as well as , P − have odd grading. For elements in the Z ) ) B ) 0 ) 0 i i X ] of the Metsaev-Tseytlin action for su- ) 0 ) 0 ( ( T R R f ˆ f G[ R R [ (3) A , 43 f T X X iX iX × ˆ G 0 + − + − ˆ G 0 0 0 0 × and F , L L L L – 4 – grading consistent with the (anti-)commutation ˆ G , where the parameter X X X X = STr[ (1) 4 (3) 1 ( ( ( ( f f S Z B F B F AB + P P P P × of the superisometry algebra K

3 being the Lie algebra of F f ˆ (2) S f respectively. The 1 2 2 1 2 1 2 1 (0) × ˆ g + f 2) for strings moving in AdS = = = = dim | 3 of grade 2, as well as supercharges onto the subspaces 1 admits a S (1) , a ) (3) (0) (1) (2) f f ˆ i × ( P X X X X ,..., 3 + P = = = = (0) We shall consider deformations of semi-symmetric space sigma mod- = 1 f X X X X = the Lie algebras corresponding to the supergroup (3) (0) (1) (2) ,A f , and define the supertrace STr[ ˆ ˆ A ˆ g G = PSU(1 P P P P ˆ g T of grade 0, ⊕ ∈ we use the standard block-diagonal matrix realisation ˆ g ab R f ˆ J = is the bosonic diagonal subgroup of the product supergroup f ˆ ,X 0 L X and ˆ g The generators generators and 3 respectively. Let us also define have even grading, while theLie subspaces algebra with introduce the projectors onto the even and odd parts of The basic Lie superalgebra relations, with the grade zero subalgebra els on supercosets of the type where F by for strings moving in AdS radii of the two three-spheres. 2.1 The action,Algebraic equations setting. of motion and kappa-symmetry In this section thepercosets two-parameter deformation with [ isometry groupof of the the deformed form modeldilaton is and R-R studied, fluxes the isbackgrounds supervielbein written are down. are Examples identified of and interest a in formula the context for of the AdS string 2 Two-parameter deformation JHEP10(2019)049 R 2, η / ) (2.5) (2.6) (2.7) (2.8) (2.9) ij (2.14) (2.10) (2.11) (2.12) (2.13)  and  L ij η γ . = ( . f . The R-matrix ˆ g ij  , R ) is the Levi-Civita ¯ η . ) η 1 Ξ projectors as . 2 2 R − ij Y ∈ ] 4 η /T  , g is the overall coupling g = R j Z and X X , − κ ∂ T and to the undeformed R f g 1 ˆ − σ ( 1 e − , η d )(1 g R 0 and τ 2 L = 1 R 1 , Y ∈ = η ) − − , − < R . g = d , κ − R O W X Y} )] η ττ , − L − σ = 0. (1 Y γ ¯ 2 η η κ ) = ( X = 2 R g d ˜ q i X η R + , q ) , ( L ∂ = . g = R 1 X 1 η = )] ] ) = [ /T C A κ L − + ¯ R η } L δ Y L g ) κ κ 43 ( η  STr[ − Y as = e ( L f = ( ˆ − κ STr[ XR R , ( = BC , κ , ) R ij – 5 – − 1 2 b ) K L  ] = 1). This new operator is still antisymmetric with X X ∈ (3) R are defined in terms of the d STr[ = Y − AB g σ Ξ ) P , + [  , κ 2 −  K R d L X d − κ ) ( κ -deformation if , q R Y} ˜ -deformation [ ] = Z R η , (1) q ) acts on , κ Y ˆ T G) P ) ˆ L ( g F depends on two real deformation parameters X ( diag( − ( κ = diag(1 R STr[ X ¯ 2 η q ( ∈ R Ad R ∓ g W ([ ] = ˜ R g R ≡ × U [ diag( 1 (2) R − g STr[  ˆ P G) ,η ( L } − R η L ) = q S = 1 Y U  = Ad ( = 1. The operators d  g R O , τσ R )  X The action of the two-parameter deformed semi-symmetric space sigma model ( is the Weyl invariant worldsheet metric with R ij [ γ To write down the equations of motion and the kappa symmetry variation it will be It has been conjectured that the symmetry of this model is, at least at the classical We use the mixed bracket notation. If the two elements in the bracket are of odd grading then the governing the shape of the deformation is antisymmetric with respect to the supertrace 1 where we have introduced respect to the supertrace, bracket is the anti-commutator. Otherwise, it is the commutator. as well as the auxiliary operator It reduces to thesigma model one-parameter when the deformation parameters useful to define the deformation parameters level, given by the asymmetrical R and solves the non-split modified classical Yang-Baxter equation and The operator We assume that the fields are inconstant the playing defining the matrix role representation. of thewhere effective string tension, d symbol with Action. for the group-valued field and reads as well as its inverse JHEP10(2019)049 , i . .  i ] i ) A ) E (2.22) (2.18) (2.19) (2.20) (2.15) (2.16) (2.17) (2.21) ( + A  g + ˜ , R = 0, with Ξ ] ( + j . E + (2) + W with , d ¯ A 2 η = 0, with i i ) g ] = 0 (2) + − , Z − j ] i ) A = A − ) (3) − with components g + (2) + ρ W κ Ξ g A A   STr[ δ . g d 1 + + ij W. ) 1 ˜ P Ξ R Ξ − E − ( ( ( ] + [( + g i κ . , (3) i g ) are yet arbitrary functions on i 1 ) ) σ δγ ρ + ) W ˜ also implies −  2 R − κ (2) + ¯ − 2 η d (3) ) can be written as O g (3) A A ρ A ) + ρ d + ] + − is then + 2 1 Z + − = E i 2.6 + ( , ) Ξ − − Ξ Ξ g 1 2 ( T  g + ) )  and W − (1) g A 2 − 2 − − A ) also has a local right-acting fermionic ρ  ˜ κ κ + [( ( R + , d i (1) i STr[( − + + 2.6 ) Ξ ) + + ρ ] = ( j + − W W 2 + 2 + + W (1) (2) − A ¯ A 2 η κ κ ρ − – 6 – ( A , d + − , i 2 κ i  ) = + ) Ξ ¯ Ξ g η = ( (2) − + g ), and ( − − ˜ i κ κ R A ˜ − 2 C ∂ δ A + ] = 0 W + [( 1 ) η − j i 2.13  η − 2¯ − ) W ) Ξ i 3 STr[ g , the action ( ( ¯ 2 η ) + ( (2) − P 0 1 ij , − + A F A − d + − } (1) [ satisfies the modified classical Yang-Baxter equation implies A + − 4(1 + ¯ ρ O ˜ − ∈ + C Ξ σ δγ h Ξ R ( 2 (  These equations of motion are supplemented by the Virasoro Ξ Defining the one-forms i W i + ( d X,Y ∂ ) i  [ but with the right hand side given by ∂ + STr ˜ C Z In addition to reparametrisation invariance and local right-acting gh, h (2) − d = = 1 ˜ σ R =  2 A 2 T  + d → − i Z − W ) Ξ g Z − T A ] = g − [ STr[( Ξ R ] = ( ,η i g has been defined under ( [ L ∂ η R − S ,η W d γ L η δ Let us consider an infinitesimal right translation of the field = S g E δ The variation of the action with respect to the field where the string worldsheet. Equivalently, this transformation can be written as kappa symmetry reducing the number of physical fermionic degrees of freedom. from which we deduce the Virasoro constraints Kappa-symmetry. gauge symmetry Virasoro constraints we focusinvariant on metric. the Its part variation of gives the action that is proportional to the Weyl- Virasoro constraints. constraints, which are obtained byric. varying the They action are with equivalent respect to to imposing the a worldsheet met- vanishing worldsheet stress tensor. To derive the The flatness condition for the undeformed currents a similar equality for Equations of motion. the equations of motion corresponding to the action ( Furthermore, the fact that JHEP10(2019)049 . i ] i , . i ] is ) j   ] ] ) (2.29) (2.27) (2.28) (2.23) (2.24) (2.25) (2.26) g j j (3) − ) ) . (1) A (1) (3) κ − i ] + κ κ j Ξ + − , Ξ ) ( ( , , , i , Ξ Ξ i ] ] ] (3) ( ( ) , i i } i i i ) j ) ) κ , i , i ) ) i i (2) + − ) ) − (3) (3) − (1) + (1) + A (2) + Ξ A ( A A (1) + (3) − + A A − − + , i Ξ + A A + i Ξ Ξ Ξ [( ) + − Ξ ( Ξ ( ( 1 , ( [( Ξ Ξ , , j (3) − P , i i ˆ F ) [( [( ) ) j j j A Υ ) (3) ) ) h (3) − − (2) + (2) − ρ (3) κ Ξ (2) − (2) + A A 1) in the defining representation. κ ]+ − [( + − i A A STr − − ) ˆ F Ξ Ξ Ξ − + i , ( Ξ Υ j i [( [( ( Ξ Ξ (1) + h ) [ i ( ( η η ˆ i i . F A { 2¯ 2¯ ) ) (2) − Υ +  − − STr = A (2) − (2) + Ξ A i − ( ] + A A j + , = diag(1 j (3) ) Ξ i ) = ) = − + ) ( ) W ˆ i G Z Z Ξ Ξ (2) + ) – 7 – + (1) ( ( ( ( Υ (2) − =   A + + (2) − A A +  + − , ρ A W W Ξ A 1 1 Ξ − } ( Ξ i ( j P P ) Ξ [( ) , = STr = STr ¯ ¯ 2 2 η η ( i ) h ), where   (2) + (1) (2) − ] ] ) is kappa-symmetric. ) can then be rewritten ˆ − ρ G i i A (1) A ) ) h ) ) + , one can show that + − κ 2.6 ,Υ STr E E σ − (1) + (3) 2.22 ( ( ˆ ) Ξ +  Ξ G ( ( STr + A A Υ   Ξ σ h , ( g g 2 − + j σ A i ) ˜ ˜ [ d 2 R R + Ξ Ξ ], the square of an elements of grade 2 yields two terms: one is ˆ F components of the projections are proportional to each other, for d ( ( + + (1) Ξ , , Υ Z ( i i belong to the grade 1 and grade 3 subspaces respectively. Using the h σ κ Z + STr 47 ) ) W W ¯ η , = diag( + 4 ¯ ¯ 2 2 ∼ η η T + (2) − (2) (3) ˆ F Ξ 46 ηT τ STr ( + − A A ) Υ and κ 2¯ i − ¯ 2 η + − + { − − − τ Ξ Ξ A = = W W ] = and + [( [( g   ] = [ ij Ξ 1 3 g (1) (1) (3) R γ [ (1) ρ ρ ρ P P ,η κ R   κ δ L ,η η L S η g STr STr S δ g δ This term can then be compensated by the following change in the metric, which shows that the action ( As discussed in [ proportional to the identity while theor other hypercharge, is proportional to theThe fermionic parity part operator, proportionalvanishes. to Finally, the the variation identity of drops the action out coming since from the the variation supertrace of of the field a commutator where fact that the instance ( As usual we then make the ansatz where, for convenience, we have defined The variation of the action ( equations of motion and the flatness condition To simplify the expression within the supertrace we use the following combination of the JHEP10(2019)049 α , + 2 } O Q 1 (2) α j − − (2.33) (2.34) (2.35) (2.30) (2.31) (2.32) 2 ) take O E ,E = 2.30 ≡ (3) i ˆ κ i (3) M { E ij ). + . , g , Ξ α κ , i δ 1 ] 1 = j Q . − ) (3) − ] = 0 g implements a Lorentz α 0 i ( 1 ) (3) A 1 (3) F ζ E − + E (2) . Defining E (1) + ∈ κ O + − = δ ≡ A A Ξ + MP ( ) = (3) + (1) , Ξ g i ( . d O (2) E ) , 1 1 i ) , h P − ) − (2) − (3) 2 − 1 g , ι ( ˆ O P κ κ − h 1 (2) − } i 1 ,E − + A (2) + − − − (2) Ad O j − Ξ P (1) + O 2 + κ δ Ξ [( κ ι ,E (2) A and the equations of motion ( ( ˆ = [( F 2 h + 2 2 P 4 Υ i (1) i ¯ η . √ (2) + ˆ κ (2) − = Ad (2) i − A ] + { A – 8 – (3) ζ P , j 1 η (2) ij 2 ) κ − − h = ¯ P ζ η Ξ − (1) 1 Let us now bring the kappa-symmetry transforma- and 1 + ¯ (1) = = E − h ] = 0 i q + ) (2) + = 1 = Ad (3) (1) Ξ A (3) − ( = ˆ i κ E , M = Ad 3 i A ζ (2) − κ that it is this normalisation that brings the torsion into its standard ) δ ,E − A (2) Ξ (1) ) as 2.2 ( (2) + is such that, for instance, ˆ κ is the bosonic supervielbein, and , i i A κ a ) MP δ + 2.29 ι P = Ξ (2) + , a (2) ) into their standard Green-Schwarz form. This in turn will allow us [( A E P (2) ˆ F (1) 2 + Υ E 2.29 ≡ κ Ξ h ) and ( ), are h [( (2) Ad STr E 2.21 , ι 2.25 ¯ ζ 1 2 η ) and ( = = = 0 2.21 ij (1) (2) γ Finally, if one identifies the supervielbein in this way, we can write the kappa trans- We start by considering the equations of motion in the fermionic sector, which, ac- ˆ κ κ E We will see later in section The interior derivative 2 3 δ κ δ ι This shows that the kappa-symmetry variationthat takes the the standard vielbein Green-Schwarz form have and been chosen appropriately. Green-Schwarz form. are the fermionic supervielbein. formations ( with normalisation In the above also still holds.the same Therefore, form assupervielbein the as equations of motion of the undeformed model if one identifies the Furthermore, To compare these expressionsneed with to the find equations of awe motion relation find, of just between as the for undeformedtransformation the on model one-parameter the we deformation, grade-2 that subspace of the superisometry algebra, tions ( to identify the supervielbein of the deformed theory. cording to ( Identification of supervielbein. JHEP10(2019)049 ], 32 and , , (2.37) (2.38) (2.39) (2.36) A } } T , as well (2) + (2) − A β 2 X ,A ,A α } . 1 , appearing in + } = (2) + (2) − S } , . A −  A A + α α + { { X A ) ) O A g g ], we shall only be − , d 12 12 + ˜ ˜ R R + , d S S 32 1 1 , d a a A − − − + − + γ γ 4 + A 2 1 O O A d { E E { g g − ( ( g ˜ ˜ a a ) ) + R R ˜ R { 1 + − , R-R bispinor E E 1 g − − 1 8 1 8 − X X + ˜ for one-forms R abc O 1 O − − } H 1)( 1)( − + + B , O } T − − abc abc a } − 5 , − b , 1 1 + A H H for readability. + − − + − A Ω T α α } − A b [ ) ) ∧ − A O O + + B ( ( E bc bc , d A Y γ γ , d − − 1 2 + + ∧ 2 } − } − A . E E , d A ( ( + − n A E − – 9 – a a µ + + } a d is A A d X + γ { E E X A + − { ] by a sign. This comes from the fact that in our conventions 2 d − { ˜ 1 8 1 8 A C ˜ g = C , d , d 32 1 E A + 1 ˜ − + , NS-NS three-form i R − + − 2 − d } − + 1 g ab A A O ab ab − + O ˜ − ∧ · · · ∧ R 1 2 + − Ω 2 1 Ω Ω 1 , O d d 1 X,Y α α µ { { + + E ) ) { + ˜ ˜ a X 2 2 C C A } } } − d γ 2 1 2 1 + − E E 1 n + + d ab ab g E + + ,A ...µ γ γ i ,A ,A ˜ 2 1 ( ( } } R − µ + + { 1 4 1 4 + − − A A 1 A A ! { 1 − n + { { = = = 1 ,A ,A 1 1 O − − a + − α α = − − + + 1 2 1 2 O A A E n O O E E { { 1 2 d A + 2 1 2 1 d d 2 1 2 1 − − − − − . Similarly, the result for d = = = B = = − T + + − B A ] a formula expressing the R-R bispinor in terms of the operators A d A A -deformed sigma model has been written down. We would like to extend those results Y d η d d 32 We have contracted spinor indicesSome of and the suppressed following the results differ from [ 4 5 = -forms, namely the exterior derivative dn is acting from the left. We also use a different convention for the components of To proceed we rewrite these expressions as also introduced the notation Y where in the last equation we have used the modified classical Yang Baxter equation. We We start by calculating the exterior derivative of with its usual Green-Schwarzbe expression, linked the to exterior the derivative spinas of connection the dilatino the and supervielbein gravitinoconcerned can field with strength extracting superfields. the In R-Rexpansion contrast fluxes in to and fermions. [ thus To will leading only order need the the relations leading take terms the in form the In [ the to the two-parameter deformation. Togeneralising achieve this, when we needed. will follow Thevariation the has same strategy allowed steps us is as to as in identify the [ follows. supervielbein. The By comparing study the of superspace the torsion kappa-symmetry 2.2 Extracting the Ramond-Ramond fluxes JHEP10(2019)049 } (3)  M. (2.46) (2.42) (2.44) (2.40) (2.43) (2.45) (2.41) ) ,A W (3)  − gives κ A + { } κ (2) . . ,  (3) , ) P } β  } 2 2 − d ,E γ (2) κ g , 1 (1) = (1+ (2) ˆ R + K ,E ,E E ˜ γ  2 + M abcde 1 1 (2) κ 2 R (2) α − h ( F E η 1 , 2 E R { 4 ¯ { η ) η − + Ad . 1  { 1 abcde } } A 1 − h W − } , γ − + − ¯ W 2) η 2 (3)  (1)  2 L ˜ (1) Ad κ O − ]. η iσ ∓ 4 + L κ − ,A ,A W/ η κ ,E 1 32 − W, + 5! − − W 1 · (2)  (2)  κ −  1 } 1 κ − (2) 2 R κ 2 } A A 1 2 − −  + g η E (3) { { + κ ˜ { 1 O } − (2)  − R κ + 1 −   ( − + ,E 1 3 1 } − h (3) 1 ,A diag abc W W − W + (3) + (1)  F − −  + do not mix the bosonic and fermionic sector (2)  Ad ,E O  E = (1 κ κ 2 ˜ ) A { – 10 – 2 L M  ,A = abc ˜ (2)  + + P 2 − { A ) η 1 2 γ κ κ κ O E  1 = 2 − (1)  1 2 2 1 W − ˜ , κ 2 2 σ Ω ¯ ¯ A η η + } − − ˜ ˜ − − ) 1 } − { M 1 3! + κ 2 + O − − (2) and  0 + (1) κ ab,c 2 + ) ) ) − ( P κ g 1 + κ ˜ , 2 − 2 − ,E 2 − 2 a { M ( ˜ 4 ,E ¯  R κ κ η 1 κ , 2 1 2 F − ] (0) + h − ] a } − (1) + + ) + − A γ − E a,b  2 ab,c } − (3)  [ η 2 + 2 + 2 + Ad is a purely bosonic group-valued field. We immediately see that [ { 2 ˜ c Ω  2 κ κ κ ˜ ˜ ¯ iσ η − { 2 1 g ˜ (3) ( ( ( M i ,A M 2 2 2 − ) should then be compared to the familiar expression 1 + ¯ − 4 4 4 ¯ ¯ ¯ η η η (2 ,E (1)   c = = 3 = 8 1 + = A − − this formula reduces to the one of [ − ¯ (0) + 2 η 2.44 E  β { S = (1 + 2 ¯ ¯ ¯ η η η 1 2 A η abc (2) α + − W  1 + H E  ∓  = − ˜ −{ S O d 1 1 ab 1 κ R ) =    + η ˜ A κ ¯ ¯ 2 2 η 1 2 η ( 2 = (3) 2 − ¯ η + ∓  L E η d = = The bispinor ( ab  Ω X In the above expression when where from which we obtain the NS-NS three-form and R-R bispinor Furthermore, we also find that We can thus identity the spin connection as Assuming that the(which operators is the case to leading order in the expansion), the projection onto where JHEP10(2019)049 and (3.1) Iα (2.51) (2.47) (2.48) (2.49) (2.50) χ , defined and five- n 3 C F 16 chiral gamma × , f ]. For the two-parameter , three-form . 1 f Y ∈ 48 , F ]) to have a supergravity solution. X 50 , . 2) and consider deformations of ∀Z ∈ . | 49 , 1 3 2) , ) , . This calculation requires going to higher | − Φ 1 + n , , Y} C O Iα , , ∇ ) ∧ n X F H SU(2) ( ] = 0 sdet( Φ PSU(1 0 − R ], a condition on the R-matrix for the one- + × Φ e }Z × 2 1 ˆ ) G = PSU(1 32 1) + [ – 11 – − − = B , 2) e n | } T n ) ( 1 C In [ = F , Y ( ,R Φ SU(1 2 = d A R − T , n e 6 F X PSU(1 ] STr[[ = [ 33 R AB b K superstring Y} , ) is a necessary condition (although see [ 4 X T [ 2.50 × superstring. 3 4 S T -deformation to be a standard supergravity solution was given. This is the × η × 3 3 is the dilaton. In analogy with the one-parameter deformation we postulate that S . Moreover, for standard supergravity backgrounds the R-R fluxes are 5 Φ × F 3 AdS Since the one-parameter deformation is a particular case of the two-parameter deformation, the uni- 6 AdS modularity condition ( In order to provecheck that that they it match the isorder spinor also derivatives in of a fermions the sufficient and dilaton, condition hence we one do would not need perform to it calculate here. the dilatinos We choose unimodularbackgrounds R-matrices in and 10 construct dimensionstorus. with the This then the embedding gives remaining supergravity of backgrounds compact for the dimensions the given two-parameter 6-dimensional deformation by of a the four- 3 Supergravity backgrounds for the two-parameter deformationIn this of section the wethe shall semi-symmetric focus space sigma on model the based case on the supercoset deformation it is natural toR-matrix postulate then the that theory if will the besupergravity deformation Weyl equations invariant is and of the governed motion. background by will a Whilewe solve unimodular our will the results standard not give provide arguments a in proof favour of of it. this claim For Lie superalgebras ofity the type condition that is we equivalentassociated are to to considering the the in R-bracket vanishing this [ of paper, this the unimodular- supertrace of the structurewhich constants is the generalisation to superalgebras of the R-bracket [ Condition for Weylparameter invariance. unimodularity property which will be supported by specificthough examples. We also define the R-R potentials matrices. Comparing the twoform expressions gives the one-form where the latter is given by valid for a type IIB supergravity background and written in terms of 16 JHEP10(2019)049 0 4 L ⊕ R 2) × L | (3.2) (3.3) (3.4) 1 2) , | 1 (1 , . , (1 need to be psu           1 } psu 4 − , 3 1 1 0 1 +1 1 0 1 +1 { − − − − − are Drinfel’d Jimbo , 1 R − and R , each of which can be      2) admits three inequiv- 1 0 1 +1 1 0 1 +1 01 +1 } | 1) subalgebra, while the 0 +1 +1 +1 0 +1 2 , − +1 +1 0− +1 − − − (2 , and (1 1           and whose explicit action on 1 0 sl 2) algebra in terms of 4 { ) | L − su p = = # 1 R , 1 01 +1 (1 − − ). We consider deformations gov- ⊗ − ⊗ − ⊗ . R 1 01 +1 +1 1 0 +1 +1 +1 psu − ⊗ − P − − − # and ,  ,       ), where Ad semi-symmetric space is to obtain the new R-matrix ij ij R 0 = j 3 ) R M M i S ,R − ⊗ ( 1 2) superalgebra that we shall refer to as the ij ij | L P − P 1 i i × δ # R , – 12 – − − 3 ,  = (1 = = ij ij = Ad ⊗ − (2) is left invariant (that is to say, the new R-matrix P psu ij ij M , P ) ) have the same action on the bosonic generators) then su ij = diag( # R 0 (2) subalgebra. The remaining entries are fermionic; see i M M ( ( R R − su 1 2 ], the complexified algebra ( P P are unimodular and hence are expected to give rise to su- = 33 1) and , − ⊗ − ij ) 2 upper left block generates the (1 2) is given by # | M 1 su × ( , ) and the right copy (subscript 0 ,R ,R L (1 R ! ! ⊗ − ⊗ − ⊗ psu respectively. ∈ R 2). We consider a realisation of the | 1 2) | , for the conventions we use. The various R-matrices can then be constructed , with two copies of the M 1 2 3 4 1 3 2 4 1 2 3 4 3 1 4 2 1 R , (1

A 2) (1 | = = psu 1 2 1 , psu Let us recall how to construct the different R-matrices associated with the superal- As already discussed in [ and the reference R-matrix P P (1 P only the permutationsconsidered. that This do leads not to the exchange two the following unimodular order R-matrices of we act with the permutation matrix Of the 24demands possible the permutations, action on 8R give rise to unimodular R-matrices. If one further lower right block generates the appendix by considering permutations of 4associated elements. with Namely, the starting distinguished from Dynkin aan diagram reference element R-matrix Dynkin diagram pergravity backgrounds, this isDynkin not diagrams. the case for R-matrices associated with thegebra other two supermatrices. The 2 and alent Dynkin diagrams, realised by a different choiceinequivalent of backgrounds. simple roots. In particular, The while associated R-matrices R-matrices associated generically to lead the to fully fermionic The superisometry algebrapsu of theleft AdS copy (subscript erned by R-matrices of theR-matrices type satisfying the non-split modified classical Yang-Baxter equation on 3.1 Choice of R-matrix JHEP10(2019)049 . . ⊕ R ! η L  (3.7) (3.8) (3.5) (3.6) 1 2) σ | ˆ F, can be ) 1 , and r , ∈ R L ! (1 g R η are arbitrary  1 R σ η psu ) 7 arcsin( and r ) for the dilaton. i 2 L and − , . R L .  ) ) η 2.48 ) 2 2 2 P P arcsin( 0 P , give rise to equivalent exp i 2 R R 3 R   − − 3 R 2) superalgebra. We focus − , , | σ , 1 2 ) 1 0 1 exp . P P , φ ) and since P  and ) associated to the completely R R R R 3 − (1 R 2). We restrict our attention to ) R | 2 σ R ϕ ) 1 ( R R , i psu 2 φ ,R , η L (1 + − L semi-symmetric space,  . Our convention makes it easier to compare ϕ R R = diag( = diag( ( R 3 = diag( L psu i η 2 4 2 S η 2  R R ⊗ − ⊗ − ⊗ × R  ( → − 3 1 ) for the R-R fluxes and ( σ R = diag( , as well as ) η 4  ρ 1 – 13 – , R 2.44 R , , σ L ) ) ) ) ) 1 ρ 1 1 ] these two R-matrices were not considered inequiv- P P P and R 33 R R arsinh( 1 − 1 2 − − , R , , 1 do not need to be identical and one can take different sets 1 2 − arsinh( P  P P 1 2 R R  ⊗ − ⊗ − ⊗ R R R ( 0 exp exp 0 exp  exp 3 and  σ case there are two copies of the 3 ) = diag( L σ 3 = diag( = diag( ψ ) 1 R S by using the formula ( , 1 3 − ψ ) R 2 t × R R case examined in [ ( + R R t 3 i ], where the bosonic background has been obtained for this choice of R-matrix. 2 2 ( , g S i 2 43 − L   g × superisometry algebra of the AdS and . This then leads to four different unimodular R-matrices on the 2 1 2 R exp exp P R 2)

| R 1 , = = diag( = or For the AdS Here by convention we use opposite signs in the left and right sectors. Of course, for the two-parameter g (1 L R 1 7 g g P deformation the combination that entersone the can action always is reabsorb diag( theour minus results sign with by [ sending In this section we extract the supergravitymatrices backgrounds corresponding to the unimodularWe R- choose the following parametrisation for the gauge-fixed group-valued field By virtue of theirsolve the unimodularity supergravity equations we of expect motion for the arbitrary corresponding deformation parameters deformed backgrounds3.2 to Supergravity backgrounds It transpires that the R-matrices backgrounds related by analytic continuationsthe and two in inequivalent unimodular the R-matrices following we shall only consider R psu The two R-matrices of positive and negative rootsR-matrices in with the the two desired copies action of on the bosonic generators and thus alent, as the backgrounds are related to each otherour by analytical attention continuation. on unimodularfermionic R-matrices Dynkin diagram, In the AdS JHEP10(2019)049  , φ , ,   (3.9) d , 2 ) φ ψ (3.12) (3.10) (3.11) ϕ φ  2 J . d d r d φ t 2 ]d ∧ d ∧ ∧ 2 J ]d − t  r ∧ φ 2 ∧ ψ d φ d φ ρ  ψ − 2 d d d )(1 r φ 2 ∧ [1 ) 2 )d ) is r d ∧ ∧ 2 . 2 t 2 2 ρ 2 [1+ t r ∧ φ 2 r ρ d 2 P ρ ψ d 2 r d + 2 ρ ψ 2 d − R ρ r κ 2 + r 2 + ) ∧ κ ) )d r − (1+ κ (1 + − 2 κ 2 t 2 + 2 , κ ρ + 2 − r − ρ d 1 κ ρ (1+ , 2 κ κ 2 κ P + − ) + ) ρ r +2 − − 2 2 ) κ κ R 2 , (1+ r r 2 φ +2 (1 + ϕ + ) + ) φ 2 2 ρ 2 + + d 2 d κ (1+ − φ ρ − κ ]d ψ r ∧ κ ϕ − d − 2 2 ∧ + ϕ d ]d κ (1 r ∧ ρ ϕ t − 2 φ . Due to their gauge symmetries (1+ r (1 + 2 − 2 + d + )d ∧ = diag( ρ 4 d − d )d φ 2 − 2 − 2 κ κ ∧ ϕ t x 2 + 2 2 ∧ r d κ 2 − t κ d r d κ r ϕ κ )d R − 2 + ∧ ∧ )d + 2 [1+ ∧ − 2 t ρ + 2 − )d 2 r (1 3 r ϕ r φ r 2 [1 2 r 2 2 )d x − + d 2 d r d ρ 2 ρ ρ − 2 + d κ ρ 2 ) = 1 + ) is ( ∧ r − r 2 + 2 ∧ r 1 r − − − 2 + 2 + ϕ r ( − κ − )(1 P (1 r 2 ϕ 2 2 κ d ψ ˜ 2 − κ − 2 F r − d ρ d R x − d are not unique. The form chosen here makes it ρ + 2 1 ρ )(1 2 − d + κ − ∧ ψ ρ 4 2 − r d – 14 – ( t + ( , 1+ d ∧ ρ 2 C ψ 2 1 ) (1+ − + 1 ∧ ρ )d ) = 1 r d + P , ϕ t κ ( − x ( r 2 2 ψ 2 t ) = 1 R ∧ κ (1+ ρ ˜ + + F )d d ρ and )]d ρ, r t κ κ + − 2 2 ( 2 + = d 2 d )]d ∧ κ κ ρ ρ, r r )+ 2 2 κ − + (1+ ( 2 t C + − ρ ρ − ψ J + d ϕ ϕ − ϕ ϕ − d 2 κ (1+ (1 d d ) d d = diag( κ ρ   ∧ 2 2 −   (1+ ) ) ∧ ∧ 1 ∧ ∧ ρ ,P ρ ) ) κ 0 0 2 − d ψ ψ 0 0 R ) ψ ψ Φ Φ κ ,P − 2 ρ, r ρ, r Φ Φ r − − )d )d ) ( ( ) ρ, r ρ, r ( κ )d )d − − e e 2 2 2 ( ( ][1+ r ˜ 2 2 (1 + e e r r ) P P F + ( 2 r r ][1+ P P ) 2 − ρ, r r ρ ˜ − − 2 2 − 2 + 2 − 2 + F ρ ( κ d ) 2 + 2 − 2 + 2 − − − ( ρ − ρ, r κ κ κ κ (1 (1 P ρ ( ∧ κ κ κ κ 2 2 F [1 ( t (1 (1 P ρ ρ 0 1+ 1+ 1+ 1+ d 2 2 , [1+ F Φ 2 + + i ρ ρ +  0 2 1 + 1 + 1 + 1 + − κ s κ s x κ − 2 − Φ ) −  ( 2 d ) = 1 + κ κ r − − + e − s s i ) ) ( − 1 ρ x ρ ρ ( − e − ˜ + − = = = 1 ( ( ρ F F 4 2 Φ = = = F F +d + 2 C C 4 2 Φ − = = 2 C C e The bosonic background is common to the two choices of R-matrix, with the metric − 2 B e s d The K¨ahlerform on the torus is the expressions for the potentials while the background corresponding to the R-matrix The dilaton and R-Rsponding to sector the depend R-matrix on the choice of R-matrix. The background corre- where and closed B-field given by JHEP10(2019)049 ] ] ) ) 3 ), 45 and 45 = 1 , , 3.15 3.12 + 3.14 (3.13) (3.15) (3.14) a . The κ 2 43 and S r . Let us . What 3 1 = 1 and 3 2 + r S κ a ) and ( φ , . × r − 3 ) + and ↔ 2 κ − 2 ], two solutions 3.11 = 0 and r ρ κ ) and supported 45 , it is possible to ↔ a − ρ 3.9 + 1 + (1 In [ parametrizes the different 2 − q κ a ψ , ϕ = 0, that the background ↔ 1 + R It has been observed [ φ , κ η q ↔ → , t = 0 or 2 r L ) are the analogues of the η − , r ) 1 ψ , ϕ 2 r 3.12 ( √ ρ ↔ 2 − 2 + F ). Thus, also the real transformations ( κ κ p – 15 – − 3.12 and then do the redefinition of ρ , reached when one of the deformation parameters, ) 1 + ) and ( 2 − 2 2 − , t ρ − q κ κ 2 κ r 3.11 proportional to the volume form on AdS = → − 1 + 2 ) and ( (1 + 2 + 1 C and 2 − κ q κ p + 3.11 ), including the dilaton and the R-R fluxes, are invariant under = d κ , r → 0 limit one recovers the undeformed background, with constant 3 1 + only contain even and odd powers of the deformation parameters 2 F 4 ρ 3.12 q → C  → 1 + ] respectively. The dilatons are indeed exactly the same, but we find κ ) , r ρ and p 45 2 ( ) and ( 2 ρ 2 − F C κ p 3.11 ). However, invariance is broken by the second set of transformations ( , ρ 1 + We then observe that ( − 1 + p κ i 3.13 8 q i ] the authors started by solving the supergravity equations of motion perturbatively in ↔ → 45 → + ρ It is only in the special case κ ρ In [ respectively. Since the bosonic roots are not simple the R-R fluxes mix the AdS and then generalized to arbitrary deformation parameter. The constant 8 = 0 solution of [  − a slightly different R-R fluxes. This is aκ consequence ofsolutions. the fact that the authors of [ Relation to previously studiedto supergravity the backgrounds. supergravityby equations a of R-R motion three-formaforementioned with flux paper bosonic have we been background shallsolutions. constructed. ( refer Using to the these same two terminology backgrounds as as in the the do not leave the supergravity backgrounds invariant but insteadeither map in between them. the leftremains or invariant right under sector, all three is maps. setare to equal, In that zero, with case constant i.e. the dilaton. two backgrounds ( where we first interchange happens to the R-Rbackgrounds ( sector under thesethe transformations map ( is summarisedwhose in effect figure is to exchange ( combine them to find a real transformation and While these two transformations involve an analytic continuation in Relation between the two supergravitythat backgrounds. the metric andthe B-field formal of transformations the two-parameter deformed model are left invariant under dilaton and a three-form also notice that κ coordinates in a non-trivial way.supergravity As equations expected, of these motion. two backgrounds satisfy the standard manifest that in the JHEP10(2019)049 ) of (3.16) (3.17) 0 0 → ∞ . It is not . = . We start =  κ ) − κ + τ κ ( κ 3.2 φ • • = along τ , φ , ) swaps the two solutions (red # 2 = 0  = 0, which corresponds to the 3.14 τ ϕ d d − κ  , r superstring. ) + 4 τ 2 and ( T is a solution of the equations of motion,  ϕ respectively. At the four end-points, one of κ t τ × 2 d ] by performing a sequence of dualities (e.g. = d 3 µτ = R S  45 in order to preserve the dimensionality of the = ) and maximal deformation limit ( + i × − – 16 – κ µ ϕ " 3 and 1 , → , ϕ 1 − ) κ R µτ , is set to zero. The backgrounds are left invariant under τ τe ( − d = ψ κ t Z = or 1 2 + = ), and thus their integrability remains an open question. κ 4 • • S , ψ 0 0 We consider the trajectory parametrised by = = = 0 ) (blue arrows), while the transformation ( − + κ κ ] mixes even and odd powers of the deformation parameters -deformation of the AdS 3.13 η 45 , ρ ) τ . The two supergravity backgrounds. The top and bottom lines represent the solu- 1 2 is the einbein. Clearly ( t R R e = t . We also study the Pohlmeyer (  where where we have introduced a mass scale Using the metric ofthis the geodesic deformed has theory, we action find that a relativistic particle moving along This limit can beκ taken for boththe backgrounds two and supergravity for backgrounds arbitraryone-parameter when deformation parameters Plane-wave limit. 3.3 Limits of theLet supergravity us backgrounds now exploreby some considering limits the of plane-wave limit theparticle corresponding supergravity moving to on backgrounds zooming a of into light-like the section geodesic geometry along seen a by great a circle of the (deformed) three-sphere. by a three-form flux,five-form flux while (or the two different deformedthree-form three-forms of backgrounds upon [ possess dimensional reduction). bothclear Accordingly, if a the one three-form can and recoverT-dualities the a on backgrounds the of torus [ T the deformation parameters, the redefinition ( arrows). were seeking solutions of the supergravity equations of motion that are only supported Figure 1 tions corresponding to the R-matrices JHEP10(2019)049 ], . 10 53 , = 0 (3.21) (3.18) (3.20) (3.19) 33 4 . ]. To take 2 − κ ,C ]. Its light- 54 , ) i 54 φ x L . 1 + d d i , ∧ q x 2 r  = 0 + , 4 superstrings [ , x − + d ) i → d 5 1 2 φ x 2 S ) r d superstring [ ∧ p i + × i 4 x x r + 5 d ,C under these transformations, and d T = ψ ) µ , r − T d + d × φ − κ 2 2 − 3 2 2 d κ L ∧ ) κ S φ + 2 + have the same plane-wave form ∧ + ) + d 4. → r κ × 2 x , x + + and AdS d , κ L brings the pp-wave metric into its canoni- 2 r 3 T d 1 + x x + , 3.2 (d 4 r 2 ∧ − d + ρ φ q µ T 2  ( )(d + 2 = 0 ρ = 0 is interesting due to its relation to x r 2 2 µ (d x × and ψ r n r ) 2 d − 2 2 + − d , ρ r → 2 − r 2 2 + S − + κ ψ + ρ + n/ 2 i, κ 1  d × µκ ρ 2 T + ( 2 µx ) – 17 – ∧ + r )( → ∼ 2 + ) 2 + 2 − = )(1 + , x n , ρ 2 κ ρ x κ , 2 d 2 + 2 C ϕ r ) + 2 d r ) ρ κ µ − 2 ψ = 2 d − x − ρ d 1 + r µL ( κ + and ) (1  ∧ )(1 + − + κ − 2 , ϕ (1 + t 2 . 2 + ρ κ r in the rescaling of + + µ κ d − T q 1 )(1 2 − 2 + µ − 2 0 ) is finite and real in this limit, giving the pp-wave background − 0 2 ) − κ x µx Φ L κ Φ 2 ρ ψ − 2 − ρ − )(1 (1 + + = e ρ + both backgrounds of section x (d 1 + 3.11 → 2 2 d 2 d − µe ρ ρ + µ q ρ T (1 + The limit d +  (1 + = 0 , − x µx + ∧ Φ 2 2 → ∞ 2 − + (1 + 4d /L + r = − , ϕ i x L x . Only ( − e − t d x 2 d + ,C − + + d = = = + 0 0 x x → 9 µL 2 . ] Φ 2 2 Φ i 2 ρ s 2 µ µκ → 4d C x + ( − d − 52  − e ρ , e + +d = = = 51 µx 2 Φ In the limit B s 2 The additional factor One should keep in mind that the string tension is rescaled = 9 d − 10 t e cone gauge fixing gives the Pohlmeyer reduced theory forcal form. strings moving in undeformed that the potentials scale with the tension as We have not includedcontribution. the This B-field, pp-wave which background is matches the a one divergent constructed closed in two-form [ with no finite and take parameter Pohlmeyer limit. the Pohlmeyer reduced model ofinto the account undeformed the AdS fact thatprecisely the twist, string the tension coordinates goes to zero in that limit we rescale, or more Discarding the total derivative B-fieldthe we observe one-parameter that deformation in of the thethe plane-wave AdS limit, deformation similarly only to enters the plane-wave background through a rescaling of the mass tions [ as well as coordinates. In order to study the geometry near this trajectory we make the transforma- JHEP10(2019)049 4 T . = 0, × (3.23) (3.24) (3.25) (3.26) (3.27) (3.22) 3 super- − S = 0 4 , × 4 T ) κ, κ 3 , φ superstring, ] is given by , , i × = d i ) 4 x 3 T, 2 ∧ x + 56 d 2 T S , i ρ d κ κ i ϕ ,C x × 35 × ) actually exhibits d x ) 3 2 → φ ). 3 r S + d d )(1 + + d T 2 3.12 × + 2 ∧ 2 φ r 3.24 3 t ψ d ψ d 2 d − d 2 r 2 i r ∧ ρ κ (1 x t 0 gives the same expression but − d ) + Φ 2 2 2 → ϕ ) + ϕ . r ρ 2 − d i ( e ρ 2 . ∧ r + d and = = 0 ) ψ + d 2 φ t φ , 0 4 + Φ d d 2 Φ ϕ 2 2 2 , x − − ϕ ρ ↔ ρ ∧ (d e ( r κ 2 0 (d − ψ r Φ 2 d 1 ) is − 1 ρ → 2 e r 1 − 2 1 + -deformation of the AdS – 18 – ) corresponds to the undeformed AdS = ρ 3.12 ,C 1 + = η r . ]. The bosonic part of the reduced theory is given ρ , ψ + 2 ) ρ , e d 2 − 2 + d ψ 55 3.12 , r ∧ ϕ ↔ d 2 ψ ∧ d d κ ) has the expected Pohlmeyer limit, while the maximal φ φ ϕ ) is r ∧ Another interesting limit is when the deformation pa- 2 ϕ d d ∧ t ρ . In this limit the metric and B-field become 2 d 2 ,C t → d r ,C r 2 3.11 ) 2 3.11 ρ ) 2 r )d ρ ) + 2 2 2 r ρ 2 ) ) + r r → ∞ 1 + + ρ 2 2 1 + r r κ φ − + , φ d 2 + + + d ρ )(1 + ρ ρ + d κ ∧ )(1 + 2 2 2 r d 2 t ρ (( ρ 2 d ϕ t ρ d ∧ 2 → ) gives the mirror model of the undeformed AdS d d ∧ − r − t − 2 − t ( d − r 0 + Investigating limits of the two supergravity solutions for (1 ( d (1 2 2 ( Φ (1 0 3.25 0 2 2 ρ ρ 2 0 − Φ Φ r r ρ 1 ρ 2 e Φ − − − = 0. r 1 2 , ρ − − − e − 1 1 e − µ t 1 1 κ , which was constructed in [ 1 − e ], 3 = = = ) and ( S = = = = = 36 → 2 Φ B 2 2 s 4 2 × t Φ B s d 2 − C 3 C 3.23 d e − e which corresponds to the one-parameter we found that the backgrounddeformation ( limit of the backgroundmirror ( theory. We will now showmirror that duality. more is true: the background ( On the other hand, this is notConclusions. the case for the other background ( in ( string [ while the maximal deformation limit of ( Further swapping The two supergravity backgrounds aremaximal deformation both limit finite of but ( remain different in this limit. The first rescaling and then taking the limit by the sum ofthe the same complex limit but sine-Gordon without modelwith twisting mass and the coordinates its sinh-Gordon counterpart.Maximal Taking deformation limit. rameter goes to infinity. More precisely, the maximal deformation limit [ AdS JHEP10(2019)049 ] 4 T 36 × (3.28) (3.29) 3 ), first S . While × iτ -deformed 3.12 3 η and assume → . ϕ σ i , x d i iσ and x ] plays an important t → 34 + d τ = 0 yields the following 2 φ − d 2 κ 2 ψ r , d κ 2 + ρ ) = 2 and then exchanging the two coordinates. + r , + ) ϕ n 2 κ − ρ ...a and 1 2 )(1 a t r 2 F d r 2 2 n )(1 + ) with i ρ 2 κ d ρ ]. It is constructed out of the original theory by 5, it was observed that at the bosonic level the = 2 – 19 – , ) and show that the mirror duality also extends κ 3.12 57 n 3 (1 + , − ...a 3.12 1 + a (1 = 2 2 ˜ F ϕ n + d = 0 in the metric, then the mirror metric is obtained by 2 For the particular example of the deformed AdS 2 t ]. This is the concept of mirror duality. In this section we 2 ρ d tϕ 2 11 ρ = 0, the mirror metric is 2 . 35 κ G r 2 − 2 r ϕϕ − κ κ 1 + − , 1 /G κ 1 + 1 + superstring, = − n + We start by constructing the mirror model of the background ( 2 and 1 = κ ], where it was shown that the latter can be seen as the light-cone gauge − tt are flat indices and we choose the labelling of the transverse space to be 2 10 s G 36 n T , d × 35 n . . . a S 1 a × n Alternatively, this can be seen as doing a T-duality in 11 common to the theorydilaton and and its the mirror. R-R Intransformation fluxes order rule we to solve to disentangle the the the supergravity background contributions equations ( from of the motion. ApplyingThis this point of view has the advantage that the transformation of the dilaton under T-duality is known. gether. The mirror fluxes (tilded quantities) are related to the original fluxes through [ where In principle the B-field also transforms but since it is a closed two-form we will drop it alto- discussing the metric. Ifthat we there denote is the nointerchanging two cross light term cone directionssuperstring by with and parameter redefinition [ construct the mirror backgroundto of the ( full background, including the dilaton and theMirror R-R model. fluxes. there exists a moretigated physical interpretation in of [ this mirrortheory of model. a This free string questionAdS on was a inves- different, mirror, background.mirror Furthermore, background for can the also be reached directly from the original background by a field this is no longer theThis case is for in a non-relativistic particular theory, truegauge whose for fixing, mirror the describes whose worldsheet a mirror theory newthermodynamics defines model. of of a an this new AdS new two-dimensional superstring QFTof quantum that upon AdS/CFT field plays light-cone a theory. using central It integrability. role is in In the solving addition the spectral to problem being a useful tool, one may wonder if The mirror model of therole light-cone in gauge-fixed the string Thermodynamic introduced Bethe in Ansatz [ tions approach in and the the context calculation of of integrableperforming finite-size models a correc- [ double Wickthis rotation in transformation the does worldsheet not coordinates, affect Lorentz-invariant theories (up to a parity reflection), 3.4 Mirror model and mirror duality JHEP10(2019)049 κ × 2) | 3 1 (4.1) , (3.32) (3.31) (3.30) ) with (1 are non- psu 3.12 −⊗ # ϕ d ⊗− ϕ ∧ d or ), up to identifying ˆ ψ ∧ # , and then interchanging )d ψ 2 r κ 3.30 , r )d /κ 2 2 r − → −⊗− r 2 . to match the expressions. ρ = 1 # (1 − 2 2 2 4 J ) and ( κ κ ρ C (1 2 # ∧ 2 − κ ρ , r

− ⊗ − ⊗ superstring 3.28 and ψ ψ − κ ϕ − φ , 4 2 # # d d φ C φ T ) = 1 d ∧ ∧ − ⊗ − ↔ → d t t ⊗ − ⊗ − ⊗ −⊗ ∧ × ∧ ρ, r t )d )d t ( 3 2 2 )d ρ ρ S )d 2 superstrings, deformations based on R-matrices 2 ρ – 20 – 0 gives the mirror model of undeformed AdS , ϕ ρ 4 × ## # # r , ψ ρ κ T 3 (1 + → − ⊗ ⊗ − − ⊗ ⊗ − ⊗ −⊗ ,P × (1 + ↔ κ + (1 + − ) (1 + → # # 2 3 2 2 ρ r φ φ ρ S r 2 − ⊗ − − ⊗ − − ⊗ − d d κ 2 × ) ) ⊗ − # # ⊗ − # ∧ ∧ . Starting from the supergravity background ( 3 ) 2 ϕ ϕ ρ, r , ρ )(1 + ρ, r R ( 2 0 ( )d )d ρ, r t κ r Φ 2 2 ( P P 2 in the torus directions, defining ˆ r r 0 − P − Φ κ → 2 − − κe − t κ /κ (1 e i 0 (1 1 + x Φ Now that we have constructed the mirror model, we can prove that the 2 1 + − + (1 √ − → √ − e i x = = = 2 4 = 0, and rescaling Φ 2 C C − − κ e 2) Dynkin diagram is the same in both copies of the algebras, while in the remaining , as expected. | , 4 12 1 κ . , T κ eter deformation of the AdS We also exploit the gauge symmetries of the potentials = (1 × 12 + 3 are expected to leadpsu to generalised supergravitythree backgrounds. cases the In Dynkin the diagrams first are two different cases in the the two copies. corresponding to the following Dynkin diagrams For completeness and tocorresponding make to the deformations governed link by non-unimodular withsuperalgebra, R-matrices. R-matrices the associated For the literature, to the let Dynkinunimodular. diagrams us Thus, also for extract the AdS the fluxes indeed gives the mirror backgroundand defined through ( 4 Examples of generalised supergravity backgrounds for the two-param- κ together with the coordinates S Mirror duality. mirror duality extends to thegenerated full background by if the one stays R-matrix within the realm of deformations One immediately sees that the limit potentials of the mirror model JHEP10(2019)049 3 S . (4.5) × 5 (4.6) (4.7) (4.2) (4.3) (4.4) 3 F ? φ = d as 5 F n ∧ µ ϕ r d X d d ∧ ∧ t ψ d d are R-matrices asso- . , r ∧ · · · ∧ . ∧ ) 2). As before, we keep R 2 ) 1  |      ρ . ρ 0 0 µ 2 1 R 1 1 ,   , J R X ) d n φ − − φ d 2 d (1 ∧ d − n r 1 1 1 0 ...ν , (1 + and 3 1 ∧ 0 ∧ ν − − − ˆ , 2 − ...µ − F psu L r 1 R A ψ κ .  µ 2 − d n d R ). The background corresponding (1 φ of A κ − − # ρ ! d ∧ ...ν ∧ 1 κ n 2 1 r 2 − 1 0 3.8 # − 2 ν ρ r ψ 0 +1 d κ , n = − +1 +1 0+1 +1 +1 1 d d − + ∧ = diag( 2 n d − r       + 2 0 0 A 2 , ), both − ⊗ − ϕ ψ φ ρr ρ ...µ 2 J = d d ) R d 1 R − − ⊗ − J − 2 r µ ∧ ∧ ( κ κ ρ ∧ ∧ ,R # 2 − ˜ 1 G + + -form F ρ − ψ L 2 ˆ κ ) n κ  κ F − d d J – 21 – , R ρ √ 2 − − + ( r ,  ) − + ∧ ## ! 13 ∧ 2 (2) algebras and hence we are left with two such κ 2 − κ 0 1 1 n F t r ij ρ ) 1 ϕ = ( κ R d d ˆ su − + d 2 + 2 F = q M 2 ) and − ) κ  1 R ∧ ∧ n ij , ? 2 + 2 − acts on a ϕ ρr − 0 + ϕ ρ − ⊗ − κ κ i d 3.2 + ? 2 + R d d )d ) is then φ 3 − κ # 2 , ...µ 0 ˆ d ∧ ∧ 1 r F (1 + 1) and 1 1 + 1 + µ = − R − ⊗ − t t , ˆ ∧ )   F − d n − s ψ ij r (1 ) d , r ) N N N d d 2 ) 0 ?A = diag( su 2 r ( given in ( ∧ M ∧ R = 2 = = = 0 ρ + (1 ( ρ − 0 5 1 3 0 0 t ϕ R N d d R = ( F F F R )d (1 r ∧ 0 2 (1 + ρ  t ρ − − ) R d r κ κ ( 1 ρ + +  ˜ F κ κ (1 + )  ρ 1 ( + − − − F κ is the determinant of the metric. The self-duality condition for the five-form reads = = G 3 1 In our conventions the Hodge star ˆ ˆ F F 13 where were we introduced the auxiliary one-form and three-form To extract the fluxes we use theto same the parametrisation ( R-matrix We can then define the two followingsupercoset, R-matrices governing the deformation of the AdS the same action onR-matrices, the namely We start by considering R-matrices associated to the Dynkin diagram In other words, whenciated constructing to the distinguished Dynkin diagram 4.1 Dynkin diagram ( JHEP10(2019)049 (4.9) (4.8) ], ob- . The (4.10) , Φ 27 ) do not  φ , = d d 3.13 ) 2 − Z r κ 2 ( r − ˜ F κ 1 + , and then doing a + 1 q , x ψ = 0 and Interestingly, one can , and a remaining part d  I ) I 2 ρ 2 J ( ), since the deformations ρ ]. These equations depend F ∧ 4.6 3 = arcsin − 28 ˆ , F  2 + ) 27 κ + 2 + are both invariant under the trans- κ κ , we find that applying the transfor- − 3 2 , θ ˆ 3 , 1 F 2  (1 + + θy 2 . − 2 J ) κ J r ∧ gives the ABF-type background of [ ( ∧ 2 + ˜ 1 F + cos + 2 + 2 ˆ κ ) F 1 κ J – 22 – ρ  2 + − ( θy ∧ ϕ κ + 1 F . Indeed, these had different dilatons and since a d 1 κ ˆ − 2 2 F q ) and three-form 3 r ) 1 r 1 ? 2 − 2 + ( . = sin ˆ − ˜ κ κ F + F 2 1  3 ) , ˆ r F (1 + + 1 1 + 1 + ( ). It is also a generalised supergravity solution and has fluxes ˆ t   ˜ F 0 0 F d s R , x 2 N N N ) ρ 2 − log ρ = 2 = = = , ( 1 2 ) and dualising along one torus direction, say 0 θy ) yields a new background, which actually corresponds to choosing 5 1 3 F 1 + N R F F F ) satisfies the generalised supergravity equations of motion with 4.6 4.6 sin  ) + ) ρ = ( 4.6 − ), this is not the case of the background ( ( 2 − do not appear on an equal footing in the fluxes. Similarly to what we 0 0 1 κ F , which can be split into a background Killing vector R ) to (  θy 3.14 κ X = 0 and changing the sign of log (1 + 3.14 1 2 − + κ  = cos κ 1 x Starting from ( While the auxiliary one-form = 2 = d I . The standard supergravity equations of motion correspond to Z from one background to the other. metric and B-field preserving SO(2) rotation Relating the twoalso backgrounds go from by one TsT backgrounddepend to transformations. on the the other by deformationsupergravity doing parameters. backgrounds TsT transformations in This on section isTsT the torus a transformation that on new the feature torus that does was not not affect true the for dilaton, the it cannot be sufficient to go On the other hand,leave contrary the to backgrounds the invariant. supergravitylutions. Rather, case, they It the give is transformationsmatrices. two not ( new clear generalised if supergravity these so- correspond to deformations based on Drinfel’d Jimbo R- the R-matrix formations ( parameters observed for the two supergravity solutionsmations in ( section full background ( tained by starting from a supergravityformally solution dualising with the a metric dilaton and linearsolves the in the some fluxes generalised isometries in and supergravity thoseon isometries. equations a of The vector motion resulting background [ Z Setting JHEP10(2019)049 ) the 4.9 2 (4.14) (4.12) (4.13) (4.15) (4.11) P 14 1) and ). This , 0 0 R R (1 ) and ( − , . su , 0           4.11 R , 1 2 − J = ( 1 1 0 1 +1 1 +1 1 0 , ∧ 0 0 ) − − − − − 2 4 R (2), while for J 1 P su ∧ − R 1 and hence we are left with − 1 0 1 +1 01 +1 +1 1 0 1 ˆ , 0 +1 +1 +1 0 +1 F 0 − − − − +1 +1 0− +1 3 ) P R ?           ) thus interpolates between the R , = = are R-matrices associated to the 1). Constructing the backgrounds (1 + , , 4.11 R = 0. Starting from the intermediate − ⊗ N (1 R 2 + − κ . # su = diag( ). = 4 ) invariant and indeed ( κ ,  ,  + x 4 5 gives the intermediate background d κ and ij ij 4.9 coordinates. In particular the K¨ahlerform on the 1 + F R 1 ∧ 4.11 y L y M M 3 q ) for R x ij ij = 0 and is thus unaffected by this TsT trans- d i i 4.6 and – 23 – , . θ − − − 2). Again, we fix the action on the 3 x ) | , 2 2 ˆ 1 ) x F ) 2 + = = − ⊗ ⊗ − , d 3 κ N P ) = arcsin ), both ij ij ∧ (1 # r ) ) 1 R R θ ( = − ⊗ = 0 point of ( x ˜ − M M 3 F psu ) associated to the second R-matrix ,R = 0 point of ( ( ( , + ) )(1 + ⊗ − F 3 4 3 L = d ρ κ − 2 of ( P P P + 4.9 4 R κ κ x R F d ⊗ − ) is rotated by q , = ( ∧ − ⊗ ) coincides with ( 1 (1 + 3 ˆ 4.6 x R F ,R ,R # d q 4.11 ) and the N = diag( − ! ! 3 2 = 0. The intermediary background ( = 2 = y 4.6 ⊗ − ) and doing the same steps, but now with rotation angle R d 1 + N ∧ κ F 1 4.11 1 2 3 4 1 3 4 2 1 2 3 4 3 1 2 4 y

= d the central bosonic node corresponds to an element of = 0 point of ( = = 2 1 J 4 3 − P P P κ R For notational convenience we identify the = 0 point of ( (2) algebras to be the same as for the reference R-matrix 14 − corresponding to the two choices torus is For central bosonic node corresponds to an element of so when constructing Dynkin diagram su two such R-matrices, namely 4.2 Dynkin diagramLet ( us now consider R-matrices associated with the Dynkin diagram we arrive at the background ( second TsT now leavescoincide the when κ The formation. Indeed, ( background ( and finally T-dualising in the new coordinate JHEP10(2019)049 , ) ) ) − − κ , 4.9 4.2 L − (4.16) R = ) or ( + associated κ 4.6 R ] corresponds R = diag( ]. of ( 58 ) up to a change ) and taking the R 33 − and = 0 or equivalently κ 2) superalgebra. We 4.11 L 3.18 R | = η 1 D . , + 4 κ (1 R ). Therefore in these two psu superstring [ ] by using a descent procedure 3.21 = 0, or equivalently 6 . 58 T L η × ) give the same background. − ⊗ 2 R S ), either via field redefinitions and/or # ,R × − 2 4.11 ). It is not, however, invariant under the ⊗ − all R-matrices of the form − κ 3.14 ) is identical to ( – 24 – = = diag( # + A proposal for the background of the two-parameter R 3.20 κ ). Therefore we shall only discuss limits of the back- 2 − ⊗ − associated to the Dynkin diagram -deformed AdS η # L ). We thus conclude that the background of [ R . R superstring has been derived in [ we find that the R-R fluxes are the same as ( 4.11 4 D 3 ) with T R R ), which generate a new generalised supergravity solution. The latter × and thus is also a generalised supergravity background. Furthermore, 3 ). ,R S L ψ 3.13 × we find the same pp-wave background as for the two supergravity solu- R denoting any R-matrix) will give rise to the same background. On the other 4.13 3 ) here. First of all, implementing the transformations ( is a diagrammatic representation of the relations between the various gener- − and all R-matrices of the form 2 t ) are all related to the background ( then the choice of R-matrix in the right copy is not relevant since it is multiplied − 4.11 → ∞ κ = diag( − = 0 in the action. Therefore at − κ L 4.13 R Let us illustrate this for the R-matrices associated to the Dynkin diagram When the deformation parameter in the right copy is set to zero, Figure R = = η + + The corresponding background will interpolate between(depending the on point the specific choice of Cartan-Weyl basis in the left copy of the algebra) and by (the symbol hand, when the deformation parameterthen in it the left is copy theκ choice of R-matrix in the left copy that does not play any role. Hence at sociated to different Dynkin diagramstake in the two copies ofto the the Dynkin diagram κ ity background. On thethe other maximal hand, deformation it limitbehaviour remains and is a in reminiscent generalised particular of supergravity does the background not in match4.4 the mirror model. Different This DynkinLet diagrams us in finish the this section two by copies commenting on deformations constructed out of R-matrices as- T-dualities on the torusground (see ( figure limit tions. Second, thelimits Pohlmeyer the limit generalised ( supergravity background actually becomes a standard supergrav- to the choice ( 4.3 Limits The generalised supergravityand ( backgrounds associated to the Dynkin diagrams ( alised supergravity backgrounds that we have constructed. Comparison with the literature. deformed AdS involving the Page forms.up It to solves signs, the agrees generalised with supergravity ( equations of motion and, of sign in it is invariant underredefinitions the of transformations ( ( is nothing else than the background associated to the R-matrix for the first R-matrix JHEP10(2019)049 , 3 3 0 0 − R R κ = = • • − + (4.17) (4.18) = − + κ κ ) leaves κ κ + 0 0 0 κ 3.14 R R , namely 2 ) (depending ) background by 4 . The transforma- − ⊗ , the background 3.13 R 0 0 − # R κ − ⊗ − and = 0 0 0 + TsT depending on TsT depending on R κ invariant. Rather it generates = = • • 0 0 3 − + κ κ R R ). In particular, it will still be a or 4.9 0 , namely 2 R ) # ) or ( ) swaps the backgrounds corresponding to ⊗ − ⊗ −⊗ invariant and swaps the two backgrounds cor- 4.6 4 − ⊗ − 3.13 R # – 25 – # and − ⊗ ⊗ − ⊗ −⊗ ) followed by the field redefinition ( 3 # R − ⊗ − 4.11 superstring. The two vertical lines in the middle represent the ⊗ − # 4 4 T 0 0 ) or ( R × )) is represented by the blue (respectively red) arrow. ( = = • • 3 − + S 4.11 − + κ κ 3.14 . The transformation ( κ κ × 0 0 3 of ( R − κ and − 0 backgrounds by a field redefinition they are also related to the ), the analysis being similar for the other choice. At the point = R 0 0 R + 4.17 κ ) (respectively ( . A window into the space of generalised supergravity backgrounds for the two-parameter 2) superalgra, but unimodular in the other copy. Let us focus on the first Dynkin | but does not leave the backgrounds associated to and . The top right and bottom right tilted lines correspond to the two backgrounds based on 1 4 4 0 0 TsT depending on TsT depending on 0 , 3.13 Also R-matrices associated to the Dynkin diagrams R R R = = (1 • • − + κ κ psu diagram ( depending on thethe specific background will choice then of coincidegeneralised Cartan-Weyl either supergravity background. with basis ( On in the other the hand, when left copy of the algebra, or are of this type. The novelty here is that the R-matrix is non-unimodular in one copy of the the point on the specific choice of Cartan-Weyl basis in the right copy of the algebra). responding to and new generalised supergravity backgroundsalso (the correspond two tilted to lines particularthe on Drinfel’d the Jimbo left). R-matrices. Of ItTsT course, is transformations since not on they clear the are if torus. obtained these from deformation of the AdS two backgrounds based on R-matricesand associated to the DynkinR-matrices diagram associated ( to thetion Dynkin diagram ( ( the two backgrounds corresponding to Figure 2 JHEP10(2019)049 4 R T 2) | (5.1) × 1 , 3 ]. The , which S (1 R 45 η -deformed × η psu 3 = L ], based on R- ⊕ η L 61 2) | 1 , superstring. Studying (1 4 T superstring. On the other psu = 1 solutions of [ 4 × superstring and constructed 3 T a 4 S 2) superalgebra. × T | , × 1 3 ). This provides an example of a × , 3 S 3 (1 S × 3.12 3 ]. It would be interesting to explore × = 0 and psu ] also extends to the full supergravity 3 60 a 35 ) or ( 4) [ | ]. They differ, however, in the specific choice 2 supergravity backgrounds we found that they 3.11 , 4 32 – 26 – (2 -deformed AdS T η × psu 3 S × 3 ]. By taking particular singular boosts of the ⊗ − ⊗ − ⊗ ⊗ − ⊗ −⊗ 60 , 59 -deformed AdS η ˆ G. We then applied this result to the AdS × ˆ G Another interesting limit of the supergravity backgrounds is the type of homoge- Let us stress that the existence of two supergravity backgrounds is not particular to The two solutions have the same metric and B-field, but different dilatons and three- neous limit considered inbackgrounds [ one can construct homogeneousmatrices Yang-Baxter solving deformations the [ classical Yang-Baxterbeen equation. applied This to boosting find procedure newunimodular has homogeneous jordanian recently R-matrices R-matrices involving odd for generators and construct hand, the other background isboth not backgrounds finite are in finite thatbackground. but limit. only Motivated For the the by maximal this latterthat deformation observation solution limit, the we gives mirror constructed the its mirror dualitybackground mirror AdS previously in theory that observed and case. in showed [ corresponds to the usual one-parameter limits of the two have different Pohlmeyer andof maximal one deformation of limits. thefixing solutions gives Taking gives the Pohlmeyer the a reduced pp-wave Pohlmeyer theory supergravity of limit background the AdS whose light-cone gauge- to understand if theyfermionic T-dualities. can Such also a be transformation shouldinvariant. related modify the by dilaton a but leave set the metric of target spacethe dualities, two-parameter deformation for — this instance property is still present after setting of Cartan-Weyl basis used in the two copies ofform the and five-form R-R fluxes.of This a provides given further bosonic new background into examples supergravity.field of The different redefinition two embeddings solutions and are related swapping by a of complex the deformation parameters. It would be interesting the non-split modifiedsuperisometry classical algebra. Yang-Baxter The equation two on R-matricesdiagram both the correspond to the fully fermionic Dynkin and satisfy the unimodularity property of [ In this paper wedeformation proposed of an the explicitform Metsaev-Tseytlin formula action for for thetwo supercosets R-R supergravity with backgrounds fluxes isometry analogous oftwo group to solutions the of the correspond two-parameter to the two different choices of Drinfel’d Jimbo R-matrices satisfying two-parameter integrable deformation that interpolates betweenand a a generalised standard supergravity supergravity solution. 5 Conclusions will then become the supergravity solution ( JHEP10(2019)049 3 3 ] for (5.2) (5.3) ] only 1) can 26 , 33 is special [0 ) coincide, geometries 3 ∈ backgrounds 3 a 3.12 4 -deformations. η T and squashed S case, a family of × 3 3 6 S ) and ( T × superstring was found, × 3 3.11 6 = 0, then the curvature 2 S or squashed S T R due to the non-compactness , the case of AdS 3 η × × 5 ρ 2 2 , . S # × superstring, finding a supergravity 2 − ⊗ 5 = 0 or S and AdS # L 2 × η − ⊗ − -deformation is a Poisson-Lie symmetric 5 η – 27 – ]. It would be interesting to analyse what happens ] but the overall dressing phases obeying unitary, ## interpolating between 0 and 1 has been constructed -deformed AdS 62 43 − ⊗ ⊗ − a η # metric. Moreover, half of the supersymmetries of the − ⊗ − -deformed AdS 4 = 0 has explicit mirror duality. However, in [ η # ⊗ − a ). It remains to be understood why this is the case. 5.3 via T-duality [ 3 S × 3 = 1 solution. It remains unclear how the other solutions ] when including fermions remains a puzzling problem. Focussing on AdS a superstring does not match the expansion of the exact light-cone gauge fixed ] obtained by a descent procedure involving the page forms, which appear to 38 5 S 58 ] for the two supergravity solutions and compare them to each other. Deformed × 5 51 ]. The background at Another direction concerns the relation of the two-parameter deformation to other The presence of a naked singularity for some value of Furthermore, the fact that the perturbative worldsheet S-matrix obtained in [ Can these results be used to answer some of the questions raised in the context of To complete our study we also explicitly constructed generalised supergravity back- 45 -deformations of other-dimensional spaces? In the AdS with constant dilaton.are Backgrounds related containing to AdS warped AdS upon including fermions. known integrable deformations. Indeed, the While this singularity cannotand be offers avoided the for intriguing AdS possibilityof of the obtaining deformation a smoothsingularity parameters deformation. disappears. vanishes, Indeed, either The whenmetric, one metric together becomes with the one theoriginal theory of flat are preserved. a T The warped two supergravity AdS backgrounds ( strings [ R-matrices have been constructedbraiding in unitary [ and crossing symmetry remains to be found. of the original AdS space is also a long standing problem in the context of background with explicit mirror duality remains an important openthe question. AdS S-matrix of [ one could calculate the light-cone gauge fixed S-matrices describing scattering above BMN matching the be generated and ifwithin there or exists outside an the R-matrixbe realm giving interesting of rise to Drinfel’d study to Jimbo consistent thedown R-matrices. truncations interpolating to of To 4 background, the gain dimensions. different more AdS insight For it the might η supergravity solutions with parameter in [ one supergravity background of the The various generalised supergravity backgroundsnitions are and/or all T-dualities related on by the (complex)results torus. field of redefi- Moreover, [ we werecorrespond able to to the make choice the ( link with the ground considered in this paper. grounds associated to the Dynkin diagrams and this type of limit for the two inhomogeneous R-matrices and associated supergravity back- JHEP10(2019)049 ]. ]. 70 68 (A.1) workshop ] and [ 67 ]. The latter -deformation λ ]. Embeddings 66 a Wess-Zumino- 32 2 Pauli matrices g . × ⊕ ! g 1 = 1. In contrast to the | − model of [ q | 1 0 0 λ

= superstrings with pure NS-NS 3 3 , σ ] that its Poisson-Lie dual in the full su- ! i 65 0 Integrability, duality and beyond − i 0

– 28 – ] for related work in the context of homogeneous -deformation, but with q = 50 2 , σ metric into supergravity have been proposed in [ ! 3 S 0 1 1 0 ×

3 = -deformation always defines a critical string theory [ The gamma matrices are constructed out of the 2 1 λ σ ]. 69 -deformed theories. ] and it has been conjectured in [ ]. This would then open the door to the exciting possibility of finding CFT η 74 64 , – -deformed AdS 63 71 λ ]. Combining the WZW term with the two-parameter deformation, one can construct 4 Last but not least, for supercosets with superisometry algebra -deformation, the Pauli matrices. A Conventions A.1 Gamma matrices in Santiago de Compostelainteresting and discussions. I Many thanks would also likeR. to to B. Borsato, Hoare thank A. for the guidance Sfondrini, throughoutand organisers A. this S. and project, Tseytlin van participants and Tongeren for L. forno. Wulff helpful 615203 for comments from valuable on the discussions the European and draft. Research B. Council This Hoare under work the is FP7. supported by grant Acknowledgments Part of this work has been presented at the Yang-Baxter deformations). This is particularlybackgrounds interesting because could these then new be purerecently NS-NS analysed been with used integrability tofluxes and investigate [ CFT the techniques, dual whichduals CFTs have of of the AdS It would be interesting tounimodular obtain R-matrices a also give closed rise formula toStarting for Weyl-invariant theories from the in the NS-NS this undeformed more and general pure R-Rdoes setting. NS-NS fluxes not background and and turn see requiring on if that R-Rdeformations the fluxes, of deformation one the could WZW then point investigate (see the [ possibility of finding marginal mations considered in this paperconstructed and in compare [ them to the multi-parameter Witten term canflux be [ added toan the integrable three-parameter action, deformation corresponding of the to semi-symmetric the space introduction sigma model of [ NS-NS perisometry algebra is related viais yet analytic another continuation instance to of the η an integrable of the It is an interesting open question to find the Poisson-Lie duals of the two-parameter defor- model [ JHEP10(2019)049 ) C 2; | (A.9) (A.6) (A.7) (A.8) (A.5) (A.2) (A.3) (A.4) . The 4 (2 1 sl i 2). | 1 , . , . , . (1 a 2 3 , 1 1 , σ σ ! Γ 1 3 ⊗ ⊗ ! C σ psu ⊗ ⊗ 0 2 2 iσ ⊗ = σ σ 1 1 3 0 0 0 0 0 t 1 ) ⊗ ⊗ ⊗ iσ ⊗ 32 dimensional gamma

a 2 2 3 ⊗

1 × 1 2 Γ σ σ σ , 1 2 1 . C ⊗ 8 (1) generated by ( = ⊗ ⊗ ⊗ , u 1 , ⊗ 1 = ! 3 1 2 3 9 5 2 3 σ 1 σ σ ⊗ , 0 ⊗ 1 8 2 , ⊗ ⊗ ⊗ ⊗ 0) , 3 . σ 7 0 , 1 2 1 2 σ ,..., , 1) generators is 16 σ − σ σ σ ! ,J ,

1 ,L = = = = = (1 αβ 8 matrices. = 0 = 6 ! = ⊗ ) 1 3 5 7 9 2 0 with 8. In particular, we have ! a su 2 × 0 γ iσ ab iσ 2 − η 0 0 ) = diag(1 σ = ,..., , a 0 0 0 αβ

, a ) . The projector a ,H C 0 (

= 2 a γ α 6789 ¯ 0 γ 1 2 – 29 – ,Γ = 1 γ δ 1 2 } γ ,Γ ( b ,Γ,Γ 1 ab = 2 α → → = 0 ,Γ +

2 = η 1 σ 2 , ⊗ ,Γ 1 σ 2 a 16 = 3 ⊗ ⊗ 16 ⊗ ⊗ 1 = 2 Γ σ a 2 Proj Proj HM ( 1 1 { 1 are then identified using a a 1 σ 2) Γ 2 1 ⊗ | (1) subalgebra defines the superalgebra σ βγ γ γ + ⊗ ⊗ a ) u 1 ⊗ ⊗ ,L 1 γ 1 3 a ,J ⊗ , 2 H 1 σ σ γ † ⊗ ! 1 σ ( Proj Proj (1 ! 0 ⊗ ⊗ = 1 1 M 2) contains the 1-dimensional ideal ⊗ ⊗ b αβ 1 2 2 Our choice for the three | Proj = 9 0 0 iσ γ ⊗ 1 σ 1 1 σ σ psu Γ , We choose the following basis for the ten 32 1 0 0 0

+ ⊗ ⊗ ⊗ ⊗ 2) is given by those elements of the complexified superalgebra (1

iσ | 1 2 (2) generators is ··· 1 2 1 2 βγ 1 8 with spinor index 2) over this 1 2 σ σ σ σ − 1 su ) , | su = b 1 Γ × = = = = = (1 γ , = 1 0 ( 4 6 8 0 2 1 L Γ (1 su 16 chiral blocks J a αβ Γ Γ Γ Γ Γ su γ = × 11 Γ and for the three The superalgebra quotient of Bosonic generators. The real form satisfying where the arrow represents the projection ontoA.2 8 Generators of projects onto a 8-dimensionalthese spinor matrices 8 subspace and thus can be used to effectively make The ten 16 and satisfy They satisfy the Clifford algebra matrices, Gamma matrices. JHEP10(2019)049 ∈ (2) (2) R su su (A.12) (A.13) (A.14) (A.15) (A.10) (A.11) X ⊕ 1) , , , and ) . ) 3 ) 2 is the . (1 2 ), elements of 2 , L ) L ,L J 3 su , . 2) 3 − | − A.7 ,J = 1 , L 1 , 3 2 122 222 , 2 8, and we define the α J . L J Q Q (1 ) α diag( = = psu 4 8 ,..., − , . , while the ones of grade 3 , iQ . α α ∈ = = diag( = diag( = diag( ! ! we use the standard block- 1 = 1 a ˆ ˆ α α Q 2 5 12 45 ˇ ˇ P Q α α L α 0 R 0 P P J J N a X αβ 2) iN ], in particular we have the anti- | ¯ γ ,Q,Q 3 i 1 3 σ 32 , t σ t = diag( = ) 121 221 (1 ) ˆ α , } α ˆ Q Q , α , ˇ α , 2 0 ) β ˇ 0 α 1) spinor index and ˆ ) ) ) with 1 ) 2 psu 1 , N 1 = = Q 1 R N ( L J Q ( 3 7 3 (1 ⊕ , ,L 3 ,J − − 1 α 1 ,X , iσ σ , su L 2 1 L J 1 L

, 2) L J – 30 – ) ) | ) X {Q ˆ ˆ α α 1 α 2) superalgebra also contains eight fermionic gen- | , − − = diag( diag( 1 α α ,Q,Q iQ (1 , (ˇ (ˇ } i i 2 is the − − − β (1 , 2 2 , 1 = diag( = diag( 112 212 1 1 psu = = α √ √ 1 4 = diag( Q Q Q 2) | ), which permits any use, distribution and reproduction in psu , Q = 1 will do so if one imposes suitable reality conditions on the P P 02 35 1 α = = ). The elements of grade 0 generate the = = J J 1 α , α X ˆ ˆ 2 6 α α Q 2.3 α α The (1 2, ˇ {Q 1ˇ 2ˇ α , θ Q Q = diag( , , psu can be gathered into a single index, ) , = 1 α ) 3 , 1 3 ) α CC-BY 4.0 I ) L 2 J ,Q,Q 2 Q . They are defined through − − This article is distributed under the terms of the Creative Commons α ,L , ,J , 2 and ˆ 2 3 111 211 2 3 Q L J L J as Q Q α , ˇ α , where = = I . Furthermore, the generators are chosen so that they belong to a specific . ˆ α Q 1 5 ˇ α α R I θ Q Q 2) Q = diag( = diag( | = diag( = diag( 1 0 3 , 01 34 P P grading as defined in ( J J (1 4 Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. are denotes by Our choice of generatorscommutation matches relations the conventions of [ The ones of grade 2 are given by Finally, the fermionic generators of grade 1 are denoted by Z subalgebra and are A.3 Two copiesFor of elements in thediagonal Lie matrix algebra realisation psu While these generators themselvesthe do Grassmann not envelope satisfy thefermions reality condition ( The indices generators erators spinor index. We choose them to be Fermionic generators. JHEP10(2019)049 , 2 4 T ]. , ] ]. Int. J. /CF T × ]. 3 , 3 ]. ]. 2 2 superstring S SPIRE ]. 5 AdS Nucl. Phys. × SPIRE IN S SPIRE , 3 IN ][ IN × /CF T SPIRE SPIRE /CF T (1998) 253 ][ 3 3 5 ][ IN IN WZW model. II: SPIRE AdS 2 ) ]. ][ ][ IN (2014) 318 Protected string ]. AdS R AdS AdS ][ hep-th/9711200 , ]. ][ (2015) 023001 S-matrix for ]. ]. WZW model. III. SPIRE SPIRE SL(2 2 (2017) 091 ) B 879 IN ]. IN R SPIRE 1) WZW model. I: The | ][ , with NS-NS flux 04 A 48 IN ][ ) (1 SPIRE SPIRE 3 The complete R ][ arXiv:1407.7475 S IN IN , su arXiv:1406.6286 (1998) 231 SL(2 SPIRE and the arXiv:1207.5531 × ][ ][ [ hep-th/0111180 2 [ IN 3 3 JHEP arXiv:1209.4049 Correspondence and Integrability [ SL(2 hep-th/0005183 , ][ 2 J. Phys. ][ Nucl. Phys. [ , , AdS AdS ]. 2 Integrability and the and and the Adv. Theor. Math. Phys. 3 3 , /CF T (2015) 303] [ A dynamic (2014) 132 3 Giant magnon solution and dispersion relation (2012) 109 arXiv:1309.5850 SPIRE /CF T arXiv:1406.0453 AdS AdS 3 [ hep-th/0001053 10 IN (2013) 003 [ 10 AdS [ Microscopic formulation of black holes in string – 31 – (2001) 2961 (2002) 106006 ][ Strings in B 895 AdS 04 arXiv:0912.1723 arXiv:1106.2558 Integrability, spin-chains and the Derivation of the action and symmetries of the An integrable deformation of the [ [ 42 hep-th/0203048 with mixed flux JHEP [ JHEP , 4 D 65 , Strings in Strings in T (2014) 066 B-field in Spinning strings in JHEP × Adv. Theor. Math. Phys. ]. (2001) 2929 (2014) 051601 ]. ]. 3 [ 10 ]. S (2010) 058 (2011) 029 42 from worldsheet integrability × Classical integrability and quantum aspects of the (2002) 549 112 SPIRE 2 3 03 08 superstring Phys. Rev. SPIRE SPIRE arXiv:1211.5119 Erratum superstring IN J. Math. Phys. 5 SPIRE , JHEP [ [ IN IN 1 , S ][ 369 , IN Corrigendum ibid. AdS S ][ ][ [ × /CFT ][ The Large N limit of superconformal field theories and supergravity (1999) 1113 × 3 JHEP JHEP 5 3 , , 38 Towards integrability for S AdS Anti-de Sitter space and holography × J. Math. Phys. (2013) 113 (2012) 133 3 , ]. S Phys. Rev. Lett. Phys. Rept. (2014) 236 04 11 , , × 3 SPIRE -deformed arXiv:1701.03501 arXiv:1406.2971 arXiv:1311.1794 IN hep-th/9802150 worldsheet S matrix JHEP in string theory in [ correspondence AdS correspondence JHEP Theor. Phys. [ [ theory action q Euclidean black hole Correlation functions [ [ Spectrum B 888 spectrum in AdS B. Hoare, A. Stepanchuk and A.A. Tseytlin, R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefa´nski, O. Ohlsson Sax and B. Stefa´nskiJr., P. Sundin and L. Wulff, R. Borsato, O. Ohlsson Sax and A. Sfondrini, A. Babichenko, B. Stefa´nskiJr. and K. Zarembo, A. Cagnazzo and K. Zarembo, J.M. Maldacena, E. Witten, F. Delduc, M. Magro and B. Vicedo, F. Delduc, M. Magro and B. Vicedo, J.M. Maldacena, H. Ooguri and J. Son, J.M. Maldacena and H. Ooguri, J.R. David, G. Mandal and S.R. Wadia, A. Sfondrini, J.M. Maldacena and H. Ooguri, R. Hern´andezand J.M. Nieto, M. Baggio, O. Ohlsson Sax, A. Sfondrini, B. Stefa´nskiand A. Torrielli, [8] [9] [5] [6] [7] [3] [4] [1] [2] [17] [18] [14] [15] [16] [12] [13] [10] [11] References JHEP10(2019)049 10 10 B and , , 32 6 5 (2015) (2007) S T Phys. JHEP 12 , × × , JHEP d 12 5 2 , 10 -models S (2019) σ Sov. Sci. Rev. × AdS Nucl. Phys. (2009) 043508 -model , JHEP ]. 2 , σ JHEP 122 ]. , 5 , 50 S (2017) 053B07 Lett. Math. Phys. (2016) 415402 Weyl Invariance of , × Sov. Math. Dokl. SPIRE (2002) 051 SPIRE 5 , IN IN -deformed Scale invariance of the 2017 ][ A 49 ]. 12 η ][ AdS -deformed strings λ -deformed AdS η SPIRE ]. JHEP and Phys. Rev. Lett. IN J. Math. Phys. , J. Phys. -model and generalized , η , ][ , σ -deformed SPIRE η -deformations of integrable IN ]. ]. q mirror model as a string -model On q-deformed symmetries as Poisson-Lie ][ σ 5 arXiv:1811.07841 S ]. arXiv:1605.04884 [ Weyl invariance for generalized supergravity × SPIRE SPIRE [ ]. 5 IN IN S-matrix for strings on Puzzles of Prog. Theor. Exp. Phys. ]. and the Yang-Baxter equation – 32 – ][ ][ , SPIRE ) AdS Double Wick rotating Green-Schwarz strings g IN ]. On classical ( SPIRE arXiv:1904.06126 ][ U IN [ SPIRE (2019) 125 ]. Triangle equations and simple Lie algebras (2016) 174 ][ IN ]. 01 SPIRE ][ arXiv:1406.2304 06 ]. ]. -symmetry of superstring IN On String S-matrix, Bound States and TBA [ ]. κ Supergravity backgrounds of the SPIRE ]. ][ -models and dS/AdS T duality SPIRE IN Target space supergeometry of σ IN (2019) 063 JHEP ][ SPIRE SPIRE superstring, T-duality and modified type-II equations JHEP , ][ SPIRE arXiv:1308.3581 arXiv:1312.3542 IN IN , 5 SPIRE 05 [ [ IN ]. S ][ ][ IN ]. ][ × ][ arXiv:1511.05795 5 arXiv:1412.5137 [ (2014) 261605 Hopf algebras and the quantum Yang-Baxter equation JHEP [ SPIRE , SPIRE Yang-Baxter On integrability of the Yang-Baxter arXiv:1608.03570 IN AdS superstrings A q difference analog of IN Generalized Supergravity Equations and Generalized Fradkin-Tseytlin [ 113 (2013) 192 (2014) 002 [ 5 arXiv:1811.10600 S 11 04 [ × (1984) 93. arXiv:0710.1568 arXiv:1507.04239 (2016) 262 5 [ [ (2015) 027 -deformed arXiv:1606.01712 arXiv:1703.09213 hep-th/0210095 arXiv:0802.3518 05 symmetries and application to[ Yang-Baxter type models AdS 024 Rev. Lett. Counterterm (2016) 045 backgrounds from the doubled formalism [ String Theories in Generalized Supergravity111602 Backgrounds η 903 supergravity equations C 4 JHEP 049 (1985) 254 [ (1985) 63 [ [ JHEP G. Arutyunov and S.J. van Tongeren, F. Delduc, S. Lacroix, M. Magro and B. Vicedo, G. Arutyunov and S. Frolov, G. Arutyunov and S.J. van Tongeren, W. M¨uck, R. Borsato and L. Wulff, B. Hoare and F.K. Seibold, J.-i. Sakamoto, Y. Sakatani and K. Yoshida, J.J. Fern´andez-Melgarejo,J.-i. Sakamoto, Y. Sakatani and K. Yoshida, G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A.A. Tseytlin, A.A. Tseytlin and L. Wulff, G. Arutyunov, R. Borsato and S. Frolov, G. Arutyunov, R. Borsato and S. Frolov, V.G. Drinfeld, M. Jimbo, A.A. Belavin and V.G. Drinfel’d, C. Klimˇc´ık, F. Delduc, M. Magro and B. Vicedo, C. Klimˇc´ık, [36] [37] [34] [35] [31] [32] [33] [29] [30] [27] [28] [25] [26] [22] [23] [24] [20] [21] [19] JHEP10(2019)049 , , ] A 5 , , S Phys. ] ] , JHEP × Int. J. Nucl. , (1983) 5 , , ]. n -model J. Phys. S σ 17 ]. (2014) 1095 , AdS × SPIRE n (2018) 018 104 IN ]. SPIRE arXiv:1104.2423 supercosets 05 ][ AdS [ IN superstrings n ]. ][ 4 S arXiv:1411.1066 arXiv:1403.6104 superstring SPIRE [ [ × M 5 JHEP IN n Penrose limits and maximal S × , ][ SPIRE 3 × S IN 5 (2011) 161 AdS Funct. Anal. Appl. ]. Superstring. Part I ]. ]. ][ × , ]. ]. 5 (2015) 23 3 AdS S (2015) 106 Lett. Math. Phys. ]. , × SPIRE ]. B 851 SPIRE arXiv:1812.07287 5 SPIRE 182 The exact spectrum and mirror duality hep-th/0201081 Strings in flat space and pp waves from SPIRE SPIRE IN [ IN IN [ IN IN ][ SPIRE ][ ][ B 891 AdS -model ][ ][ IN SPIRE σ ]. hep-th/0202021 ][ IN [ ][ ]. deformed superstrings arXiv:0912.2958 Nucl. Phys. Supergravity backgrounds for deformations of η On deformations of – 33 – SPIRE (2002) L87 [ , IN (2019) 225401 SPIRE 19 Nucl. Phys. ][ , IN (2002) 013 Quantum deformations of the flat space superstring ][ On reduced models for superstrings on Pohlmeyer reduction of Theor. Math. Phys. A 52 arXiv:0711.0155 04 , arXiv:0806.2623 arXiv:1803.07391 arXiv:1002.1097 arXiv:0802.0777 What is a classical r-matrix? [ ]. [ (2010) 094 [ Foundations of the Tree-level S-matrix of Pohlmeyer reduced form of Towards the quantum S-matrix of the Pohlmeyer reduced [ [ ]. Quantum Deformations of the One-Dimensional Hubbard Model arXiv:1510.02389 02 Marginal deformations of WZW models and the classical arXiv:1411.1266 [ JHEP [ SPIRE , J. Phys. SPIRE superstring theory IN , IN superstring 5 ][ (2008) 450 JHEP On pp wave limit for (2018) 417 S arXiv:0901.4937 η ][ (2008) 2107 , ) [ 5 × Class. Quant. Grav. arXiv:1403.5517 S (2011) 265202 (2008) 255204 5 , [ (2015) 259 ]. × B 800 Integrability of the bi-Yang-Baxter A 23 supercoset string models B 781 The Classical Trigonometric r-Matrix for the Quantum-Deformed Hubbard Chain 5 (2016) 026008 AdS Towards a two-parameter q-deformation of AdS Trivial solutions of generalized supergravity vs. non-abelian T-duality anomaly n A 44 A 41 ]. ]. ]. S AdS superYang-Mills SPIRE ( B 891 × D 93 IN n [ (2014) 002 (2009) 254003 = 4 SPIRE SPIRE SPIRE arXiv:1801.07680 IN IN arXiv:1402.2105 IN 06 Mod. Phys. Rev. supersymmetry [ Phys. Lett. Yang-Baxter equation N 42 259 AdS [ Nucl. Phys. [ Phys. [ superstring theory J. Phys. version of J. Phys. of the [ M. Grigoriev and A.A. Tseytlin, A.land Pacho S.J. van Tongeren, M. Blau, J.M. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, D. Roychowdhury, B. Hoare, R. Roiban and A.A. Tseytlin, R. Borsato and L. Wulff, D.E. Berenstein, J.M. Maldacena and H.S. Nastase, G. Arutyunov and S. Frolov, M.A. Semenov-Tian-Shansky, L. Wulff, O. Lunin, R. Roiban and A.A. Tseytlin, M. Grigoriev and A.A. Tseytlin, B. Hoare, C. Klimˇc´ık, N. Beisert, B. Hoare and A.A. Tseytlin, G. Arutyunov, M. de Leeuw and S.J. van Tongeren, B. Hoare and A.A. Tseytlin, N. Beisert and P. Koroteev, [55] [56] [52] [53] [54] [50] [51] [47] [48] [49] [45] [46] [43] [44] [41] [42] [39] [40] [38] JHEP10(2019)049 ] ] , B J. 5 , , (2011) S ] at ]. 3 × (2018) ]. SciPost 5 3 , S 05 A 44 SPIRE × AdS Phys. Lett. IN SPIRE 3 , hep-th/0510171 hep-th/9509095 ][ IN (2014) 164 [ [ JHEP -models related to ][ ]. AdS , σ 12 J. Phys. 3 and supergravity , 5 ]. arXiv:1811.00453 SPIRE [ JHEP IN (1996) 116 (2006) 288 Three-parameter integrable Superstrings on AdS , ]. ][ ]. -deformed 46 λ SPIRE ]. and the symmetric orbifold of IN ]. 3 SPIRE B 736 ][ arXiv:1504.07213 SPIRE An Integrable Deformation of the arXiv:1409.1538 IN [ (2019) 109 IN [ ][ AdS SPIRE ]. ][ The Worldsheet Dual of the Symmetric 01 SPIRE IN ]. -deformations IN ][ -deformation of CFTs and integrability λ ][ λ Jordanian deformations of the ]. SPIRE Embedding the modified CYBE in Supergravity Wrapping interactions and a new source of Nucl. Phys. JHEP (2015) 448 ]. IN Classical Integrability of the Squashed SPIRE arXiv:1606.00394 , , [ ][ IN – 34 – ][ (2014) 495402 SPIRE IN SPIRE Tensionless string spectra on AdS arXiv:1812.01007 [ B 897 ]. [ IN Nucl. Phys. Proc. Suppl. String theory on arXiv:1401.4855 ]. , On Jordanian deformations of AdS ][ [ A 47 Spacetimes for arXiv:1806.02602 (2016) 685 [ SPIRE On integrable deformations of superstring arXiv:1803.04420 Supergravity background of the ]. The most general and Schr¨odingerSpacetime via T-duality IN SPIRE arXiv:1605.03554 [ 3 Dual non-Abelian duality and the Drinfeld double [ ][ IN (2019) 103 Nucl. Phys. J. Phys. ][ B 910 , , (2014) 153 AdS SPIRE arXiv:1812.04033 04 [ IN (2018) 854 arXiv:1904.08892 Unimodular Jordanian deformations of integrable superstrings 04 ˇ permutation supercosets [ Severa, ][ (2018) 204 4 arXiv:1903.00421 hep-th/9502122 , Z [ JHEP C 78 08 ´ O. Colg´ainand H. Yavartanoo, , (2016) 434006 JHEP Poisson-Lie T duality supercosets Superstring Nucl. Phys. , (2019) 094 , n 5 ]. ]. ]. S (2019) 011 arXiv:1011.1771 S 03 [ JHEP A 49 7 × × arXiv:1803.04423 (1995) 455 5 n [ SPIRE SPIRE SPIRE = 1, IN arXiv:1410.1886 IN IN Liouville theory 085 k Product CFT JHEP deformation of [ [ supercoset [ AdS AdS Three-sphere, Warped 115401 351 Phys. superstring corrections to the spin-chain/string duality [ Eur. Phys. J. Phys. L. Eberhardt and M.R. Gaberdiel, M.R. Gaberdiel and R. Gopakumar, G. Giribet, C. Hull, M. Kleban, M. Porrati and E.L. Rabinovici, Eberhardt, M.R. Gaberdiel and R. Gopakumar, G. Georgiou and K. Sfetsos, F. Delduc, B. Hoare, T. Kameyama, S. Lacroix and M. Magro, K. Sfetsos and D.C. Thompson, Y. Chervonyi and O. Lunin, C. Klimˇc´ık, B. Hoare and A.A. Tseytlin, T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, D. Orlando, S. Reffert and L.I. Uruchurtu, P. C. Klimˇc´ıkand S.J. van Tongeren, I. Kawaguchi, T. Matsumoto and K. Yoshida, T. Araujo, E. B. Hoare and S.J. van Tongeren, J. Ambjørn, R.A. Janik and C. Kristjansen, [74] [71] [72] [73] [69] [70] [67] [68] [64] [65] [66] [62] [63] [60] [61] [58] [59] [57]