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a recently proposed model of 3d massive gravity @xcite , dubbed `` minimal massive gravity '' ( mmg ) , has bulk properties that are identical to those of `` topologically massive gravity '' ( tmg ) ( which propagates a single massive spin-@xmath0 mode @xcite ) but its boundary properties ( for ads asymptotics ) are different . specifically , mmg evades the `` bulk / boundary clash '' of tmg ; this is the impossibility ( for tmg ) of arranging for both central charges of the asymptotic conformal symmetry algebra to be positive while also arranging for the bulk mode to have positive energy .    in this paper we present a greatly simplified , and geometrical , analysis of the unitarity conditions of mmg . our results confirm those of @xcite but we also consider a slightly larger class of models by leaving free the normalisation of the parameters of the mmg action , and we cut in half the relevant parameter space by establishing equivalence under a `` duality '' transformation in the full parameter space . the final result is that unitarity restricts the parameters to a connected region of parameter space , up to equivalence . it has been proposed @xcite that the relevant asymptotic symmetry algebra for 3d gravity with flat - space asymptotics is the 2d `` galilean conformal algebra '' ( gca ) , obtained by contraction of the 2d conformal algebra . based on this proposal , flat space holography for tmg was initiated in @xcite , and it was argued that unitarity constraints on the central charges of the gca algebra could be satisfied only in a limit for which tmg degenerates to 3d conformal gravity , which has no local degrees of freedom . here we show that the same analysis , applied to mmg , leads to a flat space mmg model that is not conformal and still has local degrees of freedom , although its linearised limit coincides with linearised 3d conformal gravity , implying a linearisation instability . the tmg action can be written as the integral of a lagrangian 3-form constructed from three lorentz - vector one - forms : the dreibein @xmath1 , the ( dual ) lorentz connection @xmath2 and a lagrange multipler field @xmath3 imposing a zero - torsion constraint @xcite . using a 3d vector algebra notation for lorentz vectors we can write this lagrangian 3-form as @xmath4 = -\sigma e\cdot r + \frac{1}{6}\lambda_0\ ,   e\cdot e\times e +   h\cdot t   + \frac{1}{\mu } l_{lcs}(\omega)\ , , \ ] ] where @xmath5 and @xmath6 are the torsion and curvature 2-forms , respectively , and @xmath7 is the lorentz - chern - simons ( lcs ) 3-form for @xmath2 . this lagrangian 3-form is parametrized by the mass parameter @xmath8 and the dimensionless constants @xmath9 . we refer the reader to @xcite for details of conventions . the 1-form fields @xmath10 can be eliminated by their joint equations of motion , leading to an action for the 3-metric alone . for a particular choice of units for the 3d newton constant @xmath11 ( which has dimensions of inverse mass for unit speed of light ) , the resulting action is @xmath12 = \sigma i_{eh}[g;\lambda ] + \frac{1}{\mu } i_{lcs}[g]\ , , \ ] ] where @xmath13 is the ( 3d ) einstein - hilbert action , together with a cosmological constant term ( @xmath14 ) ; the lcs term is now the chern - simons action for the levi - civita affine connection . variation of this action yields the tmg equation @xmath15 where @xmath16 is the einstein tensor and @xmath17 the cotton tensor . the asymptotic symmetry algebra for an asymptotic ads vacuum with cosmological constant @xmath18 is the 2d conformal algebra @xcite . for tmg the ( left / right ) central charges are @xmath19 in the context of an ads / cft interpretation , this is the leading order result in a semi - classical approximation that is valid ( re - instating factors of @xmath20 ) when @xmath21 , and quantum consistency requires @xmath22 for some integer @xmath23 ( see e.g. @xcite ) . we may assume , without loss of generality , that @xmath24 given this , we see from ( [ cpm ] ) that @xmath25 will be negative and the ( putative ) dual cft non - unitary , unless @xmath26 . however , the spin-2 bulk mode is a `` ghost '' ( i.e. has negative energy ) unless @xmath27 . this is the bulk / boundary unitarity clash . the mmg action can not be written in terms of the metric alone but it can be written in terms of the @xmath28-form lorentz - vector fields @xmath29 . the lagrangian 3-form is @xcite @xmath30= l_{tmg}[e,\omega , h ]   + \frac{\alpha}{2 } e\cdot h\times h\ , , \ ] ] where @xmath31 is a new dimensionless constant ; for reasons explained in @xcite , the parameters are restricted such that @xmath32 the 1-form fields @xmath10 can not be consistently eliminated from the action but the complete set of equations of motion determine them uniquely in terms of the dreibein and its derivatives . when the resulting equation of motion is expressed in terms of the metric , one finds that @xmath33 where . ] @xmath34 and @xmath35 the @xmath36-tensor is @xmath37 where @xmath38 ( the 3d schouten tensor ) and the scalar @xmath39 is its trace . notice that the @xmath36-tensor involves only second derivatives of the metric , so the characteristics of the mmg equation will be those of the tmg equation , found by setting @xmath40 . we defer to @xcite for an explanation of how the mmg equation manages to be consistent with bianchi identities despite the fact that @xmath41 . the mmg action is proportional to the integral of the lagrangian 3-form ( [ mmgact ] ) with a proportionality constant that is itself proportional to @xmath42 . a rescaling of @xmath11 can be compensated by a rescaling of the mmg parameters . specifically , if @xmath43 , for _ positive non - zero _ constant @xmath44 , then the action is unchanged if @xmath45 and @xmath46 given that @xmath47 , this scaling allows us to choose a normalisation for the parameters such that @xmath48 , and this choice was made in @xcite . for tmg it makes sense to make this choice since setting @xmath49 leads to inconsistent field equations unless one also sets @xmath50 and then one is left with 3d conformal gravity , which has no local degrees of freedom . for mmg it is perfectly consistent to set @xmath49 , and allowing for this case leads to significant simplifications . for this reason , we impose no restriction on the parameter @xmath51 . we could use the scaling symmetry to set @xmath52 . this is especially attractive since unitarity of mmg ( with ads asymptotics ) requires @xmath53 @xcite so , anticipating this result , we could restrict to @xmath54 . however , imposing a normalisation condition on @xmath31 would complicate any discussion of the tmg limit . for this reason , we shall proceed without making any choice of normalisation for the mmg parameters . let us define a new set of fields @xmath55 for the action ( [ mmgact ] ) by setting @xmath56 this is an invertible field redefinition and therefore has no physical effect . however , the action in terms of the new fields has exactly the same form as it had for the old fields but in terms of the new parameters @xmath57 in other words , the map @xmath58 takes one mmg model into an equivalent mmg model . in particular , the mmg field equation ( [ sourcefree ] ) is unchanged because the parameters @xmath59 are @xmath60-inert . notice that the @xmath60-map is an involution ( @xmath61 ) so it relates _ pairs _ of equivalent models ; it is a `` duality '' in parameter space . notice also that @xmath62 recall that @xmath63 by definition , so @xmath60 has no fixed points . it follows that we may choose @xmath64 to have either sign , without loss of generality . every mmg model for which @xmath65 is equivalent to one with @xmath66 . this result raises a puzzle . when @xmath66 we can take the @xmath67 limit to recover tmg from mmg . given equivalence under the action of h , there should also be a limit to tmg when @xmath65 , but this can not be a simple @xmath67 limit because this would change the sign of @xmath64 . the resolution of this puzzle is that there is indeed another , less obvious , tmg limit when @xmath65 , which may be deduced by following the consequences of the @xmath60-map . one takes @xmath68 in such a way that for , non - zero @xmath31 , @xmath69 where @xmath70 and @xmath71 are finite in the @xmath67 limit . this limit now yields a tmg model with @xmath72 maximally symmetric solutions of the mmg equation ( [ sourcefree ] ) are characterized by the value of the cosmological constant @xmath73 , which can be defined ( in the context of a vacuum solution ) via the equation @xmath74 . substitution into the mmg equation yields the relation @xmath75 in the case of an ads vacuum we have a unitarity constraint arising from the requirement of positive central charges for the asymptotic 2d conformal algebra , in addition to the bulk unitarity constraints . the unitarity conditions found for mmg in @xcite are    * no - tachyon condition : @xmath76 * no - ghost condition . when combined with the no - tachyon condition this becomes the inequality @xmath77 * positivity of both central charges : @xmath78 this inequality is valid on the assumption ( which we make without loss of generality ) that @xmath79 ( recall that @xmath80 ) and saturation of it defines , in analogy to tmg @xcite , the `` chiral '' limit . properties of chiral mmg have been investigated recently in @xcite .    for tmg it is obvious that the chiral limit implies saturation of the no - tachyon condition . this is no longer obvious for mmg but it is true nevertheless , as will emerge from the analysis to follow . we now define new dimensionless variables @xmath81 by @xmath82 observe that the inequality @xmath83 follows immediately from our assumption that @xmath84 . it also follows that @xmath85 where the final equality follows from ( [ quad ] ) . the new notation also simplifies the result of @xcite for the central charges of the asymptotic 2d conformal algebra , which can now be written as @xmath86    the unitarity conditions can now be written as linear inequalities in the @xmath81-plane :    * no - tachyon : @xmath87 . * no - ghost : @xmath88 . * positive central charges : @xmath89 .    to summarise , the `` allowed '' region in the @xmath81-plane is defined by the linear inequalities @xmath90 within this allowed region , each point on the `` mmg hyperbola '' , @xmath91 corresponds to an mmg model with ads@xmath92 vacuum satisfying all ( bulk and boundary ) unitarity conditions . observe that the boundary lines @xmath93 and @xmath94 meet _ on the hyperbola _ at the `` chiral '' point with coordinates @xmath95 this confirms our earlier claim that the no - tachyon condition for mmg is saturated at the critical point . the other intersection of the two lines @xmath93 and @xmath94 is also on the hyperbola , but this corresponds to @xmath96 , and hence to @xmath97 . so boundary unitarity is violated at this other chiral point , although it is also true ( as we shall see shortly ) that some unitarity condition is violated in any neighbourhood of either chiral point . we will first use this new geometrical framework to recover standard results for tmg . we can do this by setting @xmath98 , in which case the mmg hyperbola becomes the tmg parabola @xmath99 the inequalities ( [ ineq4 ] ) simplify to ( i ) linear inequalities in the @xmath81-plane , @xmath100 which exclude the entire plane unless @xmath26 , plus ( ii ) the ( no - ghost ) restriction @xmath27 on the parabola parameters . these are contradictory requirements , so _ the unitarity conditions can not be satisfied_.      the inequalities ( [ ineq4 ] ) imply that @xmath101 and hence that @xmath102 but this contradicts the requirements that @xmath87 and @xmath83 unless @xmath103 equivalently , unitarity requires @xmath104 , in agreement with the result of @xcite .    given @xmath53 , the second inequality of ( [ ineq4 ] ) can be written as another upper bound on @xmath105 : @xmath106 and this is a stronger upper bound than @xmath87 when @xmath107 . otherwise it is weaker , so    * @xmath108 if @xmath107 . in this case @xmath109 but , since @xmath110 , the chiral point is outside the allowed region . see figure [ ychpve ] . * @xmath87 if @xmath111 . in this case @xmath112 , so the chiral point is again outside the allowed region . see figure [ ychnve ] . we learn from this that _ the chiral point is always outside the allowed region_.    ) for @xmath113 and @xmath114 . the region defined by the inequalities ( [ ineq4 ] ) is shaded in blue . the chiral point is the intersection of the hyperbola with the solid blue line , which is parallel to the left asymptote.[ychpve ] ]    ) for @xmath113 and @xmath115 . the region defined by the inequalities ( [ ineq4 ] ) is shaded in blue . note that the right - hand branch is now excluded outright ; the chiral point is at @xmath112 on this excluded branch.[ychnve ] ]    the mmg hyperbola crosses the @xmath105-axis at @xmath116 and at the hyperbola degenerates into its asymptotes , but this case is excluded by definition of mmg . ] @xmath117 for future use we record here that @xmath118    the above facts suggest that we should organize the analysis according to the signs of @xmath64 and @xmath119 . as observed earlier , we have seen that we may choose either sign for @xmath64 without loss of generality but we will ignore this fact , for now , in order to simplify comparison with the results of @xcite . this means that we have four generic cases to consider , plus the two special cases for which @xmath120 or @xmath121 . each of the four generic cases corresponds ( given @xmath122 ) to a definite sign of @xmath123 which we will also need to know .    1 . @xmath124 . in this case @xmath125 , so the right - hand branch of the hyperbola is the one passing through the origin . the chiral point is on this branch . the two conditions imply @xmath126 this corresponds to the third of the possibilities deduced in @xcite . + the allowed region in the @xmath81-plane lies to the left of the line @xmath127 , which excludes the entire right - hand branch of the hyperbola . but @xmath128 , so the entire upper - half of the left - hand branch is inside the allowed region . 2 .   @xmath129 . again @xmath125 , so the right - hand branch passes through the origin . this case occurs when @xmath130 this corresponds to the first of the possibilities deduced in @xcite . + the allowed region now lies to the left of the line @xmath93 , and under the line @xmath131 , which now passes through the chiral point on the _ lower - half _ of the right - hand branch of the hyperbola . this again means that the entire right - hand branch is excluded while the entire upper - half of the left hand branch is allowed . 3 .   @xmath132 . in this case @xmath133 , so the left - hand branch of the hyperbola is now the one passing through the origin . this case is only realised when @xmath134 this corresponds to part of the second possibility deduced in @xcite . + the allowed region lies to the left of the line @xmath93 , but since @xmath135 , the entire upper - half of the left - hand branch of the hyperbola is in the allowed region . the entire right - hand branch is excluded because the line @xmath131 passes through the chiral point on the _ lower - half _ of the right - hand branch of the hyperbola . 4 .   @xmath136 . again @xmath133 so the left - hand branch of the hyperbola passes through the origin . this case occurs when @xmath137 this corresponds to another part of the second possibility deduced in @xcite . + the allowed region lies to the left of the line @xmath127 . but @xmath138 , so the entire right - hand branch of the hyperbola is excluded while the left - hand branch is allowed . notice that @xmath139 is @xmath60-inert , so cases @xmath140 and cases @xmath141 are both `` @xmath60-pairs '' . in all cases only the upper half of the left - hand branch is allowed by unitarity . to summarise , cases 1&2 cover @xmath113 _ except _ @xmath142 and cases 3&4 cover @xmath143 _ except _ @xmath121 . we now deal with the two special cases @xmath142 and @xmath121 . as these are mapped into each other by @xmath60-duality , we can limit the discussion to @xmath142 without loss of generality . at @xmath49 the previous discussion goes through except that the two figures degenerate into a single figure with the chiral point moving to the origin and the `` chiral boundary line '' becoming @xmath144 . the upshot is that the four cases have now been reduced to just _ two _ cases , related by @xmath60-duality . these are @xmath145 and in both cases only the upper half of the left - hand branch of the mmg hyperbola is allowed by unitarity . the results of @xcite for the regions in the mmg parameter space allowed by unitarity have been recovered , and extended to include the @xmath142 case . however , each of the three allowed parameter regions found in @xcite came with a sign restriction . what we now aim to show is that this sign restriction is equivalent to the statement that the allowed segment of the mmg hyperbola is on its left - hand branch . a point on the mmg hyperbola is on the left hand branch if and only if @xmath146 but a sufficient conditon that applies in either case is @xmath147 , or @xmath148 using the definition of @xmath105 given in ( [ xandy ] ) , we may rewrite this inequality as @xmath149 on the other hand , equation ( [ quad ] ) viewed as a quadratic equation for @xmath150 has the solution , as was done there . ] @xmath151\ , .\ ] ] the sign choice appearing here enters into the unitarity analysis of @xcite . to see the implications of this sign , we substitute for @xmath150 in ( [ signineq ] ) to deduce the inequality @xmath152 as the left hand side must be positive , we conclude that the sign choice is correlated with the sign of @xmath153 :    * @xmath154 requires the _ bottom sign_. this applies to case 1 . * @xmath155 requires the _ top sign_. this applies to cases 2 , 3 and 4 . this result agrees with @xcite . cases 3&4 , together with the special case @xmath121 , jointly correspond to the second possibility deduced in @xcite . cases 1&2 correspond to the other two possibilities found in @xcite . because of the @xmath60-duality , we lose no generality by focusing on the cases @xmath156 if we also allow for @xmath49 . then @xmath66 and every point on the left - hand mmg hyperbola with @xmath83 has @xmath157 . proceeding as before we now find that @xmath158 and by squaring we deduce that @xmath159 each choice of @xmath160 satisfying this inequality corresponds to one point on the upper left - hand branch of the mmg hyperbola . we also learn that @xmath161 which confirms the reality of the square root in the expression on the left hand side of ( [ proceed ] ) . recall that @xmath162 . this shows that the points at which the mmg hyperbola crosses the @xmath105-axis correspond to flat space limits of ads@xmath92 . there are two such points , one on the right - hand branch and another on the left - hand branch . the flat space limit on the left - hand branch is on the boundary of the region allowed by bulk and boundary unitarity . strictly speaking , this boundary point in the full parameter space ; this becomes a curve after a choice of normalisation of parameters . ] is not in the allowed region . although the bulk unitarity conditions have a smooth flat - space limit , the asymptotic symmetry algebra changes .    it has been argued in @xcite that the holographic dual to a 3d bulk gravity theory with flat - space asymptotics should be a theory invariant under the 2d `` galilean conformal algebra '' ( gca ) . this is a wigner - inn contraction of the more usual virasoro@xmath163virasoro symmetry algebra . the _ non - zero _ commutators of the gca are @xmath164 & = & ( m - n ) l_{m+n } + \frac{c_1}{12 } \left(m^3-m\right ) \delta_{m+n } \nonumber \\ \left [ m_n , l_n \right ]   & = & ( m - n ) m_{m+n } + \frac{c_2}{12}\left(m^3-m\right ) \delta_{m+n}\ , , \end{aligned}\ ] ] and the central charges @xmath165 and @xmath166 are limits of linear combinations of the central charges @xmath167 of the 2d conformal algebra : @xmath168 \ , .\ ] ]    for mmg we have , using the formula ( [ cpm1 ] ) and the fact that @xmath169 as @xmath170 for fixed @xmath8 , @xmath171 where @xmath172 is one of the two values of @xmath105 on the mmg hyperbola when @xmath173 ; i.e. _ either _ @xmath174 _ or _ as explained in subsection [ subsec : duality ] , we may choose the mmg parameters so that @xmath66 without loss of generality . making this choice ( because the tmg limit is then simple ) we have @xmath125 , so that @xmath174 on the right - hand branch of mmg hyperbola and @xmath175 on the left - hand branch . thus @xmath176 as only the right - hand branch survives ( as a parabola ) in the tmg limit , we deduce that @xmath177 for flat - space tmg , in agreement with @xcite . on the left - hand branch we have @xmath178 not only is this a new possibility , not available to tmg , but also it is the value of @xmath166 found by choosing the flat - space limit on the boundary of the segment of the mmg hyperbola allowed by unitarity . it was argued in @xcite that , under certain conditions , the 2d gca algebra admits a unitary truncation to one copy of the virasoro algebra and the boundary theory becomes the chiral half of a cft . specifically , the conditions are that @xmath179 where @xmath180 is the eigenvalue of the @xmath181 generator , equal to the graviton mass . in @xcite these conditions were satisfied by taking a @xmath182 limit in which flat - space tmg degenerates to 3d conformal gravity . as just observed , mmg has _ two _ flat - space limits , one on the left - hand branch of the mmg hyperbola and one on the right - hand branch , but the condition @xmath183 can not be satisfied on the left - hand branch . however , setting @xmath142 makes both @xmath166 and @xmath180 vanish at the flat - space limit on the right - hand branch . at this point , the mmg equation degenerates to @xmath184 taking the trace we deduce that @xmath185 the left hand side of this equation is the curvature - squared scalar of new massive gravity ( nmg ) @xcite . there are some similarities here with `` new topologically massive gravity '' ( ntmg ) @xcite for which the field equation is the same as ( [ flat ] ) but with the @xmath186-tensor of nmg replacing the @xmath36-tensor of mmg ; as observed in @xcite , both tensors have the same scalar trace . the @xmath36-tensor does not contribute to linear order in the equation ( [ flat ] ) , so its linearisation ( about flat space ) is just linearised 3d conformal gravity , which does not propagate any modes . in contrast , the non - linear theory does have local degrees of freedom ; a slight modification of the hamiltonian analysis of @xcite ( to allow for @xmath49 ) shows that the physical phase - space of the non - linear theory has dimension @xmath0 per space point . there is therefore a linearisation instability of the mmg model with field equation ( [ flat ] ) , analogous to the linearisation instability of ntmg @xcite . linearisation instabilities of this type occur when the linearised theory has a gauge invariance that is explicitly broken by interactions in the full theory . this phenomenon does not occur for tmg ( the linearisation instability at the chiral point discussed in @xcite is of another type ) but it does occur for nmg at partially massless ads vacua @xcite , where the accidental gauge invariance of the linearised theory is again linearized weyl invariance . the ( unique ) ads vacuum at the `` merger point '' of mmg is `` partially massless '' , implying a linearisation instability in that case too @xcite ; in fact , equation ( [ flat ] ) is the flat space limit of the mmg equation at its merger point . we have presented a geometrical re - analysis of the bulk and boundary unitarity condition found in @xcite for the 3d `` minimal massive gravity '' ( mmg ) model with ads@xmath92 asymptotics . the four ( un - normalized ) dimensionless parameters of the mmg action are considered in two pairs ; one pair ( essentially , the cosmological constant and cosmological parameter in units of the square of the tmg mass parameter ) parametrises a plane . the unitarity conditions are linear inequalities in the plane but an ads@xmath92 vacuum exists only for points lying on a hyperbola in this plane . we have called this the mmg hyperbola ; it degenerates to a parabola in the tmg limit . both the linear inequalities and the hyperbola depend on the other two parameters , which are the coefficient @xmath51 of the einstein - hilbert term and the coefficient @xmath31 of the new mmg term , which are restricted by the condition @xmath63 . the linear inequalities are inconsistent unless @xmath53 , in which case they determine an `` allowed '' region in the plane . in all cases , only the upper left - hand branch of the mmg hyperbola is in this region , and this corresponds to a sign choice in the quadratic equation that gives the ads radius in terms of a cosmological parameter @xmath160 , which is restricted only by a simple inequality .    in our analysis , the values of @xmath51 for which @xmath187 are special , and this leads to four cases , depending on the signs of @xmath139 and @xmath64 . only three cases arose in the analysis of @xcite because @xmath49 was excluded but @xmath188 was not singled out as special . once both the @xmath49 and @xmath188 cases are included , the number of connected parameter regions allowed by unitarity is reduced to two which differ according to the sign of @xmath64 . however , we have shown that any mmg model with @xmath66 is equivalent to some mmg model with @xmath65 , and _ vice versa_. we may therefore choose either sign for @xmath64 without loss of generality . the choice @xmath66 has the advantage that it simplifies the tmg limit , and the net result for this choice is that the only restrictions on the parameter space imposed by unitarity ( given ads@xmath92 asymptotics ) are , up to equivalence , @xmath189 this defines _ a single connected region in the space of parameters of the mmg action _ of the mmg equation ; @xmath53 implies @xmath104 but to find the analogous explicit restrictions on @xmath190 and @xmath191 would require an explicit inversion of the map defined by equations ( [ bars ] ) and ( [ barlamzero ] ) , and this is known only for @xmath50 @xcite . ] . it excludes tmg . if we fix the scaling invariance ( induced by a rescaling of the 3d newton constant ) by choosing @xmath54 then the above unitarity restriction reduces ( up to equivalence ) to @xmath192 of course , this is only a necessary condition for the unitarity of any 2d cft dual to mmg , but that is all we can test at the level of semi - classical effective 3d gravity , and it is a test that tmg fails but mmg passes . we have also used our results to investigate minkowski vacua in the context of the particular proposal for flat - space holography put forward in @xcite , and to compare with the situation for tmg . one new feature of mmg is the existence of two flat space limits . one is a minkowski limit of an mmg model with ads asymptotics satisfying all unitarity conditions , but this does not appear to be useful . instead , the flat - space limit that is available to both tmg and mmg is the one for which proposed unitarity conditions on central charges of the gca algebra can be satisfied . in this limit tmg degenerates to 3d conformal gravity but mmg degenerates to a very different theory with local degrees of freedom , although this are not apparent from the linearised theory . it thus appears that mmg has implications not only for ads holography but also for flat - space holography . a.s.a . and p.k.t . acknowledge support from the uk science and technology facilities council ( grant st / l000385/1 ) . a.s.a . also acknowledges support from clare hall , cambridge , and from the cambridge trust . we are grateful to eduardo casali , elias kiritsis and alasdair routh for helpful discussions , and to eric bergshoeff and wout merbis for helpful correspondence . e.  bergshoeff , o.  hohm , w.  merbis , a.  j.  routh and p.  k.  townsend , `` minimal massive 3d gravity , '' class . quantum grav . 31 ( 2014 ) 145008 [ arxiv:1404.2867 [ hep - 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a geometrical analysis of the bulk and anti - de sitter boundary unitarity conditions of 3d `` minimal massive gravity '' ( mmg ) ( which evades the `` bulk / boundary clash '' of topologically massive gravity ) is used to extend and simplify previous results , showing that unitarity selects , up to equivalence , a connected region in parameter space . 
 we also initiate the study of flat - space holography for mmg . 
 its relevant flat space limit is a deformation of 3d conformal gravity ; the deformation is both non - linear and non - conformal , implying a linearisation instability . 
 = 4    damtp-2014 - 80