Constructing identities

G4Gb2C_{4 0}

2 0

+bG²₂C_{6 0}

2 0

−5G4C_{6 0}

2 0

−7G6C_{4 0}

2 0

. (5.140) Note that although the two graphs on theLHSare related by momen-tum conservation, this does not allow one to reduce the number of independent graphs.

Since the functionDiCSimplifyin theModular Graph Formspackage can decompose any derivative of any holomorphic Eisenstein series when the optionbasisExpandGis set to^True, the computations presented in this section can be easily performed by applying the ^CSimplify function to different representations of the same graph.

5.5 THE SIEVE ALGORITHM

With the techniques described in the last two sections, many valuable identities between modular graph forms can be derived. However, if one is interested in simplifying a particularMGF, e.g. one which has appeared as an expansion coefficient of a Koba–Nielsen integral, it is not always clear which techniques to combine to obtain the desired decomposition. In this situation, the sieve algorithm, first introduced in [16], can be used: It allows for a systematic decomposition (up to an overall constant) of arbitrary MGFs, as long as the basis for the decomposition is known.

5.5.1 Constructing identities

As a starting point, assume that we have a combinationFofMGFsof homogeneous modular weight(|A|,|B|)and we want to check whether or not it vanishes. The idea behind the sieve algorithm is to repeatedly take derivatives ofFusing the Maaß operator∇^(|A|)defined in (3.51). Due to an intricate interplay between momentum conservation identities and HSR, every derivative can be expressed as a linear combination of products of holomorphic Eisenstein series,MGFswith non-negative antiholomorphic labels for each edge,τ2with non-positive exponent,

MGFsof the formC _{k 0}

−n0

with k > n and modular invariant factors.

After taking |B| derivatives, the antiholomorphic modular weight vanishes according to (3.52) and hence each term in the derivative has to factorize, since any unfactorizedMGFswould have to have vanishing

antiholomorphic labels and therefore be amenable toHSR, leading to a factorized expression. Using the generalized Ramanujan identities from Section5.3.5, the factors of the formC _{k 0}

−n0

can be decomposed as well. Since each term is factorized, the total modular weighta+b of every leftoverMGFis strictly less than|A|+|B| and if we know all identities betweenMGFsof lower total modular weight, it is manifest if the |B|^th derivative of F vanishes or not. If F has |A| |B|, then Lemma 1 in [16] guarantees that if the derivative vanishes,F 0 up to an overall constant. If|A| ,^|B|andFcan be written as the derivative of an expression with|A| |B|, this primitive vanishes up to a constant, so F 0 as well. We conjecture that the same is true if F cannot be written as the derivative of an expression with|A| |B|, in line with all cases we tested. In this way, we can generate identities at progressively higher total modular weight.

We will now discuss in more detail how one can avoid negative antiholomorphic edge labels in the derivative of anMGF. First, note that a negative edge label in the derivative is due to a holomorphic edge in the original graph (assuming that the original graph did not already contain negative antiholomorphic labels):∇^(a) maps the labels (a,0)to the labels(a+1,−1). This −1 can be removed by the (antiholomorphic) momentum conservation identity which arises from the sameMGFwith the (a+1,−1)-edge replaced by a(a +1,0)-edge.

Since the(a+1,−1)-edge connects two vertices, both of these can be used to construct a momentum conservation identity to remove the

−1. If however the vertex we use for the momentum conservation has other holomorphic edges attached to it, there will be contributions in the momentum conservation identity in which these other edges carry negative antiholomorphic labels. These negative labels can in turn be removed by momentum conservation and so on. There is only one case, in which this procedure does not work: If the seed (it is always the same) has a closed holomorphic subgraph, we can only move the

−1 around this subgraph but never eliminate it entirely. Fortunately, this case only arises if theMGFwe applied the derivative to in the first place had a holomorphic subgraph. Thus by performingHSRbefore taking the derivative, we can avoid this problem and can be sure to be able to remove all negative entries. To summarize, sinceHSRtranslates graphs with closed holomorphic subgraphs into combinations of graphs without closed holomorphic subgraphs and holomorphic Eisenstein series, we can use momentum conservation andHSRto trade negative antiholomorphic labels for holomorphic Eisenstein series.¹⁰

The Cauchy–Riemann derivative of a holomorphic Eisenstein series has the formC_{2k+1 0}

−1 0

, i.e. it is a graph with one edge with negative

10 If we assume that all holomorphic labels of the original^MGFare at least one (as in [16]) then theHSRis the only source of holomorphic Eisenstein series. If we also allow for vanishing holomorphic labels, as we want to do here, holomorphic Eisenstein series can also arise from factorizations, e.g.∇⁽⁵⁾C_{0 2 3}

1 2 0

contains a term−3(^π

τ2)²E₂G₄ although no^HSRwas performed.

antiholomorphic weight. In this case, momentum conservation (and

HSR) cannot be used to remove the negative entry and in the original version published in [16], this fact was used to sieve the space of

MGFs for identities: After taking a derivative and trading negative antiholomorphic entries for holomorphic Eisenstein series, one subtracts the same derivative of an MGF in such a way that all holomorphic Eisenstein series cancel. Then, one can take the next derivative of the combined expression without generating irremovable negative antiholomorphic labels. After having taken|B|derivatives, the result is purely holomorphic (and still modular), so we can expand it in the ring of holomorphic Eisenstein series. By subtracting one final MGF

such that this derivative vanishes, one has constructed an identity up to an overall constant. In fact, if a combination of modular graph forms vanishes, then the holomorphic Eisenstein series have to cancel out in every derivative, as will be explained in Section5.5.2. This can however only be verified, if the prefactors of the holomorphic Eisenstein series are linearly independent. Since they carry lower total modular weight than the complete expression, this means that we need to know all identities between graphs of lower total modular weight.

As an example, consider the dihedralMGFC_{1 1 2}

1 2 1

. In order to find a simplification for this graph, start by taking its Cauchy–Riemann derivative,

Since no negative entries arise, we can directly take the next derivative,

∇⁽⁴⁾²C_{1 1 2}

where we used the notation (3.54) for the second Cauchy–Riemann derivative. The graphC_{1 2 3}

2 0 0

has to be decomposed byHSRand the negative entries in the other two graphs in the first line can be removed by antiholomorphic momentum conservation of the seedsC_{1 1 4}

1 2 0

and C_{1 2 3}

2 1 0

, respectively, and subsequentHSR. The result is

∇⁽⁴⁾²C_{1 1 2} In order to take one further derivative, we have to cancel the expression E2G4. Since

we can take one further derivative ofC_{1 1 2} gen-erating irremovable negative antiholomorphic labels. After momentum conservation andHSR, we obtain

∇⁽⁴⁾³

in (5.145) and take one final derivative,

∇⁽⁴⁾⁴

E4is also proportional to G8and we have

∇⁽⁴⁾⁴ Lemma 1 in [16] now states that this implies

C_{1 1 2} with someτ-independent constant.¹¹Using the techniques discussed in the previous sections, one can also decomposeC_{1 1 2}

1 2 1

directly and finds that the constant vanishes in this case (as expected sinceζ4is not single-valued, cf. (2.39)).

In general, findingMGFswith the correct Cauchy–Riemann derivatives to cancel the holomorphic Eisenstein series can be challenging but if we want to find a decomposition of anMGF into a set of basisMGFs, we can just take the derivatives of a linear combination of the basis elements and adjust the coefficients so that the holomorphic Eisenstein series cancel. This is what is done in the implementation of the sieve algorithm in theModular Graph Formspackage.

Instead of canceling holomorphic Eisenstein series in every derivative as described above and in [16], one can also use the generalized Ra-manujan identities discussed in Section5.3.5to perform the derivatives of the holomorphic Eisenstein series. In this way, the highest derivative of anyMGFcan be written in terms of holomorphic Eisenstein series andMGFsof lower total modular weight for which we assume that the relations are known, hence identities can be found explicitly.

11 Due to our normalization conventions, the graphs with equal total holomorphic and antiholomorphic edge labels are not modular invariant, hence the integration constant is multiplied by a suitable power of_τ^π₂.

In the example above, the resulting fourth derivatives are

Setting a linear combination of these four expressions to zero and requiring the coefficients of the various terms on theRHS to vanish leaves (5.148) as the only solution. If no solution had existed, the four

MGFsin (5.150) would have been proven to be linearly independent.

In theModular Graph Formspackage, the removal of edge labels−1 for dihedral and trihedral graphs is done by the functionsDiCSimplify

andTriCSimplify, if the optionmomSimplifyis set to^True(the default).

The sieve algorithm itself is implemented in the functionCSieveDecomp, which uses the traditional method of canceling holomorphic Eisenstein series in every step. If no further options are given, this function tries to decompose the graph given in its argument into the basis discussed in Section5.7, e.g. for the graphC_{1 1 2}

1 2 1

we considered above, we can run

In[43]:=CSieveDecomp

reproducing (5.149). The last term in the output is an undetermined integration constant, labeled by the exponent matrix of the original graph. Such a constant is added for all graphs with equal holomorphic and antiholomorphic weight. Setting the Boolean option^verboseof

CSieveDecompto^Trueprints a detailed progress report into the notebook with the expressions appearing in each derivative and the prefactors of the holomorphic Eisenstein series which are set to zero. E.g. the output for the third derivative in the computation above is

3rd derivative:

(Anti-)holomorphic Eisenstein series:

{G₄}

Coefficients that should be zero:

12 C_{3 0}

1 0

+12 bCoeff[1] C_{3 0}

1 0

Find solution for all

C_{3 0}

1 0

Solutions:

{{bCoeff[1] −_1}} .

This is the step described in (5.145) and (5.146) above. As one can see,

CSieveDecompforms a linear combination of the basis elements with coefficients^bCoeffand subtracts it from theMGFwhich is decomposed.

Then, derivatives are taken and in each step the coefficients of the holomorphic Eisenstein series are set to zero by fixing some of the

bCoeff.

If for the modular weight of theMGFno basis is implemented, the error CSieveDecomp::noBasisis issued. In general, the basis used for the decomposition is determined by the option^basisofCSieveDecomp. If^basisis an empty list (the default), the basis is determined by the function^CBasis, to be discussed in more detail in Section5.7. Otherwise, one can also supply a list of MGFsof the same weight as theMGF to be decomposed. E.g. we can reproduce the momentum conservation identity of the seedC_{1 2 2}

1 2 1

(up to an overall constant) by running

In[44]:=CSieveDecomp

c_{1 1 2}

1 2 1

, basis

c_{0 2 2}

1 1 2

, c_{1 1 2}

1 1 2

Out[44]= −_{C0 2 2}

1 1 2

−_{C1 1 2}

1 1 2

+ π^4intConst_{1 1 2}

1 2 1

τ⁴₂ .

If not all coefficients can be fixed (e.g. because the ^basis provided is not linearly independent), ^bCoeffwill appear in the output. If no decomposition could be found, the errorCSieveDecomp::noSolis issued and the derivative specified in which a holomorphic Eisenstein series could not be canceled. The calculation can then be investigated further with the^verboseoption. This can happen e.g. if the basis is not complete or if identities of lower weight for MGFs multiplying holomorphic Eisenstein series are missing. For further options and the meaning of other error messages, cf. AppendixA.

Im Dokument Modular Graph Forms and Scattering Amplitudes in String Theory (Seite 133-139)