KLT Factorization of Winding String Amplitudes

We uncover a Kawai-Lewellen-Tye (KLT)-type factorization of closed string amplitudes into open string amplitudes for closed string states carrying winding and momentum in toroidal compactifications. The winding and momentum closed string quantum numbers map respectively to the integer and fractional winding quantum numbers of open strings ending on a D-brane array localized in the compactified directions. The closed string amplitudes factorize into products of open string scattering amplitudes with the open strings ending on a D-brane configuration determined by closed string data.


Introduction
The celebrated Kawai-Lewellen-Tye (KLT) relations show that tree-level closed string amplitudes factorize into a sum of quadratic products of open string amplitudes [1]. Intuitively, the KLT relations are a consequence of the factorization of complex integrals describing closed string amplitudes into contour integrals describing open string amplitudes. The simplest incarnation of such a factorization is the classic Riemann bilinear identity.
The KLT relations have revealed elegant structures in scattering amplitudes and led to surprising connections between different theories (see e.g. [2][3][4][5][6][7][8][9][10][11]). Originally, the KLT relations were proposed as a powerful method for evaluating tree-level closed string amplitudes [1]. Intriguingly, the discovery of such relations has also led to numerous insights into quantum field theory (QFT) amplitudes by considering the infinite tension limit of string amplitudes. For example, it has been shown that Born-Infeld, non-linear sigma models, and special Galileons fit into the QFT type of KLT relations [12], via the framework of the CHY formalism [13]. The KLT relations also inspired studies of gravity as a double copy of gauge theory over the past decade [14]. The importance of the double-copy construction is justified by its use in significantly simplifying the calculation of gravity amplitudes at loop level [15,16]. A deeper understanding of these remarkable relations may hold new surprises in simplifying practical computations of amplitudes in both field and string theory.
Connecting string theory with nature requires compactifying string theory down to four spacetime dimensions. Upon compactification, the extended nature of the string clearly distinguishes string theory from local QFT. The existence of string states winding along a compact direction, which do not have a local field theory description, enriches the quantum numbers carried by string states and the collection of string amplitudes in the theory, and leads to inherently stringy phenomena, such as T-duality.
In this paper we derive a KLT-like relation for N -point scattering amplitudes of closed strings when a spatial direction is compactified on a circle and string states carry winding (and momentum) on the circle. This allows us to express any closed string amplitude in terms of open string amplitudes. But which open string amplitudes? At first sight there seems to be a puzzle. A closed string state on a circle of radius R can carry both momentum n/R and winding w along that circle, where n, w ∈ Z . In contrast, an open string state can either carry momentum or winding along the circle, depending upon whether we impose Neumann or Dirichlet boundary conditions along the circle, but cannot carry both momentum and winding. The resolution of this conundrum is that, while the winding number w i of the i-th closed string can be identified with the winding number of the i-th open string obeying Dirichlet boundary  Figure 1: Closed string amplitude with momentum and winding data (n i , w i ) obeying conservation laws N i=1 n i = 0 and N i=1 w i = 0 . We choose n 1 , . . . , n k > 0 and n k+1 , . . . , n N ≤ 0, splitting strings into incoming and outgoing states. The integer n + is the total incoming momentum. We show that the number of D-branes in the open string scattering amplitude appearing in the KLT-like factorization is n + + 1.
2π R · · · · · · n + + 1 equidistantly separated D-branes  Figure 1 shows the scattering of N closed strings with momentum and winding data (n i , w i ) . Figure 2 illustrates the corresponding D-brane configuration. The D-branes are equally separated by a distance which is T-dual to the circumference of the compactified circle. The associated open string amplitude is depicted in Figure 3, and involves open strings defined on an array of n + + 1 D-branes, with n + defined in Figure 1. Open strings can only join or split at the same D-brane. There is a total of n + +1 = 5 D-branes for this choice of closed string quantum numbers.
between two D-branes. The fractional winding number is equal to n i R/R = α n i /R 2 . The above mapping between the quantum numbers describing closed and open string states in the KLT-like relation is summarized in the table below:   closed string  open string   momentum  fractional winding  winding integer winding The precise meaning of these terms will be further clarified in §3.2. 1 This paper is organized as follows. In §2, we give an introduction to essential ingredients of winding closed strings, and evaluate in §2.3 the four closed string tachyon 2 amplitude in the presence of windings. In §3, we discuss winding open strings and derive in §3.3 the KLT relation between four closed and open string tachyon amplitudes. In §4, we generalize the KLT relation for winding strings to higher-point amplitudes that involve massless and massive vertex operators, and present the general form of the KLT relation in (4.23). We conclude our paper in §5. In Appendix A, we recast our 1 There is also a straightforward T-dual interpretation for open string scattering amplitudes with open strings ending on the D-brane configuration described in Figure 2. In the T-dual frame, an open string ends on a stack of spacetime-filling D-branes in the presence of Wilson lines. Such an open string satisfies Neunmann boundary conditions in the compactified direction and can carry nonzero momentum number (both integral and fractional, with the latter attributed to the Wilson line) but only zero winding. See §3.3 for details.
2 Although a closed string tachyon cannot simultaneously have nonzero momentum and winding along the compactified circle, the four-tachyon amplitude still provides a sufficiently interesting example that captures the salient properties of the KLT relation for winding strings.
new KLT relation in more modern terms using intersection theory, which gives rise to a compact rewriting of the KLT relation in (A.26).

Elements on Winding Closed Strings
We consider string theory described by a sigma model that maps the worldsheet Σ to a d-dimensional target space M . We take the worldsheet to be Euclidean and denote the worldsheet coordinates by σ α = (τ, σ) , with τ ∈ R and σ ∈ [0, 2π] . In the target space, we denote the coordinates by X M = (X µ , X) , where M = 0, . . . , d − 1 and µ = 0 , 1 , . . . , d − 2 . We have defined X µ ≡ X µ and X ≡ X d−1 . We compactify X over a circle of radius R , such that X ∼ X + 2πR . (2.1) A closed string that winds w times around the circle along the X-direction satisfies The mode expansion of X is where p denotes the momentum in the X-direction and α is the Regge slope. The single-valuedness of the operator exp(ipX) requires the quantization condition Here, n is the Kaluza-Klein (KK) excitation number.
To perform radial quantization, we define in terms of which the mode expansion of X is X(z , z) = X L (z) + X R (z) , where Note that p L and p R are eigenvalues of the operatorŝ where the counterclockwise oriented contour C encloses the vertex operator associated with the string.

Closed String Tachyons and Cocycles
The closed string tachyon is described by the following vertex operator [17]: The extra phase factor in (2.9) is known as the cocycle, which is important for removing the phases arising from crossing certain branch cuts when different vertex operators are interchanged with each other. To demonstrate that (2.9) is the correct vertex operator, we first note the operator product expansions (OPEs), We defined z 12 ≡ z 1 − z 2 and z 12 = z 1 − z 2 . It follows that Therefore, V C (z , z) corresponds to the eigenstate ofp L, R . Similarly, we havê We next note that the BRST invariance of (2.9) requires where k 2 = k µ k µ . It then follows that the dispersion relation is when supplemented with the level matching condition This level matching condition implies that an asymptotic tachyonic state cannot have both nonzero Kaluza-Klein and winding numbers. Finally, we consider the product of two tachyon operators, such as which is taken to be radially ordered, i.e. C 1 encloses C 2 . It follows that ∼ g 2 c e iπ w 1 n 2 z

Higher-Order Vertex Operators
More general vertex operators can be constructed by multiplying vertex operator with factors of the form ∂ m X M L and∂ m X M R , for positive integer m . BRST invariance imposes various conditions on such operators. For example, we have the BRST-invariant (up to total derivatives) and gauge-fixed vertex operators where ε I 1 ···I N J 1 ···J N is symmetric in {I 1 , . . . , I N } and in {I 1 , . . . , I N } , respectively, and The dispersion relation and the level matching conditions are Unlike closed string tachyons, the vertex operator (2.21) can simultaneously have nonzero KK and winding quantum numbers.
For an example of a BRST-invariant vertex operator that contains higher derivatives acting on X L, R , we consider the gauge-fixed operator where ε M N is a symmetric tensor that satisfies ε M M = α K M L K N L ε M N . The dispersion relation and the level matching condition in this case are Other higher-derivative vertex operators can be constructed similarly.

Four Closed String Tachyon Amplitudes
We start with the tree-level four tachyon amplitude, which exhibits the simplest nontrivial KLT relation. Due to the level matching condition (2.16), a tachyonic state can either have a nonzero KK or winding number. Later in §4, we will consider amplitudes that involve vertex operators discussed in §2.2, in which case an asymptotic closed string state can have both nonzero KK and winding numbers. The amplitude for four closed string tachyons on a spherical worldsheet is Here, χ = 2 is the Euler characteristic of the sphere and Φ 0 is the expectation value of the dilaton field. We have dropped a proportionality constant that depends on the coupling g c , which we do not need in this paper. The vertex operators V C i are ordered radially, with the integration contour C i enclosing C j , i < j . It follows that The vertex operator V C i (z i , z i ) is defined in (2.9). We have used the Möbius transformations to fix z 1 = 0 , z 3 = 1 , and z 4 = ∞ . Using the identity where This integral can be studied using intersection theory, which gives rise to a geometrical interpretation of the associated KLT relation in terms of the twisted (co)homology and intersection number. We will discuss this modern interpretation later in Appendix A.
In practice, however, we will perform the integrals following the method used in the original work in [1], without resorting to intersection theory. This allows us to focus on the novelties brought by nonzero windings. Define z 2 = x + i y (and z 2 = x − i y) for real variables x and y , and then promote the integral over y to be in the complex plane. Then, the integrand for y has its branch points at ±ix and ±i(1 − x) . We deform the contour for the y -integral that is along the real axis counterclockwise by π 2 − , for → 0 + . This deformation is captured by the following change of variables: Therefore, 3 For a fixed ζ ∈ (−∞ , 0) , the ξ -integral has branch points at Importantly, both of the branch points reside in the upper half plane. Completing the integration contour for ξ in the lower half plane, the integral over ξ yields zero. A similar argument also shows that ζ ∈ (1, ∞) contributes zero to the amplitude. Therefore, only when ζ ∈ (0, 1) , the amplitude receive nonzero contributions and where I (4) In I R , there are two branch points at the positions given in (2.34), but now with ζ ∈ (0, 1) . This implies that the branch point at i δ ξ=0 resides in the lower half plane, and the branch point at 1 + i δ ξ=1 resides in the upper half plane. We then take the part of the ξ -integral contour that lies to the right of ξ = 0 , and rotate it by 180 • across the lower half plane of the complex plane, around the branch point ξ = i δ ξ=0 . Note that this contour deformation does not cross any branch points. See [1] for more details. We find (2.37) 3 We use the convention d 2 z 2 = dx d y in (2.30).
We can alternatively take the part of the original ξ -integral contour that lies to the left of ξ = 1 to the right, and rotate is by 180 • across the upper half plane, around the branch point 1 + i δ ξ=1 . This contour deformation does not cross any branch points, either. This second choice of integration contour gives To proceed further, we introduce the generalized Mandelstam variables, The second equality in (2.40b) can be shown explicitly by using the identity .
(2.42) Therefore, The same result in (2.43a) can be confirmed by evaluating I (4) in (2.30) directly as in [18]. 4 Recall that, to derive this result, we fixed ζ while performing the integral over ξ . Alternatively, if we switch the order of integration by first performing the integral over ζ with ξ fixed, then It is also straightforward to show that the expressions in (2.43) and (2.44) are equivalent by repetitively using (2.42). For example, applying (2.42) to the beta functions in (2.43a) gives This shows explicitly that (2.43a) and (2.44a) are equivalent.

Open String Amplitude and KLT Relation
The decomposition of the four closed string tachyon amplitude into the left-moving and right-moving parts in (2.35) is already in form the same as the standard KLT relation that relates closed string amplitudes to open string amplitudes. The exceptions here include the cocycle factors and that I L and I R depend on K Li and K Ri , respectively. The kinematic data K Li and K Ri are different from each other when the winding number w i is nonzero. As we shall see the expressions of I to open string amplitudes. This will be the case by introducing appropriate D-brane configurations, which we discuss in detail below.

Winding Open Strings
Consider a stack of D-branes that extend in all the noncompactified directions and are localized in the compactified X-direction. We require that the q-th brane is located at in the compactified circle, such that the q-th and (q + 1)-th branes are separated by L . See Figure 2 for an illustration, where L = 2π R is fixed. We now take the worldsheet to be a strip, with τ ∈ R and σ ∈ (0, π) . Consider open strings that satisfy the Dirichlet boundary condition in the compactified X-direction and the Neumann boundary condition in directions along the D-branes. We fix the ends of a given open string at two of the D-branes -say, the q-th and (q + n)-th D-brane, with n ∈ Z -separated by a distance |n|L . 5 The compactified worldsheet field X(τ, σ) takes on the following mode expansion on the strip: Here, x q is the location of the q-th D-brane and x q+n = x q + nL is the location of the (q + n)-th D-brane. The winding number w denotes how many additional loops the open string winds around the compactified circle. Note that, when x q+n = x q , the open string ends on the same D-brane. The worldsheet field X(τ, σ) in (3.2) satisfies the Dirichlet boundary conditions Under the change of coordinates introduced in (2.5), the worldsheet is mapped to the upper half plane with X(z , z) = X L (z) + X R (z) and The OPE between two X's on the disk that satisfy the Dirichlet boundary conditions is It then follows that the OPEs between X L 's and X R 's are The OPEs on the boundary can be readily derived by setting z i = z i = y i ; the OPE in (3.6) vanishes on the boundary. The total winding number of the open string can be non-integer, given by Similarly, the momentum operator in the compactified direction iŝ We are interested in the open string tachyon vertex operator that corresponds to the eigenstate ofŴ with the eigenvalue W in (3.8). Such a vertex operator takes the form where y ∈ R and The vertex operator V o corresponds to an eigenstate ofŴ with eigenvalue W and ofp with eigenvalue p = 0 , consistent with that open strings carry zero momentum in the direction satisfying the Dirichlet boundary condition. We recall that X M = (X µ , X) , with X µ satisfying the Neumann boundary condition and X satisfying the Dirichlet boundary condition.
which gives the on-shell condition. As a final remark, we note that, in the degenerate case when 2πR/L is rational, there can be D-branes coinciding with each other in the compactified direction. See Figure 4 for a special case with 2πR/L = 3 . When 2πR/L is irrational, none of the D-branes will coincide after compatification and the branes densely fill the circle. Figure 4: The circle of radius R represents the compactified X-direction and the ticks represent a series of D-branes transverse to the compactified circle, equidistantly positioned along X at x q = x 0 + q L , q = 0, 1, 2, . . . , as shown in Figure 2. The diagrams above depict the special case when 2πR/L = 3 , which is rational. As a result, maximally, there are only three transverse branes in the circle. Note that these diagrams also represent the D-brane configuration when L takes other values, for example, L = 4πR/3 . An open string, shown in blue, can stretch between two D-branes as in the left diagram, or as in the right diagram by making an integer number of wrappings around the compactified direction. Generically, when 2πR/L is irrational, there is no maximal number of transverse branes in the circle.

Four Open String Tachyon Amplitudes
The Euler characteristic χ = 1 for the disk and we have specified the permutation. We apply the PSL(2, R) symmetry to fix y 1 = 0 , y 3 = 1 , and y 4 = ∞ . Then, Here, We have dropped a proportionality constant that depends on the coupling constant g o . In general, there can also be color factors from Chan-Paton charges, which we drop when the KLT relation is concerned. The total winding number W i defined in (3.8) is The conservation law of W i is imposed by the Kronecker symbol in (3.15), which gives In the case when L/(2πR) is irrational, (3.19) further implies In the degenerate case when L/(2πR) is rational, it is still possible to choose the definitions of n i and w i such that the split of the conservation law (3.19) in (3.20) is satisfied. We therefore require that (3.20) hold in the rest of the paper. Note that the amplitude in (3.15) involves only open strings, which can only join or split on a D-brane. See Figure 5 for an illustration. The total number of D-branes n i a n f a n i b n f b Figure 5: The vertical lines represent D-branes separated by the same distance L , and the blue wavy lines represent fundamental strings with their ends residing on different D-branes. The picture on the left represents the initial state and the picture on the right represents the final state of the scattering process. During the scattering process, the two strings join into one at the second D-brane from the left; then, the single intermediate string splits at the third D-brane into two. This picture captures the conservation law n 1 + n 2 + n 3 + n 4 = 0 , with n 1 = 1 , n 2 = 3 , and n 3 = n 4 = −2 . The incoming strings on the left carry the initial quantum numbers n i a = n 1 and n i b = n 2 . The outgoing strings on the right carry the quantum numbers n f a = −n 3 and n f b = −n 4 . The amplitude for this type of scattering process is given by (3.14). The total number of D-branes involved in the scattering process is equal to n 1 + n 2 + 1 = 5 . that are involved in the scattering process is given by n + + 1 , where n + is the sum of n i > 0 associated with the incoming closed strings. See Figure 1 and 2. In contrast, scatterings between open strings that do not occur on the same D-brane necessarily involve closed strings. One such example is given in Figure 6. These scattering processes with asymptotic closed string states are not captured by the open string amplitudes that we compute here and are irrelevant in the KLT relation that we consider in this paper.
Intriguingly, if we take L = 2 π R , with R = α /R the radius of the T-dual circle over which the X-direction is compactified, we find that K M i in (3.17) becomes which is identical to K Li defined for closed strings in (2.10a), and n i can be interpreted as the KK number for the closed string tachyon. 6 For the open string amplitude in (3.16) to match the expression in (2.36a), we have to make the following rescalings of α , R , and L : Here, the rescaling of R and L essentially uniformly rescales the geometry of the system.
The second identity in (3.24b) comes from the open string relation which is the stringy generalization [2,3] of the Bern-Carraso-Johansson (BCJ) relation in field theories [15]. However, in the case of nonzero winding, the field theory limit α → 0 does not converge, as winding states do not have a conventional QFT interpretation.

Four-Point KLT Relation
We are finally ready to write down the first KLT relation for winding strings. Plugging where C denotes contribution from the cocycle factors, with and applying (2.46), we find which is (3.27b) but with "L" and "R" switched, and it matches (2.44b). Similarly, we can show that (3.27b) is equal to which is (3.27a) but with "L" and "R" switched, and it matches (2.44a). From the above four tachyon amplitudes, we make the following general observations of the KLT relation for winding strings, which are also valid for amplitudes involving massless and massive vertex operators. The KLT relation in (3.27) is for string amplitudes in spacetime with a spatial direction compactified over a circle of radius R . The KLT relation maps the closed string amplitude to a sum of the products of a pair of open string amplitudes with the open strings ending on an array of D-branes transverse to the compactified direction. The D-branes are positioned equidistantly along the spatial circle, with the consecutive ones separated by L = 2π R , where R = α /R is the T-dual radius. For a closed string that carries winding number w i and KK number n i , 7 the corresponding open string wraps w i times around the compactified circle. In addition, excluding the w i times that it wraps around the circle, the open string is stretched between the q-th and (q + n i )-th branes. 8 Using (3.8), we find that 7 For tachyons we have n i w i = 0 , which implies that at least one of n i and w i is zero. However, when other vertex operators are considered, the quantity n i w i is not necessarily zero. 8 Since the series of D-branes is wrapped around the compactified circle, the shortest separation along the circle between the ends of the open string might span over fewer than n i − 1 D-branes. For example, in the diagram on the right in Figure 4, if w i = 0 , the open string starts at the q-th brane and ends at the (q + 4)-th brane in the series, which implies that n i = 4 . However, the shortest separation between the ends of the open string along the compactified circle is just between two D-branes. the closed string momentum n i /R along the compactified direction is mapped to the fractional winding number (defined in (3.8)) of the open strings. The number of D-branes that are involved in the scattering between open strings is n + + 1 , with n + the sum of n i > 0 associated with the incoming closed strings. See Figure 1  The gauge field A is associated with a U (n + + 1) Chan-Paton charge, where n + + 1 is the number of D-branes. Since we take A to be constant, one can diagonalize it to be breaking the gauge group to U (1) n + +1 . Note that θ q is T-dual to the location of the q-th D-brane. The field A is pure gauge, but a Wilson loop that winds around the compactified circle of dual radius R gains a phase factor of the form leading to a shift in the momentum n i / R along the compactified X-direction. For a string in the Chan-Paton state |q + w i , q , its canonical momentum in X is Here, n i = w i and w i = n i , with w i (n i ) the winding (momentum) number of the corresponding closed string. The above observations hold for any higher point amplitudes among general vertex operators discussed in §2.2. We prove the general KLT relation for winding strings in the next section. We also note that all such constructions straightforwardly generalize to the Narain compactification, in which more than one spatial direction is compactified. In this more case, we need to consider Dp-branes that are transverse to multiple compactified directions.

KLT Relation for Higher-Point Amplitudes
In this section, we derive the KLT relation for higher-point amplitudes, involving winding closed string vertex operators that take the general form as discussed in §2.2.

Higher-Point Closed String Amplitude
We start with an N -point tree-level closed string amplitude, which, similar to the four-point amplitude in (2.26), takes the following general form: We defined z ij ≡ z i − z j . Here, the factor F comes from contracting the derivative terms in the vertex operators that for example take the form of (2.21) and (2.24). The factor F is single valued. Fixing z 1 = 0 , z N −1 = 1 , and z N = ∞ , we find where the cocycle factor C(1 , . . . , N ) is defined by We have abbreviated F (z 1 , . . . , z N ; z 1 , . . . , z N ) z 1 =0, z N −1 =1, z N =∞ by dropping the arguments z 1 , z N −1 , and z N , as well as their complex conjugates. We write schematically the i-th closed string vertex operators as With appropriate choices of the polarization tensors (i) , this general operator reduces to the vertex operators considered in (2.21) and (2.24). In general, the tensors (i) satisfy various conditions by imposing BRST invariance on the vertex operator. The dispersion relation and the level matching condition for the state corresponding to the vertex operator (4.5) are in form the same as the expressions in (2.23). In the following calculation, we employ the standard trick in [1] and rewrite the polarization tensor as (i) Under this replacement, the F -factor in (4.3) decomposes as Note that F L, R are single-valued functions that do not contribute any branch points. Collecting terms multilinear in Li and Ri and replacing their product with (i) , we recover the original amplitude (4.3). In the following, we will take M c to denote the expression in (4.8), with the splitting of the polarization factor taken into account implicitly.

Splitting the Worldsheet Integrals
It is a standard exercise to perform the integrals over the complex plane such that it splits into two sets of contour integrals for the left-and right-movers, respectively [1], bearing caveats concerning the cocycle factor. For completeness, we will review the complex integral and articulate the differences. First, make the change of variables with z i = x i + i y i for real variables x i and y i . Then, we promote the integrals over y i to be in the complex plane, with the branch points at i x i , −i(1 − x i ) , and y j + i(x i − x j ) . Deform the contour for the y i -integral that runs along the real axis by rotating it counterclockwise by π 2 − , → 0 + . This is done by a second change of variables, We require that such deformations of the contours are done simultaneously for all the y i -integrals, in order to ensure that none of the branch points are crossed. Define in terms of which we rewrite (4.8) as We now perform the integrals over ξ i for fixed ζ i , i = 2, . . . , N − 2 . First, we define the permutation σ(2, . . . , N − 2) of the particle numbers such that the fixed values of ζ i 's are ordered as In the case when ζ σ(2) ∈ (−∞ , 0) , the branch points for the ξ σ(2) -integral are at All these branch points reside in the upper half plane. Completing the contour in the lower half plane, we find that the contribution from ζ σ(2) ∈ (−∞ , 0) to the amplitude M c is zero. Similarly, when ζ σ(N −2) ∈ (1 , ∞) , all the branch points for the ξ σ(N −2)integral are in the lower half plane. Therefore, the contribution from ζ σ(N −2) ∈ (1, ∞) to M c is also zero. Therefore, the integrals in (4.11) are only nonzero when which partitions up the associated integration region from 0 to 1. As a result, (4.11) becomes with the integration domain and When k < , for a fixed ξ σ( ) , the ξ σ(k) -integral has a branch point at 19) which resides in the upper half plane if the ξ σ(k) -contour traverses the real axis. This implies that, in order to avoid branch points, we need to shift the contour associated with the ξ σ( ) -integral to be slightly above the contour associated with the ξ σ(k) -integral in the complex plane [4]. We now elaborate on the factors that appear in (4.18a). When σ( ) < σ(k) , this factor ultimately comes from in (4.8). To pass from (4.21) to (4.20), an extra phase factor that is a function of K Li appears. The same reordering for z i 's in (4.8) also introduces a similar phase factor, which is a function of K Ri . In the case when there is zero winding, we have K Li = K Ri and there is no extra phase factor introduced after simultaneously reordering z i 's and z i 's. This is, however, not the case when winding modes are present: The extra phase factors from reordering z i 's and z i 's do not exactly cancel when K Li = K Ri . The remainder phase factor can be absorbed into the cocycle factor C(1 , . . . , N ) in (4.11), which turns the cocycle factor into C 1 , σ(2 , . . . , N − 2) , N − 1 , N in (4.16), where the arguments are permuted. The above observation is summarized by the following identity: (4.22)

Higher-Point KLT Relation
Now, the integrals over ζ i and ξ i in (4.16) can be performed in the standard way as in [1,3,4], generalizing the same procedure for four-point amplitudes in §2.3 to N -point amplitudes. We find the following N -point KLT relation:    Note that (4.23) is independent of ∈ {3 , . . . , N − 2} . This is reminiscent of the four-point amplitude that we discussed earlier in §2.3, which has two equivalent expressions in (2.37) and (2.38), depending on whether the integration contour for the ξ -integral is deformed to the left or right. To derive the N -point amplitude (4.23), we have deformed − 2 contours associated with the ξ σ(k) -integrals, k = 2, . . . , − 1 , and the remaining N − − 1 contours have been deformed to the right, without crossing any branch cuts. From the open string amplitude perspective, the existence of different ways of writing the KLT relation is due to the open string generalization of the Kleiss-Kuijf and BCJ relations [2][3][4]. 9 The KLT relation can also be cast in equivalent forms other than (4.23). When = 2 , all the contours for the ξ i -integral are deformed to the right, yielding   The expressions in (4.26) and (4.27) can be thought as two limiting cases of (4.23). 10 To show explicitly that these expressions are invariant under swapping the subscripts "L" and "R," we start with the expression in (4.26) and permute the closed string vertex operators as There is also a different representation of KLT relations as noted in [1], which can be obtained by evaluating the integrals in (4.11) directly without performing any further deformation of the integration contours. This representation has a simple combinatorial interpretation [20]. In the winding case, this representation has the benefit of making the resulting KLT relation manifestly symmetric with respect to the subscripts "L" and "R. We have swapped the dummy permutation symbol ρ and σ to get the second equality above. Furthermore, note that where  which is exactly the expression in (4.27) but with the subscripts "L" and "R" swapped. Another way to understand this invariance under switching "L" and "R" is by observing that (4.27) comes from integrating {ξ i } before {ζ i } in (4.11), and that (4.31) comes from integrating {ζ i } before {ξ i } in (4.11). Therefore, the invariance under switching "L" and "R" is the consequence of the invariance under commuting the orders of performing different integrals. Similarly, starting with the expression from switching "L" and "R" in (4.27) and then going through the analogous procedure in (4.28) ∼ (4.31), we arrive at (4.26) but with the subscripts "L" and "R" swapped. It also follows immediately that the more general KLT relation in (4.23) has to hold after switching "L" and "R."

Conclusions and Outlooks
In this paper we extended the KLT relation to winding strings in a toroidal compactification of string theory. We showed that the string amplitudes of winding closed strings factorize into quadratic products In the case of nonzero winding, there is no field theory limit by simply sending the Regge slope α to zero. However, there still is a nonsingular α → 0 limit for winding strings in the presence of a near critical Kalb-Ramond field. This type of limit was originally applied to open strings on spacetime filling D-branes, which led to noncommutative open string (NCOS) theory [21,22]. This NCOS limit was then realized for winding closed strings [23][24][25] as well as winding open strings with Dirichlet boundary conditions [26,27], leading to self-contained nonrelativistic closed and open string theory, where the string spectra enjoy a Galilean-invariant dispersion relation. It has also been shown that NCOS is T-dual to nonrelativistic open string theory [28]. Recently, there has been growing interest in nonrelativistic string theory, leading to a plethora of applications to nonrelativistic gravity and field theory (see, e.g., [27,[29][30][31][32][33][34][35][36][37]). Amplitudes for nonrelativistic closed strings have been studied in [24], while amplitudes for nonrelativistic open strings and the associated KLT relation are still in progress [38]. This endeavor may also enrich the current S-matrix program to include nonrelativistic field theories.
The KLT relation for winding string amplitudes explored in this paper opens up new possibilities of generalizing other modern techniques developed for field-theoretical amplitudes to the study of winding string states. The program of mapping out structures of winding string amplitudes may bring us new insights into string theory and quantum gravity in general.
of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.

A. KLT and Twisted Period Relation
In this appendix, we review the modern interpretation of the KLT relation using intersection theory. We will see that the KLT relation for winding strings given in §4.3 is indeed underlay by the twisted Riemann's period relation [39] (which generalizes Riemann's bilinear relation), and the inverse KLT kernel receives the geometrical interpretation as an intersection number between cycles in the twisted homology, defined on the moduli space. This is a direct generalization of the results in [9,40] for the standard KLT relation, which we follow closely in the following discussion. For the mathematical part of the following discussion, we mostly follow [41,42].

A.1. Twisted Cocycles and Closed String Amplitudes
We start with the closed string side of the KLT relation. Recall the integrals for closed string amplitudes in (4.8), and its split form in (4.16), which we choose to separate into the multi-valued function given by the Koba-Nielsen factor, and the following factors that are single valued: Note that z 1 = 0 and z N −1 = 1 are fixed. Using the above notation, the closed string amplitude (4.8) can be written as This closed string amplitude is reminiscent of the intersection number between cocycles in the twisted cohomology, which we brief in the following.
We start with the definition of the twisted cohomology [43] on the moduli space. The singular loci of the Koba-Nielsen factor (A.1) are We define a divisor D as the union of the hyperplanes in (A.4). Then, the closed string moduli space is X = C N −3 \ D . Consider the twisted logarithmic one-form of the multi-valued Koba-Nielsen factor (A.1) [9], and the covariant derivative ∇ ω = d + ω∧ on X . We require that α 2 K Ri · K Rj ∈ R \ Z . Here, E i denotes the scattering equation [5,44]. With respect to ∇ ω , we define the (N − 3)-th twisted de Rham cohomology where Ω k (X) is the space of smooth k-forms on X . 11 Consider the single-valued (N −3)forms on X , ϕ L (z 2 , · · · , z N −2 ) = f L (z 2 , · · · , z N −2 ) dz 2 ∧ · · · ∧ dz N −2 , (A.7a) ϕ R (z 2 , · · · , z N −2 ) = f R (z 2 , · · · , z N −2 ) dz 2 ∧ · · · ∧ dz N −2 , (A.7b) which are cocycles in H N −3 (X,∇ ω ) . We now consider a pairing between H N −3 (X,∇ ω ) and its dual H N −3 c (X,∇ ω ∨ ) . It is important that the dual cohomology is with compact support, and hence the subscript "c" [43]. We keep this implicit in the following expressions. These two cohomologies are dual to each other under the pairing Defining ω ∨ ≡ ω , U ∨ ≡ U , and ϕ ∨ L ≡ ϕ R , where the barred quantities are complex conjugations, we find In terms of (A.9), the closed string amplitude (A.3) becomes M c (1, · · · , N ) = C(1, · · · , N ) ϕ L , ϕ R , (A.10) Note that this is different from the pairing of two k-th twisted cohomology groups that gives rise to the intersection number of k-th twisted cocycles, which requires a different choice of the dual quantities, namely with ω ∨ = −ω and U ∨ = U −1 [43]. The duality that we are interested in here, which maps ω and U to their conjugations, has been studied in [9,42].

A.2. Loaded Cycles and Open String Amplitudes
Now we move on to the open string side. In terms of the Koba-Nielsen factor U in (A.1) and the differential form ϕ L in (A.7a), the open string amplitude M L introduced in (4.24) can be cast in the form where permutation β fixes the cyclic ordering β(1), · · · , β(N ) . We introduced the Θ-factor, Θ i 1 , · · · , i N j 1 , · · · , j N , that contributes a factor (−1) n i w j +n j w i if the ordering of i and j is the opposite in the ordered sets (i 1 , · · · , i N ) and (j 1 , · · · , j N ) . In the case when represents the (N − 3)-dimensional region 0 < z σ(2) < · · · < z σ(N −2) < 1 . The closure of this integration domain is an (N − 3)-simplex in a smooth triangulation of the real section of the moduli space X which we refer to as X(R) . Since there exist more than two coalescing punctures, a minimal blowup of the moduli space is required [42]. The blowup of a simplex is an associahedron K N −1 [46,47]. The multi-valued Koba-Nielsen factor U has its branch fixed with respect to the simplex ∆(β) in (A.11), giving rise to quantity U ∆ that is single-valued in ∆(β) . In the concrete example (A.13), we have In general, for a given permutation β in ∆(β) , there corresponds a unique associahedron, for which a particular branch of the Koba-Nielsen factor is chosen [9]. The above pairing between the simplex ∆(β) and the single-valued branch choice U (β) motivates the definition of a loaded simplex [48], Γ(β) = ∆(β) ⊗ U (β) . Using the loaded simplices we define the (N − 3)-th twisted homology group H N −3 (X, L ω ) , with coefficients in the sheaf L ω consisting of local solutions to dψ = ω ψ , where ψ is a locally holomorphic function that has the formal solution ψ ∝ U [41,43]. Then, the loading U in a loaded cycle ∆ ⊗ U is a section of L ω on ∆ . The open string amplitude (A.11) is now defined as a pairing between the twisted homology group H N −3 (X, L ω ) and the twisted cohomology group H N −3 (X,∇ ω ) , (A. 16) Here, ∆(β) ⊗ U (β) is a loaded cycle 12 that is an element of H N −3 (X, L ω ) , defined on the non-compact manifold X . Furthermore, we define the locally finite homology group H l.f. N −3 (X, L ω ) [48], and introduce a pairing between H l.f. where ∆(β) ⊗ U (β) is a loaded cycle in H l.f. N −3 (X, L ω ) .

A.3. Intersection Number and Twisted Period Relation
Finally, we consider a pairing between H N −3 (X, L ω ) and the locally finite homology group H l.f. N −3 (X, L ω ) . This pairing gives rise to a geometric interpretation for the inverse KLT kernel as an intersection number between two loaded cycles [9]. There exists an isomorphism map [41], The map "reg" can be realized by a regularization of the loaded cycles. For example, when N = 4 , and when the tachyonic vertex operators are concerned, we encounter the integral which can be generalized to arbitrary N . See [9,41,42] for further details. The intersection number between the two loaded cycles is defined with respect to the intersection points between the paths Γ and reg Γ . In general, we define the intersection number as the following pairing between reg Γ(β) ∈ H l.f. Here, ∆ and ∆ are N − 3 simplices and I p denotes the topological intersection number at the intersecting point p , which only depends on the relative orientations of ∆ and ∆ at p . It is shown in [9] that (A.22) precisely gives the inverse of the KLT kernel that is a function of K Ri . For example, see Figure 8 for intersection numbers that give rise to the inverse kernels in the KLT relations from (3.27). See [9] for detailed calculation. This intersection number between two loaded cycles also receives the physical interpretation as the α -corrected bi-adjoint scalar amplitudes [8].
instead of the expression for U in (A.1) that has been considered through this appendix, we will land on the twisted Riemann's period relation that represents the same KLT relations but with "L" and "R" swapped (which of course swaps K Li and K Ri ).