Instanton constituents and fermionic zero modes in twisted CP(n) models

We construct twisted instanton solutions of CP(n) models. Generically a charge-k instanton splits into k(n+1) well-separated and almost static constituents carrying fractional topological charges and being ordered along the noncompact direction. The locations, sizes and charges of the constituents are related to the moduli parameters of the instantons. We sketch how solutions with fractional total charge can be obtained. We also calculate the fermionic zero modes with quasi-periodic boundary conditions in the background of twisted instantons for minimally and supersymmetrically coupled fermions. The zero modes are tracers for the constituents and show a characteristic hopping. The analytical findings are compared to results extracted from Monte-Carlo generated and cooled configurations of the corresponding lattice models. Analytical and numerical results are in full agreement and it is demonstrated that the fermionic zero modes are excellent filters for constituents hidden in fluctuating lattice configurations.


Introduction
Nonlinear sigma models in two dimensions have long been used as testing ground for strongly coupled gauge theories [1]. They are scale invariant on the classical level and asymptotically free at the quantum level. The ubiquitous CP n models possess regular instanton solutions, the topological charges of which yield lower BPS-bounds on the action, they have a chiral anomaly when coupled to fermions, generate a dynamical mass by nonperturbative effects at zero temperature and a thermal mass ∝ g 2 T at finite temperature. They have numerous interesting applications to condensed matter physics (for a review see [2]) and also have been used to study the sphaleron induced fermionnumber violation at high temperature [3].
In the present work we consider CP n models at finite temperature, i.e., Euclidean models with imaginary time having period β = 1/k B T . These models possess instanton solutions with finite action and the dimension of the moduli space in a given instanton sector depends on the topology of the Euclidean space-time. For example, on the two-torus the charge-k instantons of CP n depend on as many collective parameters as the instantons of CP n+1 with one charge less [4]. In the present work we do not compactify space such that space-time is a cylinder.
For suitable field variables the selfduality equation for CP n instantons reduces to Cauchy-Riemann conditions such that all instantons are known explicitly for the plane, cylinder and torus. On the plane they are given by rational functions of the Email addresses: wieland@theorie3.physik.uni-erlangen.de (Wieland Brendel), falk.bruckmann@physik.uni-regensburg.de (Falk Bruckmann), lukas.janssen@uni-jena.de (Lukas Janssen), wipf@tpi.uni-jena.de (Andreas Wipf), christian.wozar@uni-jena.de (Christian Wozar) complex coordinate z and on the cylinder by suitable periodic generalizations thereof, see below.
In a previous work [5] one of us introduced the twisted O(3) model (which is equivalent to the CP 1 model) and showed that generically the unit charged instantons in this model dissociate into two fractional charged constituents, sometimes called 'instanton quarks'. Again there is a close analogy to the corresponding situation in Yang-Mills theories, where instantons with nontrivial holonomy along the compact direction of a fourdimensional cylinder possess magnetic monopoles as constituents [6,7,8,9].
We extend the work in [5] in several directions. First we construct the k(n + 1) constituents of CP n instantons with charge k and twisted boundary conditions and relate their positions, sizes and fractional charges to the collective parameters of the instantons. Then we calculate and analyze the zero modes of the Dirac operator for minimally coupled fermions with quasi-periodic boundary conditions in the background of the twisted instantons. We show that the zero modes can be used as tracers for the instanton constituents: they are localized to the latter, to which constituent depends on the boundary condition. Again, this has close analogies in four-dimensional Yang-Mills theories with 1 (or 2) compact dimensions [10,11,12].
CP n spaces admit a Kähler structure such that the twodimensional CP n models admit a supersymmetric extension with two supersymmetries. These models contain 4-fermi interactions and the Dirac operator is given by the linearized field equation for the fermions. We calculate the zero modes of this operator. There exists always one zero mode with squared amplitude being proportional to the action density of the twisted instanton.
We supplement our analytic studies by numerical simulations. With known algorithms we produce typical field configurations for various CP n models with twisted boundary conditions. Then we apply standard lattice cooling techniques to extract the instantons and their constituents from a given (thermalized) configuration. Again we find that an instanton of charge k consists of k(n + 1) constituents. How the corresponding instanton constituents in Yang-Mills theories emerge in the process of cooling/smearing has been studied in [13,14,15,16,17,18]. Next we compute and analyze the zero modes of the overlap Dirac operator. We find good agreement between our numerical and analytical results.
Our results demonstrate that CP n models and Yang-Mills theories share one more common feature: in both models the twisted instantons generically split into well-separated and almost static constituents and in both models the fermionic zero modes trace these constituents. In SU (N ) Yang-Mills theories the number of constituents is given by k and the rank whereas for CP n models it is given by k and by n. On the lattice the fermionic zero modes are excellent filters for the constituents. Even without much cooling the zero modes detect the constituents of the fully cooled configurations.

The CP n model in the continuum and on the lattice
The two-dimensional CP n model [19,20] can be formulated in terms of a complex (n + 1)-vector u = (u 0 , . . . , u n ) T subject to the constraint u † u = 1. The Euclidean action is given by as well as global transformations with a constant matrix U ∈ U (n + 1). The gauge field A µ can be eliminated from the action by using its algebraic equation of motion, The integer-valued instanton number, can be interpreted as the quantized magnetic flux in a fictitious third dimension. At infinity u must approach a pure gauge, u(x) → e iλ(x) c and Q is just the winding number of the map x → e iλ(x) at infinity, an element of the first homotopy group of U (1).
Configurations minimizing the action in S ≥ 4π|Q|/g 2 are called instantons. They fulfill first order self-duality equations.
The most general instanton solution with instanton number Q = k ∈ N can be written in homogeneous coordinates v j as with {v j } a set of polynomials of the complex coordinate z = x 1 + ix 2 with no common root and maximum degree k. The topological charge density of an instanton configuration then reads Lattice formulation For the bosonic model the lattice regularization can be obtained as described in [21,22]. After introducing the matrix-valued gauge invariant field which projects onto the one-dimensional subspace spanned by u, one finds This equation, valid for the model defined on a continuous spacetime, is discretized naively with the forward derivative, ∂ µ P → P x+μ − P x , such that Therefore, the action, up to an irrelevant additive constant, takes the form The simulations of the lattice models have been performed with the help of an overrelaxation algorithm [22]. In addition, to investigate the topological properties, we cooled the lattice configurations [23]. For a given configuration one cooling step consists of minimizing the action locally on a randomly chosen site x. This is achieved by constructing Q x = µ (P x+μ + P x−μ ) and replacing u x by the eigenvector corresponding to the largest eigenvalue of Q x . A cooling sweep corresponds to one cooling step per lattice site on average. Using this procedure the instanton constituents naturally emerge from the locally fluctuating fields.
For the topological charge on the lattice we used the geometric definition in [24] leading to an integer-valued instanton number. This definition and the chosen lattice action are sufficient for the analysis of global topological properties in the vicinity of classical configurations. Thus, we are not affected by the improper scaling behavior of the dynamical CP n models with n ≤ 2 [25]. x 2 Figure 1: (Color online.) Logarithm of the topological density for the 1-instanton solution of the CP 2 model (see (7) and (19)) with symmetric constituents, . The parameters λ i are chosen such that the constituents are localized according to (22)

Instantons at finite temperature
For a quantum system at inverse temperature β we identify z ∼ z + iβ. Since β is the only length scale in the problem we measure all lengths in units of β. In particular the coordinates become dimensionless and we identify z ∼ z + i. Periodic k-instanton solutions ('calorons') are given by [26,27] v By a global U (n + 1) symmetry transformation one can rotate v per such that the constant (and per assumption non-vanishing) The twisted model is only quasi-periodic in the imaginary time direction. This means that the components v j of v are periodic up to phases e 2πiµj with µ j ∈ [0, 1), i.e., the vectors v and u are periodic up to a diagonal element of the global symmetry U (n + 1). The U (n + 1)-invariants like |v| and A µ and hence also q stay periodic. Without loss of generality we assume the phases to be ordered according to µ 0 ≤ µ 1 ≤ . . . ≤ µ n .
For the general solutions of the twisted model we consider the Fourier ansatz and demand the coefficients b (s) j to be non-vanishing only for a finite range of s (for each component j). This is because the corresponding maximum and minimum of the powers then yield a finite topological charge Q. According to (7) one has to compute the following surface integrals Hence the total topological charge in the twisted model can have a fractional part, whose values are restricted by the boundary conditions. By a global transformation we enforce κ min to be taken on in the 0th component and by a (non-periodic) local transformation we further set µ 0 = 0 and κ min = 0, such that Q = κ max . According to Eq. (6), these powers also govern the asymptotic values of the fundamental fields u j . In the following we will mainly analyze twisted instantons with integer-valued instanton number Q = k ∈ N. They are obtained by κ max = k taken on in the 0th component, i.e., the Thus one can obtain the components v j by multiplying each component v per,j from (12) with exp(2πµ j z), which yields v(z) = Ω v per (z), Ω = diag e 2πµ0z , . . . , e 2πµnz . (16) For n = 1 the known twisted unit charged instanton solution [5] can be recovered in terms of the gauge invariant field We made use of µ 0 = 0 and b (1) 1 = 0 and denoted µ 1 by ω.

One-instanton sector
In order to explore the topological density of the instantons we first consider solutions with unit charge Q = k = 1. We multiply v by a constant such that b  j | with j = 0, 1, . . . , n. If, in addition, we define λ n+1 = |b (1) 0 | and µ n+1 = 1, then it can be written in the condensed form The corresponding topological charge density splits into n + 1 constituents at most. For CP 2 this is illustrated in Fig. 1 which shows ln q(x) for various choices of the parameters λ i .
For the general CP n models the occurrence of the constituents can be understood geometrically. To see this more clearly we write with In particular We compare the graphs of these n + 3 linear functions, see The dominant contribution to |v| 2 in (19) at a fixed point x 1 comes from the exponential term whose graph is above the lines defined by the other exponential terms. Hence ln |v| 2 is piecewise linear in the direction x 1 up to exponentially small corrections that are maximal in transition regions, where the highest lying graphs intersect.
Note that for a strictly linear ln |v| 2 the topological density q ∝ ∆ ln |v| 2 would vanish exactly, whereas at cusps generated by intersections of linear parts the topological density would be a Dirac delta distribution (in 3+1 dimensional Yang-Mills theory a similar singular localization can be obtained in the farfield limit [28,11]). As this is a good approximation to the actual ln |v| 2 , we conclude that the topological density of the twisted instantons splits into constituents localized at the intersection points of the lines. Because of the ordering of the µ's, the slopes of the linear functions p i are ordered as well. Note that for x 1 < −1/(2πµ 1 ) ln λ 1 the term exp(p 0 ) dominates such that ln |v| ≈ 0 on the left of all constituents. Correspondingly, ln |v| ≈ 4πx 1 on the right of all constituents.
We obtain the maximum number of constituents, if all consecutive graphs intersect separately and above the rest of the graphs, respectively. More precisely said, the twisted instanton of CP n splits into n + 1 constituents, if, and only if, a 1 ≪ a 2 ≪ · · · ≪ a n+1 , 1 whereas a i is the intersection point of the lines p i−1 and p i , i.e., in particular the twist parameters µ i are distinct, 0 = µ 0 < µ 1 < · · · < µ n < µ n+1 = 1. These positions a i are arbitrary provided the corresponding λ-parameters are chosen according to 1 Thereby we do not want to take the condition a i−1 ≪ a i too literally. It is sufficient, if a i−1 is not close to a i , whereas the required distance is determined by the slopes of p i − p i−1 and x 1 → Figure 2: ln |v| 2 and exponents p i andp as a function of x 1 , see Eqs. (19)- (20), in the CP 2 model for the case of (a 1 , a 2 , a 3 ) = (−5, 1, 4), which leads to three well-separated constituents (equivalent to 2nd example in Fig. 1). Let us discuss the case of the well-separated constituents. In the neighborhood of the intersection point a i of the lines p i−1 and p i we can approximate for x 1 not too far from the constituent i, For i = 1 the lower bound for x 1 is −∞ and for i = n + 1 the upper bound is +∞. The contribution to the topological density of the ith constituent is This shape is the same for all CP n models, cf. [5] for the CP 1 case. The constituent decays exponentially with characteristic length 2π(µ i − µ i−1 ) (measured in units of β) away from its position a i . It has a fractional topological charge Q const,i = µ i − µ i−1 and these charges add up to 1 as they should. In terms of the linear graphs the fractional topological charge is proportional to the difference of slopes of the lines that meet (which also would give the amplitude of the delta distribution mentioned above), and the total charge is the sum of all slope differences, which indeed bend the graph from p 0 with slope 0 to p n+1 with slope 4π eventually. Neighboring constituents can merge adding up the fractional topological charges. This can be understood as 'pulling down' the line that connects the two constituents in the graph of ln |v| 2 , in other words by choosing the corresponding parameter λ i small (cf. Figs. 3-4).
Under which circumstances does the time-dependence of |v| 2 contained in the last term of Eq. 19 (which is proportional to expp(x 1 )) play a role? Since the three graphs of p 0 , p n+1 andp intersect at the point (ã, 0) with 2πã = − ln λ n+1 we havep such that the time-dependentp-term can contribute to the sum in (19) only in the neighborhood ofã. Furthermore, all other lines have to lie below (ã, 0) (if one of the other lines p i lies well above that point, then the topological density becomes to a good approximation static). As the slopes of the p i (x 1 ), i = 1, . . . , n are between 0 and 1, these graphs are never dominant once they are below (ã, 0). This means that only one transition point occurs. Hence time-dependence of the instanton appears iff the topological charge is concentrated in one lump (which can be thought of as all constituents merged, cf. Fig. 4).
The case of non-distinct µ's can be understood by considering the limit µ j → µ j−1 for some j's. Then the corresponding constituent becomes flatter and broader, in the limit it will be invisible and 'massless' (i.e., without topological charge/action; this has been 'eaten' by the constituent j + 1). In this spirit we also recover the periodic solution 2 with µ 0 = µ 1 = · · · = 2 In Yang-Mills theories this amounts to the Harrington-Shepard caloron [29]. µ n = 0: The resulting topological density then consists of only one constituent with unit charge, which can, but does not have to be time-dependent (depending on where the invisible, massless constituents are localized, see also the first row in Fig. 1 of [5]). This can be demonstrated by means of Fig. 2, i.e., based on the case of well-separated constituents: The limit is taken by sending the slopes of the graphs of p 1 and p 2 to 0. If all positions a i are kept constant (by adjusting the λ-parameters), then p 1,2 → 0 (as functions) and the resulting topological density of the periodic solution is equivalent to the case of all constituents merged in the twisted model, cf. Fig. 4 and Fig. 1  (lower right panel). If the limit is taken with all λ-parameters kept constant (i.e., by sending the positions a 1,2 to −∞), then p 1,2 → 2 ln λ 1,2 can lie well above 0 in the limit and the topological density remains static, though it consists of only one unit charged constituent.
Finally, we want to mention the possibility of generating solutions with topological charge less than 1 from these instantons. Technically one has to avoid the asymptotics ln |v| ≈ 4πx 1 for large x 1 (we want to stay in the gauge where κ min = 0 and hence keep the asymptotics for small x 1 ). Hence, if the corresponding parameter λ n+1 = |b (1) 0 | is vanishing (in the Fourier ansatz there is no integer phase e 2πz ), then the total topological charge of the configuration is less than 1. A phase with κ max < 1 then gives the total topological charge (i.e., governs the asymptotics for large x 1 ). Also these configurations consist of constituents with the same formulae for locations, sizes and charges. The number of constituents varies from n down to 1, depending on how many of the remaining parameters λ are zero (in the graph the corresponding lines and intersection points are missing).
Interestingly, all these configurations have in common that the constituents in them are ordered along the noncompact direction. This has already been observed in [5] and substantiated by topological considerations. Here it can best be understood from Eq. (22). The fractional charge of the ith constituent, µ i −µ i−1 , is fixed by the twist in the boundary condition. These charges can be realized in isolation only if the ordering of their positions, a 1 ≪ · · · ≪ a n+1 , applies. If some a i do not obey this ordering, then constituents emerge with the sum of the individual fractions as their topological charge. In other words, 'pulling a constituent through a neighboring one' results not in a different ordering but in joining the constituents to a bigger one, cf. Fig. 1 (upper and lower right panels).
Notice that by giving up our choice κ min = 0 we can rearrange the constituents cyclically; this can become relevant on the lattice, where x 1 is of course periodic.
This ordering is of course related to the selfduality which dictates that all solutions are functions of x 1 + ix 2 ; antiselfdual solutions will have the opposite ordering. We therefore believe that this phenomenon is particular to 1+1 Euclidean dimensions.

k-instanton sector
The general twisted instanton solution (12). (16) with integervalued topological charge k corresponds to the norm where we introduced We encoded the two indices of b (s) j into one, i = s(n + 1) + j, and defined where s = 0, . . . , k, j = 0, . . . , n. To arrive at (28) we transformed the constant vector b (k) in the 0-direction. Similarly as for the one-instanton solution we conclude that the constituents are localized at the transition points of the piecewise linear function |v(z)| 2 . The topological density thus splits into at most k(n+1) constituents. Well-separated constituents are static and exponentially localized at a i , i = 1, . . . , k(n + 1), given in an analogous manner as in the 1-instanton case, cf. Eq. (22). The constituents carry the fractional charge µ i − µ i−1 and from the periodicity of the µ's in Eq. (30) follows that they are again ordered, see also Again we havẽ since the three graphs of p i , p i+s(n+1) andp Therefore the time-dependent term containingp

Cooling of lattice data
Charge-one instantons of the CP 2 model containing up to the maximal number of three constituents are reproduced with a cooling of our lattice data. The simulations were performed on a 6 × 100 (temporal×spatial) lattice at coupling g −2 = 2. In the spatial direction periodic boundary conditions are imposed whereas in the temporal direction the v j are twisted with prescribed µ j . Then a particular configuration is cooled. During the cooling procedure configurations with |k| = 1 and two or more well separated constituents are fairly stable even with this type of unimproved cooling (at least up to 10 5 cooling sweeps). We also observe the typical annihilation of selfdual and antiselfdual constituents. Only a small fraction of configurations is cooled to a state with three clearly separated constituents. One of these is shown in Fig. 6 at three different cooling stages. More often we end up with only two constituents. These results indicate that in a dynamical simulation topological objects with fractional charge (given by the twist parameters µ j ) may be as relevant as they are in Yang-Mills theories.

Minimal coupling to fermions
We extend the bosonic CP n model by introducing a massless Dirac fermion ψ, for the time being minimally coupled, such that the action has the form We shall use the chiral representation of the γ-matrices for which γ * = −iγ 1 γ 2 is diagonal, Splitting the Dirac spinor into chiral components ψ = (ϕ, χ) T the Dirac equation in the background of a self-dual configuration splits into the Weyl equations where ∂ and∂ denote the derivatives with respect to the complex coordinates z = x 1 + ix 2 andz = x 1 − ix 2 . It follows that the zero modes have the form with (anti-)holomorphic functions f (z) and g(z). Similarly as for the bosonic fields we impose quasi-periodic boundary conditions for the fermi field, The functions f, g can be expanded in Fourier series, These modes are only square integrable on the cylinder if the coefficients α (s) , β (s) and the twist parameter ζ fulfill certain constraints. Recall that the asymptotic behavior of the general solution with charge Q = κ max (we set κ min = 0) is Therefore there are no normalizable left-handed zero modes ϕ. 3 The quantum number s of the right-handed zero modes is constrained by 0 < s + ζ < Q. for the example of s + ζ ∈ (µ 1 , µ 2 ). For x 1 < a 1 the slope of ln |v| 2 is 4πµ 0 = 0, for a 1 < x 1 < a 2 it is 4πµ 1 , for a 2 < x 1 it is 4πµ 2 . The vertical distance between the graphs is minimal at a 2 where the zero mode is localized.
This immediately leads to the index theorem for right-handed zero modes. For integer topological charge we have to distinguish two cases for the fermionic phases ζ: ζ = 0 : Q − 1 zero modes, ζ ∈ (0, 1) : Q zero modes.
Let us further investigate the localization properties of the (right-handed) zero modes in the background of the instanton with integer charge Q = k. They have the explicit form with |v| 2 from (28). It is helpful to first consider well-separated constituents for which ln |v| 2 becomes time-independent and piecewise linear, as described in Sec. 3.2. Clearly, the zero mode has maximal amplitude at points where ln |v| 2 − 4π(s + ζ)x 1 is minimal. At these x 1 the vertical distance between the graphs of the approximately piecewise linear function ln |v| 2 and the linear function 4π(s + ζ)x 1 is minimal. For s + ζ in the interval (µ i−1 , µ i ) the minimum is at x 1 = a i where the graphs of p i−1 and p i intersect. The situation is depicted in Fig. 7.
Hence, for generic values of ζ the zero mode is localized at one constituent. The profile of the zero mode is symmetric about the constituent i for the particular value s+ζ = 1 2 (µ i−1 + µ i ), Interestingly, the profile is almost constant between the ith and (i + 1)th constituent for ζ = µ i . These 'bridges' can be understood by the fact that in this region the graphs of ln |v| 2 and 4π(s + ζ)x 1 are parallel (up to exponentially small corrections) at these values of s and the fermionic phase ζ.
Altogether the zero modes walk along the ordered set of constituents when changing the fermionic phase ζ. With phases at the bounds ζ = 0 resp. ζ = 1 (or ζ = {Q} for configurations with fractional charge) the zero modes become constants asymptotically. In other words, these zero modes have 'bridges' coming from −∞ resp. reaching out to +∞ and hence are not normalizable.
Similar arguments apply if two or more constituents merge. A few examples are given in Fig. 8.

Zero modes on the lattice
Using the overlap operator with quasi-periodic boundary conditions for the U (1) gauge field we are able to analyze its zero modes [30] in the background given by the cooled lattice configurations. Even for moderately cooled configurations do the zero modes reflect the position of the fully cooled instanton constituents for specific boundary conditions (see Fig. 9). Therefore the lattice results are in full agreement with the analytical results and in addition the fermionic zero modes are excellent tracers for instanton constituents for mildly cooled configurations.

Supersymmetric coupling to fermions
The supersymmetric CP n model [31,32] contains n + 1 Dirac fermion fields ψ j , j = 0, . . . , n, in addition to the complex scalar fields u j . Its action is where u and ψ are constrained by Introducing Weyl spinors, ψ = (ϕ, χ) T , the model is invariant under the on-shell N = (2, 2) supersymmetry transformations with the covariant derivative and anticommuting parameters ε 1,2 satisfying ε * 2 =ε 1 and ε * 1 =ε 2 . Both spinors ϕ and χ have n + 1 components. The linearized Dirac equation in an external u-field splits into two Weyl equations, For an instanton background with u = v(z)/|v| the Weyl equations simplify to where P v projects onto the holomorphic v(z), It follows that a left-handed solution reads where f (z) is an arbitrary vector of anti-holomorphic functions orthogonal to v. None of these solutions is normalizable. With the help of P v∂ P v =∂P v one shows that a right-handed solution has the form where g(z) is a vector of holomorphic functions. In order not to break supersymmetry g must fulfill the same boundary conditions as the instanton solution v. Therefore, the choice of fermionic twists is very limited here. Each function g can be constructed by linear combination of the basis elements {g (j,s) } defined by g (j,s) (z) = e 2π(s+µj )z e j , j = 0, . . . , n, s ∈ Z, (54) where e j is the unit vector pointing in direction j. For the corresponding zero modes the squared norm is Normalizability of the zero mode in the k-instanton background requires s = 0, 1, . . . , k for j = 0, 0, 1, . . . , k − 1 for j = 1, . . . , n. (56) In the case of well-separated instanton constituents we can write  where we introduced χ (i) = χ (j,s) for i = s(n + 1) + j. The linear functions p l (x 1 ) are given in (29). Again the maximum of |χ (i) | is easily found by considering the graphs of the linear functions 2p l (x 1 ) and p l (x 1 ) + 4πµ i x 1 . In a logarithmic plot both the numerator and denominator of |χ (i) | are piecewise linear. For x 1 < a i the slope of the numerator is larger and for x 1 > a i+1 the slope of the denominator is larger. This is illustrated in Fig. 10. Simple geometric arguments about these graphs reveal, that the zero modes χ (i) with 0 < i < k(n + 1) split into two constituents located at a i and a i+1 , which have the same amplitude, but decay with different lengths. The zero modes χ (0) and χ (k(n+1)) have only one maximum at a 1 and a k(n+1) , respectively. Some examples are plotted in Fig. 11. The general right-handed zero mode has the form Its (squared) norm splits into k(n+1) or less constituents. They have the same analytic form ∝ cosh −2 [2πQ const,i (x 1 − a i )] and are located at the same positions a i as the instanton constituents. There exists always a particular zero mode, whose (squared) norm is proportional to the topological density Namely, since the squared norm of χ in (53) is we obtain the exact relation for the zero mode with g = ∂v, which means that β i ∝ µ i λ i in Eq. (58). The occurrence of this particular zero mode can also be understood as follows: Any instanton background breaks half of the supersymmetry, namely the one generated by the parameter ε 2 . If we transform the configuration u inst = v(z)/ |v|, ψ inst = 0 with the broken symmetry then δψ inst is inevitably a zero mode of the Dirac operator for the action is invariant under the supersymmetry transformation. This way we obtain a non-vanishing right-handed zero mode that is a zero mode with g ∝ ∂v. Except for irrelevant prefactors, this is exactly the zero mode whose squared norm is equal to the topological density of the instanton.

Conclusions
In the present work we constructed and analyzed the integercharged instantons for twisted CP n models on a cylinder. The twisted instantons with charge k support k(n + 1) constituents. If these constituents are well-separated then they become static lumps. The fractional charges and the shapes of the constituents topological profile are governed by the phases in the boundary condition (and the scale β). The constituent positions are related to the collective parameters of the twisted instanton and hence free up to the demand that for all constituents to be present their positions must be ordered. Neighboring constituents can merge adding up their charges. If at least n + 1 constituents merge then the resulting lump becomes time-dependent. For a composite object containing multiples of n + 1 constituents time-dependent terms with higher frequencies contribute, respectively.
Our analytic findings are in complete agreement with the corresponding numerical ones. The latter were obtained by cooling lattice configurations of the twisted model with a nonvanishing topological charge.
We determined all fermionic zero modes in the background of the twisted instantons. This has been achieved for minimally coupled fermions satisfying quasi-periodic boundary conditions in the Euclidean time direction. We found that, similarly as for gauge theories, the zero modes are localized at the positions of the constituents and that they may jump from one constituent to the neighboring one if the boundary conditions for the fermions are changed. Again we compared our analytical findings to numerical results. To that aim we determined the zero modes of the overlap Dirac operator for lattice configurations with different degrees of cooling. Again we find full agreement between our analytical and numerical results, in close analogy with the corresponding situation for SU (N ) Yang-Mills theories.
In the supersymmetric CP n model the Dirac fermions transform according to the fundamental representation of the global U (n + 1) symmetry group. The linearized field equations for the n + 1 fermion-flavors define a supersymmetric Dirac operator. We studied the square integrable zero modes of this operator and showed that they generically split into k(n + 1) constituents with maxima at the locations of the instanton constituents. There exists always a particular zero mode whose norm squared is equal to the topological charge density of the supporting instanton. This zero mode is generated by the halfbroken supersymmetry. We did not elaborate on the contribution of the constituents and zero modes to the central charge of the (2, 2) SUSY algebra.
Our results are in close parallel to the corresponding findings in SU (N ) gauge theories. But since twisted instantons, their constituents and the fermionic zero modes in CP n models are much simpler as in gauge theories our results may be useful to shed further light on the relevant degrees of freedom in strongly coupled models at finite temperature. The next natural step would be to include quantum fluctuations about twisted instantons to study the quantum corrections to the constituent picture.
In the SU (N ) gauge theory there is a beautiful construction of the constituents based on the Nahm transform [6]. We believe that a similar construction, with Nahm transform as introduced in [4], could further simplify the construction of instanton constituents for twisted CP n models.
Similar aspects of twisted CP n models (such as loop groups) are discussed in a simultaneous paper [33].