Poisson equation for genus two string invariants: a conjecture
aa r X i v : . [ h e p - t h ] J a n Poisson equation for genus two string invariants: aconjecture
Anirban Basu Harish–Chandra Research Institute, HBNI, Chhatnag Road, Jhusi,Prayagraj 211019, India
Abstract
We consider some string invariants at genus two that appear in the analysis ofthe D R and D R interactions in type II string theory. We conjecture a Poissonequation involving them and the Kawazumi–Zhang invariant based on their asymp-totic expansions around the non–separating node in the moduli space of genus twoRiemann surfaces. email address: [email protected] Introduction
String invariants or modular graph forms [1,2] arise as integrands in the integrals over mod-uli space of Riemann surfaces resulting from the analysis of local interactions in the lowmomentum expansion of amplitudes in string theory. Hence understanding their detailedproperties is central to obtaining the coefficients of various interactions in the effective ac-tion. These graphs have links given by the Green function or their worldsheet derivatives,while the vertices are the positions of insertions of vertex operators on the punctured world-sheet. Obtaining eigenvalue equations satisfied by these graphs is very useful in performingthe integrals over moduli space, along with a knowledge of their asymptotic expansionsaround various degenerating nodes of the worldsheet. This has yielded various results atgenus one [1–20] and two [21–37] leading to an intricate underlying structure.The modular graph functions that arise at genus one in type II string theory are SL (2 , Z ) τ invariant functions of the complex structure τ of the torus. These graphs satisfyPoisson equations which have been derived in various cases. It is interesting to analyzethe structure of these equations simply based on the Laurent expansion of these graphsaround the cusp Im τ → ∞ . This expansion has several power behaved terms apart fromterms that are exponentially suppressed. Now simply based on the structure of the powerbehaved terms for a given graph, one can try to guess the Poisson equation it satisfies, aswas originally done in [1] hence providing a powerful tool to search for potential eigenvalueequations.The analysis of obtaining eigenvalue equations for string invariants gets considerablymore involved at genus two. The Sp (4 , Z ) invariant graph with one link that arises as theintegrand over moduli space of the D R and the D R interactions in the low momentumexpansion of the four and five graviton amplitudes respectively [21,31,32], is the Kawazumi–Zhang (KZ) invariant [25, 38, 39] which satisfies a Poisson equation on moduli space, whichhas enabled the calculation of the coefficients of the D R and D R interactions in theeffective action [26]. What about such Poisson equations for graphs with more than onelink? Unlike the analysis at genus one, there are no such known equations , though theasymptotic expansion of various graphs with more than one link around the degeneratingnodes of the genus two Riemann surface have been analyzed in detail [28, 29]. Thus inanalogy with the analysis involving their genus one counterparts, it is natural to ask ifthese asymptotic expansions can be used to guess any eigenvalue equation satisfied bygenus two string invariants.Based on the asymptotic expansions around the non–separating node of the genus twoRiemann surface of several graphs that arise in the analysis of the D R and the D R interactions, we shall argue that there is a candidate Poisson equation that arises naturallyinvolving these graphs as well as the KZ invariant, which we conjecture to be true over allof moduli space. It will be interesting to check this claim along the lines of [26, 30].After briefly reviewing relevant details about genus two Riemann surfaces and in partic-ular the non–separating node, we shall demonstrate how the Poisson equation satisfied by Differential equations involving four and six derivatives on moduli space have been obtained in [37].However, the primary equation leading to these results involves two derivatives.
We denote the genus two worldsheet by Σ , and the conformally invariant Arakelov Greenfunction by G ( z, w ). The imaginary part of the period matrix is defined by ImΩ = Y . Wedefine the inverse matrix Y − IJ = ( Y − ) IJ ( I, J = 1 , z, w ) = Y − IJ ω I ( z ) ω J ( w ) , µ ( z ) = ( z, z ) , P ( z, w ) = ( z, w )( w, z ) (2.1)which we often use, where ω I = ω I ( z ) dz is the Abelian differential one form. The integrationmeasure over the worldsheet is given by d z = idz ∧ dz .The asymptotic expansions of the various graphs around the non–separating node shallplay a central role in our analysis, which we briefly review. Parametrizing the period matrixΩ as Ω = (cid:18) τ vv σ (cid:19) , (2.2)the non–separating node is obtained by taking σ → i ∞ , while keeping τ, v fixed . Atthis node, an SL (2 , Z ) τ subgroup of Sp (4 , Z ) survives whose action on v, τ and σ is givenby [28, 29, 40] v → v ( cτ + d ) , τ → aτ + bcτ + d , σ → σ − cv cτ + d , (2.3)where a, b, c, d ∈ Z and ad − bc = 1. Here v parametrizes the coordinate on the torus withcomplex structure τ , and hence − ≤ v ≤ , ≤ v ≤ τ . (2.4)Also σ along with v and τ forms the SL (2 , Z ) τ invariant quantity t = σ − v τ (2.5)which is a useful parameter in the asymptotic expansion.We shall also make use of the SL (2 , Z ) τ invariant operators∆ τ = 4 τ ∂ τ ∂ τ , ∆ v = 4 τ ∂ v ∂ v (2.6)in our analysis. The other contribution coming from taking τ → i ∞ , while keeping σ, v fixed, is simply related to thisby τ ↔ σ exchange, and hence we consider only one of them.
2n order to motivate the Poisson equation, we shall use the expression for the Sp (4 , Z )invariant Laplacian ∆ expanded around the non–separating node. This is given by [27]∆ = t ∂ ∂t − t ∂∂t + t v + ∆ τ . (2.7)We now list various SL (2 , Z ) τ covariant expressions on the toroidal worldsheet Σ withcomplex structure τ and coordinate v that will be relevant for our purposes. They arise inthe analysis of the genus two string invariants in their asymptotic expansions around thenon–separating node, where the worldsheet is given by Σ with two additional punctures(beyond those from the vertex operators) connected by a long, thin handle whose properlength is proportional to t , hence providing a physical interpretation for this parameter.The Green function g ( v ) at genus one is given by g ( v ) ≡ g ( v ; τ ) = X ( m,n ) =(0 , τ π | m + nτ | e πi ( my − nx ) , (2.8)where we have parametrized v as v = x + τ y, (2.9)with x, y ∈ (0 , τ g ( v ) = 0 , ∆ v g ( v ) = − πτ δ ( v ) + 4 π, (2.10)where the delta function is normalized to satisfy R Σ d zδ ( z ) = 2 (note that d z = idz ∧ dz on Σ).The iterated Green function g k +1 ( v ) is defined recursively by g k +1 ( v ) ≡ g k +1 ( v ; τ ) = Z Σ d z τ g ( v − z ) g k ( z ) (2.11)for k ≥
1, where g ( v ) = g ( v ) by definition. The non–holomorphic Eisenstein series isdefined by E k +1 ≡ E k +1 ( τ ) = g k +1 (0) (2.12)for k ≥
1. They satisfy the differential equations∆ τ g k +1 ( v ) = k ( k + 1) g k +1 ( v ) , ∆ τ E k +1 = k ( k + 1) E k +1 , ∆ v g k +1 ( v ) = − πg k ( v ) . (2.13)We next consider the family of elliptic modular graph functions [29] defined by F k ( v ) = 1(2 k )! Z Σ d z τ f ( z ) k , (2.14)where k ≥ f ( z ) = g ( v − z ) − g ( z ).3rom the definition and using the results mentioned above, we get that F ( v ) = E − g ( v ) (2.15)which satisfies the equations∆ τ F ( v ) = 2 F ( v ) , ∆ v F ( v ) = 4 πg ( v ) . (2.16)Thus we see that both ∆ τ and ∆ v acting on F ( v ) yield elementary SL (2 , Z ) τ invariantquantities.The expression for F ( v ) which directly follows from (2 .
14) is not relevant for our pur-poses. However, an analysis of this expression yields the Poisson equation [32, 33, 36] (cid:16) ∆ τ − (cid:17)(cid:16) F ( v ) − F ( v ) (cid:17) = − F ( v ) (2.17)which shall be useful. Thus so far as terms involving derivatives on moduli space areconcerned, ∆ τ F ( v ) can be expressed in terms of ∆ τ F ( v ) . Also note that ∆ v F ( v ) doesnot yield anything particularly useful.These relations mentioned above will be repeatedly used in our analysis below.Figure 1: The string invariant ϕ Let us now consider the Kawazumi–Zhang (KZ) invariant which arises in the analysisof the D R and the D R interactions at genus two. This string invariant is defined by ϕ (Ω , Ω) = 14 Z Σ Y i =1 d z i G ( z , z ) P ( z , z ) (2.18)as depicted by figure 1. Its asymptotic expansion around the non–separating node is givenby [25, 27–29, 41, 42] ϕ = πt g ( v )2 + 5 F ( v )4 πt + O ( e − πt ) , (2.19)which has only a finite number of power behaved terms in t while the remaining terms areexponentially suppressed. Thus from the expression for the Laplacian in (2 . (cid:16) ∆ − (cid:17) ϕ = − π det Y δ ( v ) (2.20)up to exponentially suppressed contributions. One might guess that (2 .
20) is the exactPoisson equation satisfied by the KZ invariant over all of moduli space and not just in4n asymptotic expansion around the non–separating node keeping only the power behavedterms in t , which in fact turns out to be correct [26] . Thus we see that the asymptoticexpansion provides a good starting point for guessing the exact eigenvalue equation.As an aside, note that the asymptotic expansion of the various string invariants aroundthe separating node where v → τ, σ finite in (2 .
2) is a Taylor series in ln | λ | with afinite number of terms, along with an infinite number of corrections that are potentially ofthe form | λ | m (ln | λ | ) n for m > n ≥ λ = 2 πvη ( τ ) η ( σ ) . (2.21)Hence this asymptotic expansion is not particularly useful in trying to derive any Poissonequations as the action of the Laplacian mixes the various contributions.On the other hand, for the purposes of trying to guess Poisson equations satisfied bythe string invariants, their asymptotic expansions around the non–separating node are veryuseful. This is because such expansions involve only a finite number of terms in the Laurentseries expanded around t → ∞ , along with exponentially suppressed contributions [29].Thus acting with the Laplacian in (2 .
7) on the asymptotic series, we can simply focus onthe finite number of power behaved terms, as there is no mixing with the exponentiallysuppressed terms.
Based on the discussion above, we now analyze string invariants with more than one link,with the aim of trying to motivate a Poisson equation satisfied by them. We first considerthe graphs that arise in the integrand over moduli space of the D R interaction in the lowmomentum expansion of the four graviton amplitude, each of which has two links. Theirasymptotic expansions around the non–separating node are given in [29] up to exponentiallysuppressed contributions. One of these graphs, denoted Z , forms a closed loop on theworldsheet and its asymptotic expansion is significantly more complicated than the others,and we do not consider it.We consider the graphs Z and Z defined by Z (Ω , Ω) = − Z Σ Y i =1 d z i G ( z , z ) G ( z , z ) µ ( z ) P ( z , z ) , Z (Ω , Ω) = 18 Z Σ Y i =1 d z i G ( z , z ) G ( z , z ) P ( z , z ) P ( z , z ) (3.1)as depicted by figure 2. The asymptotic expansion around the non–separating node of Z In fact, the complete asymptotic expansion of ϕ , including the exponentially suppressed contributionsin (2 .
19) have been obtained in [27] by directly solving (2 . i) (ii) (iii) δ δ Figure 2: The string invariants (i) Z , (ii) Z and (iii) Z is given by Z = − π t − πt g ( v ) − E − g ( v ) F ( v )2+ 1 πt h − (cid:16) E − g ( v ) (cid:17) + 12 g ( v ) F ( v ) − π ∆ v (cid:16) F ( v ) + 2 F ( v ) (cid:17)i − π t h ∆ τ + 5 i F ( v ) + O ( e − πt ) (3.2)where we have kept all the power behaved terms. In (3 .
2) we have used the relation [1,9,11] D = E + ζ (3) (3.3)between modular graphs to rewrite the expression in [29], where the dihedral graph D isdefined by D ≡ D ( τ ) = Z Σ d z τ g ( z ) . (3.4)On the other hand, the asymptotic expansion of Z is given by Z = π t
18 + πt g ( v ) + g ( v ) F ( v )6 + 1 πt h − g ( v ) F ( v ) + 18 π ∆ v F ( v ) i + 18 π t h ∆ τ + 5 i F ( v ) + O ( e − πt ) . (3.5)Now our aim is to obtain a Poisson equation involving the minimal number of graphs, whichis the simplest possible setting. From (3 .
2) and (3 .
5) we see that apart from differencesinvolving various other contributions, Z contains F ( v ) in its asymptotic expansion while Z does not . Thus we immediately see that any Poisson equation (in fact, any equation)involving these two graphs must at least involve another graph which has F ( v ) in its In fact, Z is completely determined by the KZ invariant [37] Z which arises in theanalysis of the D R interaction in the low momentum expansion of the five gravitonamplitude [32]. It is defined by Z (Ω , Ω) = − π Z Σ Y i =1 d z i G ( z , z ) ∂ z G ( z , z ) ∂ z G ( z , z )( z , z )( z , z )( z , z ) (3.6)as depicted by figure 2. Thus it has three links, where two of them are given by theworldsheet (anti)holomorphic derivatives of the Green function (depicted by δ and δ infigure 2). Its asymptotic expansion around the non–separating node is given by Z = 32 πt h − (cid:16) D − D (1)3 ( v ) (cid:17) + 24 g ( v ) F ( v ) + 32 (cid:16) E − g ( v ) (cid:17) − π ∆ v F ( v ) i + 32 π t h F ( v ) − (cid:16) ∆ τ − (cid:17) F ( v ) i + O ( e − πt ) , (3.7)where we have used (2 .
17) to write the expression differently compared to [32]. Also, in(3 .
7) we have defined [29] the elliptic modular graph D (1)3 ( v ) ≡ D (1)3 ( v ; τ ) = Z Σ d z τ g ( z − v ) g ( z ) (3.8)depicted by figure 3. Thus note that D (1)3 (0) = D . v Figure 3: The elliptic modular graph D (1)3 ( v )Now in both the asymptotic expansions of Z and Z given by (3 .
5) and (3 .
7) respec-tively, the only terms involving derivatives on moduli space involve ∆ v F ( v ) and ∆ τ F ( v ) .In fact, the total contribution in either case is proportional to∆ v F ( v ) π t + ∆ τ F ( v ) π t , (3.9)and hence the combination Z + 12 Z (3.10)contains no terms with derivatives on moduli space. Furthermore, the O ( t ) and O ( t )contributions in (3 .
10) matches those in the expansion of 24 ϕ using (2 . Z + 12 Z − ϕ = − F ( v ) + 12 πt h − (cid:16) D − D (1)3 ( v ) (cid:17) + 4 (cid:16) E − g ( v ) (cid:17)i + 3 π t h F ( v ) − F ( v ) i + O ( e − πt ) . (3.11)Let us compare the terms involving F ( v ) in (3 .
2) and (3 . .
11) has only F ( v ) in its asymptotic expansion, (3 . F ( v ), ∆ v F ( v ) and ∆ τ F ( v ). Thus it is natural to ask if the action of the Sp (4 , Z )invariant Laplacian on the combination of graphs Z + 12 Z − ϕ might be related inany way to Z . Hence let us consider the action of (2 .
7) on (3 . D (1)3 ( v ) in (3 . . v on D (1)3 ( v ) is given by∆ v D (1)3 ( v ) = 4 π (cid:16) E − g ( v ) (cid:17) , (3.12)the action of ∆ τ on D (1)3 ( v ) is given by∆ τ D (1)3 ( v ) = 2 E + 4 g ( v ) + 2 g ( v ) F ( v ) − π ∆ v F ( v ) . (3.13)We present the derivation of (3 .
13) in the appendix.Putting together the various contributions, we obtain the asymptotic expansion∆ (cid:16) Z + 12 Z − ϕ (cid:17) = − πtg ( v ) − F ( v ) + 24 E + 96 g ( v ) − g ( v ) + 3 πt h (cid:16) E − g ( v ) (cid:17) − (cid:16) D − D (1)3 ( v ) (cid:17) + 8 g ( v ) F ( v ) + 1 π ∆ v (cid:16) F ( v ) − F ( v ) (cid:17)i + 3 π t (cid:16) ∆ τ + 8 (cid:17)(cid:16) F ( v ) − F ( v ) (cid:17) + O ( e − πt ) (3.14)around the non–separating node. Importantly, the right hand side of (3 .
14) contains∆ v F ( v ) which also appears in (3 . .
5) or (3 . . Z such that this contribution cancels. This leads to the asymptotic expansion∆ (cid:16) Z + 12 Z − ϕ (cid:17) + 120 Z = − π t − πtg ( v ) − F ( v ) − g ( v ) + 3 πt h g ( v ) F ( v ) − (cid:16) D − D (1)3 ( v ) (cid:17) + 48 (cid:16) E − g ( v ) (cid:17) − π ∆ v F ( v ) i + 3 π t h F ( v ) − (cid:16) ∆ τ + 8 (cid:17) F ( v ) i + O ( e − πt ) (3.15)around the non–separating node. Now subtracting 3 Z from (3 .
15) and using (3 . O (1 /t ) and O (1 /t ) contributions, giving us the asymptotic8xpansion∆ (cid:16) Z + 12 Z − ϕ (cid:17) + 120 Z − Z = − π t − πtg ( v ) − F ( v ) − g ( v ) − πt h g ( v ) F ( v ) + 112 π ∆ v F ( v ) i − π t h F ( v ) + 112 ∆ τ F ( v ) i + O ( e − πt ) . (3.16)Now the terms involving ∆ v F ( v ) and ∆ τ F ( v ) in (3 .
16) are precisely proportional to(3 .
9) and can be accounted for by the asymptotic expansion of 132 Z , leading to∆ (cid:16) Z + 12 Z − ϕ (cid:17) + 120 Z − Z + 132 Z = − π t − πtg ( v ) − g ( v ) − F ( v ) − πt g ( v ) F ( v ) − π t F ( v ) + O ( e − πt ) . (3.17)Strikingly, the right hand side of (3 .
17) is the asymptotic expansion of − ϕ around thenon–separating node on using (2 . (cid:16) ∆ − (cid:17)(cid:16) Z + 12 Z − ϕ (cid:17) = − (cid:16) Z + 7 Z (cid:17) . (3.18)Though we have deduced (3 .
18) by an analysis of the asymptotic expansions of the var-ious string invariants only around the non–separating node where we have neglected theexponentially suppressed contributions, the manner in which various simplifications occurleading to a compact expression leads us to conjecture that the Poisson equation (3 .
18) issatisfied all over the moduli space of genus two Riemann surfaces. Apart from proving ordisproving this statement, it will be interesting to try to obtain more Poisson equationsinvolving other string invariants by an analysis of their asymptotic expansions around thenon–separating node.
A The expression for ∆ τ D (1)3 ( v ) In order to calculate the action of ∆ τ on the elliptic modular graph D (1)3 ( v ), it is veryuseful to perform the analysis by varying the complex structure of the torus to obtain theeigenvalue equation [7, 10, 11, 14, 16]. The relevant variations involving the Green functionare given by [1, 7, 43, 44] ∂ µ g ( z − z ) = − π Z Σ d z∂ z g ( z − z ) ∂ z g ( z − z ) , (A.1)and ∂ µ ∂ µ g ( z − z ) = 0 , (A.2)which follows from analyzing variations with Beltrami differential µ .Since the Laplacian ∆ τ is given in terms of these variations by∆ τ = ∂ µ ∂ µ (A.3)9his enables us to calculate the action of ∆ τ on D (1)3 ( v ).Thus using ( A.
2) and ( A. τ D (1)3 ( v ) = Z Σ d z τ g ( z − v ) ∂ µ g ( z ) ∂ µ g ( z ) + h Z Σ d z τ ∂ µ g ( z − v ) g ( z ) ∂ µ g ( z ) + c.c. i , (A.4)which we now evaluate using ( A. . ∂ w ∂ z g ( z − w ) = πδ ( z − w ) − πτ ,∂ z ∂ z g ( z − w ) = − πδ ( z − w ) + πτ . (A.5)This leads to ∆ τ D (1)3 ( v ) = 2 E + 4 g ( v ) − τ π ∂ v g ( v ) ∂ v g ( v ) . (A.6)Rewriting the last term in ( A.
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