One loop tadpole in heterotic string field theory

Journal of High Energy Physics, Nov 2017

We compute the off-shell 1-loop tadpole amplitude in heterotic string field theory. With a special choice of cubic vertex, we show that this amplitude can be computed exactly. We obtain explicit and elementary expressions for the Feynman graph decomposition of the moduli space, the local coordinate map at the puncture as a function of the modulus, and the b-ghost insertions needed for the integration measure. Recently developed homotopy algebra methods provide a consistent configuration of picture changing operators. We discuss the consequences of spurious poles for the choice of picture changing operators.

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One loop tadpole in heterotic string field theory

HJE One loop tadpole in heterotic string eld theory Theodore Erler 0 1 Sebastian Konopka 0 2 Ivo Sachs 0 2 0 Theresienstrasse 37 , 80333 Munich , Germany 1 Institute of Physics of the ASCR 2 Arnold Sommerfeld Center, Ludwig-Maximilians University We compute the o -shell 1-loop tadpole amplitude in heterotic string eld theory. With a special choice of cubic vertex, we show that this amplitude can be computed exactly. We obtain explicit and elementary expressions for the Feynman graph decomposition of the moduli space, the local coordinate map at the puncture as a function of the modulus, and the b-ghost insertions needed for the integration measure. Recently developed homotopy algebra methods provide a consistent con guration of picture changing operators. We discuss the consequences of spurious poles for the choice of picture changing operators. String Field Theory; Superstrings and Heterotic Strings - 1 Introduction 2 3 2.1 2.2 3.1 3.2 3.3 4.1 4.2 4.3 4.4 5 Concluding remarks 1 Introduction Quantum closed string eld theories Closed bosonic string eld theory Heterotic string eld theory One loop tadpole in closed bosonic string eld theory General construction of cubic and tadpole vertex Case I: local PCO insertions Spurious poles Case II: PCO contours around punctures In this paper we compute the o -shell, one-loop tadpole amplitude in heterotic string eld theory. The purpose is twofold: (1) First, we would like to show that the amplitude can be computed exactly. Our success in this regard is largely due to a nonstandard choice of cubic vertex de ned by SL(2; C) local coordinate maps.1 We will take special care to provide explicit results concerning the Feynman-graph decomposition of the moduli space, the local coordinate map as a function of the modulus, and the b-ghost insertions needed for the integration measure. Actually, these data are primarily associated with closed bosonic string eld theory, but they also represent the most signi cant obstacle to explicit results for the heterotic string. Importantly, our string vertices di er from the canonical ones de ned by minimal area metrics [1]. The minimal area vertices are cumbersome for elementary calculations, though some analytic results are available 1We thank A. Sen for suggesting the utility of SL(2; C) maps. The cubic vertex we use was also discussed in [2{4]. In particular [3] describes the propagator contribution to the tadpole amplitude in closed bosonic string eld theory using this cubic vertex. We thank B. Zwiebach for pointing us to these references. { 1 { at tree level up to 4 points [5] and numerical calculations have been performed up to 5 points [6, 7].2 (2) Second, we would like to see how recent homotopy-algebraic constructions of classical superstring eld theories [9{15] may be extended to the quantum level. A signi cant issue is the appearance of spurious poles in correlation functions at higher genus [16]. We nd that the general methodology behind tree level constructions still functions in loops. Spurious poles are mainly important for the choice of picture changing operators (PCOs), which must ensure that loop vertices are nite and unambiguously de ned. We consider two approaches to inserting PCOs. In the rst approach, PCOs appear at speci c points in the surfaces de ning the vertices, in a manner similar to [17]. In the second approach, PCOs appear as contour integrals around the punctures, paralleling the construction of classical superstring eld theories. The second approach has a somewhat di erent nature in loops, however, due to the necessity of specifying the PCO contours precisely in relation spurious poles. A naive treatment can lead to inconsistencies. In this paper we are not concerned with computing the 1-loop tadpole in a speci c background or for any particular on- or o -shell states. So when we claim to \compute" this amplitude, really what we mean is that we specify all background and state-independent string eld theory data that goes into the de nition of this amplitude. It will remain to choose a vertex operator of interest, compute the correlation functions, and integrate over the moduli space to obtain a nal expression. Ultimately, however, one would like to use string eld theory to compute physical amplitudes in situations where the conventional formulation of superstring perturbation theory breaks down. The computations of this paper can be regarded as a modest step in this direction. The paper is organized as follows. In section 2 we brie y review the de nition of bosonic and heterotic closed string eld theories. To obtain the data associated to integration over the bosonic moduli space, in section 3 we compute the 1-loop tadpole in closed bosonic string eld theory. This requires, in particular, choosing a suitable cubic vertex, computing the contribution to the tadpole from gluing two legs of the cubic vertex with a propagator, and de ning an appropriate fundamental tadpole vertex to ll in the remaining region of the moduli space. In section 4 we compute the 1-loop tadpole in heterotic string eld theory. This requires dressing the amplitude of the bosonic string with a con guration of PCOs. Homotopy algebraic methods constrain the choice of PCOs to be consistent with quantum gauge invariance, but some freedom remains. We discuss two approaches: one which inserts PCOs at speci c points in the surfaces de ning the vertices, and another which inserts PCOs in the form of contour integrals around the punctures. We discuss the consequences of spurious poles for both approaches. We also present the amplitudes in a form which is manifestly well-de ned in the small Hilbert space. We end with concluding remarks. 2A new approach to the computation of o -shell amplitudes has recently been proposed based on hyIn this section we review the de nition of quantum closed string eld theories, both bosonic and heterotic versions. For the closed bosonic string, we will only sketch the basic structure of the theory as suits our needs. Everything is taken from Zwiebach's seminal paper [1], with some minor changes of notation to make contact with more recent work. The structure of heterotic string eld theory closely parallels that of the closed bosonic string. We follow [19, 20] in our treatment of the Ramond sector. 2.1 Closed bosonic string eld theory Closed bosonic string eld theory is the eld theory of uctuations of a closed string background in bosonic string theory. The background is described by a worldsheet conformal eld theory, which factorizes into a c = 26 matter component and a c = 26 bc ghost component. A string eld is an element of the state space H of this conformal eld theory. The state space has two important gradings: ghost number, and Grassmann parity, which distinguishes commuting and anticommuting states. Since the eld theory path integral requires anticommuting ghosts in the target space, we allow states in H to appear in linear combinations with Grassmann even or Grassmann odd coe cients. Quantum closed bosonic string eld theory is de ned in the framework of the BatalinVilkovisky formalism, which gives a prescription for computing amplitudes via Feynman rules derived from a gauge- xed path integral. The central object in the formalism is the quantum master action (henceforth simply the \action"), which takes the form3 HJEP1(207)56 + + 1 5! 1 3! 1 1! !( ; `1;2( ; )) !( ; `2;0) 1 2! 1 3! 1 1! S = !( ; Q ) + !( ; `0;2( ; )) + !( ; `0;3( ; ; )) + !( ; `0;4( ; ; ; )) + : : : + !( ; `1;0) + !( ; `1;1( )) Let us describe the ingredients. Dynamical string eld arbitrary state in H, but a Grassmann even element of a linear subspace characterized by states satisfying the b0 and level matching constraints: where 3We set ~ and the closed string coupling constant to 1. carries ghost number 2, but quantum mechanically it contains components at all ghost numbers which play the role of elds and anti elds in the Batalin-Vilkovisky formalism. All states in H are proportional to the operator Since L0 takes only integer eigenvalues, one can see that b i(ei L0 e i L0 ) which motivates the delta function notation. Note that (L0 ) is a projector, and b0 (L0 ) is BRST invariant. The statement that satis es b0 and level matching constraints is equivalent to where which satis es [b0 ; c0 ] = 1.4 = b0 (L0 )c0 ; 1 2 c 0 (c0 c0); Symplectic form !: the object ! is a symplectic form. It is a bilinear map from two copies of H into numbers which is graded antisymmetric, and nondegenerate. The symplectic form is concretely de ned !(A; B) = ( 1)AB!(B; A); !(A; B) = ( 1)AhA; c0 Bi; hA; Bi h I A(0)B(0)i; where the bracket h ; i denotes the BPZ inner product, which can be computed as a correlation function on the complex plane where A(0); B(0) are the vertex operators corresponding to the states A; B and I denotes conformal transformation with the map I(z) = 1=z. Often it is convenient to write the symplectic form and BPZ inner product as \double bra" states: h!j : H hbpzj : H 2 2 ! H ! H 0 0 ; 4Commutators are graded with respect to Grassmann parity. { 4 { (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) h j ! A hbpzjA B = !(A; B) B = hA; Bi: (b0 (L0 I)jbpz 1i = (I I)jbpz 1i = (I b0 )jbpz 1 i L0 )jbpz 1i: j ! 1i = b { 5 { Particularly important in the quantum theory is the inverse of the symplectic form, a Poisson bivector. We will write the Poisson bivector as a Grassmann odd \double ket" state h!j = hbpzjc0 I: j which by de nition satis es5 ! h j I I j The Poisson bivector can be constructed as follows. Since the BPZ inner product is nondegenerate, there is a Grassmann even \double ket" state satisfying hbpzj I I 0 are BPZ even, it is straightforward to show that Note that the symplectic form is only de ned operating on states in the subspace H , whereas the BPZ inner product is de ned operating on any states in H. Since the operator c0 is BPZ odd, we have the relation where I is the identity operator on the state space. We may then relate the symplectic form and BPZ inner product with the formula hbpzj(c0 I + I (2.19) The Poisson bivector must satisfy b0 and level matching constraints, so it is natural to guess the expression: 5This can also be written as h!j(12) (23)j! 1i = (3)I(1); h!j(23) (12)j! 1i = (1)I(3) where the superscripts denote copies 1; 2; 3 of the state space. Pre-superscripts denote the output, and post-superscripts the input. (2.17) (2.18) (2.20) (2.21) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) (2.22) To check that this expression is correct, compute ! h j I I j i = hbpzj = hbpzj = hbpzj = hbpzj = b0 (L0 )c0 : I I I c 0 b I b The nal operator acts as the identity on H , which establishes that ! 1 is indeed the inverse of !. Finally we note that the BRST operator Q is BPZ odd: This implies that the symplectic form and the Poisson bivector satisfy Accordingly, we say that Q is cyclic with respect to the symplectic form !. Note that because c0 does not commute with Q, in establishing the rst relation it is important to remember that ! only operates on states in H . String products `g;n: the nal and most nontrivial ingredient in the theory are the multi-string products `g;n. They are Grassmann odd multilinear maps from n copies of H into H : `g;n : H n ! H : Thus `g;n multiplies n states in H to produce a state in H . The index g refers to the \genus" of the product. We assume that `0;0 vanishes, which implies that the theory describes uctuations of a conformal background at tree level. The 1-string product `0;1 is identi ed with the BRST operator Q. `g;n carries ghost number 3 2n. The products satisfy a hierarchy of relations arising from the requirement that the action satis es the quantum Batalin-Vilkovisky master equation. These relations imply that the products realize a speci c algebraic structure, called a quantum L characterized by the following properties: 1 algebra.6 A quantum L 1 algebra is (1) The products are graded symmetric upon interchange of the arguments: `g;n(A1; : : : ; Ai; Ai+1; : : : ; An) = ( 1)AiAi+1 `g;n(A1; : : : ; Ai+1; Ai; : : : ; An): (2.35) (2) The products are cyclic with respect to the symplectic form !: !(A1; `g;n(A2; : : : An+1)) = ( 1)A1 !(`g;n(A1; : : : An); An+1): (2.36) Cyclicity of the BRST operator was already mentioned in (2.32). 6Other names include loop homotopy algebra [21] or IBL1 algebra [22]. { 6 { (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) (3) The products satisfy an in nite hierarchy of Jacobi-like identities called quantum L relations. Our main interest is the tadpole amplitude, where the following identities play an important role: 0 = (Q The rst was already given in (2.33). Since the symplectic form is nondegenerate, the string products may be de ned through the vertices HJEP1(207)56 !(A1; `g;n(A2; A3; : : : ; An+1)) ( 1)A1 fA1; A2; : : : ; An+1gg; (2.40) (2.37) (2.38) (2.39) (2.42) (2.43) (2.44) where on the right hand side we have the multilinear string functions as written by Zwiebach [1]. The multilinear string functions are graded symmetric upon interchange of any pair string elds, whereas properties (1) and (2) imply that the vertex is graded symmetric only up to a sign from moving string elds through the Grassmann odd products `g;n. This di erence accounts for the sign factor in (2.40). The multilinear string functions are concretely de ned by the formula fA1; A2; : : : ; Angg = 1 2 i dg;n=2 Z Vg;n h g;nj dg;n! b(v)dg;n A1 A2 : : : An: (2.41) There are many ingredients here, so let us describe them step by step. The number dg;n 6g 6 + 2n is the dimension of the moduli space Mg;n of genus g Riemann surfaces with n punctures. The integration is performed over a subregion Vg;n of the moduli space Mg;n. Computing an n-point amplitude at genus g requires integrating over the complete moduli space Mg;n, and much of this moduli space will be covered by Feynman diagrams composed of lower order vertices connected by propagators. The subregion Vg;n is what is left of the moduli space after these Feynman diagrams have been accounted for. The object h g;nj is an \n-fold bra" h g;nj : H n ! H 0 ; and an example of a surface state. This means that it is de ned by an n-point worldsheet correlation function on a genus g Riemann surface with prescribed local coordinates around the punctures. Speci cally: D h g;n(m)jB1 : : : Bn = h1(m) B1(0) : : : hn(m) Bn(0) Eg : m For clarity we explicitly indicate dependence on the point m 2 Vg;n. On the right hand side : : : gm denotes a correlation function on a genus g Riemann surface, with operators { 7 { inside the brackets expressed in a chosen coordinate system on the surface (for example, the uniformization coordinate). The correlation function depends on m through the moduli needed to specify the surface. The states B1; : : : ; Bn are represented by vertex operators B1(0); : : : ; Bn(0) which have been inserted at the origin of respective local coordinate disks, denoted by 1; : : : ; n with j ij < 1. The disks are mapped to the surface (in the chosen coordinate system) with the local coordinate maps h1(m; 1); : : : ; hn(m; n). The local coordinate maps in general depend on m. An important condition is that the image of the local coordinate disks on the surface do not overlap. This is because in computing amplitudes we must remove local coordinate disks and glue surfaces together on the resulting holes. Since regions of the surface cannot be \removed twice," the image of the local coordinate disks cannot overlap. Surface states are BRST invariant: h g;nj(Q I n 1 + I Q I n 2 + : : : + I n 1 Q) = 0: (2.45) In terms of the correlation function, the left hand side amounts to surrounding all vertex operators in (2.44) with a contour integral of the BRST current. This contour integral can then be deformed inside the surface and shrunk to a point, which gives zero. The nal ingredient we need for the vertex is the operator b(v), whose role is to turn the surface state h g;nj into a di erential form which can be integrated over Vg;n. The operator b(v) is a sum of b ghost insertions acting respectively on each state in the vertex: (2.46) (2.47) (2.48) n i=1 d i v(i)(m; i)b( i) + I j ij=1 2 i d i v(i)(m; i)b( i); with both contours running counterclockwise in the respective coordinates, and v(i)(m; i) is a 1-form on Vg;n. In coordinates m on Vg;n, they take the form v(i)(m; i) = dm v(i)(m; i): For simplicity, we assume that the basis 1-forms dm commute through the b ghosts and other worldsheet operators without a sign. The coe cients functions v(i)(m; i) are called Schi er vector elds. We will give more discussion later, but the basic idea is that an in nitesimal change of h g;n(m)j can be described by removing the image of the local coordinate disks j ij < 1 from the Riemann surface, slightly deforming their shape, and then gluing them back. If we change h g;n(m)j along m , the corresponding deformation of the disks will be described by vector elds v(i)(m; i) which are holomorphic in the vicinity of j ij = 1. Concretely, a deformation of the ith coordinate disk is implemented by a contour integral of the energy momentum tensor d i v(i)(m; i)T ( i) + I j ij=1 2 i d i v(i)(m; i)T ( i) (2.49) { 8 { operating on the ith state. De ning n h g;n(m)jT (v (m)): h g;nj p! b(v)p : the Schi er vector elds are de ned so that the following identity holds: This de nes the operator b(v). Given h g;nj and b(v) we can de ne a natural p-form on Vg;n: An important property is the BRST identity: h g;nj (p + 1)! b(v)p+1 (Q I n 1 + : : : + I n 1 Q) = ( 1)p+1d h g;nj p! b(v)p ; (2.53) where d denotes the exterior derivative on Vg;n. See section 7 of [1] for the demonstration. This identity plays an important role in the proof of quantum L 1 relations and the proof that cohomologically trivial states decouple from scattering amplitudes. The n-string vertex at genus g is completely speci ed once we have provided the subregion Vg;n of the moduli space and the local coordinate maps h1(m; 1); : : : hn(m; n) for each point m 2 Vg;n. This data is tightly constrained by the requirement that the products `g;n associated to the vertices de ne a quantum L 1 algebra. However, there is still some freedom in the de nition of each vertex, in particular in the choice of local coordinate maps in the interior of Vg;n. Di erent choices of vertices result in string eld theories related by eld rede nition [23]. We will present our construction of the vertices needed for the tadpole amplitude later. See [1, 24] for the classical construction based minimal area metrics. 2.2 Heterotic string eld theory Heterotic string eld theory is the eld theory of uctuations of a background in heterotic string theory. The background is described by a worldsheet conformal eld theory: the holomorphic sector is a c = 15 matter tensored with a c = 15 bc and ghost superconformal eld theory, while the antiholomorphic sector is a c = 26 matter tensored with a c = 26 bc ghost conformal eld theory. The ghosts will be bosonized to the ; ; e system [25]. A string eld is an element of the state space H of this conformal eld theory. The state space has three important gradings: ghost number, picture number, and Grassmann parity which distinguishes commuting and anticommuting states. H contains an important linear subspace, the \small Hilbert space" composed by states which are independent of the zero mode of the ghost. Therefore states in HS satisfy A = 0; A 2 HS; HS H { 9 { (2.50) (2.51) (2.52) (2.54) is the zero mode of the eta ghost. The full state space H is called the \large Hilbert space," since it contains states which depend on the zero mode. H is a direct sum of state spaces composed of Neveu-Schwarz (NS) and Ramond (R) sector states H = HNS HR: We assume that all states in H are GSO projected. Since the theory includes fermions and spacetime ghosts, we allow states to appear in combinations with Grassmann even or Grassmann odd coe cients. Quantum heterotic string eld theory is de ned in the framework of the BatalinVilkovisky formalism. The central object is the quantum master action, whose structure is quite analogous to that of the closed bosonic string: S = ( ; Q ) + ( ; L0;2( ; )) + ( ; L0;3( ; ; )) + ( ; L0;4( ; ; ; )) + : : : 1 2! 1 3! 1 1! + ( ; L1;0) + ( ; L1;1( )) 1 4! 1 2! + + 1 5! 1 3! 1 1! ( ; L1;2( ; )) ( ; L2;0) (2.56) The precise realization of this action depends on the formulation of the Ramond sector. There are two possible approaches. One approach appears in recent work of Kunitomo and Okawa [12] and subsequent papers [13{15, 26{28], and is characterized by a Ramond string eld at picture 1=2 subject to a linear constraint. The constraint is conceptually analogous to the b0 and level matching constraints of the closed string, and is well motivated from the supermoduli space perspective. However, a proper treatment requires a more sophisticated approach to the formulas for path integral [29], and we would like to take advantage of standard correlators at higher genus [16] for the purpose of locating spurious poles. Therefore we will follow a di erent approach devised by Sen [19], which has a somewhat simpler worldsheet realization. Assuming Sen's approach, the basic ingredients of the action are as follows: Dynamical string eld arbitrary state in H, but a Grassmann even element of a linear subspace, characterized by states in the small Hilbert space which satisfy b0 and level matching constraints: There is also a condition on picture number. NS states in H must carry picture 1, while Ramond states may carry picture 1=2 or 3=2. Therefore, the dynamical string eld contains an NS component and a Ramond component: 2 H HS H; b 0 = L 0 = 0: = NS + R: The NS part carries picture 1, while the Ramond part carries both pictures 1=2 and 3=2. Symplectic form . The object is a symplectic form on H . It is graded antisymmetric and nondegenerate. We will often write as a \double ket" state The heterotic symplectic form is de ned by h j : H 2 ! H 0 ; h j A B = (A; B): h j h!j(I X0P 3=2) I j! 1i: where ! is the symplectic form (2.13) of the closed bosonic string, with the understanding that the BPZ inner product is computed in the small Hilbert space of the heterotic CFT. Here P 3=2 is the projector onto states at picture 3=2 and X0 is the zero mode of the picture changing operator: The PCO is BPZ even: and satis es X0 I dz 1 [L0 ; X0] = 0; [b0 ; X0] = 0; [Q; X0] = 0: note that the operator I since it is invertible: 0 Therefore X0 preserves the b0 and level matching constraints. Generally X0 does not commute through c . However, inside the symplectic form it e ectively does, since always operates on states satisfying b0 and level matching constraints. It is important to X0P 3=2 does not spoil nondegeneracy of the symplectic form, In particular, we may construct the Poisson bivector (I X0P 3=2) 1 = I + X0P 3=2: j 1 i 2 H 2 which inverts . To do this we de ne j ! 1 i h (P 1 + P 1=2 + P 3=2)b0 (L0 ) i I jbpz 1 i Compared to (2.28) we must include explicit projections onto the appropriate pictures since otherwise sums over intermediate states result in divergences in loops. The projections are also needed so that j! 1i is a state in H 2. The heterotic Poisson bivector is then given by (2.61) (2.62) (2.63) (2.64) (2.65) (2.66) (2.67) (2.68) (2.69) (2.70) HJEP1(207)56 Following (2.29), it is straightforward to show that The nal operator acts as the identity on H . We also have h j I I j 1 i = I h j j 1 i I Accordingly, we say that Q is cyclic with respect to the symplectic form . In conclusion, we see that shares the same algebraic properties as the corresponding symplectic form ! of the closed bosonic string. By the usual argument, we can vary the free action to obtain the classical linearized equations of motion: Q = 0: This has a somewhat odd consequence: because contains states at picture 1=2 and at 3=2, the theory includes two copies of the Ramond cohomology. The main point, however, is that the interactions of the string eld can be chosen in such a way that one copy of the cohomology decouples from scattering amplitudes. Therefore, the additional states can be e ectively ignored. The constrained formulation of [12] produces the correct cohomology, and may be more suitable for understanding general coordinate invariance [ 30 ] and the relation to supermoduli space, but this will not be a central concern for us. String products Lg;n. The nal ingredient in the theory are the multi-string products Lg;n. They are Grassmann odd linear maps from n copies of H into H : The index g refers to the \genus" of the product. We assume that L0;0 vanishes, and L0;1 is identi ed with the BRST operator Q. Lg;n carries ghost number 3 2n. The quantum Batalin-Vilkovisky master equation requires that the products Lg;n de ne a quantum L algebra. The requisite properties are precisely the same as for the closed bosonic string, 1 with `g;n; ! replaced with Lg;n; . In principle, the products Lg;n could be described by of integration over subspaces of the supermoduli space of super Riemann surfaces [31, 32], but for present purposes this is not the most useful characterization. Instead we will describe the products in terms of a con guration of PCOs operating on the products `g;n of the closed bosonic string. The PCOs must be chosen in a speci c fashion so that Lg;n de ne a quantum L second condition, speci c to Sen's formulation of the Ramond sector, is ensuring that the spurious copy of the Ramond cohomology decouples from scattering amplitudes. This is implemented by requiring that all products besides the BRST operator take the form 1 algebra. A Lg;n = (I + X0P 3=2)Cg;n; (2.76) where the products have the following properties: (a) Cg;n vanishes when multiplying any Ramond state at picture (b) The output of the product Cg;n only contains NS states at picture 1 and Ramond (c) Cg;n is cyclic with respect to the bosonic symplectic form !. With these conditions, one can show that the n-string vertex at genus g takes the form HJEP1(207)56 ( ; Lg;n( ; : : : ; )) = !(b; Cg;n(b; : : : ; b)); (2.78) where b is equal to with the component at picture 3=2 set to zero. Therefore, the states at picture 3=2 decouple from the action and do not contribute to scattering amplitudes. Note that our presentation di ers slightly from Sen's, who formulates the action in terms of Cg;n rather than the quantum L 1 products Lg;n. For this reason, the proof of the quantum master equation required slight modi cation of the standard argument [20]. Analogous modi cations are needed in the proof of gauge invariance in open string eld theory [13, 14]. 3 One loop tadpole in closed bosonic string eld theory To compute heterotic tadpole we must rst specify all data associated to integration over the bosonic moduli space of the 1-punctured torus. This e ectively requires computing the 1-loop tadpole in closed bosonic string eld theory. To compute this amplitude we must x a gauge. We choose Siegel gauge: The amplitude is then given by7 i + ( 1) 1 ! ( 1; `1;0) ; where 1 is an (o -shell) state in H . The rst term arises from the Feynman diagram where two legs of the tree-level cubic vertex are tied together with a Siegel gauge propagator. The second term comes from the fundamental 1-loop tadpole vertex, which can be seen as a counterterm which \renormalizes" the tadpole amplitude computed from the tree-level action. To see that the expression is correct, let us check that BRST exact states decouple. The amplitude is characterized by contracting 1 with the 0-string product A1;0 = `0;2 I j ! 1i + `1;0: 7We do not attempt to x the normalization of the amplitude, but the sign is natural (particularly at higher points) to ensure that the amplitude is graded symmetric. For on-shell amplitudes the asymptotic states are Grassmann even, and the sign drops out. b + 0 b + 0 L+ 0 b + 0 L+ 0 = 0: (3.1) (3.2) (3.3) ∂V1,1 τ 1 = 1 + i 2. The subregion shaded in dark grey is associated with the tree level cubic vertex with two legs attached to a propagator tube of length s and twist angle . The subregion shaded light grey is associated with the fundamental 1-loop tadpole vertex. The curve @V1;1 representing the interface between these two regions depends in detail on the de nition of the tree level cubic vertex, and part of our task will be to determine this curve. Decoupling of BRST exact states is equivalent to the statement that A1;0 is BRST invariant. Let us check this: QA1;0 = Q`1;0 Q`0;2 b + 0 L+ 0 I j ! 1 : i Recalling the quantum L 1 relations (2.37) and (2.38) this simpli es to QA1;0 = Q`1;0 + `0;2(Q I + I Q) = Q`1;0 + `0;2 = Q`1;0 + `0;2 = Q`1;0 + `0;2 Q 0 Q 0 + b b L+ 0 + L+ 0 Q; 0 b + L+ 0 I I + b + 0 L+ 0 b + 0 L+ 0 b + 0 L+ 0 I j ! 1 : i I j I (I I (Q Q) j I) j ! 1 i i The commutator of Q with b0+=L0+ is the identity operator up to contributions from the boundary of moduli space which we ignore. Therefore QA1;0 = Q`1;0 + `0;2j! 1i; which vanishes according to the quantum L 1 relation (2.39). It is helpful to understand this derivation at the level of the surface states de ning the products. The amplitude can be written We expanded the propagator and level matching projector into an integral over a Schwinger parameter s 2 [0; 1] and a twist angle visualized as a cylindrical tube of worldsheet of length s twisted by an angle . The moduli space of the 1-punctured torus is two dimensional, and may be represented by the ; ]. The operator e sL0++i L0 can be fundamental region of the modular parameter of the torus, as shown in gure 1. The integration variables s; are coordinates on the part of the moduli space of the 1-punctured torus produced by the tree level cubic vertex with two legs connected by a propagator. Since the limit s ! 1 represents a degenerate torus whose handle acquires in nite length, we can anticipate that the coordinates s; will cover the part of the moduli space shown in gure 1. The remainder of the moduli space de nes V1;1, and is associated with the fundamental 1-loop tadpole vertex. Let us see how BRST exact state decouple in this language. Suppose 1 = Q and consider the propagator term. Using BRST invariance of the surface state h 0;3j and jbpz 1i we obtain b0 b0 e sL0++i L0 + I Q i i (L0+b0 b0+L0 )e sL0++i L0 I i I I b0 e sL0++i L0 + I jbpz 1i: (3.8) As explained in (2.9), the integral of the total derivative vanishes because L only integer eigenvalues. We will also see that = correspond to boundaries of the fundamental region of the moduli space which are identi ed. Therefore only the s total 0 takes derivative contributes, giving Z 1 0 ds Z d 2 h 0;3j I b0 b0 e sL0++i L0 + I Q i = Z b0 ei L0 I jbpz 1i: (3.9) Now focus on the fundamental tadpole vertex. Using (2.53) and Stokes' theorem we obtain d 2 h 0;3j I = = h 1;1j 2! b(v)2 Q d h 1;1jb(v) h 1;1jb(v) : (3.10) We require that @V1;1 corresponds the s = 0 boundary of the propagator region of the moduli space. On this boundary we can use the twist angle as a natural coordinate, and write where v ( ) is the Schi er vector eld satisfying d d h 1;1( )j = h 1;1( )jT (v ( )): The tadpole amplitude for a BRST exact state therefore takes the form These terms must cancel. To see how this cancellation can occur, note that the b ghost insertion in the rst term can be rewritten h 0;3j I ( ib0 )ei L0 I jbpz 1i = h 0;3j I ei L0 I b(v ( )) where v ( ) is the Schi er vector eld corresponding to d=d for this surface. To see why this is the case, note that from the perspective of surface state conservation laws, the b ghost is equivalent to the energy momentum tensor. If we replace b with the energy momentum tensor above, the two sides of the equation are equal to d=d of the surface state, and in particular are equal to each other. Therefore (3.13) will be zero if we assume: h 0;3j I ei L0 I I i = h 1;1( )j: (3.15) This xes the form of the 1-loop tadpole vertex at the boundary of V1;1. To construct the amplitude, we therefore proceed as follows: rst we must de ne a suitable cubic vertex. Second, we characterize the genus 1 surface states obtained by gluing two local coordinate disks of the cubic vertex with a propagator; the gluing is achieved through the standard plumbing xture relation. This determines the missing region of the moduli space V1;1 and the local coordinate of the 1-loop tadpole surface state on @V1;1. We may then make a continuous choice of local coordinate on the remainder of V1;1 and construct the appropriate 2-form to be integrated over V1;1. This speci es all data needed for evaluation of the o shell 1-loop tadpole amplitude of closed bosonic string eld theory. 3.1 Elementary cubic vertex We begin by de ning the elementary cubic vertex: !(A1; `0;2(A2; A3)) = ( 1)A1 h 0;3jA1 A2 A3 = ( 1)A1 D f0 where in the nal expression we have a correlation function on the complex plane. The global coordinate on the complex plane will be denoted z. The vertex operators A1(0); A2(0) and A3(0) are inserted at the origin of respective local coordinate disks 0; 1; 1 with j ij < 1, and the local coordinate maps transform the disks into the global coordinate z. We take the vertex operators to be inserted at 0, 1 and 1, which means that the local coordinate maps satisfy To simplify the computation of the tadpole amplitude we will assume that f0; f1; f1 take the form of SL(2; C) transformations. In particular we will not use a Witten-type vertex, so o -shell amplitudes will be di erent from the canonical ones de ned by minimal area metrics. In principle we are free to use any SL(2; C) maps in de ning the cubic vertex, but we would need to perform a sum over permutations exchanging vertex operators between punctures to ensure that the vertex is symmetric. However, it is simpler to choose local coordinate maps that result in a symmetric vertex without requiring a sum of permutations. Assuming SL(2; C) local coordinate maps, the resulting cubic vertex is unique up to an overall scaling of the local coordinate disks, as we now describe. To describe the symmetry of the cubic vertex, we introduce an S3 subgroup of SL(2; C) which interchanges the punctures at 0; 1 and 1. The subgroup has generators a(z) = 1 z; b(z) = 1=z: The group element a interchanges the punctures at 0 and 1, leaving the puncture at 1 xed, while b interchanges the punctures at 0 and 1 leaving the puncture at 1 xed. To verify that a and b generate an S3 subgroup of SL(2; C), it is su cient to check that a and b satisfy the identities of the presentation of S3: a a(z) = e(z) b b(z) = e(z) a b a b a b(z) = e(z); where e(z) = z is the identity map. The group element that interchanges the punctures at 1 and 1, leaving 0 xed, takes the form a b a(z) = z z 1 : Since SL(2; C) preserves correlation functions on the plane, we have D D f0 f0 A1(0) f1 A2(0) f1 where (z) 2 S3. The map A3(0) E = D f0 A1(0) f1 A2(0) f 1 A3(0) E; (3.24) will permute the locations of the punctures, but generally it will not permute the local coordinate maps. That is, fi will in general be di erent from f (i). However, if they happen to be equivalent, we would have (3.17) (3.18) the local coordinate disks 0; 1; 1 for the three punctures. These are transformed into regions z0; z1; z1 in the complex plane by the respective local coordinate maps f0; f1; f1. The shaded region is the \interior" of the vertex, that is, the part of the surface which is not inside any local coordinate. Since the maps f0; f1; f1 are SL(2; C), the image of the local coordinate disks are circles. The picture shows the vertex for a generic value of the stub parameter $ > 0. Towards $ = 0, the exterior circle shrinks and the interior circles grow until they touch on the real axis. and the vertex would be symmetric. Note that ei L0 jAi = ei A(0)j0i; where ei denotes the conformal transformation z ! ei z. Therefore vertex operators of level matched states are invariant under conformal transformations which rotate by a phase. Since the cubic vertex is always evaluated on level matched states, we only need to require that fi and f (i) are equal up to a phase rotation of the local coordinate disk: This identity characterizes the local coordinate maps of a symmetric cubic vertex. To solve this identity let us rst consider f0 and the permutation which switches the punctures at 1 and 1. This permutation should have no e ect on the vertex operator inserted with f0, so we should have a b a f0( ) = f0( ) f0( ) 1 = f0(ei 0;aba ): The general SL(2; C) map that preserves the origin takes the form Plugging this into (3.28) implies f 0 0 ξ ξ ∞ f 1 z1 1 f∞ z ∞ z-plane HJEP1(207)56 fi( ) = f (i)(ei i; ) f0( ) = f0( ) = + 2 : ; (3.26) (3.27) (3.28) (3.29) (3.30) where is an undetermined parameter and the required phase is ei 0;aba = loss of generality we can assume that is real and positive, since a phase can be absorbed into a trivial phase rotation of a level matched state. Next we de ne f1 and f 1 by the appropriate permutations of f0: 2 : One can then verify that the generators of S3 act on the local coordinate maps in a manner HJEP1(207)56 consistent with (3.27): a f0( ) = f1( ); a f1( ) = f0( ); 1 a f ( ) = f ( ); 1 b f0( ) = f ( ) 1 b f1( ) = f1( ) 1 b f ( ) = f0( ): Since all elements of S3 can be obtained by composing the generators, this establishes (3.27) and symmetry of the cubic vertex. An important condition is that the image of the local coordinate disks do not overlap in the global coordinate z. One may verify that this is the case if the constant satis es the inequality It is convenient to solve this inequality by writing in the form 3 where $ is real and positive number called the stub parameter. Therefore we may express the local coordinate maps z0 = f0( 0) = z1 = f1( 1) = z 1 = f1( 1 ) = 3 2e $ 0 ; e $L0+ jAi = e $ A(0)j0i; (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) (3.38) (3.39) (3.40) (3.41) (3.42) where z0; z1; z1 are the image of the local coordinate disks 0 ; 1; 1 in the z-plane. See gure 2. Note that the e ect of the stub parameter is to simply rescale the local coordinate disks before conformal transformation to the complex plane. Since the cubic vertex with $ > 0 is related to the cubic vertex with $ = 0 through h 0;3($)j = h 0;3($ = 0)je $L0+ e $L0+ e $L0+ : The operator e $L0+ can be visualized as a tube of worldsheet of length $, called a \stub," which here is attached to every leg of the cubic vertex. Stubs also appear in the cubic vertex de ned by the minimal area prescription, where canonically $ = . held xed. The local coordinate map h1 in the propagator region of moduli space can be expressed as a function of by solving (3.62) to express R+; R as functions of : R+ = 1 R = 1 e 2 i e2 i e 2 i p 1 e2 i + e4 i p 1 e2 i + e4 i With this, we can analytically continue the de nition of h1 into the tadpole vertex region of moduli space. We denote the analytic continuation as eh1, since we reserve h1 to denote the \actual" local coordinate map of the tadpole vertex. We introduce a continuous function e $( ) on the tadpole vertex region of the moduli space satisfying (3.77) (3.78) (3.79) (3.80) (3.81) (3.82) (3.83) (3.84) $e j = 1=2+ib = $e j =1=2+ib $e j =ei = 1 The local coordinate map of the tadpole vertex may then be de ned: h1( 1) = e$e $eh1( 1) $ !! $e+ i ($e $) On the right hand side is the analytic continuation of the local coordinate map in the propagator region with the stub parameter $ replaced with the function $e ( ), and the modular parameter replaced with $)= . The boundary condition (3.79) implies that h1 in the tadpole vertex region matches continuously to h1 in the propagator region. Towards j j = 1, the function $e becomes large while remains nite. From (3.68) we then know that + i ($ e $) i $ e eh1( 1) $ !! $e+ i ($e $) e $e 3 i 1 The normalization factor in (3.82) is necessary to replace e $e in this expression | which is approaching zero | with e $ as needed to get the desired behavior towards j j = 1. In this way we have obtained one consistent de nition of the local coordinate map, but it is clear that there are many other possibilities. There is nothing about this choice which suggests it would simplify computation of higher order amplitudes. This raises the broader question as to whether there is some principle, analogous to the minimal area problem, which extends the SL(2; C) cubic vertex to give a natural de nition of the tadpole and other higher order vertices. Presently we do not have an answer to this question. Let us assume a choice of h1 has been made. The next step is to determine the b-ghost insertions which de ne the measure for integration over V1;1. Conventionally the b-ghosts are expressed as a contour integrals around the puncture weighted by an appropriate vector eld. The vector eld characterizes the Schi er variation of the local coordinate patch, uniformization (−1−− ε)/2 (1− −ε)/2 (−1++ ε)/2 +ε w' 1 0 w' h1| +ε ξ1 (−1+)/2 ξ1 h1| w w1 0 (−1−)/2 (1−)/2 cut deform glue εv + . The size of the local coordinate patch is exaggerated relative to gure 5 to aid visualization. as illustrated in gure 6. Consider the tadpole surface for some represented in the uniformization coordinate. We use w1 to denote points inside the local coordinate patch, that is, points where w1 = h1( 1) for j 1j < 1 in the local coordinate disk, and we use w to denote points outside. The w and w1 regions of the tadpole surface are separated by a curve . The Schi er variation is implemented as follows. We remove the w1 region from the surface, and then change its shape by deforming the boundary curve along a vector eld v(w1) which is holomorphic in the vicinity of . This is the Schi er vector eld. We then glue the w1 region back in such a way that the deformed boundary curve + v( ) is identi ed with the original curve in the tadpole surface. This procedure e ectively de nes a nontrivial transition function between coordinates w outside the local coordinate patch and the coordinate w1: w = w1 This transition function represents some deformation of the tadpole surface state. The deformation is best understood by transforming to the new uniformization coordinate. The image of the points w; w1 in the new uniformization coordinate will be denoted w0; w10. The Schi er vector eld is determined by two conditions. First, the modular parameter of the deformed torus must be + . Second, mapping from 1 to w1, and then from w1 to w10, de nes a local coordinate map from 1 to the uniformization coordinate on the new torus. This local coordinate map is required to be identical to h1 evaluated at the deformed modular parameter + . To determine the change of the modular parameter and the new uniformization coordinate, we must compute the holomorphic 1-form. In the w coordinate, this will take the general form !(w)dw = (1 + p(w))dw: (3.86) At zeroth order in , the coordinate w is identical to the uniformization coordinate and therefore !(w) must be equal to 1. Since w satis es the identi cations w w + 1 and w w + , p(w) must be a doubly periodic meromorphic function. Since the holomorphic 1-form must be holomorphic, any poles in p(w) must lie in the w1 region. Using the transition function (3.85) we may reexpress the holomorphic 1-form in the w1 coordinate: !(w1)dw1 = !(w)dw = 1 + p(w1) dw1: (3.87) dv(w1) dw1 The right hand side must be holomorphic in the w1 region. However, p(w1) must have some singularity otherwise the holomorphic 1-form would be constant, leaving the modular parameter unchanged. Therefore the Schi er vector eld must have some singularity to cancel the singularity in p(w1). It is su cient to assume that v(w1) has a simple pole at w1 = 0: C w1 v(w1) = + holomorphic; take the form In particular with residue C. This implies that p(w) will be a doubly periodic meromorphic function with a double pole at w = 0. Up to a factor and additive constant, p(w) must then be given by the Weierstrass }-function. It is convenient to represent the Weierstrass }-function through the second derivative of the logarithm of the theta function #11. Then p(w) will The standard normalization of the holomorphic 1-form xes the additive constant to vanish. (3.88) (3.89) (3.90) (3.91) (3.92) (3.93) (3.94) Z A dw !(w) = 1 + C = 1 + C = 1: Z x+1 x d dx d 2 dw dw2 ln #11(w) ln #11(x + 1) ln #11(x) The order term drops out by the periodicity of #11. Meanwhile, integrating dw !(w) along the B cycle determines the deformation of the modular parameter. By assumption the deformed parameter is + : = = Z B + d dx d dx d dx Using the quasi-periodicity of #11, this xes the residue of the pole in the Schi er vector eld: + C ln #11(x + ) ln #11(x) : #11(z + ) = e 2 iz#11(z); C = 1 2 i : We can transform from the coordinate w1 to the new uniformization coordinate w10 using the Abel map. Assuming the map preserves the origin we obtain w10 = Z w1 0 = w1 dz 1 d 2 2 i dz2 ln #11(z) dv(z) dz In dropping the boundary term at zero we have xed the constant mode of v(w1) so that the above expression vanishes when w1 = 0. The constant mode is not uniquely determined by our analysis since it is a globally de ned holomorphic vector eld which neither changes the modulus nor the local coordinate patch. The corresponding b ghost insertion annihilates the tadpole surface state, and therefore does not e ect the nal expression for the amplitude. Therefore we are free to choose the zero mode at our convenience. If we identify w1 = h1( 1), (3.94) de nes a local coordinate map to the uniformization coordinate of deformed torus. The Schi er vector eld must be de ned in such a way that this local coordinate map is identical to h1 evaluated at + . There are actually two Schi er vector elds v1 and v2, which deform respectively the real and imaginary parts of = 1 + i 2. If we deform the real part of , the local coordinate map to the deformed torus is given by This must be compatible with (3.94): (3.95) (3.96) (3.97) (3.98) (3.99) (3.100) (3.101) (3.102) (3.103) In this way we can solve for the Schi er vector eld v1, and in a similar way v2: w10 = h1( 1) v1(w1) = v2(w1) = d d 1 1 ln #11(w1) v1(h1( 1)): w1=h1( 1) 1=h1 1(w1) 1=h1 1(w1) : We have given these expressions in the uniformization coordinate of the tadpole surface with modular parameter . However, note that the expression uses the Schi er vector elds written in the local coordinate 1 of the external state. These are related to (3.97) and (3.98) through the local coordinate map h1. With these results, the tadpole vertex can be expressed ( 1) 1 !( 1; `1;0) = 1 Z 1 Z Finally we must x the sign of the measure. In the propagator region the positive measure is given by ds ^ d , which using (3.65) is equivalent to d 2 ^ d 1 in the coordinate. Since = dw1 dw1 the positive measure must be the same in V1;1, (3.101) acquires a minus sign. Including the contribution from the propagator diagram, we therefore obtain This completes the computation of the 1-loop tadpole amplitude in closed bosonic string One loop tadpole in heterotic string In this section we discuss the tadpole amplitude in heterotic string eld theory. As in the bosonic string, we x Siegel gauge b0+ = 0. The amplitude then takes the form A1h;e1t( 1) = ( 1) 1 1; L0;2 I j 1 i + ( 1) 1 ( 1; L1;0) : (4.1) We may show that BRST exact states decouple following the same algebraic argument as presented below (3.3). Noting that 1 must be an NS state, we may express this as A1h;e1t( 1) = where j!NS1i and j!R1i contain projections onto Neveu-Schwarz and Ramond states, respectively. To compute the amplitude we must determine L0;2 and L1;0. Since we already have `0;2 and `1;0 from the closed bosonic string, much of the work has already been done. All that remains is to insert PCOs. 4.1 General construction of cubic and tadpole vertex We now give a general construction of the tree-level cubic and 1-loop tadpole vertex consistent with quantum L 1 relations. This will not completely x the choice of PCOs. We will describe two possible choices of PCOs in the following subsections. Since we are only interested in a few simple vertices, we will not need to develop the homotopy algebra formalism and other technology of [9{11] needed for constructing amplitudes to arbitrarily high order. In the formalism of [9] and subsequent work, the data about the choice of PCOs is encapsulated in the de nition of a \contracting homotopy" for the eta zero mode acting on multi-string products. Let us review this concept. Suppose we are given an n-string product bn which is well-de ned in the small Hilbert space. Let us also assume that bn is consistent with the closed string constraints, symmetric, and cyclic with respect to !. Since bn is well-de ned in the small Hilbert space, it satis es b L+ 0 [ ; bn] = 0 (4.3) where we use the shorthand bn ( 1)bn bn( Given bn we de ne a new product denoted bn satisfying bn also satis es closed string constraints, is symmetric and cyclic.11 We bn as a choice \contracting homotopy" for since it allows us to express bn in With this preparation, let us describe the cubic vertex. When L0;2 multiplies two NS 1, it requires one PCO insertion to produce an NS state at picture Given the bosonic product `0;2, we may describe this PCO though a choice of contracting Loosely speaking, the contracting homotopy inserts on the product `0;2, and the BRST operator turns into X. The advantage of this description is that L0;2 is explicitly written in BRST exact form, so we automatically have as required by L 1 relations. In addition, the contracting homotopy is de ned so that so that L0;2 is symmetric, cyclic, and satis es b0 and level matching constraints. Less obvious is that L0;2 is meaningfully de ned in the small Hilbert space. To check this, compute: which vanishes as a consequence of (2.38). This de nes L0;2 when multiplying NS states. When multiplying two Ramond states at picture 1=2, it must produce an NS state at picture 1. In this case no picture changing is required, and we may identify L0;2 = `0;2. Following the discussion of 2.2, cyclicity then xes the form of L0;2 for both NS and >>: `0;2; 1 2 1 2 L0;2 = < (I + X0)`0;2; ; (4.9) L0;2 vanishes when multiplying any Ramond state at picture 3=2. This completes the de nition of the cubic vertex up to a choice of contracting homotopy `0;2. 11To be speci c, bn is de ned in the large Hilbert space, so it should be cyclic with the symplectic form ! extended in the natural way to the large Hilbert space. (4.4) (4.5) (4.6) (4.7) (4.8) HJEP1(207)56 Continuing, we may de ne the 1-loop tadpole vertex. The tadpole vertex is de ned by a 0-string product L1;0 subject to the relation QL1;0 + L0;2j Following the general procedure of [9], we will use this equation to \solve" for L1;0. First note that the form of L0;2 in this equation will depend on whether the j an NS or Ramond state though the handle of the torus. We have more explicitly 1i propagates QL1;0 + [Q; `0;2]j!NS1i + `0;2(X0 I)j!R1i = 0: Next we factor Q out of this equation to write Therefore we must have Q L1;0 + `0;2j!NS1i I)j!R1i + Q ; where on the right hand side we add a Q-exact term. The Q-exact term is not arbitrary, but must be chosen in such a way that L1;0 is de ned in the small Hilbert space: Note that commutes through the bosonic products12 and Using [ ; 0] = I we therefore nd L1;0 = 0: ( I + I )j! 1i = 0: = `0;2j!NS1i `0;2j! 1 i `0;2j!R1i + Q( ): Q = `1;0 This will vanish if we choose = `1;0 as a consequence of (2.39). Therefore for some choice of contracting homotopy. The 1-loop tadpole vertex will then be given by This completes the de nition of the tadpole vertex up to a choice of contracting homotopies `0;2 and `1;0. 12The statement that commutes through the bosonic products amounts to the statement that we can freely deform contours through the corresponding surfaces. Since carries picture, one might worry that such contour deformations may pick up hidden residues from spurious poles. However, the sum of such residues vanishes. This follows from the fact that bosonized correlators in the large Hilbert space with some are independent of the location of . See e.g. [36]. (4.10) (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) One way to give a concrete expression for the amplitude is to choose contracting homotopies which insert PCOs at speci c points inside the surfaces de ning the vertices. We will describe this approach in this subsection. The resulting tadpole amplitude takes a form which is broadly similar to the kind of o -shell amplitudes visualized in the work of Sen [17, 35]. First let us de ne the contracting homotopy `0;2. Recall the picture of the bosonic cubic vertex in the global coordinate z, as in gure 2. As a rst guess, we may de ne `0;2 by inserting the operator at some point on the z-plane, excluding the local coordinate patches. However, the resulting vertex will not be symmetric. To obtain a symmetric vertex, generally we will need to take a sum of six terms with inserted respectively at one of six points which are mapped into each other by the S3 subgroup of SL(2; C). However, in special cases S3 may generate fewer than six points. The minimum number is two, which corresponds to inserting at z = e i =3. Therefore we de ne For simplicity of notation we will assume that all correlation functions are computed in the small Hilbert space. The above expression is therefore not directly meaningful. The intention, however, is that this expression should eventually appear in combinations where the zero mode drops out. Taking the BRST variation converts into X, and therefore the product L0;2 of NS states is given by * (ei =3) + (e i =3) ! 2 `0;2(A2; A3)) f0 A1(0)f1 A2(0)f1 A3(0) : + ; : + (4.19) (4.20) The product of Ramond states is already given in (4.9). To determine the tadpole vertex we must choose a contracting homotopy `1;0. This can be done by inserting at a point p in the uniformization coordinate of the torus representing the bosonic tadpole vertex. Generally p can be a piecewise continuous function of in the region V1;1, but it will be enough to assume that p depends continuously on . We require that p does not encounter spurious poles or enter the local coordinate patch, but otherwise it can be chosen arbitrarily.13 The contracting homotopy is then given by 13Reality of the tadpole vertex requires an additional condition !(A; `1;0) X =NS;R 1 Z D (p)b(v1)b(v2)h1 A(0) E (4.21) !(A1; L0;2(A2; A3)) = ( 1)A1 * X(ei =3)+X(e i =3) ! 2 f0 A1(0)f1 A2(0)f1 A3(0) The cubic vertex is already real since the PCOs are inserted at complex conjugate locations. p j = (pj ) : Here we sum over all spin structures, which means that both NS and R states are propagating through the A-cycle of the torus. We label spin structures according to the corresponding theta characteristics = (a; b). When NS states propagate through the A-cycle, sum the spin structures and likewise if Ramond states propagate through the A-cycle the spin structures = (0; 0); (0; 1=2); = (1=2; 0); (1=2; 1=2): Having speci ed the contracting homotopies, we can use (4.18) to nd an expression for X Z + X Z =R X =NS;R d 2 2 d D(W 2 i V1;1 * W (e i =3) ! 2 (W 1(0) + ; s=0 f0 0)(W Q 1(0) E E ; s=0 ; : (4.22) (4.23) (4.24) ; s=0 (4.25) At the moment it is not obvious how to evaluate these correlators in the small Hilbert space. However, since we know that the complete amplitude is independent of the zero mode, we may also evaluate the correlators in the large Hilbert space provided we insert Including the contribution from the propagator diagram, we obtain a complete expression for the tadpole amplitude =NS 0 Z 1 ds Z d 2 * X W (e i =3) +X W (e i =3) ! 2 X Z 1 =R 0 X =NS Z + X Z =R X =NS;R ds d 2 Z 2 d D(W 2 d D(W * W (e i =3) + W (e i =3) ! 2 f0 0)(W f0 X0)(W + ; ; + (W 1(0) E E ; s=0 ; : It is convenient to express this in terms of surface states: `1;0)) = 2 i V1;1 b(v)2 Q 1 ; where the state h 1;1j is de ned for a given 2 V1;1 To compute the BRST variation we use 1;1jA = X =NS;R D (p) h1 A(0) E ; : Q; b(v)2 2! = T (v)b(v) db(v): This relation, together with (2.51), implies the BRST identity (2.53) for the bosonic tadpole vertex. Substituting into (4.27) gives Z V1;1 b(v)2 hX b(v)2 Z V1;1 1;1jT (v)b(v) + Z V1;1 where we used and the state hX 1;1j is de ned by (4.28) with replaced by X. To proceed we must compute 1;1jQ = hX 1;1j; 1;1jT (v): (4.26) (4.27) (4.28) (4.29) 1;1jdb(v); (4.30) (4.31) (4.32) an additional . This can be achieved, for example, by replacing the vertex operator 1(0) with another operator V satisfying V = 1(0). However, it is desirable to express the amplitude in a form where the zero mode is manifestly absent. To do this we need to compute the BRST variation `1;0): If it were not for the insertion of , this would be given by the exterior derivative of h 1;1j. It is almost given by the exterior derivative of h 1;1j, but there is an additional correction needed to ensure that the location of moves in the correct way after the Schi er variation implemented by T (v). Let us look at the deformation of the real part of , implemented by the Schi er vector eld v1. The energy momentum insertion deforms the surface at by creating a nontrivial transition function between the local coordinate patch and the region outside, as described in (3.85). Passing to the new uniformization coordinate we nd T (v1) A = X =NS;R D (p0) h1j + A(0) E + ; : (4.33) The Schi er vector eld v1 is de ned so that the torus at is deformed into a torus at (with is real), and the local coordinate map h1 evaluated at has been deformed into h1 evaluated at + . The Schi er variation will move the insertion to a point p0 given by mapping p to the new uniformization coordinate. Since p is outside the local coordinate patch, to determine p0 we must rst integrate the holomorphic 1-form in the w1 coordinate (3.87) from the puncture to a boundary point of the local coordinate patch w^1; then we must integrate the holomorphic 1-form in the w coordinate from the corresponding boundary point w^ up to the point p. This gives Z w^1 0 = p p0 = dw1 !(w1) + 2 i dp ln #11(p): Therefore we can write (4.33) more explicitly X =NS;R T (v1) A D (p) h1j + 1;1j 1 T (v1) A = In the rst term is inserted at the point p evaluated at . To extract a total derivative, we must replace this with p evaluated at + . Therefore we rewrite A(0) E + ; 2 i dp A(0) X =NS;R D (pj + ) h1j + A(0) E 1 d 2 i dp ln #11(p) A(0) : Aside from a factor of i the calculation is identical for v2. We therefore nd h 1;1jT (v1) = 1;1jT (v2) = h 1;1j 1;1j; functions of de ned by 1;1j is de ned by (4.28) with replaced by @ , and Z1; Z2 are Z p w^ D + ; D (4.34) E ; E ; (4.35) : (4.36) (4.37) (4.38) (4.39) (4.40) (4.41) 1;1jb(v) : (4.42) Z Z1 = Z2 = + 1 d 2 i dp 1 d 2 dp ln #11(p) ln #11(p): 1;1jT (v) = d h 1;1j; Inserting the basis 1-forms, this becomes where Z is the appropriate 1-form. Substituting this result into (4.30) we obtain Z V1;1 b(v)2 Q = hX V1;1 b(v)2 Z Z V1;1 d h The last term is a total derivative on V1;1. Generally p need only be a piecewise continuous function on V1;1, and integrating the total derivative will produce boundary contributions at the jumps. These are precisely the corrections due to vertical integration discussed in [17, 35]. For simplicity we will assume that p is continuous on V1;1, in which case integrating the total derivative will only produce contributions from the boundary of V1;1. In particular, we assume that p respects the identi cations at the boundary of the fundamental region: pj = 1=2+ib = pj =1=2+ib; pj =ei = pj =ei( ) : (4.43) Then (4.42) implies `1;0)) X =NS;R 2 i V1;1 Z D d 1d 2 X(p)b(v1)b(v2)h1 2 i V1;1 d D (p)(W 1(0) E 1(0) E ; (4.44) HJEP1(207)56 1(0) E ; 1(0) E ; ; s=0 : In the last term we used Stokes' theorem to obtain an integral over @V1;1, which is identical to the s = 0 boundary of the propagator region parameterized with twist angle . We also used (3.14) to exchange the b ghost of the Schi er variation with a b0 insertion of the Poisson bivector. Substituting this result into (4.18) we obtain a new expression for the tadpole vertex: ( 1) 1 !( 1; L1;0) Z X =NS;R X =NS + X Z =R 2 i V1;1 1 Z 2 i V1;1 * 2 d D 2 D d 1d 2 X(p)b(v1)b(v2)h1 1(0) E 1(0) E W (ei =3) + W (e i =3) 2 W f0 0 (p) (W ; ! (p) (W 1(0) E ; s=0 : This is the desired expression for the tadpole vertex in the small Hilbert space. The zero mode is clearly absent from the rst two terms. To see that is it also absent from the second two terms, we may explicitly reexpress these correlators in terms of @ . In the third term this can be achieved by writing W (ei =3) + W (e i =3) 2 (p) = 1 Z W (ei =3) 2 p 1 Z W (e i =3) 2 p 1(0) 1(0) E + ; ; s=0 (4.45) (4.46) = 0 I I (x) dz 1 jzj=1 2 i z x dz 1 Z z ( (z) Then we have W f0 0 (p) = W f0 + Z W (f0(x)) p Therefore the zero mode disappears from the tadpole vertex, as expected. Note that the last two terms in (4.45) are necessary to compensate for a mismatch in the choice of PCOs between the tadpole vertex and propagator diagrams. However, this mismatch can be removed with the appropriate choice of contracting homotopy in the tadpole vertex. Suppose we instead de ne `1;0 by In the last term, we reexpress the operator 0 as where x is an arbitrary reference point. We can write the operator 0 = (x) + (4.47) Z A 2 Z A dw f (w) = 1 X =NS + X =R 2 i V1;1 1 Z 2 i V1;1 `1;0) = (p1) + (p2) b(v1)b(v2)h1 1(0) ; where p1; p2 and the function f (w) depend on against f (w) along the A-cycle of the torus, and we require 2 V1;1. In the second term is integrated so that ( `1;0) = `1;0. If on the boundary of V1;1 we impose the conditions Z W (ei =3) ; (p2) = W (e i =3) ; dw f (w) (w) = W f0 0 ; (4.52) the PCOs will match up continuously between the tadpole vertex and propagator diagrams, and the last two terms in (4.45) will be absent. In the language of [35], this amplitude is obtained by integration over a weighted average of continuous sections of the ber bundle Pe1;1. While it is possible to choose the PCO locations continuously as a function of the moduli, one advantage of our approach is that it automatically incorporates corrections due to vertical integration when there are discontinuities in the choice of PCOs between di erent parts of the moduli space. (4.48) HJEP1(207)56 ; ; (4.50) (4.51) 2 w = a b : We need to show that the PCOs in the tadpole amplitude do not encounter these points. The PCO con gurations in the tadpole amplitude come in three kinds: (I) Those associated with the propagator region of the moduli space with an NS internal state. internal state. (II) Those associated with the propagator region of the moduli space with a Ramond We now discuss spurious poles. Our expression for the amplitude requires computing correlators of local operators in the ; ; e system. Explicit formulas for such correlators were given in [16], where spurious poles are manifested by zeros of theta functions appearing in the denominator.14 We are speci cally interested in correlators on the torus in the small Hilbert space, and in this case the location of spurious poles may be characterized as follows. Suppose the correlator has spin structure = (a; b) and contains local operators of pictures p1; : : : ; pn inserted respectively at points w1; : : : ; wn in the uniformization coordinate. A spurious pole will appear whenever the following equality holds: (4.53) (4.54) (4.55) (III) Those associated with the tadpole vertex region of the moduli space. At the interface between the propagator and tadpole vertex regions there are contributions to the amplitude with insertions due to vertical integration. However, since carries the same picture as X, and the location of corresponds directly to the location of X in the vertex and propagator regions, these contributions do not need to be considered separately. Let us consider case (I), with an NS state in the propagator. In this part of the amplitude there is a sum of PCOs inserted respectively at w = W (e i =3) = + O(e 2$): 14The expression given in [16] is a correlator in the large Hilbert space. Therefore, to evaluate the amplitude using [16] we must insert an additional at a point x for each surface in the amplitude to saturate the zero mode. Since the amplitude is de ned in the small Hilbert space, the nal expression will be independent of x. For a xed spin structure, the correlator in the large Hilbert space generally contains spurious poles at n points, where n is the number of insertions of . The location of n 1 of these spurious poles will depend on the choice of x. However, since the nal expression for the amplitude is independent of x, these spurious poles must cancel, leaving the single spurious pole described in (4.53). p1w1 + : : : : + pnwn = a b : 2 For the tadpole amplitude there are two insertions that carry picture: the vertex operator 1(0) at picture 1, inserted at the origin, and the PCO X or at picture +1, inserted at some point w. A spurious pole will therefore appear if 2 2 pole PCO PCO 1/6 0 pole PCO 1/2 pole (3) 0 /2 pole pole 1/2 pole tude. Case (1) represents the propagator diagram with an NS state in the loop, case (2) represents the propagator diagram with a Ramond state in the loop, and case (3) represents the fundamental tadpole vertex. To leading order the PCO locations are independent of . Since we assume $ $min, the PCO locations will not deviate from 1=6 by more than a few percent as a function of . For this contribution to the amplitude we must sum over spin structures = (0; 0); (0; 1=2), and spurious poles will appear respectively at w = (1 + )=2; =2: Since the imaginary part of is never less than 1 in the propagator region, the spurious poles are quite distant from the PCOs. Next consider case (II), with a Ramond state in the propagator. Here we have a PCO integrated along the A cycle inside the propagator. For this contribution to the amplitude, we must sum over spin structures = (1=2; 0); (1=2; 1=2), and spurious poles will appear respectively at The spurious poles are far away from the PCO contour in the propagator. Finally we have case (III), corresponding to the tadpole vertex. Here a single PCO appears at and in the sum over all spin structures, spurious poles appear respectively at w = 0; 1=2; =2; (1 + )=2: However, we have complete freedom in the choice of p. We may choose p in such a way that it never encounters spurious poles. Therefore all contributions to the tadpole amplitude are free from singularities due to spurious poles. These results are summarized in gure 7. The analysis of spurious poles raises an important general question. We have de ned the tadpole vertex so that it does not encounter spurious poles. But how could we have known that the propagator contribution to the amplitude does not encounter spurious poles? Of course, if it had, we would be free to make a di erent choice of cubic vertex. But how would we know that the rede ned cubic vertex would not cause problems w = 0; 1=2: w = p; (4.56) (4.57) (4.58) (4.59) with other amplitudes? This is a general problem: it is not immediately clear whether a string eld theory action with a xed set of vertices will always de ne amplitudes that are free from singularities due to spurious poles. Sen proposed the following resolution [35]. Assuming that: states; length; (A) The fundamental vertices do not encounter spurious poles when contracted with Fock (B) The fundamental vertices come with stubs, free of operator insertions, of su cient amplitudes derived from the action should be free of singularities due to spurious poles. The basic intuition is that problems can only appear if the sum over intermediate states fails to converge when gluing vertices together in Feynman diagrams. If the vertices are equipped with su ciently long stubs, the contribution from states with large conformal weight will be exponentially suppressed, and the sum over intermediate states should converge. Our calculations appear to con rm this intuition; the e ect of long stubs is to freeze the PCO insertions to 1=6 in the NS propagator contribution to the amplitude, far away from the spurious poles. However, in this particular diagram \long" stubs are not fully necessary; even when $ < $min, the PCOs from the cubic vertex do not collide with spurious poles. Case II: PCO contours around punctures In this subsection we describe a di erent approach to inserting PCOs, given by the choice of contracting homotopy bn = 1 n + 1 0bn + ( 1)bn bn( 0 where 0 is the zero mode of the ghost: 0 = I dz 1 z=0 2 i z This choice was used in the construction of classical superstring eld theories [9{11], and gives vertices characterized by contour integrals of PCOs around the punctures. The product bn is automatically cyclic, symmetric, and consistent with the closed string constraints provided that bn possesses these properties. The appeal of this approach is that we have a concrete formula for bn which does not depend on the details of the de nition of bn. In the classical theory, this su ces to give a unique de nition of NS superstring vertices to all orders. We might hope to have similar results in the quantum theory, but spurious poles cause a complication. To fully de ne the vertex, we will need to carefully specify the choice of integration contours for the PCOs in relation to spurious poles. We will see why this is necessary in a moment. In this approach, the fundamental tadpole vertex is de ned by 1 3 `1;0 = 0`1;0; `0;2 = 0`0;2 0) (4.60) (4.61) (4.62) (4.63) and takes the form 1 3 We may compute the BRST variation in the last term: Q 0`1;0 = X0`1;0 + 0`0;2j!NS1i: 0`0;2 I + I 0) j!NS1i + `0;2( 0 (4.64) (4.65) (4.66) ; (4.67) (4.68) (4.69) (4.70) Expressed in terms of correlators, the tadpole vertex is given by, Z X =NS + X Z =R X =NS;R 2 d D 2 W W 2 i V1;1 1 3 f0 f0 0 + 1 3 1 0 2 3 f1 0 f1 0 (W f1 0 (W 1(0) 1(0) 1(0) E E : Using the decomposition of 0 in (4.47), it is easy to check that all correlators may be evaluated in the small Hilbert space. Note that the contours W f0 0 W 1 0 appearing in the rst term are very nearly the same due the identi cation w w + on the torus. However, the contours are inequivalent due to the appearance of spurious poles at =2 and (1 + )=2 for the NS spin structures. Deforming one contour into the other picks up residues from these spurious poles. This is related to the fact that while we have In deriving this expression we are anticipating that the 0 contour will not need to jump across a spurious pole as a function of the modulus. If we had needed to de ne the 0 contour in inequivalent ways on di erent regions in V1;1, we would obtain additional corrections due to vertical integration. Plugging in we therefore obtain I + I 0) j!NS1i + 0`0;2 + `0;2( 0 I) j!R1i + X0`1;0: the relation we do not have due to the picture number projections contained in j! 1i. X0 Ijbpz 1i = I X0jbpz 1i; X0 Ij! 1i = I X0j! 1i; (incorrect) pole b pole a PCO pole 0 pole 1/2 pole pole tude computed with the contracting homotopy (4.60). Case (1) represents the propagator diagram with an NS state in the loop. With a Ramond state in the loop, the PCO insertion is the same as in gure 7. Case (2) represents the fundamental tadpole vertex. Including the contribution from the propagator diagram, the tadpole amplitude is 1(0) (4.71) (4.72) expressed X =NS 0 Z 1 ds Z 2 1 3 W X Z 1 =R 0 X =NS Z + X Z X Z ds 2 d D 2 2 i V1;1 2 d D(W W 1 3 f0 W f0 0 f0 X0 + f 1 X0 + f1 X0 (W f0 b0+) (W f0 b0 ) h1 1(0) f0 X0)(W ; 1(0) E 0 + f1 1 3 1 0 2 3 f1 0 (W f0 b0 ) h1 0 (W f0 b0 ) h1 1(0) 1(0) E E d 1d 2D(W f1 X0)b(v1)b(v2) h1 We may now consider spurious poles. As before, we only need to consider the contributions from NS and R propagator regions of the moduli space, and the tadpole vertex region of the moduli space. In all cases, spurious poles are avoided, as shown in gure 8. In particular, spurious poles do not cross through the X0 contour in the tadpole vertex region of the moduli space. This justi es the simpli cation of (4.65). While we have derived a consistent expression for the tadpole amplitude, there are di culties with an overly literal interpretation of the formula (4.60) for the contracting homotopy in loop vertices. We already anticipated this when mentioning the possibility of \anomalies" due to vertical integration in the computation of Q( 0`1;0). Let us explain this in more detail. The contribution to the amplitude from the tadpole vertex region of the moduli space takes the form !( 1; X0`1;0): Since X0 is BPZ even, it is tempting to interpret this as !(X0 1; `1;0): (4.73) where X0 1 is represented as a vertex operator of picture 0. However, the amplitude expressed in this way is divergent due to a spurious pole at the puncture for the odd spin structure = (1=2; 1=2). The point is that in this context X0 1 is not really the expected picture 0 vertex operator, since the spurious pole modi es the naive result that we would obtain by computing the OPE of X(z) with 1(0). Another way of saying this is that X0`1;0 is not literally de ned by acting the operator X0 on the bosonic tadpole product `1;0, since the bosonic tadpole product itself is divergent for the odd spin structure. However, in this context the interpretation of X0`1;0 is fairly clear: it is simply given by the bosonic tadpole vertex with an integration of X(z) around the boundary of the local coordinate patch. For higher order amplitudes, there are additional subtleties. In general it is not enough to interpret 0 as integrated precisely on the boundary of the local coordinate patch, since these contours will at some point encounter spurious poles.15 Rather, it is necessary to allow some deformation of the 0 contours, and these contours must be chosen in inequivalent ways in di erent parts of the vertex region of the moduli space to cross spurious poles. This will lead to anomalous corrections due to vertical integration when computing the BRST variation. While this is consistent, what is signi cant is that there is ambiguity in how the 0 contours are speci ed in di erent regions of the moduli space. Di erent choices result in string eld theories with di erent o -shell amplitudes. Therefore, the contracting homotopy (4.60) will not de ne a unique string eld theory in loops, as it does at tree level. Nevertheless, we believe that (4.60), with an appropriate interpretation of the 0 contours, can be used to de ne consistent quantum superstring eld theories. 5 Concluding remarks In this paper we have given an exact computation of the o -shell 1-loop tadpole in quantum heterotic string eld theory. This serves as a useful test of many recent ideas behind the construction of superstring eld theories. We focused almost exclusively on the computational problem of extracting the o -shell data that goes into the de nition of the amplitude. We believe that similar analysis can be carried out for the tree-level 4-point amplitude and the 1-loop 2-point amplitude in the 1PI NS heterotic string eld theory [37]. This would be su cient to compute, for example, the leading order contributions to the dynamically shifted vacuum state in SO(32) heterotic string theory compacti ed on a Calabi-Yau [38, 39]. However, the vacuum shift is not in itself an observable. To compute genuinely new physical quantities such as renormalized masses or amplitudes around the shifted vacuum, we would need higher order amplitudes. For higher orders the SL(2; C) cubic vertex will not be enough to get exact results; we would need higher vertices with similar a nity for gluing with plumbing xture. Probably 15We encountered this issue when analyzing the 2-point amplitude in the 1PI NS heterotic string { 44 { there is a limit to how far such simpli cations can extend. However, one may observe that most of the nontrivial data that emerges from the computation of the o -shell tadpole is exponentially suppressed by the stub parameter. See (3.65) and similar equations later. This suggests that the stub parameter may be used to set up a kind of perturbation theory which could give a systematic approximation to o -shell amplitudes whose exact computation may be out of reach. This strategy may be su cient to compute physical quantities, which in any case will be independent of the stub parameter. From the perspective of the formal construction of superstring eld theories, our most signi cant conclusion is that the contracting homotopy (4.60) used to insert PCOs at tree level does not generalize in a straightforward fashion to loops. In a sense this is not surprising, since there is little reason to think that a purely algebraic construction of vertices along the lines of [9] can \know" about the global conditions on Riemann surfaces that give rise to spurious poles. However, this means that the formalism of [9] will not by itself provide a unique set of vertices for quantum superstring eld theory. The formalism must be supplemented by data specifying a choice of contracting homotopies for each vertex so as to avoid spurious poles. It remains an interesting open question whether there is a \best possible" choice of vertices de ning the action of quantum superstring eld theory to all orders. Acknowledgments 003/0000437) We would like to thank A. Sen and H. Erbin for discussions. We also thank B. Zwiebach for providing references and encouraging us to describe a concrete choice of local coordinate map in the tadpole vertex, which appears in an updated version of this paper. This work is supported in part by the DFG Transregional Collaborative Research Centre TRR 33 and the DFG cluster of excellence Origin and Structure of the Universe. The work of T.E. was also supported by ERDF and MSMT (Project CoGraDS -CZ.02.1.01/0.0/0.0/15 Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] B. 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Theodore Erler, Sebastian Konopka, Ivo Sachs. One loop tadpole in heterotic string field theory, Journal of High Energy Physics, 2017, 56, DOI: 10.1007/JHEP11(2017)056