Classical solutions of a flag manifold sigma-model

We study a sigma-model with target space the flag manifold U(3)/U(1)^3. A peculiarity of the model is that the complex structure on the target space enters explicitly in the action. We describe the classical solutions of the model for the case when the worldsheet is a sphere CP^1.


The CP 2 σ-model
We will be thinking of CP 2 as the quotient CP 2 " pC 3´t 0uq{C˚. A map v : M Ñ CP 2 from a Riemann surface M can be described by a vector-valued function vpz,zq P C 3 , where z,z are coordinates on the worldsheet M . We may assume that the vector v is The off-diagonal currents tJ mn , m ‰ nu comprise the vielbein (and are defined up to phase factors). Note that J nm "´J mn . One can define an almost complex structure on F 3 by picking any three mutually non-conjugate forms, J m1n1 , J m2n2 , J m3n3 , and declaring them holomorphic. The other three, being conjugate to these, are therefore antiholomorphic. In order to decide, which of these complex structures are integrable, a diagrammatic representation is useful. Draw three nodes and directed arrows from node m 1 to n 1 , m 2 to n 2 and m 3 to n 3 . Integrability of the so-defined complex structure is equivalent to the condition that the graph is acyclic (i.e. does not have a directed closed loop). Let us prove this. First of all, let e m , m " 1, 2, 3 be the standard unit vectors with components pe m q n " δ mn . To the holomorphic one-forms one can associate a subspace m`of the Lie algebra psup3qq C " slp3q as follows: Integrability of the complex structure is equivalent to the requirement that m`is a subalgebra: rm`, m`s Ă m`. On the other hand, the matrices E mn have the commutation relations Remembering that E mn is represented by an arrow from m to n, one sees that the closedness of m`under commutation is equivalent to the following statement: For any two consecutive arrows m Ñ n and n Ñ p (10) their 'shortcut' segment pm, pq has the arrow m Ñ p For the diagram with three vertices, i.e. for the sup3q case under consideration, it is clear that the cyclic quivers are the only ones that are ruled out.
In the general case, corresponding to the flag manifold U pN q U p1q N , suppose we have N pairwise-connected vertices, and the graph is acyclic. Then the requirement (10) is satisfied, since otherwise there would be a cycle with three vertices. Reversely, suppose the graph has a cycle. Then, using (10), one can 'cut corners' to reduce again to the cycle with three vertices, which is prohibited (see Fig. 1).
We now return to the sup3q case. Once we are given an acyclic quiver Q, the action proposed in [1] is It was also shown that the actions corresponding to three different integrable complex structures, whose associated quivers are shown in Fig. 2 in blue, differ only by topological terms: Therefore they produce the same e.o.m. In particular, it follows from (11)-(12) that a curve, holomorphic in a complex structure corresponding to one of the three quivers 1 2 3 4 5 Figure 1: The procedure showing that a cycle p1, 2, 3, 4, 5q in a graph leads to the violation of condition (10). Using (10), we replace the pair of segments p1, 2q, p2, 3q by p1, 3q, i.e. cut a corner. Then we replace p1, 3q, p3, 4q by p1, 4q, arriving at the cyclic red triangle, which violates (10). Fig. 2, is a solution to the e.o.m. What is more surprising, however, is that a curve holomorphic in any of the two non-integrable complex structures is a solution to the e.o.m. as well. To see this, one needs to write out the e.o.m. explicitly: D z pJ 12 qz " 0, D z pJ 31 qz " 0, D z pJ 23 qz " 0 and c.c. ones (13) Here D is the U p1q 3 -covariant derivative, acting as follows: DJ mn :" dJ mn`p J mmJ nn q^J mn . One sees that pJ 12 qz " pJ 31 qz " pJ 23 qz " 0 is a solution to (13), and this is precisely the defining equation of a curve, holomorphic in the almost complex structure that corresponds to the cyclic quiver Q I in Fig. 2. As regards the opposite non-integrable complex structure´I, one can rewrite the equations (13) alternatively as DzpJ 12 q z " pJ 13^J32 q zz , DzpJ 31 q z " pJ 32^J21 q zz , DzpJ 23 q z " pJ 21^J13 q zz (14) In the complex structure´I the l.h.s. vanishes, and all of the one-forms in the r.h.s. are of type p1, 0q (i.e. proportional to dz), hence their wedge products vanish as well. Note, however, that the e.o.m. written with reference to the complex structures I (13) and´I (14) are of rather different form (despite being equivalent), which is the reason that we present the corresponding quivers in Fig. (2) in different color.
Remark. In order to understand the integrable complex structures on F 3 , it is most useful to recall the following definition of the flag manifold (see, for example, [2]): Such an embedding into CP 2ˆC P 2 defines a complex structure on F 3 . In order to make contact with our previous definitions in terms of the one-forms J mn , consider for instance the complex structure corresponding to the quiver Q 1 , and a curve C The triangles indicate the complex structures, whose associated holomorphic curves are solutions of the σ-model. The three top triangles correspond to integrable complex structures, whereas the two lower ones correspond to the non-integrable ones.
holomorphic in this complex structure. The following facts are easily derived: This means that the projections of C to the CP 2 's with coordinates u 1 ,ū 3 are holomorphic curves. Moreover, u 1˝ū3 " 0. Comparing with (15), one realizes that pw, vq in (15) may be identified with pu 1 ,ū 3 q. All other integrable complex structures on F 3 are obtained by replacing pw, vq with the various pairs pu i ,ū j q and using the embedding (15).

Critical maps CP
We call a map M Ñ F 3 critical if it is a solution of the e.o.m. (13). Henceforth in this paper we will be concerned with the case M " CP 1 . From the equations (13) one deduces the following conservation equation: B z ppJ 12 qzpJ 23 qzpJ 31 qzq " 0 Note that the expression in brackets is a section of the cube of the canonical bundle K of CP 1 , and the conservation law states that it has to be anti-holomorphic, i.e. pJ 12 qzpJ 23 qzpJ 31 qz P H 0 pK 3 , CP 1 q. However, as H 0 pK 3 , CP 1 q " 0, the only such section is zero. Hence pJ 12 qzpJ 23 qzpJ 31 qz " 0 Suppose then the remaining equations (13) assume the form where α is an arbitrary (scalar) function. Hence u 2 pz,zq is harmonic (see (5)). § 3.1. Harmonic maps CP 1 Ñ CP 2 In this section we review the construction of the harmonic maps CP 1 Ñ CP 2 , which was carried out long ago [4] (using a method developed in [5] for the description of minimal maps S 2 Ñ S n ). The key property of such maps, which lies at the heart of the construction, is called 'complex isotropy': Note that this property does not hold, in general, for harmonic maps C g Ñ CP 2 , where C g is a curve of positive genus g ą 0.
hence such a section is necessarily zero, leading to (22).
Once (22) is established, consider the following sequence of maps: Combining the results of the discussion above, we arrive at the conclusion that harmonic maps CP 1 Ñ CP 2 are generically in 3 : 1 correspondence with holomorphic maps CP 1 Ñ CP 2 . Namely, for every holomorphic map v we can construct two additional harmonic descendants: w 1 " B˝v and w 2 " B˝B˝v, the second one being anti-holomorphic, so that B˝w 2 " 0. In the special case when v is not a full map, i.e. when it is a map to a proper linear subspace C 2 Ă C 3 , it turns out that w 2 " 0, so that there is a single descendant w 1 , which in this case is anti-holomorphic. The extreme case w 1 " 0 corresponds to a constant map v. § 3.2. Lift to the flag manifold In order to convert a harmonic map v " u 2 : into a critical map to F 3 , we wish to show that we can lift the former to the flag manifold, satisfying the remaining equation (20): pJ 31 qz " u 3˝Bzū1 " 0, where u 1 and u 3 are orthogonal to each other and to u 2 .
I. D z u 2 ı 0, Dzu 2 ı 0. Both of these vectors are orthogonal to u 2 (by definition) and to each other (by the isotropy property). Therefore u 1 and u 3 are linear combinations of these two vectors: Acting on u 3 with D pu2q z , we obtain (α is the scalar function from (21)): where τ is the proportionality constant from the equality pD pu2q z q 2 u 2 " τ Dzu 2 (which is derived analogously to (28)). A simple calculation shows that τ " Bzplog }Dzu 2 } 2 q .
The equation u 3˝Bzū1 " 0 then requires Together with the orthogonality conditionū 1˝u3 " 0, expressed as this leads to Bzc d´pBzd`τ dq c " 0, henceˆc d˙" λpz,zqˆf with two holomorphic functions pf pzq : gpzqq P CP 1 . It is easy to see that the remaining unknowns, such as λ, a, b can be now found from the normalization conditions u 1˝u1 "ū 3˝u3 " 1. Therefore what defines the lift to the flag manifold is a holomorphic map pCP 1 q z Ñ pCP 1 q pf :gq .
One can also think of this map as a rational function f pzq gpzq . Note that the critical map CP 1 Ñ F 3 constructed in this fashion is not holomorphic in either of the almost complex structures on F 3 , unless the matrixˆa b c dḣ as some zero elements. This is so, since D z u 2 , Dzu 2 are not orthogonal to either u 1 or u 3 , hence violating the holomorphicity conditions for all complex structures. Due to (43), the only possibilities for the above matrix to have zero elements are as follows: Ia. a " d " 0, i.e. u 1 " Dzu 2 , u 3 " D z u 2 . Thenū 1˝Dz u 2 " 0 "ū 3˝Dz u 2 . It is also easy to check thatū 3˝Dz u 1 " 0, as well asū 3˝Dz u 1 " 0. This means that the lift is a horizontal curve (with respect to the twistor fibration), which is holomorphic in complex structures Q 1 and Q I .
Ib. b " c " 0, i.e. u 3 " Dzu 2 , u 1 " D z u 2 . This is essentially a u 1 Ø u 3 reversal of the case Ia. Therefore the lift in this case is a horizontal curve, holomorphic in Q´1 and Q´I . Note that this is an exceptional case when the curve is holomorphic in the complex structure Q´1, not shown in Fig. 2. Such holomorphicity is possible due to the horizontality of the map, i.e. J 13 " 0.
II. Dzu 2 " 0. In this case u 2 is a holomorphic map. The condition is equivalent to the following two: pJ 21 qz " 0, pJ 23 qz " 0 (47) The remaining e.o.m, (20), states that Together the above equations (47)-(48) imply that we are dealing with a curve M Ñ F 3 , holomorphic in the complex structure, defined by the graph Q 3 .
The situation when Dzu 2 " D z u 2 " 0, i.e. when u 2 is a map to a point, is at the intersection of cases II and III. In this case, due to the condition pJ 31 qz " 0, pu 1 , u 3 q specify a holomorphic map to a CP 1 , orthogonal to the fixed vector u 2 . In other words, it is a map to the fiber of the fibration F 3 Ñ pCP 2 q u2 , and this map is holomorphic in two complex structures, Q 2 and Q 3 . This property was already observed in [1].
Analysis of the cases, when in place of (20) one has pJ 12 qz " 0 or pJ 23 qz " 0, goes along the same lines, with obvious permutations of u 1 , u 2 , u 3 .

Summary
In this paper we have solved the e.o.m. (13), which follow from the action (11), introduced in [1]. The solutions that we obtained correspond to the case when the worldsheet is the sphere CP 1 , and they exhaust all solutions in this case. We have shown, that, apart from various holomorphic curves, there exists a subclass of solutions that are not holomorphic in any (almost) complex structure on F 3 . The data for such solutions consist of a full holomorphic curve CP 1 Ñ CP 2 -the 'Bäcklund primitive' of (37) -and a holomorphic map CP 1 Ñ CP 1 (46).
The key property which allowed us to solve the equations is that, due to the fact that CP 1 does not have holomorphic differentials, the problem reduced to the one of finding harmonic curves in CP 2 (see (19)-(21)), and the latter problem was solved long ago [4]. This approach is not directly generalizable to other worldsheets. However, in [1] it was shown that the e.o.m. (13) can be written in terms of a oneparametric family of flat connections. For σ-models with symmetric target spaces such representation provides a method for the construction of solutions, which was developed in [7] and rigorously justified in [8]. It would be very interesting to explore, whether a suitable modification of the method would allow to obtain all solutions of the equations (13) in the case when the worldsheet is not a sphere but rather a higher-genus Riemann surface, or a cylinder.