CERN-PH-TH/2009-057[-0.2cm] LAPTH-1328/09 A New Class of \mathcal{N}=2 Topological Amplitudes

# A New Class of N=2 Topological Amplitudes

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1Department of Physics, CERN - Theory Division, CH-1211 Geneva 23, Switzerland

[0.5cm] 2Institut für Theoretische Physik, ETH Zürich, CH-8093 Zürich, Switzerland

[0.5cm] 3High Energy Section, The Abdus Salam International Center for Theoretical Physics,

[0.5cm] 4 LAPTH5, Université de Savoie, CNRS, B.P. 110, F-74941 Annecy-le-Vieux, France

[0.5cm]

We describe a new class of topological amplitudes that compute a particular class of BPS terms in the low energy effective supergravity action. Specifically they compute the coupling where , and are gauge field strengths, gaugino and holomorphic vector multiplet scalars. The novel feature of these terms is that they depend both on the vector and hypermultiplet moduli. The BPS nature of these terms implies that they satisfy a holomorphicity condition with respect to vector moduli and a harmonicity condition as well as a second order differential equation with respect to hypermultiplet moduli. We study these conditions explicitly in heterotic string theory and show that they are indeed satisfied up to anomalous boundary terms in the world-sheet moduli space. We also analyze the boundary terms in the holomorphicity and harmonicity equations at a generic point in the vector and hyper moduli space. In particular we show that the obstruction to the holomorphicity arises from the one loop threshold correction to the gauge couplings and we argue that this is due to the contribution of non-holomorphic couplings to the connected graphs via elimination of the auxiliary fields.

## 1 Intoduction

A special role in extended supersymmetric theories is played by 1/2-BPS couplings that depend only on half of the superspace, generalizing chiral supersymmetric F-terms. Usually, such interactions are easier to study because they are subject to non-renormalization theorems, while they have a variety of interesting physical applications varying from the vacuum structure all the way up to properties of supersymmetric black-holes. Moreover, in string effective field theory, these couplings are expected to be computed by topological amplitudes, depending only on the zero-mode structure of the compactification space [1, 2, 3]. An interesting property is that the half-BPS structure of these terms is broken at the quantum level. On the topological side, this breaking is due to a violation in the conservation of the BRST current described by an anomaly equation [3, 4, 5], while on the string side it is understood from the difference between the Wilsonian and ‘physical’ effective action that includes also the contribution of massless degrees of freedom [2].

The first instance of well studied 1/2-BPS couplings in supersymmetry is the series , where is the chiral (self-dual) gravitational Weyl superfield and the coefficients depend on the vector multiplet moduli in the Coulomb phase of the theory [2, 3]. ’s are computed by the genus topological partition function of an twisted -model on the six-dimensional Calabi-Yau compactification manifold of type II string theory in four dimensions, subject to a holomorphic anomaly equation that takes the form of a recursion relation. Moreover, the independence of ’s from hypermultiplets, which include the string dilaton, implies a non-renormalization theorem for their form. These results have been generalized to supersymmetric compactifications of type II string on , where two series of higher order terms were identified, computed by topological amplitudes: and , where is a superdescendent of the Weyl superfield [6, 7]. The half-BPS property leads to a harmonicity equation for the moduli dependence of the couplings [8, 9, 10], generalizing holomorphicity, up to anomalous contributions from boundary terms [10]. Despite the bigger supersymmetry, the analysis is more involved than in the case of vector multiplets, since the lack of an ordinary superspace description implies the use of on-shell harmonic superspace [11, 12, 13].

A different question is to study the corresponding couplings when one reduces the supersymmetry by half. On the string side, this can be done in two ways that are dual to each other. Either by considering the ‘semi-topological’ theory obtained by twisting the supersymmetric left-movers of the heterotic string [14, 15], or by applying a world-sheet involution on the type II amplitudes that introduces open string boundaries [16]. In the case of ’s, this generates an series of higher order F-terms of the form , where is now the gauge superfield with the gauge indices contracted in an appropriate way [14]. The holomorphic anomaly equation however does not close on ’s; it brings new objects that give rise to a double series , where denotes generically a chiral projection of a real function of chiral superfields. On the topological side, the same results are obtained upon introducing world-sheet boundaries.6

In this work, we apply the above reduction mechanism to the topological amplitudes and obtain a new series of higher order 1/2-BPS terms with supersymmetry. The novel feature of these terms is that they mix vector multiplets with neutral hypermultiplets, despite the common wisdom. Indeed, starting with , one generates the series , where is now a superdescendent of an vector superfield (with the gauge indices contracted appropriately, as before). The coupling coefficients depend in this case on both analytic vector multiplet as well as on (Grassmann analytic) hypermultiplet moduli, as dictated by the half-BPS structure. Moreover, these coupling share similar properties at the same time with the topological couplings and with the series. More precisely, the appropriate formalism for their study is again (on-shell) harmonic superspace, which complicates the analysis compared to the ’s. On the other hand, quantum corrections violate both the holomorphicity condition with respect to the vector moduli, and the harmonicity with respect to the hypermultiplets. Furthermore, the anomaly equation does not closes on ’s; it brings new objects generating the double series , where is an appropriate half-BPS projection.

The organization of the paper and the outline of the results obtained are described below. The next two sections contain the string computation of the new topological amplitudes. In Section 2, we compute the special type of topological amplitudes in type I open string theory, from the topological amplitudes , by applying a world-sheet involution.7 In fact, we evaluate a physical amplitude involving two gauge field strengths, two vector multiplet scalars (with one derivative each) and gauginos with the same four-dimensional chirality, , on a world-sheet with boundaries, and we show that it is reduced to a topological expression within the twisted -model on . Then, in Section 3, we compute the same amplitudes on the heterotic side (compactified on ), which turns out to be easier for our subsequent analysis because of the absence of the problematic Ramond-Ramond sector, exploiting heterotic – type I duality. Again, the physical amplitude is expressed as a semi-topological expression, i.e. only for the (supersymmetric) left-movers, while the bosonic part provides the gauge indices appropriately contracted (we are essentially taking products of differences of gauge groups with no charged massless matter).

These two sections are complemented by three appendices. In Appendix A, we review the main properties of the and world-sheet superconformal algebras, Appendix B contains the expressions of the three main vertex operators we use, while Appendix C contains the definitions of the theta-functions and prime forms.

The following section contain the effective field theory description of the topological amplitudes and the study of the generalized analyticity relations and anomaly equations. In Section 4, we study the interpretation of the string results, obtained in Sections 2 and 3, in the context of the effective supergravity. As mentioned above, the appropriate formulation is in terms of the harmonic superspace (for a review see [18]). We first make an analysis in global supersymmetry (subsection 4.1), introduce the harmonic variables, define the series of the effective interaction terms and derive the conditions on the moduli dependence of the couplings from their half-BPS structure. These are the usual holomorphicity with respect to the vector multiplet moduli, while the hypermultiplet moduli dependence is subject to two differential constraints, in close analogy with the equations found for the terms: the so-called harmonicity condition, expressing the property that only one combination of the four components of the hypermultiplets enter in the coupling, as well as a second-order constraint. We then study the effects of the curvature of the hypermultiplet scalar manifold (subsection 4.2), considering as an example the coset for hypermultiplets (using the harmonic description of [19]). We show in particular that the second-order differential equation is modified by an additional term linear in and proportional to a R-charge . The generalization to (conformal) supergravity is done in subsection 4.3, where the full covariantized expressions of the effective operators are obtained, as well as of the differential equations they obey.

In Section 5, we present a different derivation of the equivalence between string and topological amplitudes which is free of an ambiguity that appears in the computation we perform in Sections 2 and 3. This is achieved by evaluation of a different amplitude related by supersymmetry to the previous one, containing only fermions: two chiral and two antichiral hyperfermions, besides the gauginos. We also generalize the computation from orbifolds considered in the text, to the most general superconformal theory.

In Section 6, we verify explicitly the analyticity equations in string theory, on the heterotic side. Moreover, we evaluate the world-sheet boundary contributions for the holomorphicity and harmonicity equations that give rise to anomalous terms. In contrast to the familiar ’s and their generalizations computed by closed topological amplitudes, the anomalous terms do not generate recursion relations for the non holomorphic/harmonic dependence of the same couplings, because they involve new objects. This is similar to the case encountered in topological amplitudes, irrespectively on which string framework they are defined (heterotic or type I). The new objects involve chiral/half-BPS projections of general non-holomorphic/harmonic functions and generate a double series of higher-dimensional operators with moduli-dependent coefficients , described above. In both equations, the new quantities are proportional to the one-loop threshold corrections to the gauge couplings, on the heterotic side. We argue that the non-holomorphicity appears due to the contribution to the string amplitude (which computes the sum of all connected graphs) from via the elimination of the auxiliary fields. This section is supplemented by Appendix D, where we explicitly compute the string amplitudes generating the double series described above in a generic Calabi-Yau compactification.

## 2 Type I open topological amplitudes

In this section we will calculate a special type of topological amplitudes in type I open string theory. They are related to similar objects in the type II theory (see [6, 10, 7]) via a world-sheet involution [20, 21] which we will describe in detail first.

### 2.1 Z2 world-sheet involutions

In the type II theory, the world-sheet corresponding to a loop scattering amplitude is a compact Riemann surface of genus . This surface can be endowed with a canonical homology basis of 1-cycles , with (an example for is depicted in figure 1).

The surface can furthermore be equipped with a set of holomorphic 1-differentials , whose integrals over the homology cycles is given by

 ∫aiωj=δij, and ∫biωj=τij. (2.1)

Here the symmetric matrix is called the period matrix and it encodes all the information about the shape and size of the surface .

By viewing as a double cover we can construct an open surface by taking the quotient with respect to some involution which we will denote in the following. acts linearly on the homology cycles and we will focus on the special case

 I∗ai=Γijaj, and I∗bi=−Γijbj . (2.2)

Here is a matrix that enjoys the following properties

 Γ2=11, and detΓ=±1. (2.3)

The action of on the -differentials reads

 I∗ωi=Γij¯ωj, (2.4)

and the period matrix has to satisfy

 τ=−ΓT¯τΓ. (2.5)

The quotient constructed from the prescription (2.2) is an open Riemann surface and the boundaries are given by the fixed points of . The case which will be most important for us in the following is to choose in such a way to create as large a number of boundaries as possible (see also [16]), which is obviously

 Γ=11. (2.6)

In this way the boundaries are given by (combinations of) the -cycles of the original Riemann surface (Returning to the genus example the involution then acts as displayed in figure 2, creating a surface with 4 boundaries).

In order to calculate (open)string correlation functions on the quotient we still need to specify boundary conditions which are necessarily the same for all boundary components. In this work we will focus on the simplest case and choose either Dirichlet or Neumann conditions, in which case the open correlators on are the square root of the closed correlators on up to the following multiplicative correction factor

 (2.7)

### 2.2 Involution of N=4 topological amplitudes

The techniques described in the previous subsection have been used in [16] to compute open topological amplitudes in type I string theory as involutions of the familiar topological amplitudes (see [2, 3]). In this work, we wish to generalize the computations of [16] and apply the method of world-sheet involutions to the topological amplitudes of type II string theory on studied in [6, 10] (see also [7]). We hope to find open topological amplitudes in type I string theory preserving space-time supersymmetry.

To be more precise, there are two different types of topological amplitudes in the theory. In this work we will focus exclusively on the involution of one of them, which was called in [6].8 We recall that was shown in [6] to be computed by a -loop type II string amplitude with two graviton, two graviscalar and graviphoton insertions. Starting again from the corresponding genus (closed) world-sheet , the quotient has boundaries. In the process of calculating this involution we will consider the special case that none of the vertex operators is inserted in the bulk but all of them will be pairwise distributed over the boundary components.9 Moreover, we will choose Neumann boundary conditions for all the components. In this way we will compute a type I correlator of two gauge fields, two boundary scalars and gauginos. We will consider a torus-orbifold realization of in which case we can use a free-field representation for all vertex operators. The precise helicity combinations can then be displayed in the following table (the in the last five columns denote the charges in the various planes)

 field pos. number ϕ1 ϕ2 ϕ3 ϕ4 ϕ5 gauge field z1 1 +1 +1 0 0 0 z2 1 −1 −1 0 0 0 scalar z3 1 0 +1 +1 0 0 z4 1 0 −1 +1 0 0 gaugino xi g−2 +12 +12 +12 +12 +12 yi g−2 −12 −12 +12 +12 +12 PCO {s3} g 0 0 −1 0 0 {s4} g−2 0 0 0 −1 0 {s5} g−2 0 0 0 0 −1

Here it is understood that the PCO are also inserted on the boundary. This amplitude is precisely equal to the left-moving contribution of the corresponding type II correlator, which has already been computed in [6]. Therefore, instead of repeating the calculation again we will only state the final result

 Fopeng= ⟨∏{s3}a¯ψ3∂X3(ra)ψ3(α)⟩⋅⟨∏{s4}a¯ψ4∂X4(ra)¯ψ4(z4)⟩⋅⟨∏{s5}a¯ψ5∂X5(ra)¯ψ5(z4)⟩⟨∏3g′−4a=1b(ra)b(z4)⟩⋅ ⋅detωi(xi,z1,z3)detωi(yi,z2,z4) . (2.8)

Here is an arbitrary position on the boundaries of the world-sheet. The correlator in the denominator stems from the -ghost system while the correlators of the numerator involve the free fermions and bosons living on and their respective counterparts and coming from . Notice that no factors of appear in this expression since they have all been cancelled by the correction factor (2.7) for Neumann boundary conditions. At this stage we can use the free-field representation of the and super-conformal algebra (see Appendix A) to write

 Fopeng=detωi(xi,z1,z3)detωi(yi,z2,z4)A(ra,z4), (2.9)

where we have introduced the following shorthand notation

 A(ra,z4)=⟨∏{s3}aG−T2(ra)ψ3(α)⟩⋅⟨∏{s4,s5}aG−K3(ra)J−−K3(z4)⟩⟨∏3g−4a=1b(ra)b(z4)⟩. (2.10)

Notice that and are the supercurrents and the Kac-Moody current in the twisted internal theory and therefore are dimension 2 operators. Thus is a meromorphic scalar function of all its arguments. As two approach each other both the numerator and the denominator have a first order zero and hence has no singularity in this limit. However, when approaches any of the set the numerator is finite and non-zero (as the first correlator is independent of ), while the denominator vanishes. This means that has a pole as approaches any of the of the set . On the other hand this singularity should not be present as picture changing operators sitting at must have no singularity with the physical vertex operator at . The reason for this apparent singularity is that we have not included the full physical vertex operator at (as well as at , and ). We have only considered the fermion bilinear part of the physical vertex operators in the zero ghost picture that comes with one power of momenta. However, the full vertex operators also include at and and at and (see Appendix B). If we include all these extra terms the apparent singularity in as approaches any of the must disappear. This subtle point was overlooked in [6]. It is however very difficult to include all the possible terms, indeed it is not clear if we can choose a gauge condition for the positions of the PCO’s such that the superghost function cancels with one of the space-time functions simultaneously for all these different terms, which could enable us to do the spin structure sum explicitly. This is a problem in the RNS formulation that we are using here.

In Section 5, we show that one can compute another amplitude in the RNS formulation, involving gauginos and two chiral and two anti-chiral hyperfermions. From the discussion of the effective field theory in Section 4 it will become clear that this new amplitude is supersymmetrically related to the one considered here, more precisely the new amplitude is given by four derivatives of the latter with respect to hypermultiplet moduli. Moreover, it turns out that in this new amplitude, all the PCO’s contribute only the supercurrents of the internal theory, which will allow us to compute it explicitly in Section 5 and proof that it is indeed given by four derivatives of the topological expression given below in eq. (2.11). It will be interesting to see if the amplitude considered in the current section can be directly calculated in some other formalism, such as pure spinor formalism. In the following we assume that including all the other terms in the vertex operators amounts to antisymmetrizing with all the in the numerator of eq.(2.10).

Next we remember that this expression is still to be multiplied by Beltrami differentials folded with the -ghosts to provide the correct measure for the integration over the moduli space of a Riemann surface with boundaries and no handles. However, with our above assumption it is possible to transmute all and inside the expression of (2.10) to the positions of these ghost fields folded with the Beltrami differentials. Put differently, we can write

 A(ra,z4)(μ⋅b)3g−3=(μ⋅G−T2)g(μ⋅G−K3)2g−4(μ⋅J−−K3)ψ3(α), (2.11)

where with being some local coordinates in the moduli space and the corresponding Beltrami differentials. Besides that we have introduced the following notation

 (μ⋅G−)=3g−3∑a=1dma∫Σ(0,g+1)μaG−, (2.12)

which is therefore a one form in . The final step is the integration of the insertion points , , over the boundaries. This can be performed explicitly and yields, given the fact that the boundary components are just -cycles on which the are normalized, just a numerical factor, which we drop since it will be of no interest to us. Therefore we can write for the final amplitude

 Fopeng=∫M(0,g+1)⟨(μ⋅G−T2)g(μ⋅G−K3)2g−4(μ⋅J−−K3)ψ3(α)⟩ . (2.13)

Notice that this expression is purely topological holding only information about the number of boundaries of the world-sheet. We have therefore succeeded in linking the physical amplitude (2.8) to a topological theory.

## 3 Topological amplitudes in heterotic orbifold compactifications

In Section 2 we have been considering topological amplitudes in the type I theory. However, in order not having to deal with the problem of open string moduli (see e.g. [17]), we rather prefer to transfer the problem to a dual setup, in which the topological amplitudes again compute closed string correlators. One possibility is to exploit the duality between type I and heterotic string theory. Since this duality is perturbative in nature we expect to recover of the type I theory at the (closed) loop level in the heterotic theory. Since in the heterotic theory the bosonic right moving sector needs a slightly different treatment than the supersymmetric left moving sector, we will present the computation of this amplitude in somewhat more detail.

The field insertions we consider for the heterotic -loop correlator are two gauge fields, two scalar fields and gauginos. The helicity setup we use is identical to the type I setup, and therefore, upon using the vertex operators of Appendix B, the amplitude, which we have to compute is given by

 Fg= =⟨g−2∏i=1e−φ2ei2(ϕ1+ϕ2+ϕ3+ϕ4+ϕ5)(xi)g−2∏i=1e−φ2ei2(−ϕ1−ϕ2+ϕ3+ϕ4+ϕ5)(yi)ei(ϕ1+ϕ2)(z1)e−i(ϕ1+ϕ2)(z2)⋅ ⋅ei(ϕ2+ϕ3)(z3)e−i(ϕ2−ϕ3)(z4){s3}∏aeφe−iϕ3∂X3(ra){s4}∏aeφe−iϕ4∂X4(ra){s5}∏aeφe−iϕ5∂X5(ra)⟩⋅ ⋅⟨∏i¯JIi(¯xi)¯JKi(¯yi)¯JIg−1(¯z1)¯JKg−1(¯z2)¯JIg(¯z3)¯JKg(¯z4)⟩. (3.1)

The right moving contribution consists just of a correlator of currents, where the subscripts and with label the vector multiplets. These subscripts are also implicitly present on , but we shall not write them explicitly. We will leave this right moving correlator for the moment as it is and stick to the left moving part. Here we can perform the contractions of the various fields to obtain

 Fg= =ϑs(12∑i(xi−yi)+z1−z2)ϑs(12∑i(xi−yi)+z1−z2+z3−z4)ϑs(12∑i(xi+yi)−∑3g−4ara+2Δ)∏3g−4a

At this stage we can use the gauge-fixing condition

 12∑i(xi−yi)+z1−z2=12∑i(xi+yi)−3g−4∑ara+2Δ, ⇒3g−4∑ara=∑iyi−z1+z2+2Δ, (3.2)

which reduces the relevant part for the spin-structure sum to

 ϑs(12∑i(xi−yi)+z1−z2+z3−z4)ϑs⎛⎝12∑i(xi+yi)+z3+z4−{s3}∑ara⎞⎠⋅ ⋅5∏I=4ϑs,hI⎛⎝12∑i(xi+yi)−{sI}∑ara⎞⎠.

 ∙ ++++: ∑ixi+12∑iyi+z3+12(z1−z2)−123g−4∑ara= =∑ixi+z1+z3−z2−Δ, ∙ −−++: 12∑iyi−z3−12(z1−z2)+12{s3}∑ara−12{s4,s5}∑ara={s3}∑ara−z3−Δ, ∙ −+−+: 12∑iyi+z4−12(z1−z2)+12{s4}∑ara−12{s3,s5}∑ara={s4}∑ara+z4−Δ, ∙ −++−: 12∑iyi+z4−12(z1−z2)+12{s5}∑ara−12{s3,s4}∑ara={s5}∑ara+z4−Δ.

Multiplying the amplitude furthermore by

 1=ϑ(∑iyi+z2+z4−z1−Δ)ϑ(∑3g−4ara+z4−3Δ), (3.3)

it takes the form

 Fg=ϑ(∑ixi+z1+z3−z2−Δ)ϑ(∑iyi+z2+z4−z1−Δ)ϑ(∑{s3}ara−z3−Δ)ϑ(∑3g−4ara+z4−3Δ)∏3g−4a

Here we use bosonization identities [22] in the following way

 ϑ(∑ixi+z1+z3−z2−Δ)∏i

and write the remaining expression in terms of correlators of the internal theory to get

 Fg= detωi(xi,z1,z3)detωi(yi,z2,z4)⟨∏{s3}a¯ψ3∂X3(ra)ψ3(x)⟩⟨∏{s4}a¯ψ4∂X4(ra)¯ψ4(z4)⟩(det(Imτ))2⟨∏3g−4ab(ra)b(z4)⟩⋅ ⋅⟨{s5}∏a¯ψ5∂X5(ra)¯ψ5(z4)⟩⋅⟨∏i¯JIi(¯xi)¯JKi(¯yi)¯JIg−1(¯z1)¯JKg−1(¯z2)¯JIg(¯z3)¯JKg(¯z4)⟩.

Here we have split off which has dimension zero and was originally at and moved it to some arbitrary point since only provides a constant zero mode. Including all the possible distributions of the positions in the sets , and we find:

 Fg= detωi(xi,z1,z3)detωi(yi,z2,z4)(det(Imτ))2⋅A(ra,z4)⋅ ⋅⟨∏i¯JIi(¯xi)¯JKi(¯yi)¯JIg−1(¯z1)¯JKg−1(¯z2)¯JIg(¯z3)¯JKg(¯z4)⟩, (3.6)

where

 A(ra,z4)=⟨∏3g−4a=1G−(ra)J−−K3(z4)ψ3(x)⟩⟨∏3g−4a=1b(ra)b(z4)⟩. (3.7)

Here are the supercurrents and the current in the twisted internal theory and therefore are dimension 2 operators. Similar to the expression of (2.10) also here in the heterotic case is a meromorphic scalar function of all its arguments which develops a pole when approaches any of the . In this case the numerator is finite and non-zero when the corresponding contributes the torus part while, however, the denominator vanishes. This problem is again due to the fact that we have not included all the other possible terms in vertex operators. However, this can be resolved in precisely the same manner as in the type I case (see also Section 5). As in the open string case we assume that including all the other terms in the vertex operators amounts to an antisymmetrization of with all the in the numerator of the above equation which then cancels the zero coming from the denominator as approaches any of the .

The remainder of the argument also follows similarly to the computation in Section 2.2. We note that as a function of any or both the numerator as well as the denominator in are sections of the line bundle of quadratic differentials and have no poles or zeroes at the remaining points. Both the numerator and denominator must have additional zeroes as the degree of the divisor class of quadratic differentials is but by the Abel map generically these g additional zeroes are uniquely fixed. This implies that