6.2 Nambu-structures and Lie 3-algebroids
6.2.3 Future directions and concluding remarks
To be a proper curvature operator, the right hand side of this expression should be a section in the endomorphism bundle of F or in other words it should map sections of F into sections of F. For this it suffices to prove C∞(M)-linearity, i.e.
for a section e ∈Γ(F) we should have R(s1, s2, s3, s4)(f e) =f R(s1, s2, s3, s4)(e).
However, this is easy to show by noting for sectionssi, eand functions f:
[∇s1∧s2,∇s3∧s4] (f e) = [a(s1∧s2), a(s3∧s4)] (f)e+f [∇s1∧s2,∇s3∧s4] . (6.2.25) The first summand on the right can be rewritten by using the defining property (6.2.2). We get the following form :
[a(s1∧s2), a(s3∧s4)] (f)e=a([s1, s2, s3]∧s4+s3∧[s1, s2, s4]) (f)e . (6.2.26) But this shows that every line in (6.2.24) separately is an endomorphism and we proved the first property which one demands on a curvature operator.
In addition to that, a curvature operator should be C∞(M)-linear in each of its entries, e.g. for the first entry one has to demand R(f s1, s2, s3, s4) = f R(s1, s2, s3, s4). The proof is a lengthy but straightforward calculation using the fourth property of the covariant derivative introduced above and the Leibniz rule of the 3-bracket (6.2.2).
To sum up, we see that the notion of Filippov 3-algebroid allows to generalize the most important constructions of differential geometry to the case of a vector bundle with 3-bracket. The next step would be the introduction of a metric on F, metric compatible connections and the notion of Levi-Civita connection which finally would make it possible to formulate the analogue of an Einstein-Hilbert action for Filippov 3-algebroids. Similar to earlier chapters, it would be interesting to generalize also gauge transformations, e.g. of the NS-NS B-field and formulate corresponding string actions.
6.2 Nambu-structures and Lie 3-algebroids 131 the latter generalization of the Koszul bracket is hard to find because the defining properties 6.2.2 impose very strong algebraic and differential conditions on γ. As an example [109], demanding γ only to be a Nambu-Poisson three-tensor is not sufficient. Thus it may be possible that the choice of the Koszul 3-bracket (or even a twisted version of it) does not lead to non-trivial geometries.
Clearly there are at least two possible ways to avoid the latter problem. The first would be to choose another three-bracket than (6.2.14). The notion of a Filippov 3-algebroid is completely general and thus the total space of the bundle on which the 3-bracket (and the differential calculus) is defined needs not to be the cotangent bundle or the tangent bundle of the manifold. The correct choice of the total space and thus the three bracket is still ongoing work. Secondly we remark that the choice of the fundamental identity (6.2.2) is connected to the notion of three-associativity if one is interested in quantization questions [100]. The correct definition of the generalization of associativity to three-structures is to our knowledge an open problem. Choosing another type of fundamental identity could allow for non-trivial geometries even by using the simple generalization (6.2.14) of the Koszul bracket.
Once having found the correct Filippov 3-algebroid and the corresponding dif-ferential calculus (which then directly follows as discussed in the previous sub-section), the next question would be about the quantization of such a structure.
Setting up star-products, analyzing the corresponding associativity requirements or even trying to find a covariant star-product calculus using the differential geome-try of the previous subsection are only the first ideas in this interesting direction.
Appendix A
The Rogers dilogarithm
In this appendix, we collect some of the definitions and important properties of the Rogers dilogarithm function. This beautiful object has lots of applications in mathematics and string theory, reaching from scattering amplitudes to invariants of 3-geometries. For a more detailed analysis of the mathematical aspects of the Rogers dilogarithm function we would like to refer the reader to [110, 111], whereas its generalization is described in detail in [112, 113].
A.1 Definition and fundamental properties
The Rogers dilogarithm function L(x) for real arguments x is defined in the fol-lowing way:
L(x) := Li2(x) + 1
2 log(x) log(1−x), 0< x <1, (A.1.1) where Li2(x) denotes the Euler dilogarithm function, given by:
Li2(x) :=
∞
X
n=1
xn n2 =−
Z x 0
log(1−y)
y , 0≤x≤1. (A.1.2)
With the help of (A.1.2), the integral representation of the Rogers dilogarithm can be deduced as:
L(x) =−1 2
Z x 0
log(1−y)
y + log(y) 1−y
dy . (A.1.3)
Furthermore, from these definitions one can derive two functional relations, which in turn uniquely characterize the Rogers dilogarithm function:
L(x) +L(1−x) = L(1), L(x)−L(y) +L xy
−L 1−x1−y−1−1
+L 1−x1−y
= 0. (A.1.4)
Employing the integral representation (A.1.3), one can analytically continue L(x) to the domain C\ {0,1}. However, the resulting function L(z) is not single valued any more and one should use the universal cover ofC\ {0,1}as the domain of definition. For the complex Rogers dilogarithm the relation
L(z) +L(1−z) = L(1) (A.1.5)
still holds, but the five-term relation in (A.1.4) receives logarithmic corrections.
The systematics of those corrections can be described by the following generaliza-tion which is due to Neumann [112]:
R(z;p, q) :=L(z)− π2 6 + πi
2
plog(z−1) +qlogz
. (A.1.6)
Here,p, q are integer numbers and the constant is just a convenient normalization.
In the main text, we will not need the complicated properties of this function. How-ever, in the investigation of Virasoro-Shapiro amplitudes in the CFTH-framework, as performed in [78], its properties play an important role.
Appendix B
Tachyon correlation functions
In this appendix we review standard results on scattering theory of tachyon vertex operators which are used in the main text. For a detailed introduction into basic concepts of quantum field theory, conformal field theory and further information on tachyon vertex operators we refer the reader to the standard literature (e.g.
[114, 76] and references therein). We first consider the case of the free theory and give the results which are important for detection of open string non-commutativity in chapter 3. In a second section we focus on a special interacting case which was used in chapter 5.1.
B.1 The free case
Consider the theory ofn free bosonic fields Xi(z,z), interpreted as the embedding¯ of a 2-dimensional world sheet into flat target space. The action is given by:
S = 1 2πα0
Z
Σ
d2z δij∂Xi∂X¯ j . (B.1.1) If we specialize to the sphere Σ =P1, the propagator of the theory is given by:
hXi(z1,z¯1)Xj(z2,z¯2)i= −α0
2δijln|z1−z2|2 . (B.1.2) The energy-momentum tensor can be constructed by using the (anti-)holomorphic currents Ji(z) = i∂Xi(z) and ¯Ji(¯z) =i∂X¯ i(¯z). We denote the normal ordering of field operators by : · · · :. The holomorphic component of the energy momentum tensor is:
T(z) = 1
α0δij :JiJj: (z), (B.1.3)
and similar for the anti-holomorphic component. Finally, by using the contraction of a momentum vector pi with the coordinates Xi which we denote by p·X, the definition atachyon vertex operator carrying momentum p is as follows:
Vp(z,z)¯ def= :ei p·X(z,¯z): . (B.1.4) To begin with, we want to show that tachyon vertex operators are conformal fields and determine their conformal dimension by calculating the operator product expansion with the energy momentum tensor (B.1.3). Only the singular parts are important and we write 'if an equation holds up to regular terms.
T(z1)Vp(z2) = 1 α0
∞
X
n=0
1
n! :∂Xi(z1)∂Xj(z1) :δij : [i p·X(z2,z¯2)]n: 'α0
4
∞
X
n=2
1 (n−2)!
1
(z1 −z2)2p2 : [i p·X(z2,z¯2)]n−2: +
∞
X
n=1
1 (n−1)!
1
z1−z2i pi :∂Xi(z2) [i p·X(z2,z¯2)]n−1: 'α0
4p2Vp(z2,z¯2)
(z1−z2)2 + ∂z2Vp(z2,z¯2) z1−z2 .
(B.1.5)
In the second step, the first summand is the result of taking two Wick-contractions with : [i p·X(z2,z¯2)]n: and the second summand results by taking one contraction.
Thus the vertex operator is primary with conformal dimensionh =α0p2/4. Simi-lar results hold for the OPE with the anti-holomophic component of the energy momentum tensor.
Let us now compute the OPE of two vertex operators with momenta p1 and p2. To simplify the calculation, we first note useful identities for the calculation.
First, letA(z1), B(z2) be field operators depending on the world sheet coordinates.
Then we have the following result for the contraction with exponentials:
A(z1) :eB(z2): = A(z1)B(z2) :eB(z2):, (B.1.6) which is proven by expanding the second exponential. Using this, we can compute the OPE between two normal ordered exponentials:
:eA(z1): :eB(z2): =
∞
X
n=0
1
n! :A(z1)n: :eB(z2): '
∞
X
n=0
1 n!
n
X
k=0
n
k A(z1)B(z2) k
:A(z2)n−keB(z2): 'eA(z1)B(z2) :eA(z2)eB(z2): .
(B.1.7)
B.2 Interacting case 137