So far we have decomposed the individual strata of the Pohlmeyer-Rehren Lie algebra into irreducibleg0-modules and further decomposed those into weight spaces. While this took into account the action ofg0, it fails to illuminate the structure of the Lie brackets of all other elements. The following basic consideration of tensor products of irreducible modules of subalgebras of a given Lie algebra provides a starting point for this. We begin with a lemma highlighting the structural similarities between the tensor product and the Lie bracket.
Lemma 2.4.1. Let G be a Lie algebra with subalgebra H. Let V,V0 ⊂ G be H-modules via the adjunction. Then:
1. [V, V0]is a H-module (using the adjunction).
2. The tensor product V⊗V0is a H-module using the action
h.(v⊗v0) = h.v⊗v0 + v⊗h.v0 (2.117) for all h∈H, v∈V and v0∈V0.
3. The exterior product V∧V is a H-module using the action
h.(v∧v0) = h.v∧v0 + v∧h.v0 (2.118) for all h∈H and v,v0∈V.
4. Let further designate
ψ:V×V0→V⊗V0,
(v, v0)7→v⊗v0. (2.119)
Then the map
Ψ:V⊗V0 →[V, V0],
v⊗v0 7→[v,v0] (2.120)
is the unique linear map such that the diagram
V×V0 V⊗V0 [V, V’]
ψ
[·,·] Ψ (2.121)
commutes. Furthermore,Ψis an epimorphism of H-modules.
5. Let similarly designate
ϕ:V×V→V∧V,
(v, v0)7→v∧v0. (2.122)
Then the map
Φ:V∧V→[V, V],
v∧v0 7→[v,v0] (2.123)
is the unique linear map such that the diagram
V×V V∧V
[V, V]
ϕ
[·,·] Φ (2.124)
commutes. Φis an epimorphism of H-modules as well.
Proof. 1. This can be proved with a simple calculation using the facts that (due to the Jacobi identity), the adjunction is a derivation, equal terms with opposing signs can be eliminated, V and V0 are H-modules by assumption, and finally, again, that the adjunction is a derivation. Letv∈V, v0 ∈V0andh,h0 ∈Hand calculate
h.h0.[v,v0]−h0.h.[v,v0]
=h.([h0.v,v]+[v,h0.v0])−h0.([h.v,v]+[v,h.v0])
=[h.h0.v,v0]+[h0.v,h.v0]+[h.v,h0.v0]+[v,h.h0.v0]
−[h0.h.v,v0]−[h.v,h0.v0]−[h0.v,h.v0]−[v,h0.h.v0]
=[[h,h0].v,v0]+[v,[h,h0].v0]
=[h,h0].[v,v0].
2. A calculation very similar to the proof of 1.
3. Again, a very similar calculation.
4. Any Lie bracket is a bilinear map; in this situation the universal property of the tensor product guarantees that there is exactly one linear map such that the diagram com-mutes. It is just a matter of notation to check thatΨsatisfies this and (simply writing out the respective actions of H) is an epimorphism of H-modules. Since v⊗v0 is a preimage of [v,v0] for allv∈V,v0 ∈V0, the mapΨis surjective.
5. Analogous.
IfVandV0are finite-dimensionalH-modules, thenV⊗V0is a finite-dimensional module as well, and so is its image Ψ(V⊗V0) = [V,V0]. If furthermore H is semisimple, then we can apply Weyl’s theorem on complete reducibility (2.3.8) on bothV⊗V0 and [V,V0] and decompose them into a direct sum of irreducible H-modules. ForV⊗V0, this is known as
the Clebsch-Gordanproblem, and with the epimorphism ofH-modulesΨ(or, ifV=V0, the epimorphismΦ) we can transfer some of the information to [V,V0]. For instance, we have the estimate of dimensions
dim[V,V0] = dimΨ(V⊗V0) ≤ dimV⊗V0 = dimV·dimV0. (2.125) Note that the mapsΨ andΦ are not necessarily injective, so we only have the inequality dim[V,V0] ≤ dimV·dimV0. We only consider the simplest case of d = 3 to illustrate the method and its limitations. In this case, the Clebsch-Gordan problem has an easy solution.
Theorem 2.4.2(Clebsch-Gordan formulas forsl2). Let sV designate irreduciblesl2-modules of highest weight s. Let s1,s2∈N0with s1≥s2. Then there are isomorphisms ofsl2-modules
s1V⊗ s2V
s1+s2
M
s=s1−s2
sV (2.126)
and
sV∧ sV
s
M
n=1
2n−1V. (2.127)
Proof. This is covered in many introductory texts on representations of Lie algebras, for instance [Hal03, Theorem D.1, note different notations for the modules of a given highest
weight.]
We can use the above theorem to consider the generation ofg from certain multiplets, simplifying notation by suppressing the unnamed isomorphisms from the theorem for the remainder of this section. In subsection 2.3.3 we had seen thatg1 =2g1⊕1g1 2V⊕1V. So we calculate (note that [2g1,1g1]=[1g1,2g1]) due to antisymmetry of the Lie bracket)
[g1,g1]
=[2g1⊕1g1, 2g1⊕1g1]
=[2g1,2g1] + [2g1,1g1] + [1g1,2g1] + [1g1,1g1] (2.128) Φ(2V∧2V) + Ψ(2V⊗1V) + Φ0(1V∧1V)
2.4.2
Φ(3V⊕1V) + Ψ(3V⊕2V⊕1V) + Φ0(1V).
Comparing this to the decompositon ofg2into multiplets, by theorem 2.3.13 (cf. table 2.2)
g2 3V ⊕ 2V ⊕ 2·1V, (2.129)
it becomes obvious that not all of the morphisms ofsl2-modules Φ,Φ0 andΨare injective or that (some of) the sums are not direct.6 In other words, there are relations between the
6By actually computing the decomposition of the Lie brackets of the pairs of irreducible modules in equation (2.128), for instance [2g2,2g2]3V⊕1Vand comparing this to the results of the Clebsch-Gordan formulas (theorem 2.4.2), for instance2V∧2V3V⊕1V, one can see that the morphismsΦ,Φ0andΨare all injective in this case.
But the morphisms from lemma 2.4.1 are not neccessarily injective: take for example the multiplets2g1 2V and3g2 3V(which are uniquely determied by their highest weight and degree due to theorem 2.3.13:3). The decomposition of their Lie bracket is [2g1,3g2]4V⊕3V⊕2V⊕1V, while by Clebsch-Gordan formula (2.126),
2V⊗3V5V⊕4V⊕3V⊕2V⊕1V. In this case, the difference exists because there can be no module of type5Vin stratuml=5 due to theorem 2.3.13:5.
elements of [g1,g1] that do not follow from the axioms of the Lie algebra. So certain Lie brackets of elements ofg1can be expressed as linear combinations of others. An obvious task arises from this: find a basis of gl in terms of multiple brackets ofg1 that exposes as much structure as possible. We begin with an educated guess based on the observation that if the summands originating from the term [2g1,2g1] were missing, we would neatly be in line with equation (2.129), with injectiveΦ0andΨand direct sums.
Generalizing this to higher stratagl, we can ask the question if it is possible to obtain the entirety ofglwith injective morphisms and direct sums by iteratively adjoiningl−1 copies of1g1 tog1 = 2g1⊕1g1. Forl= 2, we unsurprisingly obtain the same number and types of multiplets that make up the entirety ofg2, but our conjecture fails forg3(again, cf. table 2.2) because
[1g1,g2]
(2.129)
[1g1,3V] + [1g1,2V] + [1g1,1V] + [1g1,1V]
Ψ(1V⊗3V) + Ψ0(1V⊗2V) + Ψ00(1V⊗1V) + Ψ000(1V⊗1V) (2.130)
2.4.2
Ψ(4V⊕3V⊕2V) + Ψ0(3V⊕2V⊕1V) + Ψ00(2V⊕1V⊕0V) + Ψ000(2V⊕1V⊕0V), but the decomposition ofg3into multiplets is
g3 4V ⊕ 2·3V ⊕ 3·2V ⊕ 3·1V ⊕ 0V, (2.131) so again, there are relations. Additionally, we do not know a priori whether the morphisms of sl2-modulesΦ,Ψ,Ψ0etc. are injective in the first place or if they cause relations themselves.
A way to approach this problem is needed, and chapter 4 provides a part of a solution.
Excursus: String Quantization and Leading Terms of Exceptional
Elements of the Invariant Algebra
After some light has been shed on the structure of the Pohlmeyer-Rehren Lie algebra in the previous chapter, in this excursion, a brief account will be given of some of the theory developed with quantization of the algebra of invariant charges of the Nambu-Goto string in mind. This serves a dual purpose: to give a motivation why the Pohlmeyer-Rehren Lie algebra merits study, and to provide some context for a result (proposition 3.4.3) and a conjecture (3.4.2) that will be introduced at the end of this chapter. The proof of this result itself is very straightforward, but to understand the terminology and relevance some explanation is required.
3.1 Pohlmeyer’s approach
Pohlmeyer’sapproach to string quantization, which motivates this thesis, goes back to the 1980s, but the reformulation using the more mainstream mathematical language of the Shuffle Hopf algebra and the Eulerian idempotent is a more recent development due to Bahnsand Meinecke. The following three sections serve the dual purpose to provide a brief overview of the physical theory motivating this thesis as well as a point of connection to previous works that use the original notions. They mostly summarize Bahns’ and Meinecke’s account [BM11] as well as Meinecke’sDiplomarbeit(in German) [Mei09, Chapters 1 and 2] and the original papers [Poh82], [Poh85], [PR86] [PR88], to which the reader is referred for more detail.
In the setting of the closed bosonic string with the Nambu-Goto (i.e. proportional to the worldsheet’s surface area) action outlined in the introduction (page 9), Pohlmeyerproposed [Poh82] the following approach. Recall from the introduction that the map
x:S1×R→Rd
(such that for allτthe mapS1 →Rd,σ7→x(σ, τ) is a spacelike closed curve) parametrizes the string’s worldsheet. Using a Laxpair, a system of linear differential equations describing the string’s dynamics is set up and solved. The expansion of a power series (in the free parameters
occurring in the monodromy matrix) of the monodromy matrix of the differential equations leads to coefficients R±
x1...xn where the indices1 x1, . . . ,xn ∈ {0, . . . ,d−1} correspond to the dimensions of spacetime for n ∈ N+. These coefficients are tensors expressed by the path ordered integrals that are known from ordinary string theory, andu±xi is theirxi-th component.
A consequence of definition (3.1) of theR±as path-ordered integrals is that they obey the multiplication rule
Since the algebraic relations between the mathematical objects introduced in this subsection only depend on the tensors’ indices which are shuffled under multiplication, Bahns and Meineckeinterpreted [BM11] the tensorsR± as words of their indices, the elements of the shuffle Hopf algebra (definition 1.1.2), and reformulated the remaining constructions – that will now be briefly introduced in their original forms – in these terms.
3.1.1 The Poisson algebra of invariant charges
The tensorsR±depend3on the parametersσandτ, but their cyclic symmetrizations, Z±
x1...xn(σ, τ) := R±
x1...xn(σ, τ)+R±
xnx1...xn−1(σ, τ)+. . .+R±
x2...xnx1(σ, τ) (3.3) could be shown to be independent ofσ and (ifxactually parametrizes the worldsheet, so that we are in theon-shellcase also)τ. Therefore, theZ±are called theinvariant charges(also known asPohlmeyer charges) of the Nambu-Goto string.
A Poisson bracket on the invariant charges can be defined as follows:
n differ only by a global sign, it has become customary to only focus onZ+and suppress the superscript+in the notation.
1Note that the indicesxiarenotcomponents of the parametrizationx. This somewhat confusing notation in this section was chosen for the sake of consistency with the rest of this thesis, where no risk of such a confusion exists.
2The left and right movers also have the property that their lightlikeness is equivalent to the constraints on the canonical coordinates on the string.
3By choosing an appropriate gauge,∂τR±=0 can be achieved, cf. [MR03, p. 73].
In [PR88, section III], Pohlmeyer and Rehren showed that the invariant charges are completein the sense that for a certain class of worldsheets, the string’s worldsheet can be reconstructed uniquely up to global translations in spacetime fromh.
If we expand the power series of the logarithm of the monodromy matrix instead of the power series of the monodromy matrix itself, we obtain the so-calledhomogeneous tensors, in particular those of homogeneity degree 1, which are called thetruncated tensors4Rt.
Since equations (3.5) and (3.6) mirror equation (1.33), and we had already identified the tensorsRwith words, we can identify homogeneous tensors with (higher) Euler idempotents of words and in particular the truncated tensors with Euler elements.
From the construction of the different types of tensors introduced in this section, several relations between them can be deduced. The tensorsRcan be written as a linear combination of homogeneous tensors (of degreesk) with the same index word,
Rx
1...xn =
n
X
k=1
R(k)x
1...xn, (3.5)
while the homogeneous tensors of degreek >1 are called that way because they are homo-geneous polynomials of degreekin the truncated tensors:
R(k)x
1...xn = 1 k!
X
0<i1<...ik−1<n
Rt
x1...xi1
· Rt
xi1+1...xi2
·. . .· Rt
xik−1+1...xn. (3.6) Remark 3.1.1. Historically, the Poisson bracket (3.4) on the invariants originally was arrived at as a Poisson bracket induced by the Poisson bracket on the space spanned by the left and right movers. That Poisson bracket was in turn induced by the one on the canonical coordinates of position and momentum. When extending the original bracket to the truncated tensors (Euler elements), one obtains an antisymmetric bilinear map which does not satisfy the Jacobi equation. Under cyclic summation however, the terms violating the Jacobi equation cancel out. Removing those terms form the Poisson bracket yields what was originally called the “modified Poisson bracket” and which is called the Pohlmeyer-Rehren Lie bracket (1.54) here. Because the removed terms vanish under the cyclic summation, derivative extension as discussed above (resulting in (3.4)) leads back to the original Poisson bracket.
Remark 3.1.2. We have now collected all the facts to give a list of reasons whyd = 3 and d=4 receive special consideration in this thesis. Since dimgl =NumLyndon(l+2,d), which is monotonously increasing withdfor a givenl(and an upper bound of which is givendl+2), cases with lowerdare generally considerably more manageable.
• One major reason lies in the application to the quantization of the Nambu-Goto string.
– Since d is the dimension of the string’s surrounding spacetime and we appar-ently live in a four-dimensional spacetime, the case of d = 4 is important to describe strings in the actual physical universe. This is especially important since the Pohlmeyer approach has the distinction of having no known obstruction to quantization in this spacetime dimension, unlike the usual conformal field theory methods of string quantization which require a critical dimension ofd = 26 (or d=10 with supersymmetry).
4In [MR03], they are called “monodromy variables”.
– d=3 can serve as a simpler toy model.
– Even lower dimensions are less well suited for this; for d = 2, the preimage of almost every point on any (spacelike) closed curve parametrizing the string for a given parameterτhas at least two elements, in contrast to the usual picture of a simple (i.e. injective except for the points on the boundary of the parametrizing interval) curve, so degeneracy is to be expected.
– Ford=1, no linearly independent time and space dimensions exist.
• The structure ofgitself, in particular as ag0so(d,C)-module (cf. remark 2.2.15) gives further reasons:
– Ford=1, there are no Lyndon words of more than a single letter, thereforegis the one-dimensional Lie algebra spanned bye(0)∈g−1, andg0=0.
– For d = 2, the stratum g0 so(2,C) is one-dimensional, hence abelian, and all irreducibleg0-modules are one-dimensional as well. Additionally, it is known that g1 does not generateg≥1(cf. proposition 1.5.3), while the opposite is conjectured ford≥3 (conjecture 4.1.2).
– So d = 3 is the lowest dimension in which a rich representation theory (of g0 so(3,C)sl2) develops.
– Regarding its representation theory, d = 4 is an unusual case since it is the only dimension for which g0 so(4,C) sl2×sl2 is nonsimple. Connected to this is the fact that the ladder operators commute and for all weights, the multiplicity of every weightµ∈Γ(cf. remark 2.3.7) isn(µ)=1.