A note on the uniqueness of the Neumann matrices in the plane-wave background

In this note, we prove the uniqueness of the Neumann matrices of the open-closed vertex in plane-wave light-cone string-field theory, first derived for all values of the mass parameter mu in hep-th/0311231. We also prove the existence and uniqueness of the inverse of an infinite dimensional matrix necessary for the cubic vertex Neumann matrices, and give an explicit expression for it in terms of mu-deformed Gamma functions. Methods of complex analysis are used together with the analytic properties of the mu-deformed Gamma functions. One of the implications of these results is that the geometrical continuity conditions suffice to determine the bosonic part of the vertices as in flat space.

As a reminder, in [1] we solved 1 for the bosonic part of the vertex |V B using the geometrical continuity conditions which are understood to hold at τ = 0, the interaction time (i = 1, ..., 8 are the transverse coordinates). Working in terms of modes allows one to deduce an expression for the vertex entirely analogous to the one in flat space 2 |V B = exp(∆)|0 (2) where and α I/II n = √ 2(α n ±α n ), ω n = sgn (n) n 2 + µ 2 and here |0 is the vacuum of the two string Fock space. The modes α n andα n are of the closed string, and β m of the open string, see [1] for the explicit modes expansions and commutators. The Neumann matrices A mn , B mn and C mn satisfy the following complicated set of coupled equations: We found that a solution to this system of equations is given by These are written in terms of deformations of the binomial coefficients u where Γ I µ (z) and Γ II µ (z) are the µ-deformed Gamma functions of the first and second kind introduced in [1], which both reduce to Γ(z) in the flat space limit µ → 0. One can find the definitions of these functions, together with some of their key properties in the Appendix.
In that paper a method was suggested for proving uniqueness of A mn , B mn and C mn , at least in flat space, however this relied on some fairly heavy-handed complex analysis. It first involved determining the singularity structure of the matrices. This is the weak point of the technique, as this could only really be motivated, albeit strongly, and not proved. The method used here however, in light of the complicated nature of the set of equations which the Neumann matrices satisfy, is remarkably simple. It relies on an elementary observation of the property of inverses of infinite dimensional matrices. Namely, if both a right and left inverse exist then they are equal and unique. Note that for infinite dimensional matrices one can have the situation where we have multiple left inverses but no right inverses, for example (see Appendix D of [10]). Fortunately it so happens that one of the Neumann matrices, A mn , is actually the right inverse of a certain infinite dimensional matrix. Thus a proof of uniqueness rests on showing that it is also the left inverse, which we prove in this letter.
In this letter we also prove existence and uniqueness of the Neumann matricesN rs mn of the cubic vertex. As a reminder [15,16] the bosonic part of the vertex is written as where a i rm are the modes of the three closed strings, normalised to satisfy the harmonic oscillator algebra [a i rm , a j † sn ] = δ mn δ ij and m ∈ Z (note a † m = a −m here). Here, the state |0 denotes the three string Fock vacuum (this satisfies a i rm |0 = 0 for m ∈ Z). The three strings have incoming momenta α r ≡ 2p (r) + , so 3 r=1 α r = 0. As in flat space we can "factorise" the Neumann matrices in terms of Neumann vectorsN r m [17,18], such that for m, n ≥ 1 3 where ω r,m = sgn (n) n 2 + α 2 r µ 2 . The Neumann vectors can be written as and in [8] we determined the vectors f which when combined give Γ + f (3) = B; these sums can be derived from the appropriate geometrical continuity conditions. See the Appendix for the definitions of the matrices A (r) , U (r) and the vector B. The solutions we found are expressed in terms of µ-deformed Gamma functions as follows: and See the Appendix for definitions of the Gamma functions and the constant M (0 + ). Note that an alternate, less direct method was first used in [12] to derive these quantities. To relate the notations of the two papers [12,8], we have Y = f (3) and k = −4K, although for convenience here we will use k.
One might ask whether a simple proof of uniqueness could be presented for the Neumann matrices of the cubic vertex as well. As stressed, in [8] we found a solution f (3) to Γ + f (3) = B. It is clear that if we could show that Γ + possesses a left inverse then the solution f (3) is indeed unique. At this point we note a welcome simplification: since Γ + is actually a symmetric matrix, existence of a left inverse implies existence of a right inverse, which in turn tells us that these inverses are equal and unique. Thus the problem of uniqueness in this case reduces to showing that Γ −1 + actually exists! It appears that no-one has actually checked existence of this matrix. In [12] an expression for the inverse is derived 4 assuming it exists -of course to complete the proof one needs to verify that the expression is the inverse directly. This is possible using the methods of [1,8]. We will present this calculation too, however it is rather more involved than than the calculation for the open-closed vertex. The great advantage of using the deformed Gamma functions in the expressions of the Neumann matrices is that the infinite dimensional matrix algebra required can be done explicitly just using elementary complex analysis. In fact what we will do is evaluate the infinite sum Γ + Γ −1 + and confirm is it equal to the identity.
A proof of uniqueness is important in the plane-wave theory. It means that the geometrical continuity conditions imposed to derive the set of equations satisfied by the Neumann matrices, using Fock space methods, actually suffice to settle the bosonic part of the vertex. This is a nice result as it means that this method can probably be used more generally. For example, one might contemplate using geometrical continuity and the oscillator methods to compute the 1-loop correction to the closed string propagator, rather than directly "gluing" two cubic vertices together. Note that in flat-space one could deduce uniqueness by appealing to the alternate Green's function method of deriving the matrices. However, these methods, at the present at least, are absent in the planewave case due to the lack of explicit conformal invariance on the worldsheet. The proofs given here actually work in flat-space too (as they are valid for all µ!) and it appears that they have not been noticed before.
Finally, we should emphasise the advantages of our method as compared to the one used in [12]: • It is not clear how one would generalise the technique used in [12] to other interaction vertices, whereas the techniques developed and used in [1,8] are clearly more generally applicable as they allow one to solve the geometrical continuity conditions directly.
• Our method is a direct generalisation of the techniques which can be used in flat space.
• Expressing the Neumann matrices in terms of the deformed Gamma functions is not merely of aesthetic value. Once armed with their analytic properties it provides a powerful calculation tool. 4 More precisely an expression for the vector Y and scalar k is derived which is enough information to deduce Γ −1 + using an identity derived in [17]. Now we present our proofs. As we have already hinted at, the proof is based on the following observation. Consider the equation where and Therefore, based on the above observation if A mn is also the left-inverse to M mn then it is the unique solution to (20). Thus we need to prove the following sum To this end consider the contour integral Recall v II k = 2Γ II µ (k+1/2) Γ I µ (k)ω 2k √ π , which tells us that A mk has zeroes for k ∈ N 0 , simple poles at k = −1/2, −3/2, −5/2, ... and another simple pole at k = m + 1/2. A mk also has a branch cut on [iµ/2, −iµ/2] and branch points at ±iµ/2. Therefore if n = m the integrand in (24) only has poles at k = 1, 2... (the rest are cancelled), in which case the remaining contributions come from the branch cuts, branch points and the integral at infinity. Using the asymptotics of v II k we see the integrand goes as k −3/2 and thus the integral at infinity does not contribute. Since v II 0 ± +iy is odd for |y| < µ/2, the integrand on either side of the branch cut is odd and thus these contributions vanish too. Finally the integrals around the branch points vanish as O(ǫ 1/2 ), where ǫ is the radius from the branch point. Thus we have verified (23) for n = m. So now we consider n = m; the argument proceeds in the same way except now the integrand also has a simple pole at k = n + 1/2. Thus we also need to include the following residue which now proves (23). We deduce that A mn is unique and we are entitled to call M mn = A −1 mn . Note it immediately follows that C mn is unique too, as it is expressed entirely in terms of A mn , see (5). To prove that B mn is also unique, we note that one of the equations satisfied by B mn is Therefore multiplying (26) by A pn and summing from n = 1, ..., ∞, and using (23) (i.e. using the fact that A is left-inverse to M ) implies It follows that B mn is also unique, thus proving the theorem. Note, one could of course check that the RHS of (27) reproduces the correct B mn by doing the sum in the usual manner. Also note the uniqueness theorem follows for flat space too by setting µ = 0.
A crucial ingredient of the Neumann matrices of the cubic vertex, is the inverse of a certain infinite dimensional matrix Γ + . It is defined as and one can find the definitions of the matrices A (r) and U (r) in the Appendix. As already explained, there is a lose end to tie up here. In [12], the existence of the inverse was assumed and using this an expression for it was derived. However no-one has checked that the final explicit expression for Γ −1 + actually is the inverse! In [8] we found a vector f (3) which satisfied Γ + f (3) = B. However we did not manage to prove it was unique. Here we will settle both of these open ends by taking the explicit expression for Γ −1 + in [12] and multiplying it into Γ + . The existence of an inverse ensures it is the unique left and right inverse (due to the matrix being symmetric), and also establishes that the solution in [8] is unique.
Thus, the candidate expression for Γ −1 + is [17] (Γ −1 The sum we are interested in is Evaluation of the RHS of (30) can be performed using two identities which we will prove. We will work in the gauge α 1 = y, α 2 = 1 − y and α 3 = −1. The first identity is: which is only valid for r = 1, 2. For r = 3 it is trivial to see we only get the first term. Note that this sum is very similar to one computed in [8], namely (15). Notice that the extra factor does not affect the asymptotics of the integrand of the corresponding contour integral, or the parity of the integrand along the branch cut. Thus we need only worry about the extra residue which occurs at p = −n. This is what gives the second term on the RHS of (31). The first term is the analogue of the RHS of (15) which comes from the residue at p = −q/α r . More explicitly, for the r = 1 case, one needs to consider the following contour integral dp 2πi 2 sin(πp) where the contour is a large circle centred at the origin. The integrand has simple poles for p ∈ N, whose residues lead to the sum we want, i.e. the LHS of (31); this is of course by construction.
Another simple pole occurs at p = −q/y and leads to the contribution f (1) q y ω q/y +µ ω q/y −ωn . This is the term that corresponds to the one which gives f (1) in (15), see [8]. The new term comes from the simple pole at p = −n as we have already pointed out. To calculate this, one needs the reflection identities a couple of times, and we find the contribution (33) Note that we choose a branch cut on [iµ, −iµ], and the line integrals on either side of this cut vanish since the integrand there is odd for |Im p| < µ. Finally, the asymptotics of the integrand are such that the integral on the large circle tends to zero as the radius of the circle tends to infinity; for this one needs the generalisation of Stirling's formula derived in [8]. Piecing all this together we deduce (31) for r = 1. The r = 2 version of the identity is easily inferred from the r = 1 by mapping This sum is very similar to another one evaluated in [8], namely (16). The summand thus differs by a factor of (ω r,q + µα r )/(ω r,q − α r ω n ), which again does not change the asymptotics or the parity along the branch cut of the corresponding integrand. Also note that the extra factor in the numerator now ensures that the contribution from q = 0 vanishes on both sides of the branch cut. To prove (34) consider the following contour integral dq 2πi 2 sin(πq) (−1) m+1 √ m sin(mπy) q 2 − y 2 m 2 qM (0 + ) ω 1,q − yω n e τ 0 (µ+ω q/y ) Γ µ(1−y) (−q(1 − y)/y) ω q/y Γ µ (−q/y)Γ µy (q) where once again the contour is a large circle centred on the origin which we will take to infinity. Note that now there is a branch cut on [iyµ, −iyµ]. Now, the residues from the simple poles at q ∈ N give us the r = 1 term on the LHS of (34). The simple poles at q(1 − y)/y ∈ N give the r = 2 term, and the simple pole at q = −my gives the r = 3 term. The simple poles of 1/(ω 1,q − yω n ) at q = ny and of 1/(q 2 − y 2 m 2 ) at q = my, for n = m, get cancelled by the factor 1/Γ µ (−q/y). For n = m however, we have a simple pole at q = ny. To evaluate its residue we need Res q=ny 1 (ω 1,q − yω n )(q 2 − y 2 m 2 )Γ µ (−q/y) = (−1) n+1 ω n 2ny 2 Γ µ (n), which after some manipulation, including the use of a reflection identity, gives the following contribution to the above contour integral: − e 2τ 0 µ M (0 + ) 2 B m = 2 π α 3 α 1 α 2 (−1) m+1 sin(mπβ) where β = α 1 /α 3 . Another definition which we use is (U (r) ) mn = δ mn (ω r,m − α r µ) m .