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according to the geometric invariant theory introduced by mumford @xcite , the _ git quotient _ @xmath0 of an action of a complex reductive group @xmath1 on a projective variety @xmath2 equipped with a polarization ( ample equivariant line bundle ) has coordinate ring equal to the @xmath1-invariant part of the coordinate ring on @xmath2 . geometrically @xmath0 is the quotient of an open _ semistable locus _ @xmath3 by an equivalence relation , where a point @xmath4 is semistable if there is a non - constant invariant section of a tensor power of the polarization that is non - zero at the point . if the action of @xmath1 on the semistable locus has only finite stabilizers , then @xmath0 is the quotient of @xmath5 by the action of @xmath1 , by which we mean here the _ stack - theoretic _ quotient , see for example @xcite . if @xmath2 is in addition smooth , then @xmath0 is a smooth proper deligne - mumford stack with projective coarse moduli space . in kempf - ness @xcite , see also mumford et al @xcite , the coarse moduli space of the git quotient is identified with the symplectic quotient of @xmath2 by a maximal compact subgroup of @xmath1 . the question of how the git quotient depends on the polarization , or equivalently , choice of moment map is studied in a series of papers by guillemin - sternberg @xcite , brion - procesi @xcite , dolgachev - hu @xcite , and thaddeus @xcite . under suitable stable = semistable conditions , the git quotient @xmath0 undergoes a sequence of weighted blow - ups and blow - downs . the class of birational equivalences which appear via variation of git is reasonably large . in fact , for a class of so - called _ mori dream spaces _ , any birational equivalence can be written as a composition of birational equivalences induced by variation of git @xcite . the question of how the cohomology of the quotient depends on the polarization is studied in kalkman @xcite , which proves a wall - crossing formula for the intersection pairings under variation of symplectic quotient . similar results , in the context of donaldson theory , are given in ellingsrud - gttsche @xcite . to state kalkman s wall - crossing formula , recall that any lie group @xmath1 has a homotopically - trivial _ universal bundle _ @xmath6 , uniquely defined up to homotopy equivalence . if @xmath2 is a @xmath1-space with a reasonable homotopy type then the equivariant cohomology @xmath7 is the cohomology @xmath8 of @xmath9 . in this paper we work with rational coefficients ; if @xmath1 acts with only finite stabilizers then the map @xmath10 has cohomologically trivial fibers and so @xmath11 by the serre spectral sequence . if @xmath2 is a smooth projective @xmath1-variety as above such that @xmath1 acts locally freely on the semistable locus ( that is , with finite stabilizers ) then there is a natural map @xmath12 studied in kirwan s thesis @xcite , given by restriction to the semistable locus @xmath13 and then descent under the quotient @xmath14 . consider the simplest case @xmath15 . let @xmath16 denote the git quotients corresponding to polarizations @xmath17 . let @xmath18 be the kirwan maps and let @xmath19 } \alpha\ ] ] denote integration over @xmath16 . formula expresses the difference between the integrals @xmath20 as a sum of fixed point contributions from components @xmath21 that are semistable for @xmath22 for some @xmath23 . each such fixed point has a contribution to the localization formula for the integral of a class @xmath24 over @xmath2 given as follows . let @xmath25 denote the normal bundle of the inclusion @xmath26 , and @xmath27 its @xmath1-equivariant euler class , or equivalently , equivariant chern class of degree equal to the real rank . we identify @xmath28 with the polynomial ring @xmath29 $ ] in a single element @xmath30 of degree @xmath31 , representing the hyperplane class in the cohomology of @xmath32 . if @xmath33 and @xmath34 is a decomposition into line bundles with weights @xmath35 then @xmath36 since @xmath1 acts with no non - trivial fixed vectors on @xmath37 , the euler class has an inverse @xmath38\ ] ] after inverting the equivariant parameter @xmath30 . if one has a splitting as in then the inverted class admits an expansion @xmath39 the contribution from a fixed point component @xmath21 is the integral of the restriction @xmath40 times the inverted euler class , denoted @xmath41 , \quad \alpha \mapsto    \int_{[x^{g , t } ] } \iota_{x^{g , t}}^ * \alpha \cup    { { \operatorname}{eul}}_g(\nu_{x^{g , t}})^{-1 } .\ ] ] let @xmath42 denote the residue , that is , the coefficient of @xmath43 : @xmath44 \to { \mathbb{q } } , \quad \sum_{n \in { \mathbb{z } } } a_n \xi^n \mapsto a_{-1 } .\ ] ]    [ kalk1 ] let @xmath15 and let @xmath2 be a smooth projective @xmath1-variety equipped with polarizations @xmath45 . suppose that stable = semistable for both polarizations @xmath46 . then @xmath47    in other words , failure of the following square to commute is measured by an explicit sum of wall - crossing terms :    @xmath48    the formula also holds for certain quasi - projective varieties , such as vector spaces whose weights are contained in an open half - space , see the more general theorem [ kalk2 ] below . ( integration over projective space ) [ projspace ] the following simple example illustrates the notation involved . let @xmath15 act on @xmath49 by scalar multiplication , so that @xmath50 $ ] . suppose that polarizations @xmath51 correspond to the characters @xmath52 , so that the git quotients @xmath53 and @xmath54 with a unique singular value @xmath55 corresponding to the point @xmath56 . the kirwan map @xmath57 sends the generator @xmath58 to the hyperplane class @xmath59 . we compute the integrals @xmath60 for @xmath61 via wall - crossing . on one side , the integral is zero , since the quotient is empty . by the kalkman formula @xmath62 }    \xi^a \cup { { \operatorname}{eul}}_g({\mathbb{c}}^k)^{-1 } \\   & = & { { \operatorname}{resid}}_\xi   \xi^a / \xi^{k }    =   \begin{cases }   1 & a = k- 1 \\                        0   & \text{otherwise } \end{cases } \end{aligned}\ ] ] confirming that @xmath63 is the dual of the fundamental class . this ends the example . the main result of this paper is a generalization of theorem [ kalk1 ] to the setting of gromov - witten theory , that is , quantum cohomology . the formula is the same , but the kirwan map , trace , and fixed point contributions are _ quantized _ in the sense that each is replaced by a formal map depending on a formal variable @xmath64 whose restriction to @xmath65 gives the classical version above . first we recall the definition of quantum cohomology of a smooth polarized projective @xmath1-variety @xmath2 . the _ equivariant novikov field _ @xmath66 associated to a @xmath67-variety @xmath2 with polarization @xmath68 consists of linear combinations of formal symbols @xmath69 for @xmath70 : @xmath71 we write @xmath72 if we wish to emphasize the dependence on @xmath73 . the _ equivariant quantum cohomology _ is the tensor product @xmath74 or @xmath75 if we wish to emphasize the dependence on @xmath73 . a more standard definition in algebraic geometry would use the cone of effective curve classes , but the definition give here has better invariance properties , for example , under hamiltonian perturbation . below we will need several variations on this definition . let @xmath76 denote the space of finite linear combinations of the symbols @xmath69 . we denote by @xmath77 the subspace @xmath78 . let @xmath79 be the space consisting of expressions involving only powers @xmath69 with @xmath80 . we denote by @xmath81 the subspace @xmath82 .    the quantum cohomology @xmath83 has the structure of a frobenius manifold , in particular , a family of products @xmath84 defined by equivariant virtual enumeration of genus zero _ stable maps _ , that is , maps @xmath85 from projective nodal curves @xmath86 of arithmetic genus zero to @xmath2 with no infinitesimal automorphisms . the moduli stack @xmath87 of such maps of homology class @xmath88 and arithmetic genus @xmath89 is a proper deligne - mumford stack equipped with a perfect relative obstruction theory and evaluation map @xmath90 the action of @xmath1 on @xmath2 induces an action of @xmath1 on @xmath87 by translation ; then the evaluation maps induce a map @xmath91 the quantum product at @xmath92 is defined by restricting to @xmath93 and defining for @xmath94 @xmath95 where the push - forwards are defined using the behrend - fantechi virtual fundamental classes @xmath96_g$ ] , and the definition is extended to @xmath97 by linearity over @xmath98 , see @xcite . note that we take the virtual fundamental classes to lie in the homology @xmath99 of the coarse moduli space , rather than in the chow ring as in @xcite . the quantum cohomology @xmath83 can also be defined using the smaller ring @xmath100 but not @xmath101 , because of the infinite sum . the orbifold quantum cohomology @xmath102 of the quotient @xmath0 is defined by virtual enumeration of stable maps from orbifold curves , as follows . for any element @xmath103 let @xmath104 be the centralizer of @xmath103 , @xmath105 $ ] its conjugacy class , and @xmath106 the subgroup generated by @xmath89 . we denote by @xmath107 the _ rigidified inertia stack _ from abramovich - graber - vistoli @xcite , given by @xmath108 } x^{g,{{\operatorname}{ss } } } / ( z_g/ { \langle}g { \rangle}).\ ] ] denote by @xmath109 the quantum cohomology of the git quotient @xmath0 defined using the same novikov field @xmath98 ; this larger ring contains @xmath110 by virtue of kirwan s injection @xmath111 . virtual enumeration of representable morphisms from orbifold curves satisfying certain conditions to @xmath0 defines a family of products @xmath112    a quantum version of the kirwan map @xmath113 ( we keep the same notation as in the classical case ) was constructed in @xcite . the map @xmath114 is a formal , non - linear map with the property that each linearization @xmath115 is a @xmath116-homomorphism , defined by virtual enumeration of _ affine gauged maps_. such a map is by definition a representable morphism @xmath117 from a weighted projective line @xmath118 to the quotient stack @xmath119 mapping the stacky point at infinity @xmath120 to the semistable locus @xmath121 . ( more precisely , the domain is a smooth deligne - mumford stack of dimension one with a single stacky point with automorphism group of order @xmath122 . ) these are the algebro - geometric analogs of the _ vortex bubbles _ considered in gaio - salamon @xcite . the compactified moduli stack @xmath123 of affine gauged maps of homology class @xmath70 is , if stable = semistable for @xmath2 , a proper smooth deligne - mumford stack with a perfect relative obstruction theory over the complexification of stasheff s multiplihedron , and evaluation maps @xmath124 the quantum kirwan map is defined for @xmath125 by @xmath126 it is defined over the smaller equivariant novikov ring @xmath100 , but one obtains good surjectivity properties only using the novikov field @xmath98 , see @xcite . the map @xmath114 is a quantization of kirwan s in the sense that @xmath127 is the map studied in @xcite . a quantization of the classical integration map over @xmath0 is given by the _ graph potential _ in givental @xcite @xmath128 defined by virtual enumeration of genus zero orbifold stable maps to @xmath129 of homology class @xmath130 for some @xmath131 . let @xmath132 denote the stack of such maps of class @xmath133 , which we view as an element of @xmath134 via the inclusion of the semistable locus . the graph potential is defined for @xmath135 by @xmath136 } { { \operatorname}{ev}}^ * ( \alpha , \ldots , \alpha ) .\ ] ] again @xmath137 is defined over the equivariant novikov ring @xmath138 . the graph potential @xmath137 is related via the quantum kirwan map to a _ gauged gromov - witten potential _ @xmath139 defined by virtual enumeration of _ gauged maps _ , by which we mean morphisms from @xmath86 to the quotient stack @xmath119 , satisfying a mundet stability condition @xcite generalizing semistability for vector bundles on curves :    ( mundet semistability ) [ mundetsemistable ] a _ gauged map _ from a smooth projective curve @xmath86 to the quotient stack @xmath119 is a morphism @xmath140 , consisting of a pair @xmath141 , see @xcite . we suppose that @xmath142 is the complexification of @xmath143 , and @xmath143 is equipped with an inner product invariant under the action of @xmath144 , inducing an identification @xmath145 .    1 . ( associated graded morphism ) given a reduction of @xmath146 to a parabolic subgroup @xmath147 given by a section @xmath148 and an antidominant , central element @xmath149 in the lie algebra @xmath150 of @xmath151 , there is an _ associated graded _ morphism given by a bundle @xmath152 whose structure group reduces to the levi subgroup of @xmath146 and stable section @xmath153 with domain a nodal curve @xmath154 . the bundle @xmath155 is the limit of the bundles @xmath156 associated to the homomorphisms @xmath157 for @xmath158 , while the section @xmath159 is the gromov limit of the family of sections @xmath160 of @xmath161 induced by @xmath149 . ( hilbert - mumford weight ) the principal component @xmath162 of @xmath154 is the irreducible component such that the restriction @xmath163 of @xmath164 to @xmath162 maps isomorphically to @xmath86 . the principal component @xmath165 of the associated graded section @xmath159 takes values in the fixed point set of the automorphism induced by @xmath149 . the _ hilbert - mumford weight @xmath166 _ determined by the polarization @xmath73 , is the usual hilbert - mumford weight , that is , the weight of the @xmath167-action on @xmath73 over a generic value of @xmath165 . 3 .   ( ramanathan weight ) assume that @xmath168 is a weight of @xmath151 . the ramanathan weight of @xmath169 of @xmath170 with respect to @xmath171 is given by the first chern number of the line bundle determined by @xmath149 via the associated bundle construction : if @xmath172 is the one - dimensional representation of @xmath151 with weight @xmath149 then the line bundle is @xmath173 and the _ ramanathan weight _ is @xmath174 }     c_1 ( \sigma^*p \times_r { \mathbb{c}}_\lambda ) .\ ] ] 4 . ( mundet weight ) the _ mundet weight _ is the sum of the hilbert - mumford and ramanathan weights : @xmath175 5 . ( mundet semistability ) the morphism @xmath170 is _ mundet semistable _ if @xmath176 for all such pairs @xmath171 @xcite , and _ mundet stable _ if the above inequalities are satisfied strictly .    more generally , a _ nodal @xmath177-marked gauged map _ consists of an @xmath177-marked nodal projective curve @xmath178 and a morphism @xmath179 of class @xmath180 , consisting of a principal component @xmath181 which maps under @xmath164 and projection to the first factor isomorphically to @xmath86 , and a collection of _ bubble components _ @xmath182 . the datum @xmath183 is _ mundet semistable _ if the restriction @xmath184 is mundet semistable , and each bubble component @xmath185 on which @xmath164 is given by a trivial @xmath1-bundle with constant section has at least three special ( marked or nodal ) points . let @xmath186 denote the stack of mundet semistable @xmath177-marked gauged maps . taking @xmath2 and @xmath1 to be points , and @xmath187 to be trivial , one obtains the _ fulton - macpherson space _ associated to @xmath86 , that is , the moduli space @xmath188)\ ] ] of stable maps to @xmath86 of class @xmath189 \in h_2(c,{\mathbb{z}})$ ] . we denote by @xmath190 the stack of pre - stable maps of class @xmath189 $ ] , that is , without the stability condition . the category of mundet - semistable gauged maps from @xmath86 to @xmath119 of homology class @xmath187 and @xmath177 markings forms an artin stack @xmath186 @xcite , which if stable = semistable is a proper smooth deligne - mumford stack with perfect relative obstruction theory over @xmath190 , evaluation map @xmath191 and forgetful morphism @xmath192 .    [ toric ] ( toric case ) suppose that @xmath1 is a torus acting on a vector space @xmath2 with weight decomposition @xmath193 so that @xmath1 acts on the one - dimensional representation @xmath194 with weights @xmath195 . suppose that @xmath2 is equipped with a polarization given by a trivial line bundle with character @xmath196 . let @xmath86 be a curve of genus @xmath197 . then @xmath198 where the polarization on @xmath199 is the trivial line bundle with character @xmath200 , see @xcite . for example , if @xmath15 acts on @xmath49 by scalar multiplication then @xmath201 for @xmath202 and positive character @xmath200 . ( gauged gromov - witten potential ) suppose that stable = semistable for all gauged maps . the gauged potential @xmath203 is the formal map defined by @xmath204 } { { \operatorname}{ev}}^ * ( \alpha,\ldots , \alpha ) \ ] ] for @xmath24 , and extended to @xmath83 by linearity .    ( graph potential and gauged potential as quantizations of integration ) both @xmath137 and @xmath203 are quantizations of integration over the quotient in the following sense : given @xmath205 and a fulton - macpherson class @xmath206 , consider the integral @xmath207 } { { \operatorname}{ev}}_n^ * \alpha \cup f^ * \beta .\ ] ] if @xmath208 is the point class then @xmath209 } \alpha .\ ] ] indeed any map from the curve @xmath86 to @xmath0 of homology class @xmath197 is constant . it follows that @xmath210 in particular , for @xmath211 we obtain @xmath212 which implies the claim . a similar discussion holds for @xmath203 , that is , @xmath209 } ( d_0 \kappa_x^g |_{q = 0 } ) \alpha .\ ] ] the reader who is confused by the appearance of the fulton - macpherson class should keep in mind that a similar phenomenon appears in the fukaya category setting : ( co)units are defined by integrals over the moduli space of holomorphic disks with a single marking where the _ position of the marking is fixed_. this ends the remark .    the gauged potential and the graph potential of the quotient are related by the _ adiabatic limit theorem _ of @xcite ( which is a generalization of an earlier result of gaio - salamon @xcite ) : let @xmath213 be a real _ stability parameter _ ; we consider mundet stability with respect to the polarization @xmath214 with @xmath215 . [ adiabatic limit theorem ] [ largerel ] if @xmath0 is a locally free quotient then stable = semistable for @xmath213 sufficiently large ( more precisely , for any class @xmath216 there exists an @xmath217 such that @xmath218 implies stable = semistable ) and @xmath219    in other words , the diagram @xmath220{e , t}{{\kappa}_x^g } \node { } \node{qh ( x { /\kern-.7ex/}g ) }    \arrow{sw , b}{\tau_{x { /\kern-.7ex/}g } } \\ \node { } \node{\lambda_x^g } \node { }     \end{diagram}\ ] ] commutes in the limit @xmath215 . there is more general version of the adiabatic limit theorem for _ euler class twistings _ : the universal curve @xmath221 admits a universal gauged map @xmath222 for any @xmath1-equivariant bundle @xmath223 we denote by @xmath224 the index of the bundle @xmath225 ; this is an object in the bounded derived category of @xmath226 , since @xmath227 is proper . the euler class @xmath228 is well - defined after passing to the equivariant cohomology of @xmath226 for the trivial @xmath167-action corresponding to scalar multiplication on the fibers and inverting the equivariant parameter .    in order to study the dependence of the graph gromov - witten potential of the quotient on the choice of polarization , suppose that @xmath229 are two polarizations of @xmath2 and @xmath230 is the family of rational polarizations given by interpolation . let @xmath16 denote the git quotients , @xmath231 the _ quantum _ kirwan maps ( note that we do not introduce new notation for the quantum version ; the classical kirwan map is obtained by setting @xmath65 ) and @xmath232 the graph potentials . the main result of this note is a formula for the difference @xmath233 as a sum of fixed point contributions given by gauged gromov - witten invariants with smaller structure group . namely , for any @xmath234 such that @xmath235 is non - empty , there exists a stack @xmath236 of _ @xmath1-fixed @xmath177-marked gauged maps _ from @xmath86 to @xmath119 consisting of a principal component mapping to @xmath237 and bubbles mapping to @xmath119 . we denote by @xmath238 the moving part of the normal complex @xmath239 defined in , with euler class @xmath240 .\ ] ] the mundet stability condition in general depends on a choice of equivariant khler class ; here we are interested in the _ large area limit _ in which the gauged gromov - witten invariants are related to the git quotient . we denote by @xmath241 the space of @xmath242-valued functions on @xmath243 . note that @xmath244 has no ring structure extending that on @xmath98 . the space @xmath244 can be viewed as the space of distributions in the quantum parameter @xmath64 . [ fixed point potential ] let @xmath245 be as above , and @xmath23 such that @xmath235 is non - empty . the _ fixed point potential _ associated to this data is the map @xmath246   \\ \quad \alpha \mapsto \sum_{d \in h_2^g(x,{\mathbb{z } } ) } \sum_{n \ge 0 }   \int_{[{\overline}{{\mathcal{m}}}_n^{g}(c , x , l_t , d)^g ] } ( q^d / n ! )   { { \operatorname}{ev}}^ * ( \alpha , \ldots ,   \alpha ) \cup { \epsilon}_+(t(x / g))^{-1}\end{gathered}\ ] ] for @xmath24 , extended to @xmath247 by linearity . we may now state the main result of the paper , in the case of circle actions . [ quantum kalkman formula , circle group case ] [ qkalkcirc ] suppose that @xmath15 and @xmath2 is a smooth projective @xmath1-variety equipped with polarizations @xmath45 , and stable = semistable for @xmath51 . then @xmath248    in other words , failure of the following square to commute is measured by an explicit sum of wall - crossing terms given by certain gauged gromov - witten invariants :    @xmath249    the diagram is somewhat more complicated than in the classical case in , because the maps @xmath250 are defined using different the novikov rings @xmath251 . if a symbol @xmath69 appears in one novikov ring @xmath252 but not the other then the corresponding contribution to @xmath253 must be equal to the @xmath69-term in the wall - crossing contribution on the right . see example [ threepoint ] .    the wall - crossing formula for gromov - witten invariants theorem [ qkalkcirc ] should be considered mirror to various results in on the behavior of the derived category of bounded complexes of coherent sheaves under variation of git quotient appearing recently in segal @xcite , halpern - leistner @xcite and ballard - favero - katzarkov @xcite , and more generally for crepant birational transformations , earlier in @xcite . more precisely , one would hope that these results have extensions to open - closed gromov - witten theory , in particular , semi - orthogonal decompositions of fukaya categories similar to those for the derived categories of matrix factorizations in the works above . we have stated the formula in its simplest form ; there are various extensions which include :    1 . ( insertions from fulton - macpherson spaces ) the moduli stacks @xmath226 resp . @xmath254 admit _ forgetful morphisms _ @xmath255 to the moduli stack @xmath256 of homology class @xmath189 $ ] stable maps to @xmath86 . for any class @xmath257 , one may insert @xmath258 in the integral defining the potentials as in . the wall - crossing formula then holds in the sense of _ homotopy compositions of traces and morphisms of cohft algebras _ discussed in @xcite . in fact , without the fulton - macpherson insertions the potentials @xmath259 will often vanish , unless one interprets them equivariantly in the case of a curve @xmath86 of genus zero with a circle action , in which case they are closely related to givental s _ @xmath260 and @xmath261 functions _ ( fixed point potential as a quantization of kalkman s ) the fixed point potential @xmath262 quantizes the fixed point contributions in kalkman s formula , in the following sense : let @xmath86 be a genus zero curve and consider the @xmath177-th taylor coefficient derivative @xmath263 . consider the integral with insertion of the fulton - macpherson class @xmath264 given by @xmath265 } { { \operatorname}{ev}}_n^ * \alpha \cup f^ * \beta \cup    { \epsilon}_+(t(x / g))^{-1}.\ ] ] if @xmath266 is the point class then @xmath267 } \alpha \cup { { \operatorname}{eul}}_g(\nu_{x^{g , t}})^{-1}\ ] ] which appears in kalkman s wall - crossing formula . indeed any map from the genus zero curve @xmath86 to @xmath119 of homology class @xmath197 consists of a trivial bundle and constant section . it follows that @xmath268 in particular , for @xmath211 we obtain @xmath269 which implies the claim . furthermore , the euler class @xmath270 is the euler class of the normal bundle to @xmath235 . ( twistings by euler classes ) one can introduce _ twisted gromov - witten invariants _ as follows . for any equivariant bundle @xmath271 on @xmath2 , inserting the euler class @xmath272 of gives twisted gauged gromov - witten invariants . introducing similar twistings in the definition of @xmath114 , the wall - crossing formula extends to this case as well . 4 .   ( wall - crossing for individual gromov - witten invariants ) although we have written the formula as a difference of potentials , after unraveling the definitions one obtains wall - crossing formulas for individual gromov - witten invariants , or at least finite combinations of them . in fano cases , one can often obtain wall - crossing formulas for individual gromov - witten invariants . ( wall - crossing for non - convex actions ) in same cases , even though the action of @xmath1 on @xmath2 is not convex at infinity ( e.g. @xmath1 is a torus acting on a vector space @xmath2 with weights not contained in any half - space ) the moduli spaces @xmath226 are non - compact , but the moduli spaces admit an auxiliary group action with proper fixed point loci , and the maps @xmath273 etc . can be defined via localization . the wall - crossing formula holds in this case as well , as long as the auxiliary group action extends to the various master spaces involved and the fixed point loci on these master spaces is proper ( after fixing the homology class of the gauged map . ) see example [ simpleflop ] below . ( three - point gromov - witten invariants of projective space ) [ threepoint ] the following simple example illustrates the notation involved in theorem [ qkalkcirc ] . as in example [ projspace ] , let @xmath274 acting on @xmath49 by scalar multiplication , so that @xmath50 $ ] . suppose that polarizations @xmath51 correspond to the characters @xmath52 , quotients @xmath275 and @xmath276 , and a unique singular value @xmath277 corresponding to the point @xmath56 . let @xmath278 denote the hyperplane class . for integers @xmath279 we compute genus @xmath197 @xmath280 @xmath281 invariants @xmath282 of @xmath283 via wall - crossing ; this was already covered in cieliebak - salamon @xcite . since @xmath284 , the minimal chern number of @xmath2 is @xmath285 . for dimensional reasons , there are no quantum corrections in @xmath286 and so @xmath287 the adiabatic limit theorem [ largerel ] with fulton - macpherson insertions implies that the genus zero , three - point invariants @xmath288 of class @xmath289 in the quotient @xmath0 equal gauged gromov - witten invariants : @xmath290 } { { \operatorname}{ev}}_1^ * \xi^a \cup { { \operatorname}{ev}}_2^ * \xi^b \cup   { { \operatorname}{ev}}_3^ * \xi^c \cup f^ * \beta\ ] ] where @xmath291 is the projective line and @xmath292 is the point class , that is , the class fixing the location of the three marked points . consider the wall - crossing formula theorem [ qkalkcirc ] , applied to the third order derivatives of the potential in the directions @xmath293 with the fulton - macpherson insertion described above . there are no holomorphic spheres in @xmath2 , so the moduli stack @xmath294 is a point , consisting of the bundle @xmath146 with first chern class @xmath295 with constant section equal to zero . thus the fixed point stack is @xmath296 the index bundle of @xmath297 at the fixed point is @xmath298 it has euler class @xmath299 by the wall - crossing formula [ qkalkcirc ] @xmath300 } \frac { { { \operatorname}{ev}}_1^ * \xi^a \cup    { { \operatorname}{ev}}_2^ * \xi^b \cup { { \operatorname}{ev}}_3^ * \xi^c \cup f^ * \beta}{{\epsilon}_+(t(x / g ) ) } \\ & = & { { \operatorname}{resid}}_\xi \xi^{a + b + c } / \xi^{2k }   \\ & = & \begin{cases } 1 & a    + b + c = 2k- 1 \\ 0 & \text{otherwise } .\end{cases}\end{aligned}\ ] ] thus @xmath301 as is well - known . for example , taking @xmath302 we obtain that there is a unique line in projective space passing through two generic points and a generic hyperplane . we give another fano example in example [ c1 ] , where we compute the change in a coefficient in the fifth quantum power of the first chern class for a blow - up of the projective plane . this ends the example . the most interesting case of the wall - crossing formula theorem [ qkalkcirc ] is the _ crepant case _ by which we mean that the sum of the weights at any fixed point vanishes ( definition [ crepant ] ) ; the term _ crepant _ was introduced by reid @xcite as the opposite of discrepant in the context of the minimal model program . in the last section we give a proof of a version of the crepant transformation conjecture of li - ruan @xcite , bryan - graber @xcite , coates - corti - iritani - tseng @xcite , coates - ruan @xcite on equivalence of gromov - witten theories in this case :    [ wall - crossing for crepant birational transformations of git type ] [ cytype ] suppose that @xmath303 are as in theorem [ qkalkcirc ] , and @xmath86 has genus zero . if all the wall - crossings are crepant then @xmath304 almost everywhere ( a.e . ) in the quantum parameter @xmath64 .    1 . theorem [ cytype ] implies many of the special cases already known in the literature , although actually computing the transformations @xmath305 relating the graph potentials @xmath306 can be a difficult task . note that iwao - lee - lin - wang @xcite , lee - lin - wang @xcite extend the invariance to cases not necessarily related by variation of git , while the results here are more general than that of @xcite , @xcite since we allow `` weighted flops '' . 2 .   almost everywhere equality in the formal parameter @xmath64 means the following : considering both sides as elements in @xmath307 , the space of distributions on @xmath308 , the difference is a tempered distribution in at least one direction and its fourier transform in that direction has support of measure zero , see section [ cy ] . the proof of theorem [ cytype ] uses an action of the picard stack of the curve on the fixed point stacks . in the crepant case the ( almost ) invariance under this action implies that , after summing over degrees , the wall - crossing term is a sum of derivatives of delta - functions in the quantum parameter @xmath64 . [ simpleflop ] ( simple three - fold flop , cf . li - ruan @xcite , iwao - lee - lin - wang @xcite , lee - lin - wang @xcite ) the following simple example may help to explain the notion of `` almost everywhere vanishing '' of the wall - crossing contributions in the quantum parameter . let @xmath309 acting on @xmath310 with weights @xmath52 each of multiplicity @xmath31 . then @xmath311 where @xmath291 is the projective line and we abuse notation by denoting by @xmath312 the total space of two copies of the tautological line bundle @xmath313 . then @xmath314 and @xmath315 are isomorphic but the birational transformation relating them , induced by the variation of git quotient , is a simple flop . we consider the wall - crossing formula corresponding to the three - point invariants with each insertion given by @xmath316 ; that is , the third derivative of @xmath306 with a fulton - macpherson class @xmath258 fixing the positions of the three marked points . in order to make sense of the non - compact integration in the @xmath317 case one must take an equivariant extension of with respect to the action of @xmath167 acting by scalar multiplication , and use localization for the residual @xmath167-action . thus if @xmath318 is the equivariant parameter for the auxiliary @xmath167 then @xmath319 where @xmath320 is the symplectic class integrating to @xmath321 on the zero - section @xmath322 . the fixed point set of the residual @xmath167 action is the zero section of @xmath323 , which acts with weights @xmath324 ( with multiplicity two ) on the normal bundle . the degree zero moduli space is @xmath16 itself , and integration over this non - compact space can be defined via localization as @xmath325 the auxiliary circle action extends to the various master spaces involved , and the wall - crossing formula holds in this case as well , defining each contribution via localization . there is a unique fixed point @xmath56 , with normal bundle isomorphic to @xmath2 . thus the wall - crossing term is @xmath326 thus kalkman s wall - crossing formula reduces to @xmath327    moving on to non - zero classes @xmath328 , there is a unique gauged map of class @xmath187 mapping to the fixed point @xmath56 , with normal complex having weight @xmath321 with multiplicity @xmath329 and weight @xmath330 with multiplicity @xmath331 , by riemann - roch . thus the wall - crossing term class @xmath187 is @xmath332 the class @xmath187 occurs in the expression @xmath333 iff @xmath334 resp . @xmath335 . indeed , the contribution of @xmath69 is the contribution of @xmath69 to the gauged gromov - witten potential in theorem [ largerel ] , which by definition is an integral over mundet stable maps which in this case is the git quotient of @xmath336 by the @xmath167 action corresponding to the polarization @xmath51 , where @xmath337 denote the one - dimensional representations with weight @xmath52 . for @xmath338 , only one factor @xmath339 is non - vanishing and the git quotient is empty for the polarization @xmath340 corresponding to the character @xmath341 . since each @xmath342 appears in only one of the novikov rings @xmath343 we must have , after composing with the quantum kirwan map , that the higher degree integrals for @xmath344 are @xmath321 for @xmath345 and @xmath197 for @xmath346 resp . for @xmath347 are @xmath348 for @xmath349 and @xmath197 for @xmath345 , each corresponding to an integral over multiple covers of the zero section . summing over classes @xmath187 the wall - crossing formula in theorem [ qkalkcirc ] becomes @xmath350 the reader may compare with the treatment of simple flops in @xcite , ( * ? ? ? * corollary 3.2 ) which contains essentially the same computation . note that here we have not given an explicit description of the maps @xmath305 which relate the two graph potentials . this ends the example . in this section we give a proof of kalkman s formula [ kalk1 ] and give a generalization to non - abelian quotients . the wall - crossing formula is somewhat simpler for the case of a circle group , so we begin with that case . let @xmath15 and @xmath2 a smooth projective @xmath1-variety as above , equipped with polarizations @xmath45 . the proof of the wall - crossing formula is by localization on a proper deligne - mumford stack @xmath351 whose fixed points include @xmath16 and the fixed point components @xmath235 with @xmath23 . from the point of view of symplectic geometry , this is the _ symplectic cut _ construction in lerman @xcite , but we need the algebraic approach here given in thaddeus ( * ? ? ? * section 3 ) . first recall from the introduction that @xmath352 denotes the family of rational polarizations interpolating between @xmath51 . we denote by @xmath353 the corresponding git quotients , by which we mean the stack - theoretic quotient of the semistable locus for @xmath354 by the action of @xmath1 . this means that if @xmath355 are moment maps for action of a maximal compact of @xmath1 on @xmath51 with respect to a unitary connection then @xmath356 is a moment map for the action of @xmath1 on @xmath354 . even more concretely , if @xmath355 are equal up to a constant @xmath357 then @xmath358 are all equal up to a constant depending on @xmath359 . for the following see thaddeus ( * ? ? ? * section 3 ) . [ existence of a master space ] [ master ] suppose that stable = semistable for @xmath51 and for any @xmath23 and any @xmath359-semistable point @xmath360 , @xmath361 acts with finite stabilizer on the fiber @xmath362 . then there exists a smooth proper deligne - mumford @xmath1-stack @xmath351 equipped with a line bundle ample for the coarse moduli space whose git quotients @xmath363 are isomorphic to those @xmath364 of @xmath2 by the action of @xmath1 with respect to the polarization @xmath354 and whose fixed point set @xmath365 is given by the union @xmath366 where @xmath235 is the component of @xmath367 that is semistable for parameter @xmath359 . the projectivization @xmath368 of the direct sum @xmath369 of the polarizations @xmath45 has a natural polarization given by the relative hyperplane bundle @xmath370 of the fibers , with stalks @xmath371 }    = { \operatorname}{span}(l_- + l_+)^{\vee}.\ ] ] let @xmath372 denote the projection . ( a word of warning : the notation @xmath373 will be used for a number of different projections in this paper . ) the space of sections of @xmath374 has a decomposition under the natural @xmath167-action with eigenspaces given by the sections of @xmath375 with @xmath376 . it follows that the semistable locus in @xmath377 is the union of @xmath378 where @xmath379 is the semistable locus for @xmath380 $ ] . let @xmath381 denote the geometric invariant theory quotient , by which we mean the stack - theoretic quotient of the semistable locus . the assumption on the action of the stabilizers implies that the action of @xmath1 on the semistable locus in @xmath368 is locally free , so that stable = semistable for @xmath368 . it follows that @xmath351 is a proper smooth deligne - mumford stack . the quotient @xmath351 contains the quotients of @xmath382 with respect to the polarizations @xmath51 , that is , @xmath16 . the @xmath1-action with weights @xmath52 on @xmath383 induces a @xmath1-action on @xmath351 with fixed point set as described above . indeed , the fixed points are pairs @xmath384 $ ] such that either @xmath385 , or @xmath386 are both non - zero but the projection to @xmath2 is @xmath1-fixed .    [ cond ] ( condition for stable = semistable ) the proof shows that the @xmath1-action on @xmath377 has stable = semistable iff the condition of the lemma holds , that is , stable = semistable for @xmath51 and for any @xmath23 and @xmath359-semistable point @xmath360 , @xmath361 acts with finite stabilizer on the fiber @xmath362 . indeed , this is the condition for @xmath1 to act locally freely on the semistable locus in @xmath377 .    the projection @xmath387 induces a @xmath1-invariant map @xmath388 . composing with the kirwan map one obtains a canonical map @xmath389 . its composition with @xmath390 is equal to @xmath391 . taking the residue of the localization formula for the @xmath1-action on @xmath351 gives theorem [ kalk1 ] , see lerman @xcite . indeed for any equivariant class @xmath24 of top degree , its pullback to @xmath377 descends to a class @xmath392 whose restriction to @xmath323 is @xmath393 , and whose restriction to @xmath394 is @xmath395 . since @xmath396 the integral of @xmath397 over @xmath351 vanishes . on the other hand , by localization the integral is @xmath398 } \alpha = \int_{[x { /\kern-.7ex/}_- g ] } \frac{\kappa_{x,-}^g(\alpha)}{{{\operatorname}{eul}}_g(\nu_- ) } + \int_{[x { /\kern-.7ex/}_+ g ] } \frac{\kappa_{x,+}^g(\alpha)}{{{\operatorname}{eul}}_g(\nu_+ ) } + \sum_{t \in ( -1,1 ) } \int_{[x^{g , t } ] } \frac{\iota^*_{x^{g , t } } \alpha}{{{\operatorname}{eul}}_g(\nu_{x^{g , t}})}\ ] ] where @xmath399 are the normal bundles to @xmath400 in @xmath351 . the inverted euler classes are @xmath401 hence taking residues one obtains @xmath402 } \frac{\iota^*_{x^{g , t } } \alpha } { { { \operatorname}{eul}}_g(\nu_{x^{g , t}})}\ ] ] as claimed . kalkman s wall - crossing formula can be used to study the intersection pairings for variation of git for the action of an arbitrary connected complex reductive group @xmath1 on a smooth polarized projective variety @xmath2 as follows . let us examine the structure of the fixed point set of @xmath167 on @xmath351 . for any @xmath403 , we denote by @xmath404 the stabilizer of @xmath405 under the adjoint action of @xmath1 . let @xmath406 be a maximal torus and @xmath407 the corresponding cartan algebra . let @xmath408 denote the weyl group . for any @xmath409 , we denote by @xmath410 resp . @xmath411 the group of @xmath412 that fix the element @xmath403 resp . line @xmath413 . thus the quotient @xmath414 is either isomorphic to @xmath415 or to @xmath416 , depending on whether or not there is a weyl group element acting as @xmath330 on @xmath417 . [ fplem ] suppose that stable = semistable for the @xmath1-action on @xmath418 , so that @xmath351 is a smooth proper deligne - mumford stack with @xmath167 action , constructed in the proof of lemma [ master ] . .   any fixed point @xmath419 with @xmath420 $ ] for some @xmath421 has the property that for all @xmath158 , @xmath422 for some unique @xmath423 . 2 .   for each @xmath409 there is a morphism @xmath424 with fiber @xmath425 . the images of @xmath426 cover @xmath427 , disjointly after passing to equivalence classes of one - parameter subgroups . 3 .   for any @xmath24 , the pull - back of @xmath428 under @xmath426 is equal to image of @xmath429 under the restriction map @xmath430 4 . the pull - back of the normal bundle @xmath431 of @xmath427 under @xmath426 is canonically isomorphic to the image of @xmath432 under the quotient map @xmath433 , by an isomorphism that intertwines the action of @xmath434 on @xmath435 with the action of @xmath167 on @xmath431 .    if @xmath436 \in { \tilde}{x}^{{\mathbb{c}}^\times}$ ] , with @xmath421 then @xmath436 \in { \tilde}{x}^\xi$ ] , where @xmath30 is a generator of the lie algebra of @xmath167 and @xmath437 is the zero set of the vector field @xmath438 generated by @xmath30 . since @xmath351 is the quotient of @xmath377 by @xmath1 , if @xmath439 denotes the vector field on @xmath377 generated by @xmath30 then @xmath440 for some @xmath403 . since @xmath1 acts locally freely @xmath405 must be unique . integrating gives @xmath441 for all @xmath442 . by uniqueness of @xmath405 , this holds for every point in the component of @xmath437 containing @xmath436 $ ] . the fiber of the morphism @xmath443 is the set of @xmath444 fixing both @xmath445 and @xmath446 under the adjoint action . the set of @xmath447 fixing @xmath445 is @xmath448 , while the set of @xmath444 fixing @xmath434 and @xmath445 is isomorphic to @xmath425 . hence the fiber is @xmath425 . the remaining claims are left to the reader . the group @xmath449 acts trivially on @xmath450 , which is therefore a @xmath451-space . this is the `` smaller structure group '' referred to in the abstract . for any fixed point component @xmath452 that is @xmath359-semistable , we denote by @xmath453 the normal bundle of @xmath454 modulo @xmath455 , quotiented by @xmath451 let @xmath456 , \quad \alpha \mapsto \int_{[x^{\zeta , t } { /\kern-.7ex/}(g_\zeta/{\mathbb{c}}^\times_\zeta ) ] } \alpha \cup { { \operatorname}{eul}}_{{\mathbb{c}}^\times_\zeta}(\nu_{x^{\zeta , t}})^{-1}\ ] ] where we have omitted the map @xmath457 to simplify notation , and @xmath30 is the equivariant parameter for @xmath434 . [ kalkman wall - crossing ] [ kalk2 ] let @xmath2 be a smooth projective @xmath1-variety and @xmath45 polarizations with stable = semistable for the @xmath1 action on @xmath377 . then @xmath458 } \frac{|w_\zeta|}{|w_{{\mathbb{c}}\zeta}| } { { \operatorname}{resid}}_\xi \tau_{x,\zeta , t}^{g_\zeta}\ ] ] where the sum over @xmath459 $ ] represents a sum over one - parameter subgroups of @xmath1 , up to conjugacy .    [ classexamples ]    1 .   [ blowupp2 ] [ eight ] ( blow - up of the projective plane as a quotient of a product of projective lines by a circle action ) let @xmath460 with polarization @xmath461 with @xmath462 . let @xmath463 . since @xmath297 is the same as @xmath464 plus a trivial line bundle , we have @xmath465 furthermore , since all weights are @xmath321 , we have @xmath466 . consider the wall - crossing from the chamber @xmath467 to the first non - empty chamber @xmath468 . there is a unique fixed point with weights @xmath469 on the normal bundle ; hence @xmath470 indeed , @xmath471 is three times the generator of @xmath472 , so @xmath473 } c_1({\mathbb{p}}^2)^2 = 9 .\ ] ] consider next the wall - crossing from the chamber with quotient @xmath474 to the chamber with quotient @xmath475 , where @xmath476 is the blow - up of @xmath477 at a point . letting @xmath478 denote the blow - down map we have @xmath479 where @xmath480 are the hyperplane class and class of the exceptional divisor respectively . hence @xmath481 to compare this result with the wall - crossing formula , note that the fixed point set @xmath235 consists of a unique point @xmath482,[1,0],[1,0])$ ] which is semistable exactly for @xmath483 , with tangent weights @xmath484 . it follows that the first chern class squared and euler classes are @xmath485 hence the wall - crossing term is @xmath486 } \iota_{x^{g , t}}^ * \alpha \cup { { \operatorname}{eul}}_g(\nu_{x^{g , t}})^{-1 } \right ) = { { \operatorname}{resid}}_\xi \frac{\xi^2}{-\xi^3 } = -1 .\ ] ] the wall - crossing formula reduces to @xmath487 in fact each blow - up of @xmath477 lowers @xmath488 by @xmath321 , leading to the 9 del pezzo surfaces , each of which contains a simple simply - laced root system corresponding to classes of self - intersection @xmath489 etc . @xcite . 2 .   ( blow - up of the projective plane as a quotient of affine four - space by a two - torus ) let us do the same example in a different way , namely as a quotient of an affine space . @xmath310 with @xmath490 acting with weights @xmath491 . consider the path of polarizations in @xmath492 from @xmath493 to @xmath494 so that @xmath495 is isomorphic to @xmath477 via the map @xmath496 = [ ( a , b , cd^{-1},1 ) ] \mapsto [ a , b , cd^{-1 } ] .\ ] ] on the other hand @xmath314 is isomorphic to the blow - up of @xmath477 with the map to @xmath477 blowing down the exceptional divisor given by @xmath497 \mapsto [ a , b , cd^{-1 } ]   .\ ] ] as one moves in a line from say @xmath498 to @xmath499 the symplectic quotients are in order @xmath500 where the initial and final contractions are projective bundles over the fixed point sets @xmath501 resp . @xmath291 for the residual action of @xmath167 on the quotient of @xmath2 by the diagonal action . see figure [ twochamber ] , where the two chambers for the possible quotients are shown together with the moment polytopes of the quotients in each chamber . +    action on @xmath502,height=192 ] + consider first the wall - crossing from the empty chamber to the non - empty chamber with quotient @xmath477 in figure [ twochamber ] . we have @xmath503 . hence the wall - crossing term is @xmath504 } c_1({\mathbb{p}}^2)^2 = 9 $ ] as expected . + the wall - crossing term for passing from the chamber with quotient @xmath477 to that with quotient @xmath476 is @xmath505 this implies that @xmath506 } c_1({\mathbb{p}}^2)^2 = 8 $ ] as we already computed in item above . + passing from the chamber with quotient @xmath476 to the adjacent chamber with empty quotient , note that the symplectic quotient at the wall - crossing point @xmath507 equal to @xmath291 and a wall - crossing term of @xmath508 } \kappa_{x}^{{\mathbb{c}}^\times } \frac { ( 3\xi_1 +    2\xi_2)^2 } { \xi_2 ( \xi_1 + \xi_2)}\ ] ] where @xmath509 is the kirwan map for the quotient by the first factor of @xmath167 , so that @xmath510 has @xmath167-quotients @xmath323 . under @xmath511 we have that @xmath512 maps to the generator @xmath513 of @xmath514 while @xmath515 maps to the parameter @xmath30 for the residual @xmath167-action . the wall - crossing term is @xmath516 } \frac { ( 3\omega + 2\xi)^2 } { \xi ( \omega +    \xi ) } & = & { { \operatorname}{resid}}_\xi \int_{[{\mathbb{p } } ] } \frac { ( 3\omega + 2\xi)^2 } {    \xi^2 } ( 1 - \omega/ \xi + \omega^2/\xi^2 - \ldots ) \\&= &    { { \operatorname}{resid}}_\xi \int_{[{\mathbb{p } } ] } 12 \omega/\xi - 4 \omega/\xi = -8 . \end{aligned}\ ] ] thus as expected the wall - crossing formula reduces to @xmath517 3 .   ( resolution of a crepant singularity ) the following example illustrates the application of the wall - crossing to orbifolds , in which the integrals become rational , and also the non - compact case , in which the integrals must be defined via localization . suppose that @xmath518 and @xmath15 acts with weights @xmath321 with multiplicity @xmath285 and @xmath519 with multiplicity @xmath321 , so that @xmath520 ( by our conventions , a stack with trivial canonical bundle ) while @xmath495 is isomorphic to the total space of @xmath521 . let @xmath522 ( the next - to - highest chern class of @xmath523 ) so that @xmath393 is the top chern class of @xmath16 . then @xmath524 } { { \operatorname}{eul}}({\mathbb{c}}^k/{\mathbb{z}}_k ) = 1/k .\ ] ] indeed , interpreting this integral via localization using the @xmath167-action given by scalar multiplication , there is a unique fixed point with stabilizer of order @xmath285 which contributes @xmath525 to the integral . on the other side of the wall , @xmath526 } { { \operatorname}{eul}}({\mathbb{p}}^{k-1 } ) = k .\ ] ] the wall - crossing term is @xmath527 as expected . ( a del pezzo surface as a quotient of a product of projective lines by a non - abelian group ) let @xmath528 with the diagonal action of @xmath529 , equipped with the polarization @xmath530 corresponding to varying the symplectic class on the first factor . from the symplectic point of view , this is the symplectic quotient of @xmath531 by the @xmath532-action with radii of the spheres @xmath533 . the moment map for the action is @xmath534 the quotient @xmath0 is homeomorphic to the _ moduli space of polygons _ @xmath535 a point in the zero level set of the moment map is fixed by a one - parameter subgroup iff the points @xmath536 are colinear , in which case the subgroup fixing them is rotation around the corresponding line . one sees easily from this ( or the algebraic description ) that the singular values of @xmath359 are @xmath537 corresponding to a single orbit of colinear configurations @xmath538 when @xmath539 and four orbits of colinear configurations @xmath540 when @xmath541 . thus the chamber structure for @xmath359 is @xmath542 . since the quotient in the last chamber is empty , and all the weights are one , the quotient @xmath353 for the last chamber must be @xmath477 , while passing from the second to first chamber the quotient undergoes a blow - up at @xmath543 points . consider the wall - crossing given by passing between chambers so that @xmath544 and @xmath545 we compute the square of the first chern class by wall - crossing : let @xmath463 , so that @xmath546 . the fixed points of @xmath167 action for the given singular value are the @xmath547 configurations with @xmath540 with all colinear . consider the case that @xmath548 is fixed by the maximal torus action ; the tangent bundle of this point has weights @xmath549 hence @xmath550 at @xmath551 , while the normal bundle modulo @xmath455 has weights @xmath552 . taking into account the factor @xmath553 ( corresponding to the fact that the configurations related by multiplication by @xmath330 have the same image in @xmath365 ) we have @xmath554 } c_1(x { /\kern-.7ex/}_+ g)^2 & = & \int_{[x { /\kern-.7ex/}_- g ] } c_1(x { /\kern-.7ex/}_- g)^2 + 4 { { \operatorname}{resid}}_\xi \frac{\xi^2}{-\xi^3 } \\ & = & 9 -4 = 5 \end{aligned}\ ] ] which matches the fact that each blow - up of @xmath477 lowers @xmath488 by @xmath321 .      our main result will be derived from a virtual extension of wall - crossing formulas , carried out by kiem - li @xcite using the virtual localization formula of graber - pandharipande @xcite ; we assume that the reader is familiar with the concepts of equivariant perfect obstruction theories etc . from those papers . in this section we explain the virtual extension and give an application to a simple complete intersection . let @xmath555 be a deligne - mumford @xmath1-stack equipped with a @xmath1-equivariant perfect obstruction theory which admits an embedding in a smooth deligne - mumford @xmath1-stack . this means that @xmath555 is equipped with an object @xmath271 of the bounded derived category of @xmath1-equivariant coherent sheaves , together with a morphism from @xmath271 to the cotangent complex , satisfying certain axioms , see @xcite , @xcite ; a typical example is an invariant complete intersection as in example [ nodal ] below . if @xmath403 is any element then the fixed point stacks @xmath556 also have equivariant perfect obstruction theories compatible with that on @xmath555 ; we denote by @xmath557 the virtual normal complex for the embedding @xmath558 . the euler class @xmath559 is well - defined in the equivariant cohomology @xmath560 , after inverting the equivariant parameter @xmath30 . from the virtual localization formula one obtains the following virtual version of the wall - crossing theorem [ kalk2 ] , which can be considered a corollary of kiem - li @xcite :    [ virtual kalkman wall - crossing ] [ virkalk ] let @xmath555 be a proper deligne - mumford @xmath1-stack equipped with a @xmath1-equivariant perfect obstruction theory which admits an embedding in a smooth deligne - mumford @xmath1-stack . let @xmath561 be a @xmath1-line bundles that are ample for the coarse moduli spaces , so that the stable = semistable for @xmath377 . let @xmath562 resp . @xmath563 denote integration resp . equivariant integration over @xmath564 resp . @xmath565 times @xmath566 . then @xmath567 } \frac{|w_{\zeta}|}{|w_{{\mathbb{c}}\zeta}| }   { { \operatorname}{resid}}_\xi \tau_{{\mathcal{x}},\zeta , t}^{g_\zeta}\ ] ] where the sum over @xmath459 $ ] represents a sum over unparametrized one - parameter subgroups of @xmath1 , up to conjugacy . [ nodal ] ( wall - crossing over a nodal fixed point ) the following simple example may help illustrate the notation . suppose that @xmath2 is a nodal projective line with a single node , equipped with the standard @xmath167-action on each component , so that the weights of the action on the tangent spaces the node are @xmath568 . we equip @xmath2 with a polarization so that the weights are @xmath52 at the smooth fixed points , and @xmath197 at the nodal point . then @xmath569 is a point for @xmath23 , and is singular for @xmath570 . since @xmath2 is complete intersection , @xmath2 is virtually smooth and the virtual wall - crossing formula of theorem [ virkalka ] applies . we examine the wall - crossing for the trivial class @xmath571 at the singular value @xmath572 . the virtual normal complex at the nodal point is the quotient of @xmath573 , the sum of one - dimensional representations with weights @xmath574 , modulo their tensor product @xmath575 , which has weight zero . hence the normal complex has inverted euler class @xmath576 the integrals on the left and right hand sides are @xmath321 ( being the integrals over points ) while the wall - crossing term is @xmath577 }   { { \operatorname}{eul}}_g ( \nu_{x , g , t})^{-1 }   = { { \operatorname}{resid}}_\xi 0 = 0 \end{aligned}\ ] ] as desired . in this section , we prove a quantum generalization of kalkman s wall - crossing formula theorem [ kalk2 ] . in the first two subsections , we define the wall - crossing terms as integrals over moduli spaces of gauged maps fixed by a central subgroup . the last two subsections contain a construction of a master space for moduli spaces of gauged maps , and a proof of the wall - crossing formula via virtual localization on the master space . the construction of the master space is obtained from one for a different compactification of gauged maps introduced by schmitt @xcite , pulled back under a relative version of givental s morphism from stable maps to the quot scheme . schmitt s compactification has the advantage that it is constructed by git methods so the classical techniques apply . let @xmath1 be a complex connected reductive group and @xmath578 a central subgroup . for any principal @xmath1-bundle @xmath579 , the right action of @xmath580 induces an action on the associated bundle @xmath581 , and so on the space of sections . the fixed point set of @xmath580 on @xmath581 is @xmath582 , the associated bundle with fiber the fixed point set @xmath583 . let @xmath584 be as above . the action of @xmath580 on the space of sections of @xmath581 preserves mundet semistability and so induces an action of @xmath580 on @xmath585 . the action of @xmath580 on @xmath146 induces an action on parabolic reductions , preserving the associated graded bundles and morphisms and so the mundet weight of definition [ mundetsemistable ] . the following is straight - forward , and similar to the case of stable maps in kontsevich @xcite :    let @xmath584 be as above . the fixed point locus for the action of @xmath580 on @xmath226 is the substack whose objects are pairs @xmath586 such that    1 . @xmath164 takes values in @xmath587 on the principal component @xmath162 ; 2 .   for any bubble component @xmath588 mapping to a point in @xmath86 , @xmath164 maps @xmath589 to a one - dimensional orbit of @xmath580 on @xmath581 ; and 3 . any node of @xmath154 maps to the fixed point set @xmath587 .      in this subsection we explain the wall - crossing terms in the main result theorem [ qkalkcirc ] and state the generalization to arbitrary group actions theorem [ kalk3 ] below . let @xmath2 be a smooth projective @xmath1-variety equipped with polarizations @xmath229 . [ fixed gauged maps ] [ fixedef ] for each @xmath403 , let @xmath590 denote the stack of mundet - semistable morphisms from @xmath86 to @xmath591 that are @xmath434-fixed and take values in @xmath592 on the principal component . via the inclusion @xmath593 the universal curve over @xmath590 admits a morphism to @xmath119 , and the tangent complex @xmath239 defines an object @xmath594 in the bounded derived category of @xmath590 with an action of @xmath434 . we denote by @xmath595 the moving part of @xmath594 with respect to the action of @xmath434 and @xmath596 its euler class , as in , but in the more general setting considered here . [ fixedlargearea ] ( large - area limit of fixed gauged maps ) the mundet semistability condition for fixed gauged maps simplifies somewhat in the limit @xmath597 . suppose that @xmath454 is the component of @xmath450 that is @xmath354-semistable . for any class @xmath187 there exists a @xmath598 such that for @xmath599 , any mundet - semistable fixed map for polarization @xmath354 must consist of a principal component mapping to @xmath600 and bubbles mapping to @xmath591 . indeed , mundet semistability for @xmath214 implies that the hilbert weight @xmath166 is at most @xmath601 times minus the ramanathan weight @xmath602 , for any @xmath603 . in particular , this holds @xmath604 for @xmath405 antidominant and @xmath605 the trivial parabolic reduction , in which case the ramanathan weight for a pair @xmath170 with respect to @xmath171 is simply @xmath606 . the latter is bounded by a constant @xmath607 depending on @xmath608 , since @xmath609 is the projection of @xmath187 onto @xmath610 . so the hilbert weight @xmath611 is less than @xmath612 . it follows for any @xmath599 where @xmath613 and any @xmath354-semistable pair @xmath170 , the section @xmath164 takes values in @xmath614 . ( in algebraic language , this is @xmath607 times the inverse of the smallest difference in hilbert - mumford weights for @xmath405 for different fixed point components . ) to summarize , in the large - area limit @xmath215 , the set of values of @xmath359 for which @xmath615 is non - empty coincides with the set of values of @xmath359 for which @xmath454 is non - empty , and the principal component of any object @xmath170 of @xmath615 takes values in @xmath614 . conversely , suppose that stable = semistable for the @xmath1-action on @xmath377 . suppose that @xmath170 is @xmath451-semistable and @xmath164 takes values in @xmath614 on the principal component . then @xmath170 is @xmath616-semistable for @xmath445 for sufficiently large @xmath213 for a unique @xmath617 in a small neighborhood of @xmath359 . indeed , the stable = semistable assumption implies that hilbert - mumford weight for the action of @xmath434 on @xmath454 is non - constant ( that is , @xmath454 is semistable for a single value of @xmath359 , see remark [ cond ] ) . it follows that @xmath618-semistability of @xmath170 for the holds for @xmath619 close to @xmath359 and @xmath213 sufficiently large so that @xmath620 , while @xmath451-semistability holds by assumption . in general , semistability for a product of groups is not equivalent to semistability for the factors but this is true if one of the factors acts trivially , as in the present case . we will construct in the proof below a perfect relative obstruction theory on @xmath621 . this construction allows the following definition :    [ fixed point contributions to wall - crossing for gromov - witten invariants ] [ contrib ] virtual integration over the stacks @xmath622 defines a `` fixed point contribution '' @xmath623 , \\ \quad \alpha \mapsto \sum_{d \in    h_{g}^2(x,{\mathbb{z } } ) } \sum_{n \ge 0 } \int_{[{\overline}{{\mathcal{m}}}_n^{g_\zeta}(c , x , l_t,\zeta , d ) ] } ( q^d / n ! )   { { \operatorname}{ev}}^ * ( \alpha , \ldots , \alpha ) \cup { \epsilon}_+(t(x / g))^{-1 } \end{gathered}\ ] ] for @xmath24 , extended by linearity over @xmath624 , where we omit the restriction map @xmath625 to simplify notation . [ quantum kalkman formula , arbitrary group case ] [ kalk3 ] suppose that @xmath2 is equipped with polarizations @xmath51 so stable = semistable for the action of @xmath1 on @xmath377 . then the gromov - witten invariants of @xmath16 are related by a sum of twisted gauged gromov - witten invariants for subgroups @xmath626 @xmath627,t \in ( -1,1 ) } \frac{|w_{\zeta}|}{|w_{{\mathbb{c}}\zeta}| }    { { \operatorname}{resid}}_\xi \tau^{g_\zeta}_{x,\zeta , t } , \ ] ] where the sum over @xmath459 $ ] is over equivalence classes of unparameterized one - parameter subgroups generated by @xmath403 . we already gave a simple example [ threepoint ] of this formula in the introduction . we give another fano example :    [ c1 ] ( quantum powers of the first chern class for the blow - up of the projective plane ) suppose that @xmath628 acts on @xmath310 with weights @xmath629 . consider the path from @xmath498 to @xmath499 in @xmath630 crossing through the chambers with git quotients @xmath500 . we denote by @xmath315 resp . @xmath314 the second resp . third quotient . the quantum kirwan morphism for @xmath631 has no quantum corrections , since these varieties are fano , and so @xmath632 we consider the wall - crossing formula for invariants with @xmath633 fixed markings , corresponding to the small quantum product @xmath634 . by example [ toric ] , the moduli space of gauged maps is , after fixing the locations of the markings , the quotient of the space of sections @xmath635 by @xmath1 , with stability condition corresponding to the stability condition for @xmath2 , see @xcite . we take @xmath636 , so that @xmath637 ( or @xmath638 for short ) where for any weight @xmath639 , @xmath640 denotes the one - dimensional representation with weight @xmath639 . the moduli spaces of gauged maps corresponding to the various chambers are therefore the empty set , @xmath641 , its blow - up along a @xmath291 , and the empty set again . we consider the sequence of polarizations corresponding to the vectors @xmath498 , @xmath493 , @xmath494 , @xmath499 . the first wall - crossing term for degree @xmath642 invariants corresponds to the direction @xmath643 , for which there is a unique @xmath405-fixed stable map with normal weights @xmath642 with multiplicity 4 and @xmath644 with multiplicity @xmath31 . the wall - crossing term is @xmath645 using this implies that @xmath646 $ ] which is shorthand for saying that the coefficient of @xmath647 $ ] is @xmath648 ; here @xmath116 is the small quantum product , as expected since @xmath649 .\ ] ] the second wall - crossing term corresponds to the change of quotient from @xmath477 to @xmath476 with @xmath650 is ( after fixing the five points on @xmath291 ) an integration over @xmath651 , the quotient of the @xmath405-fixed summand @xmath652 by @xmath167 , @xmath653 } \kappa_{{\mathbb{c}}^7}^{{\mathbb{c}}^\times } \frac { ( 3 \xi_1    + 2\xi_2)^5 } { \xi_1 ^ 4 \xi_2}\ ] ] where @xmath654 is the descent map in equivariant cohomology from @xmath638 . using that @xmath655 is the generator @xmath513 of @xmath514 this gives @xmath656 } \frac { ( 5 \omega + \xi)^5 } { ( \omega +    \xi)^4 ( \omega - \xi ) } & = & { { \operatorname}{resid}}_\xi \int_{[{\mathbb{p } } ] } - \xi^{-5 } ( 25 \omega \xi^4 + \xi^5)(1 - 4 \omega/\xi ) ( 1 + \omega/\xi ) \\ & = & 11 .\end{aligned}\ ] ] it follows that @xmath657 .$ ] the last wall - crossing term for the degree @xmath642 invariant is @xmath658 } \kappa_{{\mathbb{c}}^7}^{{\mathbb{c}}^\times } \frac { ( 3\xi_1 +    2\xi_2)^5 } { \xi_2(\xi_1 + \xi_2)^2 } = 232 .\ ] ] this is as expected since the quotient is empty in the last chamber . one can verify that the expansion of @xmath659 contains @xmath660 $ ] using the known quantum multiplication table for @xmath476 from crauder - miranda @xcite : @xmath661 where @xmath662 are the exceptional resp . fiber resp . point resp . fundamental classes respectively , @xmath663 , and using a little help from mathematica . ( we thank eric malm for teaching us how to get mathematica to compute these coefficients . )      by the adiabatic limit theorem [ largerel ] , to prove the wall - crossing formula [ kalk3 ] it suffices to prove a formula for the difference of gauged potentials . the following result is a algebro - geometric generalization of a wall - crossing formula of cieliebak - salamon @xcite for gauged gromov - witten invariants of quotients of vector spaces defined using symplectic geometry . we will deduce our main result theorem [ kalk3 ] by taking the large area limit @xmath664 of the following theorem :    [ gwall ] let @xmath2 be a smooth projective @xmath1-variety . suppose that @xmath229 are polarizations such that semistable = stable for the stack @xmath665 of [ masterprop ] . then the gauged gromov - witten potentials are related by @xmath666,t \in ( -1,1 ) }   \frac{|w_{\zeta}|}{|w_{{\mathbb{c}}\zeta}| } { { \operatorname}{resid}}_\xi   \tau^{g_\zeta}_{x,\zeta , t}\ ] ] where the sum is over equivalence classes @xmath459 $ ] of unparametrized one - parameter subgroups generated by @xmath403 , and the same mundet semistability parameter should be used to define the potentials on both sides of the equation . the proof of the wall - crossing formula [ gwall ] depends on the construction of _ master space _ in the sense of [ master ] whose quotients are the moduli spaces of mundet stable gauged maps . [ existence of a master space ] [ masterprop ] under suitable stable = semistable conditions , there exists a proper deligne - mumford @xmath167-stack with perfect relative obstruction theory @xmath667 whose @xmath167-fixed substack is the union of @xmath668 and the images of the stacks @xmath615 for @xmath669 of definition [ contrib ] . furthermore , @xmath667 admits an embedding in a non - singular deligne - mumford stack . the proof will be given after several constructions . first recall the quot - scheme compactification of mundet semistable morphisms from @xmath86 to @xmath119 constructed by schmitt @xcite :    [ quot - scheme compactification of gauged maps ] in the case @xmath670 and @xmath671 , a _ projective bump _ over a smooth projective curve @xmath86 is a datum consisting of a vector bundle @xmath672 ; a line bundle @xmath673 ; and a surjective morphism @xmath674 . more generally , if @xmath1 is an arbitrary group embedded in @xmath675 , and @xmath2 is a smooth projective @xmath1-variety embedded in @xmath676 then a projective bump is a projective @xmath675-bump and a reduction of the frame bundle @xmath677 to @xmath1 such that the corresponding injection @xmath678 takes values in the associated @xmath2-bundle @xmath679 . a bump is _ mundet semistable _ if it satisfies the inequality of definition [ mundetsemistable ] for every pair @xmath171 consisting of a parabolic reduction @xmath605 and antidominant coweight @xmath149 . let @xmath680 denote the stack of mundet semistable projective bumps of class @xmath608 over @xmath86 with target @xmath2 . in the toric case discussed in example [ toric ] there are no non - constant holomorphic maps from curves to @xmath2 and so @xmath681 that is , the moduli space of maps from irreducible domain is already compact and does not require compactification . the following summarizes results of schmitt @xcite :    [ git construction of projective bumps ] [ schmittgit ] @xcite @xmath680 admits a presentation as a git quotient of a _ framed moduli space _ @xmath682 ( consisting of bundles that can be written as quotients of a given trivial bundle on @xmath86 , which we omit from the notation ) by the action of a group @xmath683 acting on the space of framings , with respect to a polarization @xmath684 on @xmath685 depending on the choice of polarization @xmath68 . if stable = semistable , then @xmath680 is a proper deligne - mumford stack with a projective coarse moduli space . the stack @xmath680 seems not to be virtually smooth in general and so , it seems , can not be expected to admit a good virtual fundamental class . we denote by @xmath686 the stack - theoretic quotient , @xmath687 which contains @xmath680 as an open substack . in order to study the dependence on polarization , suppose we are given two polarizations @xmath45 and denote by @xmath688 the relative hyperplane bundle as in . consider the git quotient @xmath689 we call an object of @xmath690 a _ polarized bump _ , since generically an object of it is equivalent to a bump together with a point in the line bundle over it . from the geometric invariant theory construction , it follows :    [ stack of polarized bumps ] if stable = semistable then the stack @xmath690 is a smooth proper deligne - mumford stack with a projective coarse moduli space . we denote by @xmath691 the stable locus of the bundle @xmath692 for @xmath693 depending on @xmath694 $ ] , and by @xmath695 the artin quotient stack . by lemma [ master ]    [ master space for bumps ] the residual action of @xmath167 on the stack @xmath690 has git quotient with respect to the polarization @xmath692 isomorphic to @xmath696 . we wish to obtain similar results for the kontsevich - style compactification , which is the one having a virtual fundamental class with the expected properties . the projection @xmath697 is a @xmath291-bundle over @xmath698 . let @xmath699 ) \subset { \overline}{{\mathfrak{m}}}_n^g(c , x ) $ ] denote the artin stack of gauged maps that are semistable with respect to the stability condition defined by @xmath354 for some @xmath700 $ ] : @xmath701 ) : = \bigcup_{t \in [ -1,1 ] } { \overline}{{\mathcal{m}}}_n^g(c , x , l_t ) .\ ] ]    there is a _ relative givental morphism _ from @xmath699)$ ] to @xmath702 extending the forgetful morphism on the open stratum ( if non - empty ) . the morphism , roughly speaking , replaces each bubble tree which maps to a point in @xmath86 with a base point for the linear system corresponding to the quotient . the relative givental morphism is constructed more generally in popa - roth @xcite . each object in @xmath703 ) $ ] is a pair of a bundle @xmath704 with stable section @xmath705 , where @xmath706 is a nodal curve ; the givental morphism replaces @xmath164 with the corresponding quotient . there is no morphism from @xmath707 to @xmath708 since the domain has objects with infinitely many isomorphism classes of bundles , while the isomorphism classes of bundles in the target form a finite set . [ polarized gauged maps ] a _ mundet semi - stable polarized gauged map _ is an object in the fiber product of the relative givental morphism with the projection from the master space for projective bumps : @xmath709 ) \times_{{\overline}{{\mathfrak{m}}}^{g,{{\operatorname}{quot}}}(c , x ) } { \overline}{{\mathcal{m}}}^{g,{{\operatorname}{quot}}}(c , x , l_+,l_-)\ ] ] that is , a polarized gauged bump together with a lift of the underlying bump to the moduli stack of stable gauged maps that are mundet - semistable for some @xmath700 $ ] .    if stable = semistable for polarized gauged maps , then for any @xmath608 , @xmath667 is a smooth proper deligne - mumford stack . since @xmath710 is proper and deligne - mumford , and the codomain is proper deligne - mumford under the stable = semistable assumption . smoothness is immediate from the fiber product description , since the second morphism is an open embedding .      in this subsection , we deduce the quantum wall - crossing formula by applying virtual localization to the master space @xmath667 constructed in the previous subsection . under the given hypotheses the stack @xmath667 is a proper deligne - mumford stack and an open substack of artin stack @xmath711 defined by dropping the semistability condition . [ obstruction theory for polarized gauged maps ] for any @xmath608 , the artin stack @xmath712 has a relative obstruction theory over the moduli stack @xmath190 of parametrized prestable curves . if stable = semistable then the restriction of this obstruction theory to @xmath667 is perfect . the complex in the relative obstruction theory is denoted @xmath713 in the diagram below , constructed from the following commutative diagram of complexes of coherent sheaves . we use @xmath714 resp . @xmath715 for the cotangent complex resp . relative cotangent complex for the morphism to avoid conflict with previous notation . @xmath716 }   \arrow{w } \arrow{s } \\ \node{t^{\vee}_\pi } \node{t^{\vee}_{{\overline}{{\mathfrak{m}}}_n^{g}(c , x , l_-,l_+ ) } } \arrow{w }   \node { \pi^ *    t^{\vee}_{{\overline}{{\mathfrak{m}}}_n^g({c},x,[-1,1 ] ) } } \arrow{w } \node{t^{\vee}_\pi[1]}\arrow{w } \end{diagram}\ ] ] where the horizontal lines are exact triangles and the second vertical arrow @xmath717 exists by the axioms of a triangulated category . the map @xmath717 satisfies the axioms of an obstruction theory by the five lemma applied to the cohomology of the above diagram . if the automorphisms are finite then the relative obstruction theory is perfect . indeed for the base @xmath718 is identified with the dual of the lie algebra of the group of automorphisms ( * ? ? ? * example 6.7(c ) ) , and this carries over to the total space by the long exact sequence .    for any @xmath608 , @xmath667 admits an embedding in a non - singular deligne - mumford stack . consider an embedding @xmath719 and @xmath1-equivariant embedding @xmath720 for some @xmath285 . let @xmath721 denote the universal bundle over the moduli space of framed bumps @xmath722 , equipped with the action of @xmath683 by changing the framings as in proposition [ schmittgit ] . then @xmath723 embeds in the projectivization @xmath724 . the latter is projective by schmitt s construction ( it is the pull - back of the universal bundle on quot scheme ) and so embeds @xmath683-equivariantly in some @xmath725 . then @xmath667 is an embedded substack of @xmath726 , with objects given by stable maps that are compositions of stable sections of @xmath727 with the inclusion into @xmath728 . since @xmath729 is a non - singular deligne - mumford stack , the quotient @xmath726 is a non - singular artin stack . the group @xmath683 acts locally freely on an open subset @xmath730 by the stable = semistable assumption and the quotient @xmath731 is deligne - mumford . [ wall - crossing for gauged gromov - witten invariants , preliminary version ] [ prewall ] let @xmath2 be a smooth projective @xmath1-variety equipped with polarizations @xmath45 . suppose that stable = semistable for the master space @xmath667 . then for any @xmath205 , @xmath732 } { { \operatorname}{ev}}^ * \alpha - \int _ { [        { \overline}{{\mathcal{m}}}_n^g(c , x , l_-,d ) ] } { { \operatorname}{ev}}^ * \alpha \\ = { { \operatorname}{resid}}_\xi \left ( \sum_{f } \int_{[f ] } \iota_f^ * { { \operatorname}{ev}}^ *    \alpha \cup { { \operatorname}{eul}}_{{\mathbb{c}}^\times}(\nu_f)^{-1 } \right)\end{gathered}\ ] ] where @xmath733 ranges over the fixed point components of @xmath167 on @xmath734 not equal to @xmath735 with normal complex @xmath736 .    by virtual localization applied to the @xmath167-action on @xmath737 , as in the kiem - li fixed point formula theorem [ virkalk ] . next we describe the moduli spaces of @xmath167-fixed gauged maps in terms of gauged maps with smaller structure group . let @xmath738 denote the category of @xmath405-fixed gauged maps whose principal component is semistable for the polarization given by @xmath354 for some @xmath23 , as in definition [ fixedef ] . [ fixed points as reducible gauged maps ] [ red ] any @xmath167-fixed component of @xmath739 is in the image of @xmath738 in @xmath740 for some @xmath23 where @xmath741 is a non - zero element , @xmath445 is stabilizer , and @xmath449 the unparametrized one - parameter subgroup generated by @xmath405 , consisting of maps @xmath742 taking values in @xmath743 on the principal component , and @xmath591 on the bubbles . furthermore , for @xmath744 the map @xmath745 has fiber isomorphic to @xmath746 . any fixed point of @xmath167 on @xmath747 is necessarily in the fiber over a gauged map @xmath748 with infinite automorphism group . the infinitesimal automorphism generated by the action of @xmath167 is a section of the adjoint bundle @xmath749 , given by an element @xmath750 at a base point @xmath751 . the structure group of @xmath146 reduces to the centralizer @xmath445 , and being fixed by the automorphism the section @xmath164 takes values in the fixed point set @xmath752 of @xmath405 on the principal component . the fiber of the morphism is @xmath753 for @xmath409 , corresponding to pairs @xmath170 related by @xmath1-isomorphisms that are not @xmath445-isomorphisms . [ bubbles ] we may describe the fixed point locus as a stack of `` bubble trees '' as follows : there is an isomorphism @xmath754 }    \left ( { \mathcal{m}}^{g_\zeta,{{{\operatorname}{fr}}}}_r(c , x^{\zeta , t } ) \times_{(x^\zeta)^r }    \prod_{j=1}^r { \overline}{{\mathcal{m}}}_{|i_j|+1}(x ) \right)^\zeta / ( g_\zeta)^r\ ] ] where the right - hand - side denotes the fixed point set of the action @xmath434 on the moduli stack of gauged maps to @xmath454 with bubbles in @xmath2 . indeed , by definition each object of @xmath755 consists of a principal component mapping to @xmath600 and a collection of bubble trees in @xmath2 fixed ( up to reparametrization ) by the action of @xmath434 . [ obstheory ] ( obstruction theory for the fixed point components ) @xmath756 is an artin stack , and if every automorphism group is finite modulo @xmath434 , each substack with fixed homology class @xmath757 is a proper deligne - mumford stack with a relatively perfect obstruction theory over @xmath758 whose complex is @xmath759 . the relatively perfect obstruction theory @xmath756 is pulled back from that on the @xmath167-fixed point set in @xmath665 in proposition [ red ] , which exists by the construction of relatively perfect obstruction theories on fixed point loci in @xcite . let @xmath760 $ ] denote the virtual fundamental class in the homology of the coarse moduli space resulting from corollary [ obstheory ] . integration with respect to these classes yields _ @xmath405-fixed gauged gromov - witten invariants _ of definition [ contrib ] . these gauged gromov - witten invariants satisfy splitting properties similar to the gauged gromov - witten invariants defined by integration over @xmath226 constructed in @xcite , @xcite , that we will not discuss here . the @xmath405-fixed gauged gromov - witten invariants that appear in the wall - crossing formula involve further twists by euler classes of index bundles @xmath761 , that is , the moving part of @xmath762 . this completes the construction of the fixed point potential in definition [ contrib ] . we remark that the resulting fixed point potential @xmath763 takes values in @xmath764 rather than in @xmath765 because gromov compactness fails for gauged maps in the case that a central subgroup @xmath434 acts trivially . indeed , in this case , the energy @xmath766 of a gauged map of class @xmath187 does not determine the isomorphism class of the bundle , since twisting by a character of @xmath434 does not change the energy . theorem [ kalk3 ] now follows from theorem [ prewall ] and proposition [ red ] , and the observation that stable = semistable for @xmath2 implies stable = semistable for gauged maps in the limit @xmath664 , discussed in @xcite and , for fixed gauged maps , in remark [ fixedlargearea ] . the right - hand - side of theorem [ kalk3 ] can also be re - written using the quantum kirwan map for @xmath767 , using the adiabatic limit theorem for @xmath768-gauged maps . however , in our examples the gauged gromov - witten invariants are always easier to compute than the gromov - witten invariants of the git quotients , so we have left the formula as written . ruan and others , see @xcite , conjectured that _ crepant resolutions _ induce equivalences in gromov - witten theory . more generally , suppose that @xmath769 are smooth proper deligne - mumford stacks with projective coarse moduli spaces related by a birational equivalence @xmath770 . such a birational equivalence is called _ crepant _ ( or a _ @xmath771-equivalence _ ) if @xmath770 factors into morphisms @xmath772 from a smooth stack @xmath580 with projective coarse moduli space such that the pullbacks of the canonical divisors to @xmath580 are equal , as in kawamata @xcite . it is conjectured by li - ruan @xcite , bryan - graber @xcite and others ( perhaps motivated by physics papers such as witten @xcite ) that in such a situation ( not necessarily arising from geometric invariant theory ) the gromov - witten theories of @xmath773 and @xmath774 are equivalent , in a sense to be made precise . many special cases have been proved , see for example iwao - lee - lin - wang @xcite , lee - lin - wang @xcite , boissire - mann - perroni @xcite , bryan - gholampour @xcite , @xcite , bryan - graber - pandharipande @xcite , coates - corti - iritani - tseng @xcite and coates - iritani - tseng @xcite . we prove a version theorem [ cytype ] of ruan s conjecture for crepant birational equivalences induced by variation of git . suppose that @xmath2 is a smooth projective @xmath1-variety , and @xmath16 are git quotients obtained from polarizations @xmath45 . suppose that stable = semistable for @xmath377 so that the master space @xmath775 is a smooth proper deligne - mumford stack . [ crepant ] a birational transformation of git type @xmath776 will be called _ crepant _ if the sum of the weights of @xmath167 on @xmath777 , counted with multiplicity , vanishes . more generally , in the case of twisted gromov - witten theory associated to a @xmath1-vector bundle @xmath223 , we say that @xmath778 is crepant resp . weakly crepant if the same holds for @xmath779 where @xmath780 is the index of @xmath271 as in . it is easy to check that the definition of crepant transformation of git type is a special cases of the definition of crepant transformation ( @xmath771-equivalence ) in kawamata @xcite etc . indeed , kempf s descent lemma ( * ? ? ? * theorem 2.3 ) and the crepant condition together imply that the canonical bundle ( possibly twisted by the top exterior power of @xmath780 ) descends to each singular quotient , from which the canonical bundles on @xmath16 are pulled back . the proof of invariance follows from a symmetry of the fixed point contributions under an action of the picard stack @xmath781 of line bundles on @xmath86 defined as follows . the lie algebra @xmath782 has a distinguished factor generated by @xmath405 , and using an invariant metric the weight lattice of @xmath782 has a distinguished factor @xmath783 given by its intersection with the lie algebra of @xmath434 . after passing to a finite cover , we may assume that @xmath784 . [ pic ] ( action of the picard stack on fixed loci ) in the limit @xmath215 there is a canonical action of @xmath785 on the moduli stack @xmath621 for any @xmath23 . the isomorphism preserves the relative obstruction theories and so the behrend - fantechi virtual fundamental classes . furthermore , the isomorphism preserves the class @xmath786 for any @xmath787 . an object of @xmath621 consists of a tuple @xmath788 where @xmath579 is a @xmath1-bundle and @xmath705 is @xmath405-fixed , in particular , the restriction of @xmath164 to the principal component of @xmath86 maps into the fixed point locus @xmath450 . let @xmath449 denote the subgroup of @xmath445 generated by @xmath789 . then @xmath785 acts on @xmath621 by @xmath790 where we use that @xmath791 since the action of @xmath434 on @xmath450 is trivial . because the mundet weights @xmath792 approach @xmath166 as @xmath215 , the limiting mundet is unchanged by the shift by @xmath793 in the limit @xmath215 and so mundet semistability is preserved , see remark [ fixedlargearea ] . the action of an object @xmath793 of @xmath785 induces an isomorphism @xmath794 where @xmath795 . the action of @xmath785 lifts to the universal curves , denoted by the same notation . since the obstruction theory on @xmath615 is the @xmath405-invariant part of @xmath594 , the isomorphism preserves the relative obstruction theories and so the behrend - fantechi virtual fundamental classes . since the evaluation map is unchanged by pull - back by @xmath796 ( up to isomorphism given by twisting by @xmath793 ) , the class @xmath797 is preserved . [ distrib ] in preparation for the proof of theorem [ cytype ] we briefly recall a few items from the schwartz theory of _ distributions _ for which the standard reference is hrmander @xcite . recall that if @xmath798 is the circle then we denote by @xmath799 the space of continuous linear functionals on the smooth functions on @xmath798 , and by @xmath800 the space of _ tempered distributions_. fourier transform defines an isomorphism of @xmath801 with the space of functions on @xmath783 with polynomial growth . we view @xmath64 as a coordinate on the punctured plane @xmath167 . any formal power series in @xmath802 defines a distribution on @xmath798 , which is tempered if the coefficient of @xmath69 has polynomial growth in @xmath187 . in particular @xmath803 is the delta function at @xmath804 , and has fourier transform the constant function with value @xmath321 . any distribution of the form @xmath805 , for @xmath806 polynomial , is a sum of derivatives of the delta function ( since fourier transform takes multiplication to differentiation ) and so is almost everywhere zero . we study the dependence of the fixed point contributions @xmath807 with respect to the picard action defined in lemma [ pic ] . suppose that @xmath793 is a @xmath434-bundle of first chern class the generator of @xmath808 , after the identification @xmath809 . we denote the corresponding class in @xmath810 by @xmath811 . consider the action of @xmath783-subgroup of @xmath785 generated by @xmath793 . the contribution of any component @xmath615 of class @xmath812 differs from that from the component induced by acting by @xmath813 , of class @xmath814 , by the ratio of euler classes @xmath815 which we now compute . let @xmath454 be the component of the fixed point set @xmath450 which is semistable for @xmath23 . for simplicity , we assume that @xmath454 is connected ; in general , one should repeat the following argument for each connected component . consider the decomposition into @xmath167-bundles @xmath816 where @xmath167 acts on @xmath817 with non - zero weight @xmath35 . then @xmath818 is canonically isomorphic to @xmath819 on the bubble components , since the @xmath1-bundles are trivial on those components . because the pull - back complexes are isomorphic on the bubble components , the difference @xmath820 is the pullback of the difference of the restrictions to the principal part of the universal curve , that is , the projection on the second factor @xmath821 these restrictions are given by @xmath822 where @xmath823 is the map from the universal curve to @xmath86 . the projection @xmath824 is a representable morphism of stacks given as global quotients @xcite in which case by grothendieck - riemann - roch @xcite , @xcite @xmath825 where @xmath826 is a constant section of @xmath824 . the equality of chern characters above implies an isomorphism in rational @xmath771-theory ; by the splitting principle we may assume that the @xmath817 are line bundles . the difference in euler classes is therefore given by the euler class of the last summand in @xmath827 let @xmath828 be the sum of weights of the action at the fixed point component @xmath454 . expanding out the product we obtain @xmath829 and @xmath830 indicates further terms with the property that he coefficient of @xmath831 is polynomial in @xmath122 . by the crepant assumption in definition [ crepant ] , the sum of the weights is @xmath832 summing the terms from @xmath833 we obtain a distribution almost everywhere vanishing : for any singular value @xmath234 , @xmath834 } \\ \left ( 1 +   \xi^{-1 } \sum_{i=1}^k r c_1(\nu_{x^{\zeta , t},i } ) + \ldots \right )   { { \operatorname}{ev}}^ *   ( \alpha , \ldots , \alpha )   \end{gathered}\ ] ] where as before the terms @xmath830 are polynomial in the @xmath122 . now for any polynomial @xmath835 in @xmath122 , @xmath836 vanishes almost everywhere in @xmath64 , being a function times a sum of derivatives of delta functions in @xmath64 , see remark [ distrib ] . since @xmath837 is a sum of such wall - crossing terms , this completes the proof of theorem [ cytype ] .    1 . the proof shows that if in fact the action of @xmath1 on @xmath2 is calabi - yau in the sense that @xmath838 , then the wall - crossing term is not only almost everywhere vanishing but also a sum of _ at least second _ derivatives of delta functions . 2 .   the standard formulation of the crepant transformation conjecture in coates - ruan @xcite etc . uses analytic continuation . the above results say nothing about convergence of the gauged gromov - witten potentials , so it is rather difficult to put the version above in this language . however , if the potentials have expressions as analytic functions with overlapping regions of definition on the torus @xmath839 with coordinate @xmath64 , then they are equal on that region . ( crepant resolution of @xmath840 ) let @xmath15 act on @xmath518 with weights @xmath841 , and let @xmath51 denote the trivial line bundles with characters @xmath52 . then @xmath495 is isomorphic to the quotient stack @xmath840 and @xmath314 is the crepant resolution of its coarse moduli space . theorem [ cytype ] implies that the gromov - witten theories of @xmath16 are equivalent . even in the case of an @xmath842-singularity ( @xmath843 ) the transformation relating the two equivalent gromov - witten theories is non - trivial ; see coates - corti - iritani - tseng @xcite and bryan - gholampour @xcite for more on quantum cohomology of resolutions of ade singularities . ( flop ) let @xmath15 act on @xmath844 with weights @xmath574 each with multiplicity @xmath285 , and let @xmath51 denote the trivial line bundles with characters @xmath52 . then @xmath353 undergoes a simple flop as @xmath359 passes through zero ( blow - down of a @xmath283 followed by a blow - up ) and theorem [ cytype ] implies that the gromov - witten theories of @xmath323 are equivalent . see iwao - lee - lin - wang @xcite for the invariance under simple flops in general and example [ simpleflop ] . ( quantum mckay correspondences for abelian three - fold singularities ) let @xmath151 be the regular representation of @xmath845 and consider the affine scheme of representations of the mckay quiver with relations , @xmath846 we denote by @xmath847 the set of isomorphism classes of irreducible representations of @xmath845 , for any @xmath848 let @xmath849 denote the corresponding irreducible representation , and @xmath850 the representation ring . the group of equivariant automorphisms of the regular representation @xmath851 acts on @xmath852 by conjugation and induces a faithful action of @xmath853 . any _ stability parameter _ @xmath854 induces a character @xmath855 \mapsto \prod_{\rho \in { \operatorname}{irr(\gamma ) } } \det(g_\rho)^{\theta(\rho)}\ ] ] and so a linearization of the trivial bundle . denote @xmath856 the stack - theoretic quotient of the semistable locus in @xmath852 with respect to @xmath318 . a result of bridgeland , king , and reid @xcite shows that @xmath857 is a crepant resolution of the coarse moduli space of @xmath858 . craw - ishii @xcite show further that any two projective crepant resolutions of ( the coarse moduli space of ) @xmath858 are related by variation of git . by theorem [ cytype ] and the results of craw - ishii @xcite , any two quotients @xmath859 for which stable = semistable have equivalent gromov - witten theories in the sense of theorem [ cytype ] assuming that the git quotient at the wall being crossed is a smooth stack . this already follows from the results of iwa - lee - lin - wang @xcite , since as craw - ishii @xcite shows , the transitions are all simple flops .    it would be interesting to understand , given a quiver with relations , for which stability parameters the git quotients are smooth stacks . for example , ishii - ueda @xcite have given a non - degeneracy condition for smoothness of the moduli spaces associated to dimer models for generic stability parameters . for any codimension one wall whose associated git quotient is smooth ( that is the semistable locus associated to the polarization lying in a wall is smooth ) one obtains an equivalence of gromov - witten theories . however , we do not know for which walls this condition holds . m.  brion and c.  procesi . action dun tore dans une varit projective . in a.  connes et  al . , editors , _ operator algebras , unitary representations , enveloping algebras , and invariant theory _ , volume  62 of _ progress in mathematics _ , pages 509539 , paris , 1989 , 1990 . birkhuser , boston .                                                            g.  kempf and l.  ness . the length of vectors in representation spaces . in k. lnsted , editor , _ algebraic geometry _ , volume 732 of _ lecture notes in mathematics _ , pages 233244 , copenhagen , 1978 , 1979 . springer - verlag , berlin - heidelberg - new york . i. manin . , volume  4 of _ north - holland mathematical library_. north - holland publishing co. , amsterdam , second edition , 1986 . algebra , geometry , arithmetic , translated from the russian by m. hazewinkel .
we prove a quantum version of kalkman s wall - crossing formula @xcite , @xcite comparing intersection pairings on geometric invariant theory ( git ) quotients related by a change in polarization , under certain stable = semistable assumptions . 
 the formula is the same , but each expression in the formula is quantized in the sense that it is replaced by an integral over moduli spaces of certain stable maps ; in particular , the wall - crossing terms are gauged gromov - witten invariants with smaller structure group . as an application , 
 we show that the graph gromov - witten potentials of quotients related by wall - crossings of crepant type are equivalent up to a distribution in the quantum parameter that is almost everywhere zero . 
 this is a version of the crepant transformation conjecture of li - ruan @xcite , bryan - graber @xcite , coates - ruan @xcite etc . in cases where the crepant transformation is obtained by variation of git . 
 = 8.5 in