Superstring Vertex Operators and Pictures

A string state is not just a vector in a Hilbert space. In the worldsheet path integral it is represented by a local operator inserted on the surface. This is the state-operator correspondence applied to the conformal field theory of $X^\mu$, $\psi^\mu$, the $bc$ ghosts, and the $\beta\gamma$ superghosts.

For the bosonic string, the first examples are familiar:

$$V_T=c\,e^{ik\cdot X}, \qquad V_A=c\,\zeta_\mu\partial X^\mu e^{ik\cdot X}.$$

In the RNS superstring there is one new ingredient that becomes central: the same physical state has many representatives labeled by picture number. A massless open-string gauge boson, for example, may be written in the $-1$ picture as

$$V_A^{(-1)}=c\,e^{-\phi}\,\zeta_\mu\psi^\mu e^{ik\cdot X},$$

or in the $0$ picture as

$$V_A^{(0)} = \zeta_\mu\left(\partial X^\mu+i k_\nu\psi^\nu\psi^\mu\right)e^{ik\cdot X},$$

in the common $\alpha'=2$ CFT normalization. These two operators are not different particles. They are different BRST representatives of the same physical state. The operation that relates them is picture-changing.

This page develops the dictionary systematically. The guiding principle is

$$\boxed{ \text{physical string states} \quad\Longleftrightarrow\quad \text{BRST cohomology classes of vertex operators}. }$$

The vertex operator must also have the correct ghost number and picture number for the worldsheet amplitude in which it appears.

In the RNS formalism a physical state has several picture representatives. The $-1$ NS and $-1/2$ Ramond representatives are often the simplest; picture-changing produces equivalent representatives in other pictures.

Conventions for dimensions

The local matter OPEs are normalized as on the previous pages,

$$X^\mu(z)X^\nu(w)\sim -\eta^{\mu\nu}\ln(z-w), \qquad \psi^\mu(z)\psi^\nu(w)\sim {\eta^{\mu\nu}\over z-w},$$

so that the holomorphic matter stress tensor is

$$T_{\rm m} =-{1\over2}:\partial X^\mu\partial X_\mu: -{1\over2}:\psi^\mu\partial\psi_\mu:.$$

For a closed-string chiral factor, this convention gives

$$h\!\big(e^{ik\cdot X}\big)={\alpha'k^2\over4},$$

with $\alpha'=2$ in the displayed OPE. For open-string boundary vertices, the doubling trick gives twice the chiral contraction, and the boundary scaling dimension of the plane wave is

$$h_{\partial}\!\big(e^{ik\cdot X}\big)=\alpha' k^2.$$

Since $k^2=-M^2$ in mostly-plus spacetime signature, these formulas reproduce the open-string RNS mass formula

$$\alpha'M^2=N_{\rm NS}-{1\over2}$$

in the NS sector. For example, the unprojected NS ground-state vertex contains $e^{-\phi}e^{ik\cdot X}$, whose boundary dimension is

$${1\over2}+\alpha'k^2.$$

Requiring boundary dimension $1$ gives $\alpha'k^2=1/2$, or $\alpha'M^2=-1/2$, the NS tachyon. The GSO projection removes this state in the supersymmetric open string.

Superghost bosonization and picture number

The commuting superconformal ghosts are bosonized as

$$\boxed{ \beta=e^{-\phi}\partial\xi, \qquad \gamma=\eta e^\phi. }$$

Here $\eta$ and $\xi$ are anticommuting fields of weights $1$ and $0$, while $\phi$ is a boson with background charge. The stress tensor of the $\phi$ system is conventionally written

$$T_\phi=-{1\over2}:\partial\phi\partial\phi:-\partial^2\phi,$$

which implies

$$\boxed{ h(e^{q\phi})=-{1\over2}q(q+2). }$$

Several values should become automatic:

$$\begin{array}{c|c|c} \text{factor} & q & h \\ \hline e^{-\phi} & -1 & 1/2 \\ e^{-\phi/2} & -1/2 & 3/8 \\ e^\phi & 1 & -3/2 \end{array}$$

The picture number is the charge measured by the zero mode of the picture current. For the simple exponentials above, it is just the exponent $q$:

$$P(e^{q\phi})=q.$$

Thus NS vertices are naturally written in integer pictures, such as $-1$ and $0$, while Ramond vertices are naturally written in half-integer pictures, such as $-1/2$ and $+1/2$.

A crucial global fact is that the $\phi$ background charge produces a picture-number anomaly. On a genus-$g$ closed Riemann surface, each chiral sector must carry total picture number

$$\boxed{P_{\rm total}=2g-2.}$$

On the sphere this is $P=-2$ for the left-movers and $\widetilde P=-2$ for the right-movers. On the disk, an open-string tree amplitude also requires total picture number $-2$. This is why a tree-level open-string amplitude with $n$ NS external states is often computed with two vertices in the $-1$ picture and the remaining $n-2$ vertices in the $0$ picture.

Integrated and unintegrated vertices

A vertex operator has two closely related forms.

For an open string on the disk or upper half-plane, an unintegrated vertex has the form

$$\boxed{ \mathcal V=c\,V_{\rm matter+superghost}, }$$

where $V_{\rm matter+superghost}$ has boundary dimension $1$. Since $c$ has dimension $-1$, the full unintegrated insertion has dimension $0$. This is exactly what is needed for an insertion at a fixed point on the boundary.

An integrated open-string vertex is

$$\boxed{ \int_{\partial\Sigma} dx\, U(x), }$$

where $U$ has boundary dimension $1$ and no $c$ ghost. Integrated vertices are used for punctures whose positions are not fixed by conformal symmetry.

For closed strings on the sphere, an unintegrated vertex has the form

$$\boxed{ \mathcal V=c\widetilde c\,V(z,\bar z), }$$

where $V$ has weights $(1,1)$. An integrated closed-string vertex is

$$\boxed{ \int_\Sigma d^2z\,V(z,\bar z). }$$

The rule of thumb for tree amplitudes is simple:

$$\begin{array}{c|c|c} \text{worldsheet} & \text{conformal Killing group} & \text{fixed insertions} \\ \hline \text{disk} & PSL(2,\mathbb R) & 3\ \text{open vertices carry }c \\ \text{sphere} & PSL(2,\mathbb C) & 3\ \text{closed vertices carry }c\widetilde c \end{array}$$

The remaining vertices are integrated. In BRST language, the relation between the two forms follows from descent. If $\mathcal V=cV$ is BRST closed, then the corresponding integrated operator is obtained by acting with the $b$-ghost zero mode along the modulus direction. Schematically,

$$[Q_B,U]=\partial(\cdots),$$

so the integral of $U$ is BRST invariant up to boundary terms on moduli space.

Open-string NS vertices

Before the GSO projection, the NS ground state is tachyonic. Its simplest unintegrated vertex is

$$\boxed{ \mathcal V_T^{(-1)}=c\,e^{-\phi}e^{ik\cdot X}. }$$

The matter-superghost part has dimension

$${1\over2}+\alpha'k^2.$$

The condition that this be $1$ gives

$$\alpha'k^2={1\over2}, \qquad \alpha'M^2=-{1\over2}.$$

The GSO projection keeps the opposite NS fermion parity from the tachyon, so this operator is absent in the supersymmetric open-string spectrum.

The first GSO-even NS state is the massless vector. In the $-1$ picture its unintegrated vertex is

$$\boxed{ \mathcal V_A^{(-1)} =g_o\,c\,e^{-\phi}\,\zeta_\mu\psi^\mu e^{ik\cdot X}\,T^a. }$$

Here $T^a$ is a Chan—Paton generator, $g_o$ is the open-string coupling, and $\zeta_\mu$ is the polarization. The dimension of $e^{-\phi}\psi^\mu$ is

$${1\over2}+{1\over2}=1,$$

so dimension one requires

$$\boxed{k^2=0.}$$

BRST closure also imposes transversality,

$$\boxed{k\cdot\zeta=0.}$$

The gauge redundancy is

$$\boxed{\zeta_\mu\sim \zeta_\mu+\lambda k_\mu.}$$

This is not merely a spacetime guess. In the worldsheet theory, the longitudinal polarization is BRST exact. The photon gauge symmetry is therefore a statement about the BRST cohomology of open-string vertex operators.

The corresponding $0$-picture representative is

$$\boxed{ \mathcal V_A^{(0)} =g_o\,\zeta_\mu \left(\partial X^\mu+i k_\nu\psi^\nu\psi^\mu\right)e^{ik\cdot X}\,T^a, }$$

again in the $\alpha'=2$ local CFT normalization. Depending on whether one writes the vertex on the boundary of the upper half-plane or in a purely holomorphic doubled convention, harmless factors of $2\alpha'$ may be redistributed between $k$ and $\partial X$. The invariant content is the pair of terms

$$\partial X^\mu \qquad\text{and}\qquad k\cdot\psi\,\psi^\mu.$$

The first term is the bosonic coupling to the embedding coordinate; the second is required by worldsheet supersymmetry.

Massless vertices organize the spacetime fields: open-string vectors live on boundaries, while closed-string fields are products of left- and right-moving sectors inserted in the bulk.

Open-string Ramond vertices

Ramond ground states are spacetime spinors. The local operator that creates a Ramond branch cut for the worldsheet fermions is the spin field $S_\alpha$. In ten dimensions its holomorphic dimension is

$$h(S_\alpha)={10\over16}={5\over8}.$$

The natural massless Ramond vertex is in the $-1/2$ picture:

$$\boxed{ \mathcal V_\lambda^{(-1/2)} =g_o\,c\,e^{-\phi/2}\,u^\alpha S_\alpha e^{ik\cdot X}\,T^a. }$$

The superghost factor has dimension $3/8$, so

$$h(e^{-\phi/2}S_\alpha)= {3\over8}+{5\over8}=1.$$

Therefore the plane wave must have zero dimension:

$$\boxed{k^2=0.}$$

The Ramond physical-state condition $G_0|u;k\rangle=0$ becomes the massless Dirac equation. In vertex-operator language it is the condition that the supercurrent OPE have no forbidden singularity:

$$\boxed{k_\mu\Gamma^\mu u=0.}$$

The GSO projection selects a definite ten-dimensional chirality,

$$\Gamma_{11}u=\pm u,$$

where the sign depends on the theory and on the chosen open-string sector. Together, the GSO-projected NS vector and Ramond spinor form the ten-dimensional $N=1$ super Yang—Mills multiplet on a stack of D-branes.

Closed-string vertices

Closed-string vertices are left-right products. The simplest massless NS—NS vertex is

$$\boxed{ \mathcal V_{\rm NSNS}^{(-1,-1)} =g_c\,c\widetilde c\, e^{-\phi}e^{-\widetilde\phi}\, \epsilon_{\mu\nu}\psi^\mu\widetilde\psi^\nu e^{ik\cdot X}. }$$

The left-moving factor $e^{-\phi}\psi^\mu$ has dimension $1$, and the right-moving factor $e^{-\widetilde\phi}\widetilde\psi^\nu$ has dimension $1$. Thus the plane wave must be massless:

$$\boxed{k^2=0.}$$

BRST closure gives the transversality conditions

$$\boxed{k^\mu\epsilon_{\mu\nu}=0, \qquad \epsilon_{\mu\nu}k^\nu=0.}$$

BRST exact states implement the gauge equivalences

$$\epsilon_{\mu\nu} \sim \epsilon_{\mu\nu}+k_\mu\xi_\nu+k_\nu\widetilde\xi_\mu.$$

The polarization tensor decomposes into the familiar NS—NS spacetime fields:

$$\epsilon_{\mu\nu} = \underbrace{\epsilon_{(\mu\nu)}^{\rm traceless}}_{\text{graviton }G_{\mu\nu}} + \underbrace{\epsilon_{[\mu\nu]}}_{\text{two-form }B_{\mu\nu}} + \underbrace{{1\over D-2}\eta_{\mu\nu}\epsilon^\rho{}_{\rho}}_{\text{dilaton }\Phi},$$

up to the usual refinement that the dilaton polarization must be chosen transverse modulo gauge transformations.

The mixed sectors give spacetime fermions. In schematic notation,

$$\mathcal V_{\rm RNS}^{(-1/2,-1)} =c\widetilde c\,e^{-\phi/2}S_\alpha\, e^{-\widetilde\phi}\widetilde\psi^\mu \,u^\alpha{}_{\mu}e^{ik\cdot X},$$

and similarly for NS—R. These contain the gravitino and dilatino.

The RR sector is represented by a bispinor vertex,

$$\boxed{ \mathcal V_{\rm RR}^{(-1/2,-1/2)} =c\widetilde c\,e^{-\phi/2}e^{-\widetilde\phi/2} S_\alpha\widetilde S_\beta\, \mathcal F^{\alpha\beta}e^{ik\cdot X}. }$$

The bispinor $\mathcal F^{\alpha\beta}$ is equivalent, via gamma matrices, to a sum of differential-form field strengths:

$$\mathcal F^{\alpha\beta} \sim \sum_n {1\over n!}F_{\mu_1\cdots\mu_n} \left(C\Gamma^{\mu_1\cdots\mu_n}\right)^{\alpha\beta}.$$

The allowed degrees are fixed by the chiral GSO projections. Equivalently, type IIA has odd RR potentials $C_1,C_3,\ldots$, while type IIB has even RR potentials $C_0,C_2,C_4,\ldots$, with the five-form field strength self-dual.

Picture-changing

Picture-changing is generated by the BRST commutator with $\xi$:

$$\boxed{ X_{\rm PCO}(z)=\{Q_B,\xi(z)\}. }$$

The full picture-changing operator contains matter and ghost terms. Its most important term for simple on-shell vertices is

$$X_{\rm PCO}(z)=e^\phi T_F^{\rm m}(z)+\text{ghost terms}.$$

The ghost terms are essential for exact BRST invariance, but the leading matter term explains the familiar conversion from the $-1$ picture to the $0$ picture.

For a BRST-closed small-Hilbert-space vertex $V^{(-1)}$, the picture-changed vertex is

$$\boxed{ V^{(0)}(w)=\lim_{z\to w}X_{\rm PCO}(z)V^{(-1)}(w). }$$

The key OPEs are

$$e^\phi(z)e^{-\phi}(w)\sim z-w,$$

and

$$T_F^{\rm m}(z) \left[\zeta_\mu\psi^\mu e^{ik\cdot X}\right](w) \sim {\zeta_\mu\left(\partial X^\mu+i k_\nu\psi^\nu\psi^\mu\right)e^{ik\cdot X}(w) \over z-w}.$$

Multiplying the two OPEs gives a finite limit:

$$\boxed{ X_{\rm PCO}\cdot \left(e^{-\phi}\zeta\cdot\psi e^{ik\cdot X}\right) = \zeta_\mu\left(\partial X^\mu+i k_\nu\psi^\nu\psi^\mu\right)e^{ik\cdot X}. }$$

This is precisely the $0$-picture vector vertex.

The picture-changing operator raises picture number by one. For the massless vector, the factor $e^\phi e^{-\phi}\sim z-w$ cancels the simple pole in the matter supercurrent OPE and leaves the $0$-picture vertex.

Picture-changing is often described as an isomorphism of BRST cohomologies at different pictures. This statement is correct for the standard on-shell cohomology, but it comes with a practical warning: picture-changing operators are local insertions. If a PCO collides with another operator or with a boundary of moduli space, one can encounter contact terms. For elementary tree amplitudes this is usually avoided by choosing convenient fixed pictures from the start.

How BRST closure encodes the physical-state conditions

The old covariant quantization conditions were

$$L_n|\Phi\rangle=0, \qquad G_r|\Phi\rangle=0 \qquad (n>0,\ r>0),$$

with an additional $G_0$ constraint in the Ramond sector. In the operator language these conditions become statements about singular terms in OPEs with $T$ and $T_F$, and finally about BRST cohomology.

For the open NS vector, the matter part of the $-1$ picture operator is

$$W_A^{(-1)}=e^{-\phi}\zeta\cdot\psi e^{ik\cdot X}.$$

BRST closure of $cW_A^{(-1)}$ requires $W_A^{(-1)}$ to have dimension $1$ and to be a superconformal primary. The dimension condition gives

$$k^2=0.$$

The supercurrent condition gives

$$k\cdot\zeta=0.$$

The remaining equivalence relation

$$\zeta_\mu\sim\zeta_\mu+k_\mu\lambda$$

comes from quotienting by BRST-exact operators. Thus the BRST cohomology contains precisely the transverse photon polarizations.

For the Ramond vertex, the same logic gives

$$k^2=0, \qquad k_\mu\Gamma^\mu u=0,$$

and quotienting by exact states removes unphysical spinor components. In light-cone language this leaves the $8_s$ or $8_c$ of $SO(8)$, paired by spacetime supersymmetry with the $8_v$ of the vector.

Useful vertex-operator dictionary

The following table collects the most commonly used on-shell representatives. For open-string vertices, the displayed operators are boundary insertions. For closed-string vertices, holomorphic and antiholomorphic factors are multiplied.

$$\begin{array}{c|c|c|c} \text{sector} & \text{state} & \text{picture} & \text{vertex, without coupling constants} \\ \hline \text{open NS} & \text{tachyon, removed by GSO} & -1 & c e^{-\phi}e^{ikX} \\ \text{open NS} & A_\mu & -1 & c e^{-\phi}\zeta_\mu\psi^\mu e^{ikX} \\ \text{open NS} & A_\mu & 0 & \zeta_\mu(\partial X^\mu+i k\cdot\psi\,\psi^\mu)e^{ikX} \\ \text{open R} & \lambda_\alpha & -1/2 & c e^{-\phi/2}u^\alpha S_\alpha e^{ikX} \\ \text{closed NS--NS} & G_{\mu\nu},B_{\mu\nu},\Phi & (-1,-1) & c\widetilde c e^{-\phi-\widetilde\phi}\epsilon_{\mu\nu}\psi^\mu\widetilde\psi^\nu e^{ikX} \\ \text{closed R--R} & F_n & (-1/2,-1/2) & c\widetilde c e^{-(\phi+\widetilde\phi)/2}S_\alpha\widetilde S_\beta\mathcal F^{\alpha\beta}e^{ikX} \end{array}$$

The table suppresses Chan—Paton factors for open strings and the distinction between integrated and unintegrated closed-string vertices. It also suppresses cocycle factors, which are needed for fully precise mutual locality of spin fields but do not change the physical-state conditions.

Amplitude bookkeeping

A well-defined amplitude must satisfy three separate bookkeeping rules.

First, the $bc$ ghost zero modes must be soaked up. On the disk or sphere this means using three unintegrated vertices. For example, an open-string color-ordered disk amplitude can be organized as

$$\mathcal A_n =\left\langle \mathcal V_1(x_1)\mathcal V_2(x_2)\mathcal V_3(x_3) \prod_{i=4}^n\int dx_i\,U_i(x_i) \right\rangle_{\rm disk},$$

with the three fixed positions removing the volume of $PSL(2,\mathbb R)$.

Second, the total picture number must match the superghost anomaly. For an open-string disk amplitude with NS external states,

$$\sum_i P_i=-2.$$

A standard choice is

$$(-1,-1,0,0,\ldots,0).$$

For an amplitude with two Ramond fermions and any number of NS bosons, a common choice is

$$\left(-{1\over2},-{1\over2},-1,0,\ldots,0\right).$$

Third, the vertices must be BRST closed, and any BRST-exact insertion decouples from the amplitude unless there are boundary contributions in moduli space. This is the worldsheet origin of spacetime gauge invariance and of the Ward identities obeyed by string scattering amplitudes.

Exercises

Exercise 1: dimensions of superghost exponentials

Using

$$h(e^{q\phi})=-{1\over2}q(q+2),$$

compute the dimensions of $e^{-\phi}$, $e^{-\phi/2}$, $e^\phi$, and $e^{-2\phi}$.

Solution

Substitute the four values of $q$.

For $q=-1$,

$$h(e^{-\phi})=-{1\over2}(-1)(1)={1\over2}.$$

For $q=-1/2$,

$$h(e^{-\phi/2})=-{1\over2}\left(-{1\over2}\right)\left({3\over2}\right)={3\over8}.$$

For $q=1$,

$$h(e^\phi)=-{1\over2}(1)(3)=-{3\over2}.$$

For $q=-2$,

$$h(e^{-2\phi})=-{1\over2}(-2)(0)=0.$$

The last result is useful in inverse picture-changing and in discussions of the small versus large Hilbert space.

Exercise 2: mass shell of the open NS vertices

Use boundary dimensions to show that $c e^{-\phi}e^{ikX}$ has $\alpha'M^2=-1/2$, while $c e^{-\phi}\zeta\cdot\psi e^{ikX}$ is massless.

Solution

For an unintegrated open vertex $cW$, the operator $W$ must have boundary dimension $1$.

For the NS ground-state vertex,

$$W_T=e^{-\phi}e^{ikX},$$

and

$$h(W_T)={1\over2}+\alpha'k^2.$$

Setting $h(W_T)=1$ gives

$$\alpha'k^2={1\over2}.$$

Since $k^2=-M^2$,

$$\alpha'M^2=-{1\over2}.$$

For the vector,

$$W_A=e^{-\phi}\zeta\cdot\psi e^{ikX},$$

so

$$h(W_A)={1\over2}+{1\over2}+\alpha'k^2.$$

Setting $h(W_A)=1$ gives $\alpha'k^2=0$, hence $M^2=0$.

Exercise 3: picture-changing the vector

Using the leading term $X_{\rm PCO}=e^\phi T_F^{\rm m}+\cdots$, show that picture-changing the $-1$ picture vector gives the $0$ picture vector.

Solution

The relevant OPEs are

$$e^\phi(z)e^{-\phi}(w)\sim z-w$$

and

$$T_F^{\rm m}(z)\,\zeta_\mu\psi^\mu e^{ikX}(w) \sim {\zeta_\mu\left(\partial X^\mu+i k\cdot\psi\,\psi^\mu\right)e^{ikX}(w) \over z-w}.$$

The factor $z-w$ from the superghost OPE cancels the simple pole from the matter supercurrent OPE. Taking $z\to w$ gives

$$X_{\rm PCO}\cdot \left(e^{-\phi}\zeta\cdot\psi e^{ikX}\right) = \zeta_\mu\left(\partial X^\mu+i k\cdot\psi\,\psi^\mu\right)e^{ikX}.$$

This is the zero-picture vector vertex, up to convention-dependent overall normalization.

Exercise 4: Ramond dimension and the Dirac equation

Show that the open Ramond vertex $c e^{-\phi/2}u^\alpha S_\alpha e^{ikX}$ is massless by dimension counting. Then state the additional physical-state condition on $u$.

Solution

In ten dimensions the spin field has dimension

$$h(S_\alpha)={10\over16}={5\over8}.$$

The superghost factor has dimension

$$h(e^{-\phi/2})={3\over8}.$$

Therefore

$$h(e^{-\phi/2}S_\alpha)=1.$$

For the full matter-superghost factor to have boundary dimension $1$, the plane wave must have zero dimension:

$$\alpha'k^2=0.$$

Thus the state is massless. The Ramond zero-mode constraint gives the spacetime Dirac equation

$$k_\mu\Gamma^\mu u=0.$$

The GSO projection further chooses one ten-dimensional chirality for $u$.

Exercise 5: picture-number balance

What picture assignments would you use for the following tree amplitudes?

1. A disk amplitude with four external open-string gauge bosons.
2. A disk amplitude with two open-string Ramond fermions and two open-string gauge bosons.
3. A sphere amplitude with three external NS—NS closed-string states.

Solution

On the disk the total open-string picture number must be $-2$.

For four gauge bosons, a standard assignment is

$$(-1,-1,0,0).$$

For two Ramond fermions and two gauge bosons, a standard assignment is

$$\left(-{1\over2},-{1\over2},-1,0\right).$$

The two Ramond vertices contribute $-1$ total, so one NS vertex must be placed in the $-1$ picture and the other in the $0$ picture.

On the sphere the left-moving and right-moving total picture numbers must each be $-2$. For three NS—NS states, one convenient assignment is

$$(-1,-1),\quad (-1,-1),\quad (0,0).$$

Equivalently, one may distribute picture-changing operators in other ways, as long as the total is $(-2,-2)$.

Exercise 6: NS—NS gauge redundancies

Starting from the massless NS—NS polarization tensor $\epsilon_{\mu\nu}$, explain how the graviton, two-form, and dilaton arise, and identify the corresponding linearized gauge redundancies.

Solution

The polarization decomposes as

$$\epsilon_{\mu\nu} =\epsilon_{(\mu\nu)}^{\rm traceless} +\epsilon_{[\mu\nu]} +{1\over D-2}\eta_{\mu\nu}\epsilon^\rho{}_{\rho}.$$

The symmetric traceless part is the graviton polarization, the antisymmetric part is the Kalb—Ramond two-form polarization, and the trace is the dilaton polarization.

The BRST-exact polarizations generate

$$\delta\epsilon_{\mu\nu}=k_\mu\xi_\nu+k_\nu\widetilde\xi_\mu.$$

Splitting the parameters into symmetric and antisymmetric combinations gives the linearized diffeomorphism of the graviton and the two-form gauge transformation

$$\delta B_{\mu\nu}=k_\mu\Lambda_\nu-k_\nu\Lambda_\mu.$$

The dilaton is the gauge-invariant scalar combination left after imposing transversality and quotienting by these redundancies.