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Son of gauge fixing on the lattice
Nuclear Physics B (Proc. Suppl.) 26 (1992) 432-434 North-Holland
SON OF GAUGE FIXING ON THE LATTICE * J.
. Hetrick, Ph. de Forcrand, A. Nakamurat, and R. Sinclair
Interdisciplinary Project Center for Supercomputing, ETH- Zentrum, CH-8092 Zurich The gauge fixing (Gribov) ambiguity on the lattice is investigated in simple models . The global understanding of this phenomenon comes from Singer's analysis of the bundle of connections . This bundle in continuum gauge theory is infinite dimensional and of non-trivial topology, however on the lattice it is finite dimensional and tractable. Theoretical expectations from Morse theory are supported by numerical evidence, and the structure of lattice gauge orbit space is displayed. Finally, shortcomings of standard gauge fixing prescriptions are addressed.
1. MATHEMATICAL BACKGROUND Assigning a copy of a gauge group G to each point in (euclidean) spacetime h (complete with a projection and transition functions), defines P(X, G), the G-bundle over X. A gauge field configuration is a connection in this bundle . In the path integral for this gauge theory, we sum over all such connections and this space too is a fiber buudóe, P, the bundle of connections. Its fiber is tg, the set of all gauge transformations, hence an element of 9 is a local gauge transformation G(x). The base space of *P is P/9, the physical configuration space of the theory . Gauge fixing is the process of choosing in P, one representative connection for each element of P/g. Locally this is always possible by the Fadeev-Popov delta function, which selects from the integration sum fp DA, only those configurations which lie on a section, s, of P, ie. only gauge fixed configurations . Singer [1] however, showed that due to topological obstructions this is not always possible globally, hence we will find situations where the gauge fixed section is not well defined and s has multiple solutions. These are *
°RESENTED BY .1 . HETRICK
0920-5632/92/$05 .00 0 1992 - Elsevier Science Publishers B.V
exactly Gribov's copies . Here our analysis continues [2] by examining the fiber g for which we use some basic ideas of Morse theory (see [31). Suppose we have an n dimensional manifold M and wish to learn something about its topology. Consider any function f on M with isolated critical points p (p : Vf lp = 0). The eigenvalues A of VVf Ip determine the local curvature of f at p. Now define the Morse index pp as the number of Vs < 0 and Ck as the number of critical points with p = k . Then the topology of M is related to the C's through the Morse inequalities : 1
.J
i-0
i-0
where the bi are the Betti numbers of M, with equality for j = n . This formalism is relevant to gauge theory since one can fix the gauge by extremizing a function [4] over ç, for instance, I(A, g) = xdx J1A911
(2)
where JJA9 1) is the Hilbert norm of g t A,.g -igt 8g . For a given Au , 9o E 9 which extremizes I satisfies á.Ag° = 0 . Like f above, I(A, g) is a Morse All rights reserved .
J E. Hetrick et al. ISon ofgaugefixing on the lattice
function on the fiber 9 whose critical points are gauge transformations to gauge fixed configurations. Further, the Hessian of I is the FadeevPopov matrix, hence the Morse index u,. is the number of negative eigenvalues of the FP operator. Similarly on the lattice I(A, g) is replaced by I(U, g) = E ReTr{ 1 - gt (x)U (x)g(x + P)} (3) x,P Extrema of (3) are gauge fixed lattice configurations to be labeled by their Morse index.
2. SURVEY of RESULTS First we consider 1 + 1 dimensional gauge theories, where X is a torus [5] . We gauge fix to Coulomb gauge by restricting the sum in eq. (3) to a timeslice, ie. a circle, composed of it sites. We can write down all gauge fixed configurations by hand as those which preserve the Wilson loop W around the circle . The lattice fiber GL is isomorphic to: C9L=GxGx . . .xG/G-Gn -1 .
(4)
In the continuum, Coulomb gauge implies A1 = A1(t) . We can always diagonalize Al(t) by a time-like gauge transformation which preserves Coulomb gauge. We do the same for Ul on the lattice. We then diagonalize the Fadeev-Popov matrix for each configuration, and compute the Morse index y, of the configuration as the number of negative eigenvalues . This generates histograms Ck . Examples are shown below where W = etc/io : U(1), n = 7 sites lc 0 1 _2 'Cj, 4 4 1 42
_3 35
4 21
_5 14
433
SU(2), n = 5 sites p 0 1112 13T-4T51 6 718191101 c,] 1 0 ti 5 0]10
Notice that the Ck's always satisfy the Morse inequalities, (1), with the Hetti numbers of GL in (4), including equality for j = N. One sees here vestiges of the difference in the topological structure between U(1) and SU(N) on a circle . U(1) theory has 8 vacua labeled by the winding q of the gauge transformation A = exp{i27rgx/L} between copies. These configurations are precisely the n copies with p = 0 or p = n, ie. all relative minima or maxima of (3), while in SU(2) where such a transformation has no topology, these transformations are distributed in p and only one configuration has p = 0. Still these configurations are important in SU(2) as residual symmetries of the Coulomb gauge (see conclusion) . When we go to two spatial dimensions (or Landau gauge in 1+1 din-is.), the situation already becomes complicated . It is no longer possible to write down all solutions explicitly and so we find configurations which extremize (3) numerically. The fiber JCL for a 3 x 3 lattice is an 8-dimensional torus, and we find that gauge fixed configurations are not always isolated . A 3-dimensional projection of this space is shown in figure 1 . We have computed the Morse indices of these points and are analyzing this structure .
3. CONCLUSIONS In the continuum for U(1) on S', the copies A1 = A1 + 2rn/L, the 0 vacua, are all within the Gribov region, ie. they have only positive Fadeev-Popov eigenvalues, whereas on the lattice as we have seen, the 7t 0-vacua are distributed roughly evenly with p = 0 or p = n. Thus if
J.E. hietricket al. /Son ofgouge fuing on the lattice
.4
Fig . 1 . A 3-dimensional projection of the U(1) fiber over the vacuum for a 3 x 3 lattice . Points in this figure are saddles of eq. (3) and lie on the section of gauge fixed configurations.
one takes only copies p = 0 into account in summing the partition function, one undercounts these topological excitations . Furthermore, (3) is not degenerate even with respect to the p = 0 copies, hence summing over absolute minima singles out just one configuration, missing the 0 structure altogether . The problems associated with identifying the correct extrema are not limited to situations involving 0 vacua . SU(2) on a circle is homotopically trivial, however there still remain after fixing to Coulomb gauge, residual transformations of the above type where A1 shifts by a constant . This residual freedom is important since it ensures that all operators are periodic in A1 with period 2-,. IL, by O(A l ) _
k
DA O(A + 27rk/L) e -=S(a)
= 0(Ai + n2Tr/L)
dn
as required by the fact that Yang-Mills theory on a S1 is a purely topological theory. However the correct summation is required in order for the above property to emerge in the path integral formulation . Unlike the 0 copies in the case of U(1) above, these important configurations are not all of Morse index y = 0 (or n), but are generally saddle points of (3) . They are thus overlooked in standard gauge fixing algorithms which extremize (3) . Further, consider a quantity like < A1 > . With different summations this operator will yield conflicting results. The problem is rather more difficult in a physical Yang-Mills theory, ie. 4-d SU(3), since in addition to having to sort physically important saddle points from lattice artifacts, which even in our models numbered 0(107), just finding saddle points of a high dimensional function is a numerically difficult task. As we have seen, choosing as gauge fixed configurations the global minima of (3) or all relative minima selects only a subset of the relevant configurations . While these choices are certainly in the right direction, it is imperative to find some clear signal for physical configurations. References Present Address: FESt Heidelberg and Inst. Theor. Phys. Univ. Heidelberg, D-6900 Heidelberg, Germany I. Singer, Comm. Math. Phys. 60 (1978) 7 Ph. de Forcrand et al., Nucl. Phys. B (Pros . Suppl.) 20 (1991) 194
P. van Baal, ITP Utrecht preprint THU-91 (Thoughts on) Gribov Copies
More
[4] D. Zwanziger, Comm. Math. Phys. 138 (1991) 291 [5] J. E. Hetrick and Y. Hosotani, Phys. Lett . B230 Phys. Rev. D38 (1988) 2621; E. Witten, IAS preprint IASSNS-HEP-91/3 On Yang-Mills The(1989) 88,
ories in Two Dimensions
SON OF GAUGE FIXING ON THE LATTICE * J.
. Hetrick, Ph. de Forcrand, A. Nakamurat, and R. Sinclair
Interdisciplinary Project Center for Supercomputing, ETH- Zentrum, CH-8092 Zurich The gauge fixing (Gribov) ambiguity on the lattice is investigated in simple models . The global understanding of this phenomenon comes from Singer's analysis of the bundle of connections . This bundle in continuum gauge theory is infinite dimensional and of non-trivial topology, however on the lattice it is finite dimensional and tractable. Theoretical expectations from Morse theory are supported by numerical evidence, and the structure of lattice gauge orbit space is displayed. Finally, shortcomings of standard gauge fixing prescriptions are addressed.
1. MATHEMATICAL BACKGROUND Assigning a copy of a gauge group G to each point in (euclidean) spacetime h (complete with a projection and transition functions), defines P(X, G), the G-bundle over X. A gauge field configuration is a connection in this bundle . In the path integral for this gauge theory, we sum over all such connections and this space too is a fiber buudóe, P, the bundle of connections. Its fiber is tg, the set of all gauge transformations, hence an element of 9 is a local gauge transformation G(x). The base space of *P is P/9, the physical configuration space of the theory . Gauge fixing is the process of choosing in P, one representative connection for each element of P/g. Locally this is always possible by the Fadeev-Popov delta function, which selects from the integration sum fp DA, only those configurations which lie on a section, s, of P, ie. only gauge fixed configurations . Singer [1] however, showed that due to topological obstructions this is not always possible globally, hence we will find situations where the gauge fixed section is not well defined and s has multiple solutions. These are *
°RESENTED BY .1 . HETRICK
0920-5632/92/$05 .00 0 1992 - Elsevier Science Publishers B.V
exactly Gribov's copies . Here our analysis continues [2] by examining the fiber g for which we use some basic ideas of Morse theory (see [31). Suppose we have an n dimensional manifold M and wish to learn something about its topology. Consider any function f on M with isolated critical points p (p : Vf lp = 0). The eigenvalues A of VVf Ip determine the local curvature of f at p. Now define the Morse index pp as the number of Vs < 0 and Ck as the number of critical points with p = k . Then the topology of M is related to the C's through the Morse inequalities : 1
.J
i-0
i-0
where the bi are the Betti numbers of M, with equality for j = n . This formalism is relevant to gauge theory since one can fix the gauge by extremizing a function [4] over ç, for instance, I(A, g) = xdx J1A911
(2)
where JJA9 1) is the Hilbert norm of g t A,.g -igt 8g . For a given Au , 9o E 9 which extremizes I satisfies á.Ag° = 0 . Like f above, I(A, g) is a Morse All rights reserved .
J E. Hetrick et al. ISon ofgaugefixing on the lattice
function on the fiber 9 whose critical points are gauge transformations to gauge fixed configurations. Further, the Hessian of I is the FadeevPopov matrix, hence the Morse index u,. is the number of negative eigenvalues of the FP operator. Similarly on the lattice I(A, g) is replaced by I(U, g) = E ReTr{ 1 - gt (x)U (x)g(x + P)} (3) x,P Extrema of (3) are gauge fixed lattice configurations to be labeled by their Morse index.
2. SURVEY of RESULTS First we consider 1 + 1 dimensional gauge theories, where X is a torus [5] . We gauge fix to Coulomb gauge by restricting the sum in eq. (3) to a timeslice, ie. a circle, composed of it sites. We can write down all gauge fixed configurations by hand as those which preserve the Wilson loop W around the circle . The lattice fiber GL is isomorphic to: C9L=GxGx . . .xG/G-Gn -1 .
(4)
In the continuum, Coulomb gauge implies A1 = A1(t) . We can always diagonalize Al(t) by a time-like gauge transformation which preserves Coulomb gauge. We do the same for Ul on the lattice. We then diagonalize the Fadeev-Popov matrix for each configuration, and compute the Morse index y, of the configuration as the number of negative eigenvalues . This generates histograms Ck . Examples are shown below where W = etc/io : U(1), n = 7 sites lc 0 1 _2 'Cj, 4 4 1 42
_3 35
4 21
_5 14
433
SU(2), n = 5 sites p 0 1112 13T-4T51 6 718191101 c,] 1 0 ti 5 0]10
Notice that the Ck's always satisfy the Morse inequalities, (1), with the Hetti numbers of GL in (4), including equality for j = N. One sees here vestiges of the difference in the topological structure between U(1) and SU(N) on a circle . U(1) theory has 8 vacua labeled by the winding q of the gauge transformation A = exp{i27rgx/L} between copies. These configurations are precisely the n copies with p = 0 or p = n, ie. all relative minima or maxima of (3), while in SU(2) where such a transformation has no topology, these transformations are distributed in p and only one configuration has p = 0. Still these configurations are important in SU(2) as residual symmetries of the Coulomb gauge (see conclusion) . When we go to two spatial dimensions (or Landau gauge in 1+1 din-is.), the situation already becomes complicated . It is no longer possible to write down all solutions explicitly and so we find configurations which extremize (3) numerically. The fiber JCL for a 3 x 3 lattice is an 8-dimensional torus, and we find that gauge fixed configurations are not always isolated . A 3-dimensional projection of this space is shown in figure 1 . We have computed the Morse indices of these points and are analyzing this structure .
3. CONCLUSIONS In the continuum for U(1) on S', the copies A1 = A1 + 2rn/L, the 0 vacua, are all within the Gribov region, ie. they have only positive Fadeev-Popov eigenvalues, whereas on the lattice as we have seen, the 7t 0-vacua are distributed roughly evenly with p = 0 or p = n. Thus if
J.E. hietricket al. /Son ofgouge fuing on the lattice
.4
Fig . 1 . A 3-dimensional projection of the U(1) fiber over the vacuum for a 3 x 3 lattice . Points in this figure are saddles of eq. (3) and lie on the section of gauge fixed configurations.
one takes only copies p = 0 into account in summing the partition function, one undercounts these topological excitations . Furthermore, (3) is not degenerate even with respect to the p = 0 copies, hence summing over absolute minima singles out just one configuration, missing the 0 structure altogether . The problems associated with identifying the correct extrema are not limited to situations involving 0 vacua . SU(2) on a circle is homotopically trivial, however there still remain after fixing to Coulomb gauge, residual transformations of the above type where A1 shifts by a constant . This residual freedom is important since it ensures that all operators are periodic in A1 with period 2-,. IL, by O(A l ) _
k
DA O(A + 27rk/L) e -=S(a)
= 0(Ai + n2Tr/L)
dn
as required by the fact that Yang-Mills theory on a S1 is a purely topological theory. However the correct summation is required in order for the above property to emerge in the path integral formulation . Unlike the 0 copies in the case of U(1) above, these important configurations are not all of Morse index y = 0 (or n), but are generally saddle points of (3) . They are thus overlooked in standard gauge fixing algorithms which extremize (3) . Further, consider a quantity like < A1 > . With different summations this operator will yield conflicting results. The problem is rather more difficult in a physical Yang-Mills theory, ie. 4-d SU(3), since in addition to having to sort physically important saddle points from lattice artifacts, which even in our models numbered 0(107), just finding saddle points of a high dimensional function is a numerically difficult task. As we have seen, choosing as gauge fixed configurations the global minima of (3) or all relative minima selects only a subset of the relevant configurations . While these choices are certainly in the right direction, it is imperative to find some clear signal for physical configurations. References Present Address: FESt Heidelberg and Inst. Theor. Phys. Univ. Heidelberg, D-6900 Heidelberg, Germany I. Singer, Comm. Math. Phys. 60 (1978) 7 Ph. de Forcrand et al., Nucl. Phys. B (Pros . Suppl.) 20 (1991) 194
P. van Baal, ITP Utrecht preprint THU-91 (Thoughts on) Gribov Copies
More
[4] D. Zwanziger, Comm. Math. Phys. 138 (1991) 291 [5] J. E. Hetrick and Y. Hosotani, Phys. Lett . B230 Phys. Rev. D38 (1988) 2621; E. Witten, IAS preprint IASSNS-HEP-91/3 On Yang-Mills The(1989) 88,
ories in Two Dimensions