Linear T-duality Research Articles

Abelian S-matrix elements of D-brane and T-duality

Recently, it has been suggested that the elements of the S-matrix on the worldvolume of an Abelian D-brane might be in accordance with the Ward identity associated with the T-duality. This shows that by applying linear T-duality, a group of S-matrix elements could be found invariant under such transformations. In this work, we apply the T-duality transformations on the S-matrix elements of one B-field and three NS Abelian gauge fields and find some other Abelian S-matrix elements of one closed and three open string states. Also, we will show that the predicted S-matrix elements are reproduced exactly by explicit calculations.

Higher derivative corrections to the DBI action at α′2 order

We use the compatibility of D-brane action with linear off-shell T-duality and linear on-shell S-duality as guiding principles to find all world volume couplings of one massless closed and three massless open strings at order $\alpha'^2$ in type II superstring theories in flat space-time.

Higher derivative corrections to the Wess-Zumino action

In the first part of this paper, we calculate the disk-level S-matrix elements of one RR, one NSNS and one NS vertex operators, and show that they are consistent with the amplitudes that have been recently found by applying various Ward identities. We show that the massless poles of the amplitude at low energy are fully consistent with the known D-brane couplings at order $\alpha'^2$ which involve one RR or NSNS and two NS fields. Subtracting the massless poles, we then find the contact terms of one RR, one NSNS and one NS fields at order $\alpha'^2$. Some of these terms are reproduced by the Taylor expansion and the pull-back of two closed string couplings, some other couplings are reproduced by linear graviton in the second fundamental form and by the B-field in the gauge field extension $F\rightarrow F+B$, in one closed and two open string couplings. In the second part, we write all independent covariant contractions of one RR, one NSNS and one NS fields with unknown coefficients. We then constrain the couplings to be consistent with the linear T-duality and with the above contact terms. Interestingly, we have found that up to total derivative terms and Bianchi identities, these constraints uniquely fix all the unknown coefficients.

On T-duality of R 2-corrections to DBI action at all orders of gauge field

Recently, it has been observed that in a T-duality invariant world-volume theory in flat spacetime, all orders of gauge field strength and all orders of the D-brane velocity appear in two specific matrices. Using these two matrices, we construct the world-volume couplings of two massless NSNS states at order $\alpha'^2$ and all orders of the velocity and the gauge field strength, by requiring them to be invariant under the linear T-duality. The standard extension $F\rightarrow F+P[B]$, then produces all orders of the pull-back of B-field into the action. We compare the resulting couplings for zero velocity and gauge field strength, with the $\alpha'^2$ terms of the disk-level S-matrix element of two massless NSNS vertex operators in the presence of a constant background B-field. We have found an exact agreement.

D-brane action at orderα′2

We use compatibility of D-brane action with linear T-duality, S-duality and with S-matrix elements as guiding principles to find all world volume couplings of one massless closed and two open strings at order $\alpha'^2$ in type II superstring theories. In particular, we find that the squares of second fundamental form appear only in world volume curvatures, and confirm the observation that dilaton appears in string frame action via the transformation $\hat{R}_{\mu\nu}\rightarrow \hat{R}_{\mu\nu}+\nabla_{\mu}\nabla_{\nu}\Phi$.

Higher-derivative corrections to type II supergravity: Four Ramond-Ramond terms

It is known that the sphere-level S-matrix element of four type II superstrings has one kinematic factor. At the low energy limit, this factor produces the kinematic factor of the corresponding Feynman amplitudes in the supergravity. It also produces higher-derivative couplings of four strings. In this paper, we explicitly calculate the kinematic factor of four Ramond-Ramond (RR) states in the supergravity. Using this factor, we then find the eight-derivative P-even and P-odd couplings of four RR fields, including the self-dual RR five-form field strength. We show that the P-even couplings are mapped to the standard ${\overline{R}}^{4}$ couplings by linear T-duality and S-duality transformations. We also confirm the P-even couplings with direct calculations in type II superstring theories.

Generalized Riemann curvature corrections to type II supergravity

We observe that the replacement of the Riemann curvature with the generalized Riemann curvature into the corrections to the type II supergravity at order $\alpha'^3$ which are in terms of the contractions of four Riemann curvatures $R^4$, is not fully consistent with the S-matrix elements in the superstring theory. In particular, they produce non-zero S-matrix elements for odd number of B-field strengths which are not consistent with the string theory results. Using the consistency of the couplings with the linear T-duality as a guiding principle, we consider all T-duality invariant couplings and fix their coefficients by requiring them to be consistent with the S-matrix elements. The new Lagrangian density is then equivalent to the replacement of the generalized Riemann curvature into the expression $t_8t_8R^4$.

Ramond-Ramond corrections to type II supergravity at order α′3

Recently, it has been shown that the NS-NS corrections to the type II supergravity given by Gross and Sloan are invariant under the linear T-duality. In this paper, we study the invariance of this action under sequence of S-duality and linear T-duality to find the R-R corrections to the supergravity at order $\alpha'^3$, up to field redefinition.

S-matrix elements from T-duality

Recently it has been speculated that the S-matrix elements satisfy the Ward identity associated with the T-duality. This indicates that a group of S-matrix elements is invariant under the linear T-duality transformations on the external states. If one evaluates one component of such T-dual multiplet, then all other components may be found by the simple use of the linear T-duality. The assumption that fields must be independent of the Killing coordinate, however, may cause, in some cases, the T-dual multiplet not to be gauge invariant. In those cases, the S-matrix elements contain more than one T-dual multiplet which are intertwined by the gauge symmetry.In this paper, we apply the T-dual Ward identity on the S-matrix element of one RR (p−3)-form and two NSNS states on the world volume of a Dp-brane to find its corresponding T-dual multiplet. In the case that the RR potential has two transverse indices, the T-dual multiplet is gauge invariant, however, in the case that it has one transverse index the multiplet is not gauge invariant. We find a new T-dual multiplet in this case by imposing the gauge symmetry. We show that the multiplets are reproduced by explicit calculation, and their low energy contact terms at order α′2 are consistent with the existing couplings in the literature.

Ricci curvature corrections to type II supergravity

We examine the Neveu Shwarz-Neveu Shwarz corrections to the type II supergravity given by Gross and Sloan, under the linear T-duality transformations. We show that the gravity and the B-field couplings are invariant under off-shell T-duality, whereas the dilaton couplings are invariant under on-shell T-duality. Requiring the compatibility of the dilaton couplings with off-shell T-duality as a guiding principle, we extend them to include the appropriate Ricci curvature and the divergence of the B-field strength.

T-duality of the Riemann curvature corrections to supergravity

We examine the sigma model Riemann curvature corrections to the supergravity action under T-duality transformations. Using the compatibility of the effective action with on-shell linear T-duality and with the S-matrix calculations as guiding principles, we have incorporated in this action the couplings of four B-field strengths and the couplings of two Riemann curvatures and two B-field strengths at order α′3. Using the S-matrix calculations we have also found new dilaton couplings in the string frame at this order.

Four-derivative brane couplings from string amplitudes

We evaluate the string theory disc amplitude of one Ramond-Ramond field C^(p-3) and two Neveu-Schwarz B-fields in the presence of a single Dp-brane in type II string theory. From this amplitude we extract the four-derivative (or equivalently order (alpha')^2) part of the Dp-brane action involving these fields. We show that the new couplings are invariant under R-R and NS-NS gauge transformations and compatible with linear T-duality.

T-duality of anomalous Chern–Simons couplings

It is known that the anomalous D p -brane Chern–Simons couplings are not consistent with the standard rules of T-duality. Using compatibility of these couplings with the linear T-duality transformations, the B-field gauge transformations and the general coordinate transformations as guiding principles we find new couplings at order O ( α ′ 2 ) for C ( p − 3 ) , C ( p − 1 ) , C ( p + 1 ) and C ( p + 3 ) .

Towards extending the Chern-Simons couplings at order O(α′2)

Using the compatibility of the anomalous Chern-Simons couplings on D p -branes with the linear T-duality and with the antisymmetric B-field gauge transformations, some couplings have been recently found for C (p−3) at order O(α′2). We examine these couplings with the S-matrix element of one RR and two antisymmetric B-field vertex operators. We find that the S-matrix element reproduces these couplings as well as some other couplings. Each of them is invariant under the linear T-duality and the B-field gauge transformations.

On RR couplings on D-branes at order O(α′2)

Recently, it has been found that there are couplings of the RR field strength F (p) and the B-field strength H on the world volume of Dp-branes at order $ \mathcal{O}\left( {{\alpha^{{\prime}2}}} \right) $ . These couplings which have both world-volume and transverse indices, are invariant under the linear T-duality transformations. Consistency with the nonlinear T-duality indicates that the RR field strength F (p) in these couplings should be replaced by $ {\mathcal{F}^{(p)}} = d{\mathcal{C}^{\left( {p - 1} \right)}} $ where $ \mathcal{C} = {e^B}C $ . This replacement, however, produces some non-gauge invariant terms. On the other hand, the nonlinear terms are invariant under the linear T-duality transformations at the level of two B-fields. This allows one to remove some of the nonlinear terms in $ {\mathcal{F}^{(p)}} $ . We fix this by comparing the nonlinear couplings with the S-matrix element of one RR and two NSNS vertex operators. Our results indicate that in the expansion of $ {\mathcal{F}^{(p)}} $ one should keep only the B-field gauge invariant terms, e.g., B ∧ dC (p−3) where both indices of B-field lie along the brane. Moreover, in this case one should replace B with B + 2πα′f to have the B-field gauge invariance.

Ramond-Ramond field strength couplings on D-branes

By examining in details the disk level S-matrix element of one massless RR and one NSNS states at order O(\alpha'^2), we find the coupling of one RR and one NSNS fields on the world volume of a D$_p$-brane. The non-zero couplings involve the first derivative of the RR field strengths F^{(p)}, F^{(p+2)} and F^{(p+4)}. We then fix the on-shell ambiguity of the couplings by requiring consistency with the linear T-duality transformations. Moreover, consistency with the non-linear T-duality requires that the RR field strength in the above couplings to be {\cal F}=d{\cal C} where {\cal C}=e^{B}C.