Topological objects and chiral symmetry breaking in QCD

Transcription

1 Manfried Faber Diplomarbeit Topological objects and chiral symmetry breaking in QCD Ausgeführt am Atominstitut der Technischen Universität Wien unter der Anleitung von Prof. Dr. Manfried Faber und Dr. Roman Höllwieser durch Thomas Schweigler Horngasse 5, 314 Pottenbrunn 7. September 212 Thomas Schweigler

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3 In this master thesis, topological objects in SU2 gauge theory are investigated. Besides investigating the objects in general, I also tried to find out more about their importance for spontaneous chiral symmetry breaking. The lattice gauge object, whose properties have been mainly investigated, is the spherical vortex [1 3]. As already mentioned in the cited papers, the spherical vortex seems to be some sort of squeezed instanton. In this document, a very detailed investigation of the spherical vortex is performed. The investigation takes place partly in the continuum and partly on the lattice. In previous work, only spherical vortices with temporal extent of one lattice unit have been investigated. For such objects, one gets vanishing topological charge but nonvanishing difference n n + = 1 of left and right chiral zeromodes. In this document, it is shown that this discrepancy is simply a discretization effect. For growing temporal extent of the vortex, the lattice topological charge approaches 1 and the index theorem Q T = n n + is fulfilled again. Moreover, the action and topological charge density of the spherical vortex have been calculated analytically in the continuum. The spatial and temporal localization of the zeromodes for the spherical vortex have been investigated. Another investigation concerned the lowest nonzero eigenvalues of the Dirac operator for the spherical vortex. It was demonstrated that, for the vortex getting smaller and smaller, these eigenvalues approach the eigenvalues of the free Dirac operator. The zeromode occurs no matter how small the vortex becomes. Moreover, the claim made in [4] that self-dual/anti-self-dual gauge field contributions attract the negative/positive chiral components of the eigenmodes was checked for the lowest non-zero-modes. The results for the localization of the scalar and the chiral density are in agreement with this claim. The same investigation was also done for the lowest non-zero-modes for the instanton. It is interesting to see, that in case of the the instanton consisting only of self-dual field contributions only negative chiral components are attracted by the object, in case of the spherical vortex consisting of both self-dual and anti-self-dual contributions also positive chiral components are attracted by the object. Last but not least, the interaction between a spherical vortex and a spherical antivortex object with Q T = 1 has been investigated. The transformation of the two would-be zeromodes into two near-zero modes was demonstrated.

4 Contents 1. Introduction Spontaneous symmetry breaking Chiral symmetry in continuum QCD Chiral symmetry in lattice QCD Vacuum expectation values from eigenvalues and eigenvectors Local chiral condensate Derivation using the path integral formalism Derivation using Wick s theorem The Banks-Casher relation Vector current Interactions between topological objects Eigenvalues of the free Dirac operator The spherical vortex The spherical vortex as vacuum to vacuum transition The action of the vacuum to vacuum transition The topological charge density of the vacuum to vacuum transition Putting the vacuum to vacuum transition onto the lattice The spherical vortex on the lattice Gluonic results Zeromode for antiperiodic boundary conditions Zeromodes for periodic boundary conditions The low lying non-zero-modes The continuum limit of the spherical vortex in one time slice Ginsparg-Wilson topological charge density of the spherical vortex Interactions between two spherical vortices Separation in temporal direction Separation in spatial directions Vector Current Instantons Conclusions and outlook 82 A. Appendix to Chapter 5 83 A.1. Chern-Simons number under local gauge transformations A.2. Proving the identity lim k 1 f k t f k t f k t = 1 6 δt

5 Contents A.3. Action and topological charge density expressed with anti-self-dual field contributions A.4. Eigenvalues for configurations of the form of

6 1. Introduction For my master thesis I investigated topological objects in SU2 gauge theory. Besides investigating the objects in general, I also tried to find out more about their importance for spontaneous chiral symmetry breaking. Let me give a quick overview of the topics covered in this report. In the remaining part of this chapter, I will give a quick introduction to spontaneous symmetry breaking as well as chiral symmetry in continuum and lattice QCD. In Chapter 2 I will discuss how vacuum expectation values can be calculated by using the eigenvalues and eigenvectors of the Dirac operator. The discussion will be hold for lattice QCD with Ginsparg-Wilson fermions. A particular important result will be the Banks-Casher relation From the Banks- Casher relation the importance of topological objects for spontaneous chiral symmetry breaking becomes clear. Before some of this objects are investigated on the lattice, I will give a general discussion of the interaction of topological objects in Chapter 3. In Chapter 4 I will calculate the eigenvalues of the free continuum and the free Wilson Dirac operator. Later this eigenvalues will be compared to the eigenvalues obtained for topological gauge field configurations. In Chapter 5 I will closely examine the properties of a topological object called spherical vortex. The investigation will partly be performed analytically and partly on the lattice. In Chapter 6 some results for instantons will be presented. The thesis will conclude with Chapter 7. It contains a summary of the results and some ideas for future research. Note that throughout this document we will only have one quark flavor. In the following, the Einstein summation convention is used, underlined indices are not summed over Spontaneous symmetry breaking In QCD, the so-called chiral symmetry is hidden and non-exact. We speak of a hidden, spontaneously broken or dynamically broken symmetry when the Lagrangian density and therefore the equations of motion is invariant under the symmetry transformation, but not all vacuum expectation values vev are. The vev of an operator that is not invariant under the symmetry transformation can serve as a so called order parameter. If this order parameter is nonvanishing, then the symmetry is broken spontaneously. The converse clearly doesn t hold. If the order parameter is zero, there still can be another nonvanishing non invariant vev. Anyhow, note that an infinite system is necessary for spontaneous symmetry breaking. Therefore, to get meaningful order parameters we have to perform some limits. The starting point is a system, where the symmetry is broken explicitly. The magnitude of this explicit breaking is determined by some parameters in case of the chiral symmetry by the quark masses. For some values of these parameters lets call it the symmetric limit the symmetry is restored. In order to infer whether the symmetry is spontaneously broken or not one has to at first take the infinite volume limit 6

7 1.2. Chiral symmetry in continuum QCD and then the symmetric limit for the order parameter. If this gives a nonzero value, then the symmetry is broken spontaneously. The Goldstone theorem tells us that if a continuous symmetry is broken spontaneously, then massless scalar particles appear in the spectrum. In case of the chiral symmetry, this particles called Goldstone bosons are the Pions. As already mentioned above, the chiral symmetry is not an exact symmetry of the action, it is broken explicitly by the rather small quark masses. This is the reason for the nonvanishing Pion mass Chiral symmetry in continuum QCD In this section the chiral symmetry in continuum QCD will be discussed. For simplicity, we will assume to have only one quark flavor. In this case, the Lagrangian density reads L ψ, ψ, A µ = ψ γ µ µ + i γ µ A µ + m ψ, 1.1 where m is the fermion mass. The operator γ 5 = γ 1 γ 2 γ 3 γ 4 is the generator of the so called chiral rotations expi αγ 5 where α R. Under this chiral rotations, the fermion fields transform as ψ ψ = e i αγ 5 ψ, ψ ψ = ψ γ 4 = ψ e i αγ 5, 1.2 where we used the anti-commutation relations {γ µ, γ 5 } = to get the transformation of ψ. With 1.2 we can write the Lagrangian density for the transformed fields as L ψ, ψ, A µ = ψ γ µ µ + i γ µ A µ + m ψ = ψ e i αγ 5 γ µ µ + i γ µ A µ + m e i αγ 5 ψ = ψ e i αγ 5 γ µ e i αγ 5 µ + i A µ ψ + ψ e i αγ 5 m e i αγ 5 ψ = ψ γ µ µ + i γ µ A µ + m e i 2αγ 5 ψ. 1.3 We see, that the Lagrangian density for massless fermions remains unchanged under chiral rotations. This is what one calls chiral symmetry. With D = γ µ µ + i A µ, chiral symmetry can be expressed through the simple equation D γ 5 + γ 5 D =. 1.4 Note that chiral symmetry occurs only in the massless limit which is often referred to as the chiral limit. As can be easily seen from 1.3, the mass term does not stay invariant under chiral rotations, it explicitly breaks the chiral symmetry. By looking at the mass term, it is easy to see that the so called chiral condensate ψψ 1.5 represents a suitable order parameter for chiral symmetry breaking. Clearly, the operator ψψ is not invariant under chiral rotations. A nonvanishing chiral condensate therefore implies that the chiral symmetry is broken. The Lagrangian density 1.1 can also be expressed with the left- ψ L and right-handed ψ R components of the fermion fields. One gets the left- and right-handed components by applying the appropriate projection operators P L or P R to ψ. I.e. ψ L = P L ψ, ψl = ψ P R, ψ R = P R ψ, ψr = ψ P L

8 1. Introduction The projection operators are given by P L = γ 5 and P R = γ With 1.6 and D = γ µ µ + i A µ the Lagrangian density can be written as L ψ, ψ, A µ = ψ L D ψ L + ψ R D ψ R + m ψl ψ R + ψ R ψ L Chiral symmetry in lattice QCD In this section, I will only state the most important equations without giving a detailed explanation. For a more detailed discussion see for example [5]. On the lattice 1.4 is replaced by the Ginsparg-Wilson GW equation D γ 5 + γ 5 D = a D γ 5 D. 1.9 For Dirac operators D fulfilling the GW equation in the following addressed as GW Dirac operators, the Lagrangian density L = ψ D ψ is invariant under the transformation ψ ψ = exp i αγ 5 1 a 2 D ψ, ψ ψ = ψ exp i αγ 5 1 a 2 D. 1.1 If one wants to obtain an expression similar to 1.8 for GW operators, then the definition for the left- and the right-handed components of the fermion fields has to be changed to ψ L = ˆP L ψ, ψl = ψ P R, ψ R = ˆP R ψ, ψr = ψ P L 1.11 with ˆP L = γ 5 1 a D and ˆPR = γ 5 1 a D P L and P R are again given by 1.7. With these definitions and the GW equation 1.9 the validity of ψ D ψ = ψ L D ψ L + ψ R D ψ R 1.13 can be shown. A sensible choice for the mass term is m ψl ψ R + ψ R ψ L = m ψ P L ˆPL + P R ˆPR ψ = m ψ 1 a 2 D ψ As can easily be seen, this expression goes to the continuum mass term for a. With 1.13 and the mass term 1.14, the Lagrangian density on the lattice has the continuum structure 1.8. I.e. L = ψ D m ψ = ψ L D ψ L + ψ R D ψ R + m ψl ψ R + ψ R ψ L. The massive GW Dirac operator D m therefore is D m = D + m 1 a 2 D = ω D + m

9 1.3. Chiral symmetry in lattice QCD with D fulfilling the GW equation 1.9. As can be easily checked, the abbreviation ω is given by ω = 1 a m From the mass term 1.14 we see that the chiral condensate for GW fermions should be defined as ψ 1 a 2 D ψ Let us now discuss some properties of the eigenvalues and eigenmodes for γ 5 -hermitian Dirac operators fulfilling the GW equation. The eigenvalues of such an operator come in complex conjugate pairs and lie on a circle in the complex plane. The midpoint of the circle is 1/a + i, the radius is 1/a as well. Because of the restriction to this circle in the complex plane, one can prove with basic trigonometry that the relations Re[λ] = a λ 2, Im[λ] = ± λ 1 a2 λ are valid. The ± in the expression for the imaginary part simply accounts for the fact that the complex eigenvalues come in complex conjugate pairs. Another very important equation relating the eigenmodes of the complex conjugate eigenvalues is ψ λ = γ 5 ψ λ with Dψ λ = λψ λ and Dψ λ = λ ψ λ. Clearly, the above discussed circle crosses the real axis at the two points λ = zeromodes and λ = 2/a doubler modes. Zero as well as doubler modes can be chosen to have exact chirality, i.e. to be eigenvectors of γ 5. This can easily be shown by using the GW equation 1.9. On the other hand, for eigenmodes ψ λ corresponding to a complex eigenvalue λ the identity ] tr [ψ λ γ 5 ψ λ = follows from the γ 5 -hermiticity of the Dirac operator. 9

10 2. Vacuum expectation values from eigenvalues and eigenvectors In this chapter we will discuss how to calculate the vacuum expectation values vev of observables starting from the eigenvalues and eigenvectors of the Ginsparg-Wilson GW Dirac operator see Section 1.3 for a short discussion of GW fermions. To make the discussion easier, I will quickly introduce some notation used in this chapter. In the following, the vev of an Operator O is is denoted by O. As well known, the vev can be written as path integral O = 1 Z D[U] D[ψ, ψ] S[ψ, ψ,u] e O[ψ, ψ, U] 2.1 over the fermion fields ψ, ψ and the gauge fields U. The action S[ψ, ψ, U] consists of the fermion action S F [ψ, ψ, U] and the gauge action S G [U]. The path integral 2.1 can be split into a fermionic and a gauge field part. The fermionic part is given by with O F [U] = 1 Z F [U] D[ψ, ψ] e S F [ψ, ψ,u] O[ψ, ψ, U] 2.2 Z F [U] = D[ψ, ψ] e S F [ψ, ψ,u]. 2.3 The gauge field part reads A U = 1 Z D[U] e S G[U] Z F [U] A[U]. 2.4 With A[U] = O F [U] the vev O can be written as O = O F U. In this chapter, we will often evaluate only the fermionic part of the path integral, i.e. we will calculate the ground state expectation value for fixed gauge field U. In the following, the Einstein summation convention is used, underlined indices are not summed over Local chiral condensate In this section we will discuss how to express the chiral condensate for fixed gauge field in the following denoted by Σ F with the eigenvalues and eigenvectors of the massless GW Dirac operator D for this fixed gauge field configuration. From 1.17 we see that Σ F is given by Σ F a, m, N, n, U = ψn 1 a 2 Dn, m ψm. 2.5 F 1

11 2.1. Local chiral condensate It depends on the lattice constant a, the quark mass m, the lattice size N and the gauge configuration U. The expression 2.5 can be evaluated to Σ F = 1 a 4 1 ω ψλ 1 nα j a ψ λj nα a ωλ j + m a, j where the sum runs over all eigenvalues λ j. Let us quickly explain the indices of ψ λj nα a. The index λ j means that it is the eigenmode to the eigenvalue λ j. The index in brackets stands for the lattice side, α is the Dirac and a the color index. The result 2.6 can be obtained in two different ways. First we will discuss how to derive it directly from the path integral, second it will be derived with the help of Wick s theorem. Note that in Section 2.2 we will derive the Banks-Casher relation starting from 2.6. This chapter deals only with lattice QCD, for a discussion of the local chiral condensate and the Banks-Casher relation in continuum see for example [6]. Before starting to derive 2.6, let us also give the expression in lattice units. With m = m 1/a and therefore ω = 1 m /2 and λ j = λ j 1/a we get Σ F = 1 a 3 1 ω ψλ 1 nα j a ψ λj nα a ωλ j m 2 j Derivation using the path integral formalism Eq. 2.5 can be written as path integral over the fermion fields. We have Σ F = 1 Z F d ψnγb... d ψ111 d ψ11... d ψnγ 1 B e S F ψnαa 1 a 2 Dn, mαβ ab ψmβ, b 2.8 where the Grassmann variables are marked with a tilde, N is the number of lattice points, Γ and B represent the range of the Dirac respectively color indices. S F is the fermion action, Z F the fermion determinant. We can now perform a change of the generators of the Grassmann algebra. The transformation matrix is given by the eigenbasis of the Dirac-Operator. Dirac operators which obey the Ginsparg-Wilson equation and are also γ 5 -hermitian are normal [5], which means that there exists an orthonormal eigenbasis. Therefore, the transformation matrix is unitary. Explicitly the transformation reads ψmα a ψmα a = i = i ψ λi mα a ψλi 2.9 ψ λ i mα a ψλi 2.1 where Dn, mαβ ab ψ λi mβ b = λ i ψ λi nα a

12 2. Vacuum expectation values from eigenvalues and eigenvectors We can now rewrite the action with the transformed Grassmann numbers: S F = a 4 ψpαa D m p, qαβ ab = a 4 ψpαa ω Dp, qαβ ab ψqβ b = a 4 ψ λ i pα a ψλi ω Dp, qαβ ab + m δp, qαβ ab ψqβ + m δp, qαβ ab = a 4 ψλi ψλj ωλ j + m ψ λ i pα a ψ λj pα a = a 4 ψλi ψλi ωλ i + m b ψ λj qβ b ψ λj 2.12 D m denotes the massive Dirac operator, which is given by D m = ω D + m 1 with ω = 1 a/2 m. In the third line we just inserted Eq. 2.9 and 2.1, in line four we used Eq and to get from the expression of line 4 to line 5, the orthonormality of the eigenvectors was used. With 2.12 we can now calculate the fermion determinant Z F = = = i = i = i d ψnγb... d ψ111 d ψ11... d ψnγ exp a 4 ψpαa 1 B d ψλm... d ψλ1 d ψ λ1... d ψ λm exp d ψλi d ψ λi a 4 ψλi ψλi ωλ i + m exp a 4 ψλi ψλi ωλ i + m d ψλi d ψ λi 1 a 4 ψλi ψλi ωλ i + m 1 a 4 ωλ i + m. D m p, qαβ ab ψqβ b 2.13 In the second line, we just performed the change of Grassmann variables given in Eq. 2.9 and 2.1. Note that the transformation is unitary, therefore no additional factors originating from the determinant of the transformation-matrix occur. The total number of eigenvalues is denoted by M = NΓB. In the third line we used that pairs of Grassmann numbers commute, therefore we can write the exponential of the sum as a product of exponentials. Similar we can perform the path integral given in 2.8: Σ F = 1 Z F = 1 Z F d ψλm... d ψλ1 i d ψ λ1... d ψ λm e S F ψ λ j nα a ψλj 1 a 2 Dn, mαβ ab d ψλi d ψ λi 1 a 4 ψλi ψλi ωλ i + m ψ λj ψλk ψλ nα j a ψ λk nα a 1 a 2 λ k j,k ψ λk mβ b ψ λk 12

13 2.1. Local chiral condensate For the further calculation we split the sum over the λ j and λ k in diagonal and off-diagonal terms. First we have a look at the diagonal terms 1 Z F j=k = j 1 Z F i d ψλi d ψ λi 1 a 4 ψλi ψλi ωλ i + m ψλj ψλk ψ λ j nα a ψ λ k nα a 1 a 2 λ k i j 1 a 4 ωλ i + m ψλ nα j a ψ λj nα a 1 a 2 λ j = 1 1 a j 4 ψ λ nα j a ψ λj nα a 1 a 2 λ j ωλ j + m = 1 a 4 1 ω ψλ 1 nα j a ψ λj nα a ωλ j + m a. 2 j 2.14 In line three the expression for Z F given in Eq was used. The off-diagonal terms don t give any contribution: 1 Z F j k i d ψλi d ψ λi 1 a 4 ψλi ψλi ωλ i + m ψλj ψλk ψ λ j nα a ψ λ k nα a 1 a 2 λ k = 1 1 a 4 ωλ i + m d ψλj d Z ψ λj 1 a 4 ψλj ψλj ωλ j + m ψλj F j k j i k d ψλk d ψ λk 1 a 4 ψλk ψλk ωλ k + m ψλk ψλ nα j a ψ λj nα a 1 a 2 λ j = 2.15 As easy to see, both the integration over ψλj ψλj and ψλk ψλk gives. Combining 2.14 and 2.15 gives Derivation using Wick s theorem In index notation, 2.5 reads Σ F a, m, N, n, U = ψnαa = δn, mαβ ab δn, mαβ a ab 2 a 2 Dn, mαβ ab Dn, mαβ ab ψmβ b ψnα a ψmβ b F F Using Wick s theorem ψnα a ψmβ b F = 1 a 4 D 1 m m, nβα ba 2.17 we get Σ F a, m, N, n, U = 1 a 4 δn, mαβ ab a Dn, mαβ Dm 1 m, nβα 2 ab ba

14 2. Vacuum expectation values from eigenvalues and eigenvectors D m = ω D + m 1 again denotes the massive Dirac operator. To get to the result 2.6 we now simply have to diagonalize the Dirac matrix. We will do this by inserting the completeness relation for the eigenvectors δ ab δ αβ δm, n = i ψ λ i nβ b ψ λi mα a into This gives Σ F a, m, N, n, U = 1 δn, a 4 pαɛ a ac 2 = 1 a 4 δn, pαɛ a Dn, pαɛ ac ac 2 i,j = 1 a 4 i,j 1 a 2 λ i ψ λ i mβ b Dn, pαɛ ac ψ λ i mβ b ψ λi pɛ c δp, mɛβ cb D 1 m m, qβη bd D 1 m m, qβη bd ψ λ j nα a ψ λi nα a ωλ j + m 1 ψ λ j nα a ψ λj mβ b = 1 a 4 1 a 2 λ i ωλ i + m 1 ψλ nα i a ψ λi nα a i = 1 a 4 1 ω ψλ 1 nα i a ψ λi nα a ωλ i i + m a. 2 δq, nηα da ψ λj qη d 2.19 Note that to get from the third to the fourth line, the orthogonality of the eigenvectors has been used. The last line of 2.19 again represents The Banks-Casher relation In this section, the Banks-Casher relation will be derived starting from 2.6. The chiral condensate Σ is related to Σ F as defined in 2.5 in the following way: Σ a, m, N = 1 ψn N 1 a 1 2 Dn, m ψm = Σ F a, m, N, n, U N n U 2.2 The first thing we will do in order to derive the Banks-Casher relation is to sum 2.6 over n. Remembering that all eigenvectors are normalized, this gives n Σ F a, m, N, n, U = 1 a 4 1 ω λ j 1 ωλ j + m a. 2 14

15 2.2. The Banks-Casher relation Next, we will split the sum over all eigenvalues into a sum over λ j = zeromodes, λ j = 2/a doubler modes and the complex eigenvalues. We get 1 1 Σ F a, m, N, n, U = N n ω a 4 m a N D 2 a 4 ω 2/a + m a a 4 1 ω 1 ωλ j + m + 1 ωλ j + m a = 1 m a 4 N + 1 a 3 Im[λ j]> Im[λ j ]> m 1 λ j 2 4 m 2 + λ j 2 1 m Here N denotes the number of zeromodes and N D the number of doubler modes which is the same as the number of zeromodes. It is well known that the number of left n and right n + chiral zeromodes can be related to the topological charge density Q T via the index theorem Q T = n n +. It turns out that for Monte Carlo gauge field configurations one never finds both n and n + [5]. This fact is called absence of fine tuning. Therefore, one can identify N = n + n + with Q T. To get the second term in the last line of 2.21 the relations 1.18, m = m 1/a and λ j = λ j 1/a have been used. Note that expression 2.21 is not defined in the chiral limit m. In order to get the Banks-Casher relation, we have to calculate Σa = lim lim m N Σ a, m, N I.e., as discussed in Section 1.1 we first have to take the infinite volume limit N and then the symmetric limit m. It is important that the two limits are not exchanged. Inserting 2.21 into 2.2 and performing the infinite volume limit we get 1 Q T lim Σ a, m, N = lim N N N a 4 m + 1 a 3 d λ ρ λ with 1 ρ λ = lim N N 1 λ 2 4 Im[λ j]> π 1 π m m 2 + λ 2 1 Om 2 δ λ λ j U Note that a ρ λ is the eigenvalue density. Let us now discuss the fate of the first term in We know that the topological susceptibility χ T = 1 N a Q 2 4 T stays finite in the limit N [5]. Therefore, terms of the form 1 N Q 2 2 T have to vanish for N. This means that also the first term in 2.23 vanishes. To perform the chiral limit m in 2.22 we use the identity [5] and get Σa = 1 a 3 1 lim m π m m 2 + λ 2 1 Om 2 = δ λ d λ ρ λ 1 λ 2 4 π δ λ = 1 a 3 π ρ. 15

16 2. Vacuum expectation values from eigenvalues and eigenvectors Denoting the eigenvalue density per unit volume by ρ λ = 1 a 4 a ρ λ 2.25 we finally get the Banks-Casher relation Σa = πρ Vector current As discussed in [5, 7], it is difficult to calculate a conserved current with Ginsparg-Wilson fermions. In this section, we will have a look at the vector current given by V µ n = ψn iγ µ δn, m a 2 Dn, m ψm The so defined current is not conserved, but it is invariant under chiral rotations 1 [7]. Moreover, as can be easily seen, it converges to the continuum expression ψ iγ µ ψ for the vector current in the limit a. Similar to the calculation in Section 2.1, we can calculate the ground state expectation value of 2.27 for a given gauge field: j µ = ψn iγ µ δn, m a 2 Dn, m ψm F = i d ψnγb... d ψ111 d Z ψ11... d ψnγ F 1 B e S F ψnαa γ µ αβ δn, mβγ a ab 2 D n, mβγ ab ψmγ b 2.28 Performing the transformations given in 2.9 and 2.1, and using 2.12 and 2.13 we can evaluate the path integral given in 2.28 to j µ = i Z F = i Z F d ψλm... d ψλ1 i d ψ λ1... d ψ λm e S F ψ λ j nα a ψλj γ µ αβ δn, mβγ a ab 2 d ψλi d ψ λi 1 a 4 ψλi ψλi ωλ i + m j,k D n, mβγ ab ψ λj ψλk ψλ nα j a γ µ αβ ψ λk nβ 1 a a 2 λ k ψ λk mγ b Again, the off-diagonal terms j k give no contribution. The diagonal terms evaluate to j µ = i ψ nα λ j a γ µ αβ ψ λj nβ a a 4 1 a ωλ j + m 2 λ j chiral rotations as defined in 1.1 j ψ λk 16

17 2.3. Vector current Note that the zeromodes don t give any contribution. This follows from the fact that zeromodes are eigenstates of the chirality operator γ 5, i.e. γ 5 ψ = ±ψ. Using the anti-commutator relations {γ µ, γ 5 } =, we get ψ γ µ ψ = ψ γ µ γ 5 2 ψ = ψ γ 5 γ µ γ 5 ψ = ψ γ µ ψ. This can only be fulfilled by ψ γ µ ψ =. 2.3 Using this and the fact that the nonzero eigenvalues come in complex conjugate pairs with Dγ 5 ψ λ = λ γ 5 ψ λ we can simplify 2.29 to j µ = i a 4 Im[λ j ]> ψ λ j 1 a 2 γ µ ψ λ j λj ωλ j + m 1 a 2 λ j ωλ j + m In the massless limit m, ω 1 we get j µ = lim m j µ = i a 4 = 1 a 4 Im[λ j ]> Im[λ j ]> ψ λ j ψ λ j γ µ ψ λj 2 Im[λ j ] λ j 2. γ µ ψ λj 1 a 2 λ j λ j 1 a 2 λ j λ j This expression can be simplified further by using 1.18 and λ j = λ j 1/a. One gets j µ = 1 a 3 Im[λ j ]> ψ 4 λ j γ µ ψ λj λ j 2 In the so called chiral representation the euclidean gamma matrices are given by γ i = i σi i σ i and γ 4 = where σ i are the Pauli matrices. The eigenvectors ψ λj can be written as ψ λj = ψλj R ψ λj L 2.34 where ψ λj L and ψ λj R stand for the left and right handed components. Inserting

18 2. Vacuum expectation values from eigenvalues and eigenvectors and 2.34 into 2.32 gives j i = i a 3 = i a 3 Im[λ j ]> Im[λ j ]> = 2 a 3 j 4 = 1 a 3 Im[λ j ]> Im[λ j ]> = 2 a 3 Im[λ j ]> ψ λj R σ i ψ λj L ψ λj L σ i ψ λj R ψ λj R σ i ψ λj L ψ λj R σ i ψ λj L [ ] Im ψ λj R σ i ψ λj L 4 λ j ψ λj R ψ λ j L + ψ λj L ψ λ j R [ ] Re ψ λj R σ i ψ λj L This shows that the current is always real. 2 1, 4 λ 1. j 2 4 λ j λ j λ j

19 3. Interactions between topological objects In the following I will discuss the interactions between topological objects in continuum QCD. Under topological objects I understand gauge field configurations with nonvanishing topological charge Q T. For an introduction about topological charge and Atiyah- Singer index theorem in continuum see for example [8]. The situation I want to discuss is that of a gauge field A µ = A 1µ + A 2µ consisting of the two fields A 1µ and A 2µ which are separated in euclidean space and have nonvanishing topological charge Q T. Therefore, the Dirac operators D 1 for A 1µ alone and D 2 for A 2µ alone would have at least one zeromode. In the following, the zeromodes of D 1 and D 2 will be called would-be zeromodes. Lets discuss the case in which A 1µ has Q 1T = 1 and A 2µ has Q 2T = 1. For simplicity I will assume that D 1 has only one left handed zeromode ψ 1 and D 2 only one right handed zeromode ψ 2. Clearly, this two would-be zeromodes are orthogonal, they can be part of an orthogonal basis. Lets now have a look at the Dirac operator in this orthogonal basis. In particular we are interested in the upper left 2 2 Block ψ1 D ψ 1 ψ 1 D ψ ψ 2 D ψ 1 ψ 2 D ψ 2 of the Dirac matrix. The continuum Dirac operator D is given by D = γ µ µ + i A µ x = γ µ µ + i A 1µ x + i A 2µ x = D 1 + γ µ i A 2µ x = D 2 + γ µ i A 1µ x. 3.2 The first element of 3.1 therefore evaluates to ψ 1 D ψ 1 = ψ 1 D 1 ψ 1 + ψ 1 γ µ i A 2µ x ψ 1 =. 3.3 One can see that ψ 1 γ µ i A 2µ x ψ 1 vanishes from ψ 1 γ µ i A 2µ x ψ 1 = ψ 1 γ µ i A 2µ x γ 5 2 ψ 1 In the same way one can prove = ψ 1 γ 5 γ µ i A 2µ x γ 5 ψ 1 = ψ 1 γ µ i A 2µ x ψ ψ 2 D ψ 2 =. 3.5 Let us now calculate the off-diagonal terms. The first off-diagonal term evaluates to ψ 1 D ψ 2 = ψ 1 D 2 ψ 2 + ψ 1 γ µ i A 1µ x ψ 2 = + c = c. 3.6 Here c stands for the overlap integral ψ 1 γ µ i A 1µ x ψ 2. In general, c will be large for eigenmodes that overlap a lot, and small for eigenmodes that overlap only a little. 19

20 3. Interactions between topological objects Note that it is crucial that ψ 1 and ψ 2 have different chirality. Otherwise, by the same argument as used in 3.4, the overlap integral would have to vanish. The second off-diagonal term of the upper-left 2 2 Block is ψ 2 D ψ 1 = ψ 1 D ψ 2 = ψ1 D ψ 2 = c. 3.7 Here it was used that the continuum Dirac Operator D is antihermitian, i.e. D = D. Combining 3.3, 3.5, 3.6 and 3.7, the upper left 2 2 block of the Dirac matrix reads c c. 3.8 This 2 2 block can easily be diagonalized. The eigenvalues λ 1,2 and the normalized eigenvectors ψ 1,2 of 3.8 are λ 1,2 = ± i cc = ± i c, ψ 1,2 = 1 ±i c c This means that the interaction transforms the two zeromodes into two near-zero modes. The strength of the interaction quantified by the overlap integral c determines the size of the near-zero eigenvalue. Note that the new near-zero modes consist in equal parts of the would-be zeromodes ψ 1 and ψ 2. Therefore the scalar and chiral densities of the near-zero modes are simply an average of the densities of the would-be zeromodes. This can also be seen from the results presented in Section So far we have ignored everything except the upper left 2 2 block of the Dirac matrix. To get exact eigenmodes we clearly have to diagonalize the whole Dirac matrix and not only this 2 2 block. However, 3.9 represents a legitimate approximation to the exact eigenvalues and eigenmodes if the elements of the form ψ j D ψ 1,2 = ψ j γ µ i A 2,1µ x ψ 1,2 with j > 2 are a lot smaller than the overlap integral c = ψ 1 γ µ i A 1µ x ψ 2. Usually this will be the case, because ψ 1 is localized at A 1µ and ψ j with j > 2 is not. Note that for the results presented in Section 5.7.1, 3.9 seems to be a good approximation. However, the results presented in Section are very different from 3.9. The reason for this seems to be the small value of the overlap integral for the gauge configurations investigated in this section. Therefore, the upper left 2 2 block of the Dirac matrix is not dominating any more and there is no reason for the validity of 3.9. Let us also have a quick look at what happens when we have two would-be zeromodes with the same chirality. In this case also the off-diagonal terms of the upper left 2 2 matrix vanish and the would-be zeromodes are actual zeromodes. To me it is still unclear what happens in case of GW fermions. For the GW Dirac operator I can t show that the diagonal terms of the upper left 2 2 block of the Dirac matrix vanish. However, from the results for GW fermions presented in Section 5.7 it seems that the discussion held in this chapter is also valid for GW fermions. Note that the mechanism discussed in this chapter is the basis for the instanton liquid model of spontaneous chiral symmetry breaking see [9] for a detailed review. In the instanton liquid model the QCD-vacuum consist of an ensemble of instantons and antiinstantons whose would-be zeromodes split into near-zero modes because of interactions. 2

21 Therefore, one gets a nonvanishing eigenmode density ρλ for λ =. This means that according to the Banks-Casher relation 2.26 the chiral condensate is nonvanishing too and the chiral symmetry therefore broken. Clearly, one can construct such a model also with other topological objects. It seems like center vortices are dominating the QCD vacuum and are also relevant for spontaneous chiral symmetry breaking [1 12]. It was discussed in [13, 14] how vortex intersections and so-called writhing points contribute to the topological charge density. Moreover it is discussed in [1 3] and Chapter 5 how the color structure of vortices can contribute to the topological charge density too. Therefore, the chiral symmetry breaking might simply come from the splitting of would-be zeromodes corresponding to such topological charge contributions. 21

22 4. Eigenvalues of the free Dirac operator In this section we will analytically calculate the eigenvalues of the massless free Dirac operator. These eigenvalues will come handy later when we compare the eigenvalues for different gauge field configurations to the eigenvalues for the free case. Moreover, it is important to know the multiplicity of the eigenvalues of the free Dirac operator as it helps to choose sensible cutoffs when calculating quantities as a series over eigenvectors see Section 5.6. In the following, we will analytically calculate the eigenvalues of the free continuum Dirac operator in euclidean space, of it s naively discretized version and also of the Wilson Dirac operator. Unfortunately we haven t been able to analytically calculate the eigenvalues of the free overlap Dirac operator, nor have we found such a calculation in literature. In the following, the Einstein summation convention is used, underlined indices are not summed over. Let us first calculate the eigenvalues of the free continuum Dirac operator. The free continuum Dirac operator for massless fermions in euclidean space is given by D = γ µ µ, 4.1 where γ µ are the euclidean Dirac matrices. What we want to do now is to find the eigenvalues λ of this operator. That means, we want to find out for what values of λ, the equation D αβ ψ β x = λ ψ α x 4.2 has non-trivial solutions ψ α x. To do so, we will use the following ansatz for the eigenfunctions. Inserting this into 4.2 we get ψ α x = u α expip µ x µ 4.3 i γ µαβ p µ λ δ αβ u β expip µ x µ =. This equation has to be fulfilled for every x. We therefore can cancel the exponential function and get i γ µαβ p µ λ δ αβ u β =, 4.4 which means, that by inserting ansatz 4.3, the eigenproblem in space and Dirac indices has simplified to an eigenproblem in Dirac indices only. Note that the matrix i γ µαβ p µ of the reduced eigenproblem is antihermitian the euclidean gamma matrices are hermitian. Therefore, the eigenvalues will be purely imaginary. Another important implication of the antihermiticity is, that the matrix i γ µαβ p µ possesses a complete set of eigenvectors. Together with the well know completeness of the set of plane wave functions { expip µ x µ p µ R }, this ensures, that functions of the form 4.3 represent a complete basis of the space of functions in space and Dirac indices. Clearly, this implies 22

23 that they also span the eigenspace of the Dirac operator. Therefore, we don t overlook any eigenvalues by making ansatz 4.3. Let s now go back to finding the eigenvalues. Clearly 4.4 has only non-trivial solutions u β when det [i γ µαβ p µ λ δ αβ ] = is fulfilled. After a somewhat lengthy calculation or with a computer algebra program, the determinant can be evaluated to p µ p µ + λ 2 2. Setting this equal to gives λ = ±i p µ p µ. 4.5 Note that both the eigenvalue for the plus and the minus sign have a degeneracy of 2, which gives in total 4 eigenvalues for the 4 x 4 Dirac matrix 4 x 4 in Dirac indices. Additional degeneracy comes from the color indices. The free Dirac operator is color blind. Therefore, the degeneracy of the different eigenvalues is multiplied by n col, where n col is the range of the color indices. Which values for p µ are allowed in equation 4.5 depends on the boundary conditions. This will be discussed later after the eigenvalues for the naively discretized and the Wilson Dirac operator have been calculated. Let us now have a look at the eigenvalues of the naively discretized Dirac operator given by 4 1 D naive m, n = γ µ δm + ˆµ, n δm ˆµ, n a µ=1 Here m and n stand for the lattice side, ˆµ represents the unit vector pointing from one lattice side to the neighboring in direction µ. Note that the only difference between 4.1 and 4.6 is that the differential has been replaced by a central difference. In order to solve the eigenproblem D αβ m, n ψ β n = λ ψ α n 4.7 we can again make an ansatz similar to the one for the continuum Dirac operator. It reads ψ α n = u α expi a p µ n µ. 4.8 For simplicity, let us from now on set the lattice constant a = 1. In the end result, a will be inserted again by dimensionality considerations. With the ansatz 4.8 and a = 1, the eigenproblem reads 1 γ µαβ e ipµ e ipµ λ δ αβ u β expip ν n ν =. 2 For this equation to have nontrivial solutions u α, again the determinant [ 1 ] det γ µαβ e ipµ e ipµ λ δ αβ 2 has to vanish. With the help of a computer algebra program the determinant was evaluated to cos2 p µ 2 λ µ=1 23

24 4. Eigenvalues of the free Dirac operator Setting this equal to gives λ = ±i cos2p µ. µ=1 Let us now insert the lattice constant a by dimensionality considerations. p µ has the same dimension as 1/a. It therefore has to be accompanied by a, because the argument of the cosine has to be dimensionless. λ has the dimension of a mass i.e. the dimension of 1/a, which means, we have to introduce an overall factor 1/a. Inserting a as discussed gives λ = ±i a 2 cos2p µ a. 4.9 Note that ansatz 4.8 is periodic, not only in lattice indices n µ, but also in p µ. This means, that p µ and p µ + 2πzµ a with z µ Z correspond to the same eigenfunction. Therefore, in order to get the correct multiplicities for the eigenvalues, we have to restrict the range of p µ to µ=1 π a < p µ π a. 4.1 This restriction applies in addition to the restrictions originating from the boundary conditions. In the limit a assuming that p µ stays finite, expression 4.9 becomes 4.5. To see this, we have to expand the cosine function into a Taylor series. This gives λ = ±i a 2 1 2p µ a 2 + Oa 4 = ±i p µ p µ + Oa 2, µ=1 from which the claim made above follows. Last but not least, we will calculate the eigenvalues of the free Wilson Dirac operator. The Wilson Dirac operator for free fermions is given by µ=1 D W m, n = 4 4 δm, n + a µ=1 1 [ γµ δm + ˆµ, n δm ˆµ, n 2a δm + ˆµ, n + δm ˆµ, n ] In addition to the naively discretized Dirac operator, the Wilson Dirac operator contains a term, which is a discretization of a/2 µ µ. For solving the eigenproblem 4.7 we will use the same ansatz that was used in case of the naively discretized Dirac operator, i.e. we will use the ansatz given in 4.8. For the calculation, we will again set the lattice constant a = 1. With this, the eigenproblem for the free Wilson Dirac operator reads γ µαβ e ipµ e ipµ δ αβ e ipµ + e ipµ + 4 λ δ αβ u β expip ν n ν =. 2 2 For this equation to have nontrivial solutions u α, again the determinant det γ µαβ e ipµ e ipµ δ αβ e ipµ + e ipµ + 4 λ δ αβ 2 2 µ=1 24

25 has to vanish. With the help of a computer algebra program, the determinant can be evaluated. The result is somewhat lengthy and therefore not explicitly given here. Setting the determinant equal to and solving for λ gives 4 λ = 4 cosp µ ± i 4 sin 2 p µ. µ=1 µ=1 Inserting the lattice constant a by dimensionality considerations gives the end result for the eigenvalues of the free Wilson Dirac operator: λ = 1 4 a 4 cosap µ ± i 4 sin 2 ap µ µ=1 µ=1 Expanding the trigonometric functions in 4.12 into a Taylor series gives λ = 1 4 a 4 1 a 2 p µ p µ + Oa 4 ± i a 2 p µ p µ + Oa 4 = Oa ± i p µ p µ + Oa 2. µ=1 Therefore expression 4.12 becomes 4.5 in the limit a assuming that p µ stays finite. Also note that, because we used ansatz 4.8, restriction 4.1 also applies to p µ in As already mentioned above, which values for p µ are allowed in 4.5, 4.9 and 4.12 depends on the boundary conditions BC for the fermion fields. Usually, one uses periodic boundary conditions in the spatial directions and periodic or anti-periodic boundary conditions in the temporal direction. Therefore, the allowed values for p µ are p i = 2nπ { 2nπ, p 4 = an t for periodic BC 2n+1π an sp an t for antiperiodic BC, n Z where N sp is the spatial and N t the temporal extent of the lattice. As one can easily see from 4.5, 4.9 and 4.12, there exist at least two different p µ for every eigenvalue λ. Therefore, the total multiplicity of an eigenvalue λ is given by n mult λ = 2 n col n p λ, where n p λ is the number of different p µ corresponding to this particular λ. The factor 2 comes from the Dirac indices. In order to make this a bit more clear, let us now have a look at the multiplicity of the eigenvalues for one particular lattice size. The discussion will be made for the free continuum and the free Wilson Dirac operator. For the naively discretized free Dirac operator, the discussion is a bit more complicated as we also get doubler modes. The lattice we will have a look at is of the size N sp = 12, N t = 12. We will assume antiperiodic boundary conditions in temporal direction, a = 1 and a SU2 gauge field theory i.e. n col = 2. The momentum vectors corresponding to the lowest eigenvalues are then given by p i = and p 4 = ±π/12. Therefore, we have n p λ 1 = 2 for the lowest eigenvalues λ 1. That means, we get a multiplicity n mult λ 1 = 2 n col n p λ 1 = 8. For the second lowest eigenvalues λ 2 we have p i=j = ±π/6, p i j = with j {1, 2, 3} 25

26 4. Eigenvalues of the free Dirac operator Λ.4 Λ a b no. Figure 4.1.: Eigenvalues of the free continuum blue triangles, the free Wilson red crosses and the naively discretized free Dirac operator green discs on a lattice with periodic boundary conditions in spatial and antiperiodic boundary conditions in temporal direction. The doubler modes of the naively discretized free Dirac operator are not plotted. The lattice constant a = 1. and p 4 = ±π/12, i.e. n p λ 2 = 12. This gives a multiplicity n mult λ 2 = 48. For the third lowest eigenvalue λ 3, one has to distinguish between continuum and Wilson Dirac operator. For the continuum Dirac operator, λ 3 corresponds to p i=j = ±π/6, p i=k = ±π/6, p k i j = and p 4 = ±π/12 or to p i = and p 4 = ±3π/12. Therefore, we get n p λ 3 = = 26. Let us explain the first term. The factor 3 comes from the binomial coefficient 3 2 which stand for the possibilities of picking two elements out of a pool of 3. The factor 4 comes from the different sign possibilities for one pair of p i, 2 from the sign possibilities for p 4. The second term consist only of the sign possibilities for p 4. In total, we get a multiplicity of n mult λ 3 = 14 for the third lowest eigenvalue of the free continuum Dirac operator. For the discretized Dirac operators, the two different kinds of momentum vectors don t give the same eigenvalue, we therefore get n mult λ 3 = 8 and n mult λ 4 = 96. In Fig. 4.1 the eigenvalues for the case discussed in this paragraph are plotted. Note, that the multiplicity of the eigenvalues of the naively discretized free Dirac operator is 16 times doubler modes the multiplicity of the corresponding eigenvalues of the free Wilson Dirac operator. In Fig. 4.1b however, the doubler modes are not plotted. Let us conclude this section by having a look at how good the eigenvalues of the free overlap Dirac operator calculated numerically can be approximated by the eigenvalues of the free continuum, free Wilson or the naively discretized free Dirac operator. As can be seen from Figs. 4.2 and 4.3a, the best approximation is given by the Wilson Dirac operator. Comparing Fig. 4.3a with Fig. 4.3b, one can see that the approximation gets better for larger lattice sizes. This is consistent with the fact, that in the continuum limit a both the eigenvalues of the overlap and the eigenvalues of the Wilson Dirac operator have to go to the eigenvalues of the continuum Dirac operator. 26

27 Λ.4 Λ a no b no. Figure 4.2.: Eigenvalues of the free continuum blue triangles, the free overlap black diamonds, calculated numerically and the naively discretized free Dirac operator green circles on a lattice with periodic boundary conditions in spatial and antiperiodic boundary conditions in temporal direction. The doubler modes of the naively discretized free Dirac operator are not plotted. The lattice constant a = 1. Λ Λ a no b no. Figure 4.3.: a Same as Fig. 4.2 but for the free Wilson red crosses and the free overlap Dirac operator black diamonds, calculated numerically. b Same as a but for a lattice. The lattice constant a = 1. 27

28 5. The spherical vortex In this chapter I will investigate an object called the spherical vortex. This object has previously been investigated in [1 3]. It possesses unit topological charge as well as the structure of a center vortex. In Sections 5.1 to 5.3 the spherical vortex will be investigated in the continuum. First I will discuss in Section 5.1 that the spherical vortex can be seen as transition between a trivial gauge field and a pure gauge field with winding number 1. The transition occurs in temporal direction and can be of arbitrary temporal extent. Subsequently, the action and topological charge density of the spherical vortex in continuum will be calculated in Sections 5.2 and 5.3. In Section 5.4 we will discuss how to put the continuum object onto the lattice. In Section 5.5 some results for the spherical vortex on the lattice are presented. First the gluonic results are discussed in Section In previous work only spherical vortices with temporal extent of one lattice unit have been investigated. For such configurations a discrepancy between the lattice topological charge and the difference n n + of left and right chiral zeromodes has been found. From the results in Section one can see that the lattice topological charge grows monotonously with the temporal extent of the spherical vortex. This demonstrates that the above mention discrepancy is simply a discretization effect. In the rest of Section 5.5 the localization of various eigenmodes is investigated. In Section 5.6 the results for an approximated GW topological charge density are presented. The interaction between spherical vortices is investigated in Section 5.7. To conclude this chapter, the vector current for the spherical vortex is discussed in Section 5.8. Note that the interesting thing about the spherical vortex is that it shows that center vortices can contribute to the topological charge density without having any intersection or writhing points. However, it is not clear yet if such topological charge contributions are important or even exist in Monte Carlo gauge field configurations. In the following, the Einstein summation convention is used, underlined indices are not summed over. If not otherwise stated, all results are presented for lattice constant a = 1. All fermionic results are calculated with the overlap Dirac operator. The boundary conditions for the fermion fields in time direction are always antiperiodic with the exception of Section where they are periodic The spherical vortex as vacuum to vacuum transition The first part until eq. 5.9 of this section more or less follows the presentation in [9]. With the field strength tensor given by F µν = σj 2 F j µν 5.1 the expressions for the topological charge Q T and the topological charge density qx 28

29 5.1. The spherical vortex as vacuum to vacuum transition read Q T = d 4 x qx, qx = 1 32π 2 F a µν F a µν, F a µν 1 2 ɛ µναβ F a αβ, ɛ 1234 = To give this expressions a meaning, we have to assume that the fields have no singularities. Then the topological charge Q T is related to the number of left and right chiral zeromodes of the Dirac operator by [8] Q T = n n This relation is a special case of the more general Atiyah-Singer index theorem [15]. In 5.3, n is the number of left chiral zeromodes and n + is the number of the right chiral zeromodes. Under left/right chiral zeromodes, I will understand eigenmodes with γ 5 ψ = ψ respectively γ 5 ψ = +ψ, with γ 5 = γ 1 γ 2 γ 3 γ 4. Note, that for the expressions in 5.2 and 5.3, different sign conventions are possible. I will use the convention given above, which gives positive/negative topological charge for self-dual/self-anti-dual fields. Another equivalent convention that leaves 5.3 unchanged is given by qx = 1 32π 2 F j µν F j µν with F j µν 1 2 ɛ µναβ F j αβ and ɛ 4123 = 1. It turns out that the topological charge density qx can be written as a full derivative of a four vector K µ : qx = µ K µ x, K µ = 1 16π 2 ɛ µαβγ A a α β A a γ 1 3 ɛabc A a αa b βa c γ 5.4 Using this formula, we can rewrite the topological charge in the following way: Q T = d 4 x qx = d 4 x µ K µ = dt d 3 x i K i + d dt d 3 x K With the help of the divergence theorem we can write the integral d 3 x i K i as S 2 ds i K i. Assuming that the fields A i µ vanish sufficiently fast at spatial infinity, also K µ vanishes sufficiently fast at spatial infinity and the integral gives. Therefore we get Q T = dt d d 3 x K dt Let us now define the so-called Chern-Simons number N CS = d 3 x K Note that K 4 t depends only of the spatial fields A i and their spatial derivatives i A j at time t see 5.5. It doesn t depend on A 4 or any temporal derivative 4 A µ. Therefore also the Chern-Simons number is independent of these quantities. What this means is, that there are per se no restrictions on the time dependence of the Chern-Simons number. With the Chern-Simons number, we can rewrite 5.6 as Q T = N CS + N CS. 5.8 The Chern-Simons number N CS can change by integer values under local gauge transformations gx. Note that the transformed A i at fixed t depends only on the original A i, 29

30 5. The spherical vortex the gauge transformation g and at its spatial derivatives i g at t. This can be seen from the usual formula for a gauge transformation: A µ = g A µ i µ g Therefore, the change in N CS at time t depends only on g at time t. Assuming again that the field A µ vanishes sufficiently fast at spatial infinity, the change in the Chern-Simons number at given time t is given by the winding number of the gauge transformation at given time t: N CS = N CS + N w, N w = 1 24π 2 ] d 3 x ɛ ijk Tr C [g i g g j g g k g 5.9 I will not give a proof of 5.9 for the general case. In Appendix A.1 however, I will treat the special case in which the gauge field A µ = before the gauge transformation. Also note that it is not possible to create a topological charge by a gauge transformation. One can see this by noticing that the winding number is a continuous function of the gauge transformation. Therefore, by continuously changing the gauge transformation, the winding number N w must also change in a continuous fashion. As only integer values for the winding number are possible, the winding number stays constant under continuous changes of the gauge transformation. If we have a pure gauge field, that means A µ = g i µ g = ig µ g = i µ g g, then the Chern-Simons number is given by the winding number of the gauge transformation. Therefore, a transition between two pure gauge fields at t = and t = +, generated by gauge transformations with winding number N w1 respectively N w2, gives the topological charge Q T = N w2 N w1. Suppose that we have a gauge transformation g with winding number N w = 1 and a function ft with f = and f+ = 1. Then the field A µ = f t i µ g g 5.1 clearly has topological charge Q T = 1. For the hedgehog gauge transformation given by g = cos αr1 + i e r σ sin αr, 5.11 the winding number can be evaluated to [9] N w = 1 [ αr sin2αr ] r= π 2 r= Note that by r the radius in 3 dimensional space is meant, i.e. r = x 2 + y 2 + z 2. With αr = = π and αr = =, the hedgehog gauge transformation possesses winding number N w = 1 as can easily be checked by insertion into The field 5.1 with g given by the hedgehog gauge transformation can also posses the structure of a center vortex. An object is called a center vortex, if the Wilson loops topologically linked to the object are given by a non-trivial center element of the gauge group. In case of the SU2 gauge group used here, the only non-trivial center element is given by 1. In continuum, the Wilson loop along a closed curve C is defined as the path-ordered exponential given by UC = P exp i dx µ A µ C 3