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Natural Science Forum / Physics / Research / October 2007D!=26 anomalies
is there a nice and simple way to see how worldsheet weyl anomaly translates into target spacetime lorentz anomaly, in bosonic string theory? > is there a nice and simple way to see how worldsheet weyl anomaly > translates into target spacetime lorentz anomaly, in bosonic string > theory? Lorentz anomaly visible only in lightcone gauge. In the BRST quantisation we obtain anomaly in the algebra generated by BRST charge Q. What you are meaning in phrase "anomaly translates"? > > is there a nice and simple way to see how worldsheet weyl anomaly > > translates into target spacetime lorentz anomaly, in bosonic string [quoted text clipped - 3 lines] > quantisation we obtain anomaly in the algebra generated by > BRST charge Q. What you are meaning in phrase "anomaly translates"? I don't know. The phrase was used in Polchinski page 94 under 'Discussion'. It seems to me that Polchinski says that it can be seen (in lightcone gauge) that Weyl anomaly causes Lorentz anomaly. I would like to understand that better... > is there a nice and simple way to see how worldsheet weyl anomaly > translates into target spacetime lorentz anomaly, in bosonic string > theory? I don't know if this is nice and simple, but the idea is this, at least in part. The Weyl anomaly implies that the group of conformal transformations (the group of diffeomorphisms which have the effect of rescaling the metric) is no longer a symmetry of the theory. The Lorentz anomaly occurs in the light cone gauge (the gauge group is the group conformal transformations). In the light cone gauge a particular reference frame is singled out. Under a Lorentz tranformation this reference frame changes, and you consequently have to make a gauge transformation (i.e., a conformal transformation) to stay in the light cone gauge. Consequently, the generators of the Lorentz group need to include terms generating particular conformal transformations. SInce the conformal symmetry is anomalous this leads to anomalous Lorentz symmetry. charlie torre > The Weyl anomaly implies that the group of conformal transformations > (the group of diffeomorphisms which have the effect of rescaling the > metric) Conformal transformations is not subgroup of diffeomorphisms. > is no longer a symmetry of the theory. The Lorentz anomaly occurs in > the light > cone gauge (the gauge group is the group conformal transformations). In the lightcone gauge we fix only reparametrisations of wordsheet. > to...@cc.usu.edu wrote: > > The Weyl anomaly implies that the group of conformal transformations > > (the group of diffeomorphisms which have the effect of rescaling the > > metric) > > Conformal transformations is not subgroup of diffeomorphisms. Actually, it is. Given a smooth manifold, diffeomorphisms form a transformation group of the manifold. Given a metric on that manifold, one can ask for the set of diffeomorphisms which act (by pull-back) on the metric such that the pulled-back metric is proportional to the original metric. It is easy to see that such transformations form a subgroup of the diffeomorphism group. For a generic manifold and metric, this subgroup is trivial (just the identity). For a two-dimensional manifold with simple topology (e.g., the plane, or the cylinder) this subgroup is non-trivial; it is infinite-dimensional in fact. It is this group I was calling "the conformal group" - using "the Weyl group" to denote the group of rescalings of the metric by functions. Classically, one can trace the exisitence of the conformal (diffeomorphism) symmetry group to the existence of the Weyl rescaling group. In the quantum theory, when the Weyl group becomes anomalous, so does the conformal group (it becomes only projectively represented on the state space). > > is no longer a symmetry of the theory. The Lorentz anomaly occurs in > > the light > > cone gauge (the gauge group is the group conformal transformations). > > In the lightcone gauge we fix only reparametrisations of wordsheet. Yes, the light cone gauge fixes the world sheet diffeomorphism gauge group. The light cone gauge is a specialization of the conformal gauge. The conformal gauge fixes the world sheet metric to be conformal to the 2-d Minkowski metric; this gauge does not interfere with target space Lorentz invariance. The conformal gauge leaves an infinite-dimensional residual gauge group - the group of conformal transformations. One uses this residual group to fix the gauge completely - to the light cone gauge. The light cone gauge is not target space Lorentz invariant, and this is why the Lorentz generators need terms which generate conformal transformations to stay in the light cone gauge. This is analogous to, say, electrodynamics where one can fix the Lorentz invariant Lorentz gauge, which leaves an infinite-dimensional residual gauge group. This residual gauge group can be used to completely fix the gauge, e.g., to the Coulomb or radiation gauge. This latter gauge is not Lorentz invariant; the Lorentz generators in the Coulomb gauge must include generators of residual gauge transformations to preserve this non-covariant gauge. In the quantum theory of the string, the breaking of the Weyl - hence conformal symmetry by anomalies renders the Lorentz algebra anomalous. charlie torre >>Conformal transformations is not subgroup of diffeomorphisms. > [quoted text clipped - 6 lines] > It is easy to see that such transformations form a subgroup of the > diffeomorphism group. We have here a different treatment of term "conformal transformation". Manifold with metric can be viewed as pair (manifold,metric). Morphism also can be viewed as pair (diffeomorphism, ...) Conformal transformation can be defined as such morphism wich only deform metric. But we know that the usual diffeomorphism induce metric transformation. In same cases it gives metric "proportional to the original metric". In this picture conformal transformations explore possible symmetries of our space. In string theory worldsheet metric is independent variable. And we must study metric changes wich is not diffeomorphisms. > For a two-dimensional > manifold with simple topology (e.g., the plane, or the cylinder) this > subgroup is non-trivial; it is infinite-dimensional in fact. It is > this group I was calling "the conformal group" - using "the Weyl > group" to denote the group of rescalings of the metric by functions. Hm. I always thinks that by definition: local conformal group = Weyl group conformal group is subgroup of local conformal group. > > Conformal transformations is not subgroup of diffeomorphisms. > > Actually, it is. IFAIU, this is just a distinction between the active and passive interpretation of conformal transformations. You can rescale the metric keeping the coordinates fixed - this is a Weyl transformation. Or you can perform a coordinate transformation, such that the metric in the new coordinates is proportional to the metric in the old ones - this is a conformal diffeomorphism. The difference between these viewpoints is a matter of interpretation, not a matter of substance. I prefer the second viewpoint, because it led me to answer an important question: what extension of the diffeomorphism algebra reduces to the Virasoro central term upon restriction to its conformal subalgebra. On Oct 5, 4:57 am, thomas_larsson...@hotmail.com wrote: > You can rescale the metric keeping the coordinates fixed - this is a > Weyl transformation. [quoted text clipped - 5 lines] > The difference between these viewpoints is a matter of interpretation, > not a matter of substance. I disagree. You can change the curvature of a metric by a Weyl transformation, e.g.,make a flat metric non-flat. But you cannot change the curvature of a metric with any kind of a diffeomorphism, e.g., a flat metric stays flat after a diffeo (conformal or otherwise). charlie torre On Oct 5, 9:41 pm, to...@cc.usu.edu wrote: > On Oct 5, 4:57 am, thomas_larsson...@hotmail.com wrote: > [quoted text clipped - 15 lines] > > charlie torre I agree with Charlie. Here I give some dictionary: In Wald's book 2 things are defined: 1) conformal isomorphism (end of appendix c) 2) conformal transformation (appendix d) [Clearly, conformal isomorphism does not change lengths (because change in metric is compensated by change in coordinates); conformal transformation changes lengths (because metric is changed i.e. rescaled, while coordinates are untouched).] Charlie refers to these 2 things as (respectively): 1) sometimes conformal diffeomorphism, sometimes conformal transformation 2) Weyl transformation In Polchinski's book (page 44) the term "conformal transformation" is used for holomorphic change of coordinates while at the same time keeping the metric fixed. (Or in other words it is conformal isomorphism followed by Weyl transformation which just rescales the metric back to its original value.) Hope everybody now agrees. ---------- Also, Charlie mentioned previously that in light cone gauge, the gauge group is the group of conformal transformations. I don't understand why that is so. Charlie, could you please give us some more details (formulas or references)? > Also, Charlie mentioned previously that in light cone gauge, the gauge > group is the group of conformal transformations. > I don't understand why that is so. Charlie, could you please give us > some more details (formulas or references)? I probably was not sufficiently clear here. This will probably be my epitaph, unfortunately. The point I was trying to make was this. The light cone gauge is designed to gauge fix the full 2-d diffeomorphism gauge group (i.e., coordinate transformations, reparametrizations, whatever). However, when considering issues of target space Poincare (or, more to the point, Lorentz) symmetry, it is useful to think of the light cone gauge as arising from a sequence of two gauge fixings. First we go to the conformal gauge, where the worldsheet metric is taken to be a function times the appropriate 2-d flat metric. This gauge does not interfere with target space Lorentz symmetry, which is a symmetry of the theory both classically and quantum mechanically. The conformal gauge does not fix the gauge completely. Any conformal isometry (diffeo which rescales the worldsheet metric) will keep the metric in conformal gauge. The 2-d conformal group in this setting is infinite-dimensional. One can use this residual symmetry to further specialize the conformal gauge to the light cone gauge. But this gauge is defined relative to a particular target space Lorentz frame. If you perform a target space Lorentz transformation you will not preserve the gauge - you will be in a light cone gauge defined relative to a different target space Lorentz frame. Now, all the light cone gauges associated with different target space Lorentz frames are specializations of the conformal gauge. So given two light cone gauges (i.e., light cone gauges defined relative to two target space Lorentz frames), there will always be a conformal isometry which relates them. One incorporates this conformal isometry into the definition of the infinitesimal Lorentz generators so that one can get the Lorentz group acting as a symmetry group of the light cone gauge-fixed theory, classically at least. The conformal anomaly then makes these generators anomalous. I do not think I will try to show you formulas. And I don't know what is the best reference, either. But let me recommend Marc Henneaux's contribution to the book "Principles of String Theory". There he gives a pretty careful and detailed discussion of these issues. charlie torre > First we go to the conformal gauge, where the worldsheet metric is > taken to be a function times the appropriate 2-d flat metric. This [quoted text clipped - 3 lines] > isometry (diffeo which rescales the worldsheet metric) will keep the > metric in conformal gauge. Let me say this in other words. Initially action has gauge symmetry group with 3 degree of freedom. Diffeomorphism equal to two arbitrary functions and rescaling metric - one function. Metric rescaling is not commute with diffeomorphism, but possible exist some subgroup which commute. Selecting conformal gauge we fix two degree of freedom. But its neither pure diffeomorphisms nor pure rescaling. > The 2-d conformal group in this setting is > infinite-dimensional. One can use this residual symmetry to further > specialize the conformal gauge to the light cone gauge. It is not right to say that residual symmetry is conformal group. Actually it is subgroup of symmetry group of the action. When we go to the quantum theory we want to realize symmetry as unitary operators and check its algebra for any anomaly. In not complete fixed theory (when we allow gauge and unitary ghosts) anomaly arise most simple as violations of gauge symmetry algebra. Here I am not understand how to perform full checking of the arbitrary theory on anomaly existence. And which anomalies are admissible, which is not. |
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