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it has been studied years ago that the supersymmetric generalizations of the bosonic cp(n-1 ) sigma model is automatically an @xmath2 supersymmetric theory by the virtue of that the target space cp(n-1 ) is a khler manifold @xcite . recently edalati and tong suggested and built an @xmath3 cp(n-1 ) heterotic sigma model by study of the low - energy dynamics of vortex strings in @xmath4 four - dimensional gauge theories@xcite . later shifman and yung formulated a geometric representation for this heterotic model with the cp(n-1 ) target space for bosonic fields and an extra right - handed fermion which couples to the fermion fields of the @xmath5 cp(n-1 ) model@xcite , and thus breaks the supersymmetry to @xmath3 .    in this paper , we follow the geometric formulation to discuss the fermionic zero modes of @xmath3 heterotic cp(1 ) sigma model in the instanton background . nontrivial homotopy group structure of target space cp(1 ) allows for the bosonic instanton background of the form of a holomorphic function @xmath6 , where @xmath7 and @xmath8 are instanton size and center respectively . under such background , the fermionic zero modes can be obtained by acting supercharges and superconformal charges to the instanton solution . since the heterotic deformation of the standard cp(1 ) sigma model only preserves half of four original supercharges , the supersymmetries and superconformal symmetries are partly broken . the remaining fermionic generators are not sufficient to generate all fermionic zero modes from the instanton background . therefore we will find zero modes by solving dirac equations . in section 2 , we will calculate the chiral anomaly at first to give the number of zero modes . in section 3 we will explicitly solve dirac equations of zero modes and use the result of section 2 to pick up those linear independent modes . in the last section we generalize this result to cp(n-1 ) heterotic sigma models . we start from the chiral anomaly of the standard cp(1 ) sigma model . the notations are in accordance with the reference @xcite . the lagrangian of cp(1 ) is given as follows : @xmath9\nonumber\\ & = & \frac{2}{g_{0}^2\chi^2}\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi+\frac{2i}{g_{0}^2}\bar{\xi}\gamma^{\mu}(\partial_{\mu}\xi - ia_{\mu}\xi)+\frac{2}{g_{0}^2}(\bar{\xi}\xi)^2\end{aligned}\ ] ] where @xmath10 , @xmath11 and @xmath12 for convenience to calculate the chiral anomaly . it has a chiral symmetry and thus an axial current : @xmath13 the chiral anomaly is well - known@xcite as : @xmath14 this anomaly term corresponds to the tadpole diagram : +     now we are going to consider the chiral anomaly of heterotic cp(1 ) sigma model . the lagrangian of the heterotic model for bilinear fermion terms is deformed from eq.(1 ) by adding terms with an extra right - handed fermion field @xmath15 @xcite . since only bilinear terms of fermions need to be considered when calculating one loop diagrams , we just write them up : @xmath16 + \frac{2i}{g_{0}^2\chi^2}\bar{\psi}\gamma^{\mu}(\partial_{\mu}\psi-\frac{2}{\chi}\phi^{\dagger}\partial_{\mu}\phi\psi)\nonumber\\ & = & i\bar{\zeta}\gamma^{\mu}\frac{1+\gamma_5}{2}\partial_{\mu}\zeta+(\bar{\zeta}\gamma^{\mu}\frac{1+\gamma_5}{2}b_{\mu}\xi+h.c.)+ \frac{2i}{g_{0}^2}\bar{\xi}\gamma^{\mu}(\partial_{\mu}\xi - ia_{\mu}\xi)\end{aligned}\ ] ] where we define @xmath17 and @xmath18 . it is easy to see @xmath19 and the lagrangian is invariant under @xmath20 and @xmath21 . therefore the corresponding axial current is : @xmath22    we will show that the chiral anomaly of the heterotic model will not be corrected by the extra @xmath23 terms , and thus prove that the number of zero modes is the same as the standard cp(1 ) model . actually , to calculate this anomaly one needs to consider diagrams as follows : +   + the diagrams b and c cancel with each other because both of them are finite , and thus free to shift integral variables . the diagrams d and e also cancel each other . it can be proved by using momentum @xmath24 times the sum of these two diagrams , @xmath25 +   + by multiplying @xmath26 , the original converged digrams d and e become two divergent parts respectively . however they are all logarithmically divergent , and thus still free to shift integral variables to prove the cancellation . therefore the anomaly of heterotic case is still only from the tadpole diagram , and is the same as in the standard cp(1 ) case , see equation ( 3 ) . for the standard cp(1 ) model , @xmath27 has four zero modes , under the instanton background @xmath28@xcite . since the chiral anomaly is not corrected in the heterotic case , the number of zero modes should be still four . we now proceed to derive the dirac equations for @xmath27 and @xmath23 from the heterotic lagrangian and wick - rotate them to euclidean space by @xmath29 . we only keep linear terms in the equations of motion and solve them . we will see that the solutions are also satisfied with the equations of motion when we add nonlinear terms . since we have already shown that the number of independent zero modes is four , the solutions solved in the equations of motions up to linear terms are all zero modes . from eq . ( 4 ) and instanton background @xmath30 , in spinor components we have : @xmath31 notice that eqs . ( 8) ( 9 ) and ( 11 ) are the same as in the standard cp(1 ) case . therefore the solutions to these three equations should be the same as before . they give all four zero modes . since we have argued that the total number of zero modes is four , there is no extra zero modes raised from field @xmath32 , although the equation of motion of @xmath32  ( 10 ) is deformed by the @xmath33 term . therefore we have : @xmath34 by the same argument , fermionic field @xmath23 and @xmath35 can not provide extra zero modes . from eqs . ( 6 ) ( 7 ) and ( 10 ) , we get : @xmath36 at first sight , the solutions seem inconsistent with the anomaly calculation , which gives the number of the difference between left and right handed fermions under instanton background , @xmath25 @xmath37 from solutions ( 12)-(15 ) , it seems that the first two solutions are right handed fermions while the last two are left handed , thus the number of the difference is zero . however , because we wick - rotated to euclidean space to perform the calculation , in which the complex conjugate is not a well - defined involution for chiral fermions , the field @xmath38 is thus totally independent of @xmath27 . moreover in euclidean space , the lagrangian is @xmath39 invariant . therefore the field @xmath38 transformed under @xmath39 is the same as @xmath40 . bearing this in mind , we can be back to the lagrangian ( 4 ) : @xmath41 now it is clear that , when wick - rotated to euclidean space , the field @xmath42 and @xmath43 we solved in eqs . ( 10 ) and ( 11 ) are in fact the fields @xmath44 and @xmath45 respectively , while the field @xmath38 plays the same role as @xmath40 in euclidean space . therefore the four solutions are all right handed fermions . it is consistent with the anomaly calculation .    as for the field @xmath23 , solution ( 16 ) shows that the extra field @xmath23 mixes with field @xmath27 , however it is linearly dependent of field @xmath27 , and thus does not produce extra zero modes as what we also expected from the result of the chiral anomaly calculation ( 17 ) .    at last , one more worthy to mention is the solutions of field @xmath27 . they are of the same form as in the standard cp(1 ) zero modes . it means that not only the number of zero modes does not change , the zero modes of @xmath27 also receive no correction in the heterotic model . in fact it is easy to understand this result from equation ( 8) and ( 9 ) . although the supersymmetries are partly broken by deformation in the heterotic case , the classical equation of motion of @xmath27 up to linear terms are still @xmath1 supersymmetic invariant . therefore in this sense the zero modes of @xmath27 still can be generated by supercharges and superconformal charges from the instanton background , even though half of the symmetries are broken in the heterotic model . in addition , one can check that these solutions are satisfied with the equations of motion when adding nonlinear terms of fermions . our conclusion to the heterotic cp(1 ) model is that the zero modes of @xmath27 are the same as in the standard cp(1 ) case , while the extra fermion @xmath23 does not provide extra zero modes . this result can be simply generalized to the heterotic cp(n-1 ) model . for the heterotic cp(n-1 ) case , the lagrangian containing bilinear terms of fermions is@xcite : @xmath46 + ig_{i\bar{j}}\bar{\psi}^{\bar{j}}\gamma^{\mu}(\partial_{\mu}\psi^{i}+\gamma^{i}_{lk}\partial_{\mu}\phi^{l}\psi^{k})\end{aligned}\ ] ] hence , the equations of motion up to linear terms under instanton backgroud @xmath47 are : @xmath48 it is not hard to see the solutions are : @xmath49 where @xmath50 is the khler potential . from these solutions we see that the number of zero modes in the heterotic cp(n-1 ) model is still @xmath51 , while @xmath23 can be linearly expressed by the zero modes of @xmath27 and thus not extra zero mode . i am grateful for m. shifman who led me to the subject , and a. vainshtein for very useful and fruitful discussions and reviewing the final version of the manuscript . i also want to thank m. sasseville for proofreading the text . +      b. zumino , phys . lett . * b87 * , 203 ( 1979 ) m. edalati and d. tong , jhep * 0705 * , 005 ( 2007 ) m. shifman and a. yung , phys . rev . * d77 * , 125016 ( 2008 ) v. a. novikov , m. a. shifman , a. i. vainshtein and v. i. zakharov , phys . rept . * 116 * , 103 ( 1984 ) m. shifman , a. vainshtein and r. zwicky , j. phys . * a39 * , 13005 ( 2006 )
we discuss instanton fermionic zero modes in the heterotic @xmath0 modification of the cp(1 ) sigma model in two dimensions . by calculating its chiral anomaly 
 we prove that the number of fermionic zero modes is same as in the standard @xmath1 cp(1 ) case , and determine their explicit form .