INLO-PUB-9/98

Monopole Constituents inside

SU(n) Calorons

Thomas C. Kraan and Pierre van Baal

Instituut-Lorentz for Theoretical Physics, University of Leiden,

PO Box 9506, NL-2300 RA Leiden, The Netherlands.

Abstract: We present a simple result for the action density of the charge one periodic instantons - or calorons - with arbitrary non-trivial Polyakov loop at spatial infinity. It is shown explicitly that there are lumps inside the caloron, each of which represents a BPS monopole, their masses being related to the eigenvalues of . A suitable combination of the ADHM construction and the Nahm transformation is used to obtain this result.

## 1 Introduction

Instantons and BPS monopoles are self-dual finite action solutions to the Yang-Mills equations of motion. Their origin is topological and related to windings in the gauge transformations describing their behaviour at infinity, where these solutions approach vacua, necessary for the action to be finite. The action of these solutions is proportional to these winding numbers or charges, which suggests that they are composed out of elements with unit charge. We will find by computing the action density that even charge one periodic instantons (calorons) for the gauge group are further composed out of constituents. The constituents are basic BPS monopoles [1], whose magnetic charges cancel exactly.

Periodic instantons were first discussed in the context of finite temperature field theory [2, 3]. Motivated by issues of T-duality [4] and D-brane constructions [5] in string theory, periodic instantons with a non-trivial Polyakov loop were recently constructed. The ingredients of the Nahm transformation for calorons [6], can be naturally presented in terms of monopole constituents [7]. This has been the basis of the approach followed by Lee and co-workers [8, 9]. Our work relies on a more direct approach, combining the ADHM construction of multi-instanton solutions [10] with the Nahm construction [6], related by Fourier transformation. The constituent monopoles appear as explicit lumps in the action density [11].

The composite nature of the periodic instantons can also be appreciated by an argument due to Taubes [12] on how to build out of monopoles, configurations with non-trivial topological charge. It has far reaching consequences, which go beyond the existence of exact caloron solutions [11]. Its relevance for QCD has motivated us to extend our work to .

Finite action solutions on should approach vacua at spatial infinity. Due to the topology of this base manifold, these vacua can be non-trivial. This endows the caloron with extra parameters - labeling these vacua - which are studied in terms of the eigenvalues of the Polyakov loop around at spatial infinity. The Polyakov loop is defined in the periodic gauge, , as

(1) |

where is the period and denotes path-ordering. At infinity the value of the Polyakov loop does not change under continuous deformations of the loop and its eigenvalues are topological invariants.

In extending our work to the case it turned out to be natural to generalise to (see also the appendices of ref. [9]). We will present the formula for the action density in section 2. The derivation is outlined in section 3. A detailed description will be given elsewhere. In section 4 we discuss the properties of the solution.

## 2 The result

We consider the calorons with no net magnetic charge, in which case the Polyakov loop (holonomy) at spatial infinity becomes constant. Its eigenvalues will play an important role in the construction,

(2) |

Making use of the gauge symmetry, we can choose the eigenvalues such that

(3) |

assuming maximal symmetry breaking for the moment. We define , related to the mass of the constituent monopole. Standard arguments, summarised below, gives instanton parameters for fixed , including the global gauge transformations that do not change . We will see that parameters can be interpreted as the positions () of the constituents. The remaining parameters in this interpretation are the phases related to the unbroken gauge group , on which the action density does not depend and the position of the caloron in time, which we fix to be 0 by translational invariance. Also we will use the scale invariance to set . Where needed, the proper dependence can be reinstated on dimensional grounds. We find the following surprisingly simple formula

(4) |

where the positive scalar potential is defined as

(5) |

Here denotes the center of mass radius of the monopole. The order of matrix multiplication is crucial, .

## 3 The construction

In our description of the caloron with non-trivial Polyakov loop, we pick the so-called algebraic gauge,

(6) |

which is related to the periodic gauge by the non-periodic gauge transformation . In the algebraic gauge all gauge field components approach zero at infinity. The technique we use is to interpret the ADHM data as the Fourier coefficients of the functions that appear in the Nahm transformation. This is to solve the quadratic ADHM constraint, which is non-trivial for a periodic array of instantons, twisted in colour space going from one time slice to the next.

We summarise the ADHM formalism for charge instantons [10], to fix our notation. It employs a dimensional vector , where is a two-component spinor in the representation of . Alternatively, can be seen as an complex matrix. In addition one has four complex hermitian matrices , combined into a complex matrix , using the unit quaternions and , where are the Pauli matrices. With abuse of notation, we often write . Together and constitute the dimensional matrix , to which is associated a complex dimensional normalised zero mode vector ,

(7) |

Here the quaternion denotes the position (a unit matrix is implicit) and can be solved explicitly in terms of the ADHM data by

(8) |

As is an positive hermitian matrix, its square root is well-defined. The gauge field is given by

(9) |

For to be a self-dual connection, has to satisfy the quadratic ADHM constraint, which states that (considered as complex quaternionic matrix) has to commute with the quaternions, or equivalently

(10) |

defining as a hermitian Green’s function. The self-duality follows by computing the curvature

(11) |

making essential use of the fact that commutes with the quaternions, and being self-dual ( is anti-selfdual). The quadratic constraint can be formulated as , where , and one obtains

(12) |

where is the spinorial trace. Note that this implies that vanishes on the diagonal for . To count the number of instanton parameters we observe that the transformation , , with leaves the gauge field and the ADHM constraint untouched. Taking this symmetry into account, we find the dimension of the instanton moduli space to be dimensional. Global gauge transformations are realised by , with . Those that leave invariant reduce the dimension of the gauge invariant parameter space (by for maximal symmetry breaking). Finally, we quote an elegant result [13] for the action density in terms of

(13) |

The charge one caloron with Polyakov loop at infinity is built out of a periodic array of instantons, twisted by . This is implemented in the ADHM formalism by requiring (suppressing colour and spinor indices)

(14) |

with . Using that 9) leads to the required result, eq. (6). Demanding , eq. (

(15) |

suitably implements eq. (14) and is partially solved by imposing

(16) |

with still to be determined to account for eq. (12), which also constrains the spinor in the representation of to

(17) |

It is useful to introduce the projectors , with and , such that and .

We now perform the Fourier transformation to the Nahm setting [6], which casts into a Weyl operator and into a singularity structure on ,

(18) | |||

Introducing the vector , it is standard to show

(19) |

and the quadratic ADHM constraint, which takes the form

(20) |

leads to the (for abelian) Nahm equation

(21) |

The symmetry in the ADHM construction translates into a gauge symmetry on , which leaves invariant and allows one to set , being the position in time, which we absorb in . Since , it follows that , see eq. (17), such that we may introduce (), with . In terms of these we find

(22) |

where for and 0 elsewhere, taking into account has period 1. Note that is only fixed up to a constant , related to the freedom of adding a constant to the solution of the Nahm equation, eq. (21). The 4-vector describes the position of the caloron. Also note that the are independent of the phases of , affected by the residual gauge symmetry, of which are independent due to the gauge group being rather than .

The vector is interpreted as the position of the constituent. On each sub-interval , is constant, which is precisely the Nahm datum for a single BPS monopole located at [14]. The length of the Nahm interval for the single BPS monopole corresponds to its asymptotic Higgs value, and thereby to its mass. Thus, the subinterval corresponds to a BPS monopole at , with mass proportional to . Together, the BPS monopoles form the caloron.

The Green’s function , central in the ADHM construction, is found after a Fourier transformation of eq. (10), introducing as the solution to the differential equation

(23) |

where the radii are given by , to be interpreted as the center of mass radii of the constituent monopoles. The solution of a quantum-mechanical problem on the circle with a piecewise constant potential and delta function impurities is obtained by solving it on each sub-interval, where is of simple exponential form. Starting at and matching properly at so as to account for the scattering by the impurity, we can go full circle to return at where one last matching accounts for the delta function at the rhs. of eq. (23).

## 4 Discussion

We briefly discuss the properties of the charge one caloron. From eq. (5) we see that the constituent monopole can be located at arbitrary , with arbitrary mass , subject only to the constraint , choosing , and appropriately. As the action density, eq. (4), is expressed as a total derivative, the action is easily found by partial integration, with the expected result of . The size of the instanton is related to the differences in position of the constituent monopoles. As we work in units of , the situation of a small scale (nearby constituents), corresponds to large , i.e. to an instanton on . At the other extreme one has well separated lumps for small , i.e. in the static limit. In figure 1 we present a typical caloron for decreasing values of using eq. (4). We will resist the temptation of showing results for other , as eq. (4) can be readily implemented.

When the lumps are far apart, they do not deform each other and become spherically symmetric. Since the solution is self-dual, the constituents have to be basic BPS monopoles. This can be proven by carefully analysing eq. (4) for the limit where for all , in which case the action density approaches 15]. The other constituents need not be well-separated from each other for the above argument to hold. In particular sending the constituent to infinity (i.e. ) suffices to make the caloron static. What remains are monopole constituents with a combined magnetic charge opposite to the magnetic charged of the constituent monopole that has been removed. As the solution is static in this limit one is left with an BPS monopole. Indeed, for we see from the solution of the Nahm equation, eq. (21), that lives on an interval, rather than on the circle, as is appropriate for the monopole [16]. One readily obtains the energy density of this monopole by taking the limit in eq. (5), verifying that it decays as , as opposed to without removing the constituent. . This is precisely the behaviour of the BPS monopole [

Our results have been derived for the case of maximal symmetry breaking, . The situation of non-maximal symmetry breaking corresponds to a constituent obtaining zero mass, . In that case its center of mass radius drops out of eq. (5), using

(24) |

This was also observed for , in which case non-maximal symmetry breaking corresponds to a trivial Polyakov loop, , and the solution becomes that of Harrington and Shepard [2]. Hence our formula for the action density should also be valid for non-maximal symmetry breaking.

Although our formalism can be extended easily to higher topological charges [9], the appropriate Nahm equation (i.e. solving the quadratic ADHM constraint) becomes a non-abelian problem, and finding solutions requires more powerful tools. Nevertheless, it is interesting to note that it is natural to conjecture that instantons (i.e. an instanton of charge ) can be built from monopoles, since each instanton can be considered as being built from BPS monopoles. The monopole constituents are only well separated when is small, where the instanton parameters can be interpreted as positions and phases (including ). We will not speculate further on these matters here, but want to emphasise that the monopole constituent picture has some interesting phenomenological implications for the description of the long distance properties of QCD, discussed in detail in ref. [11], and which will be the subject of further investigations.

## Acknowledgements

TCK was supported by a grant from the FOM/SWON Association for Mathematical Physics.

## References

- [1] E.B. Bogomol’ny, Yad. Fiz. 24 (1976) 861; Sov. J. Nucl. 24 (1976) 449; M.K. Prasad and C.M. Sommerfield, Phys. Rev. Lett. 35 (1975) 760.
- [2] B.J. Harrington and H.K. Shepard, Phys. Rev. D17 (1978) 2122; ibid. D18 (1978) 2990.
- [3] D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1983) 43.
- [4] T.C. Kraan and P. van Baal, Exact T-duality between calorons and Taub-NUT spaces, Phys. Lett. B428 (1998) 268 (hep-th/9802049).
- [5] K. Lee and P. Yi, Phys. Rev. D56 (1997) 3711 (hep-th/9702107);
- [6] W. Nahm, Self-dual monopoles and calorons, in: Lect. Notes in Physics. 201, eds. G. Denardo, e.a. (1984) p. 189.
- [7] H. Garland and M.K. Murray, Commun. Math. Phys. 120 (1988) 335.
- [8] K. Lee, Instantons and magnetic monopoles on with arbitrary simple gauge groups, hep-th/9802012; K. Lee and C. Lu, SU(2) calorons and magnetic monopoles, hep-th/9802108.
- [9] K. Lee and P. Yi, Dyons in supersymmetric theories and three-pronged strings, hep-th/9804174.
- [10] M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld, Yu. I. Manin, Phys. Lett. 65 A (1978) 185; M.F. Atiyah, Geometry of Yang-Mills fields, Fermi lectures, (Scuola Normale Superiore, Pisa, 1979).
- [11] T.C. Kraan and P. van Baal, Periodic Instantons with non-trivial Holonomy, hep-th/9805168; New Instanton Solutions at Finite Temperature, hep-th/9805201.
- [12] C. Taubes, Morse theory and monopoles: topology in long range forces, in: Progress in gauge field theory, eds. G. ’t Hooft et al, (Plenum Press, New York, 1984) p. 563.
- [13] H. Osborn, Nucl. Phys. B159 (1979) 497.
- [14] W. Nahm, Phys. Lett. 90B (1980) 413.
- [15] P. Rossi, Nucl. Phys. B149 (1979) 170.
- [16] W. Nahm, All self-dual multimonopoles for arbitrary gauge groups, CERN preprint TH-3172 (1981), published in Freiburg ASI 301 (1981); The construction of all self-dual multimonopoles by the ADHM method, in: “Monopoles in quantum field theory”, eds. N. Craigie, e.a. (World Scientific, Singapore, 1982), p.87.