$\Omega$ versus Graviphoton

I study the deformation of the topological string by $\bar\Omega$, the complex conjugate of the $\Omega$-deformation. Namely, I identify $\bar\Omega$ in terms of a physical state in the string spectrum and verify that the deformed Yang-Mills and ADHM actions are reproduced. This completes the study initiated recently [1] where we show that $\bar\Omega$ decouples from the one-loop topological amplitudes in heterotic string theory. Similarly to the N=2* deformation, I show that the quadratic terms in the effective action play a crucial role in obtaining the correct realisation of the full $\Omega$-deformation. Finally, I comment on the differences between the graviphoton and the $\Omega$-deformation in general and discuss possible $\bar\Omega$ remnants at the boundary of the string moduli space.


Introduction
Little attention has been devoted to the study of the Ω-background as a non-holomorphic deformation at the string level. Indeed, from the gauge theory point of view, the Ω-deformation is a background that twists space-time in a particular way allowing for a path-integral derivation of the partition function through localisation [2][3][4]. Viewed as a background, it is non-holomorphic in the sense that it has a holomorphic as well as an anti-holomorphic part denotedΩ. However, the latter decouples from physical observables and this is clear from the localisation perspective whereΩ is a Q-exact deformation of the effective action.
From the string theory point of view, there is a natural topological limit that one can impose on the Ω-background. If one denotes 1,2 the two parameters of the latter, then, for 1 + 2 = 0 the Ω-deformed gauge theory partition function is the field theory limit of the topological string partition function F g [5] which computes a class of higher derivative gravitational couplings in the effective action [6]. The connection between string amplitudes and supersymmetric gauge theories has been further extended beyond topological limit [7][8][9]. In all these studies, the anti-holomorphic partΩ is implicitly set to zero. This is understandable from the gauge theory side because of the decoupling ofΩ. In string theory, however, it is not clear, a priori, that the same property remains true. A particular instance where the breakdown, in string theory, of gauge theory properties is the moduli dependence of topological amplitudes. Indeed, as shown in [10], the topological amplitudes

The stringyΩ
In [1], starting from heterotic string theory compactified on T 2 × K3, we postulated thatΩ can be identified as a constant background for a self-dual field strength F T which is the vector partner of the Kähler modulus T of T 2 . Here, I repeat the same analysis in the context of the dual type I string theory.
Recall that, in ten dimensions, type I and heterotic string theories are S-dual to each other.
However, in four dimensions, there are regions in the moduli space where both theories are weakly coupled [13]. Hence, I focus on those particular regions and consider a type I theory compactified on T 2 × K3. Here, I realise K3 as a T 4 /Z 2 orientifold which admits both D9-and D5-branes.
Tadpole cancellation restricts the number of such D-branes. However, I keep their number generic as one could consider a non-compact K3 as well. In table 1, I summarise the mapping between the universal scalar fields of the vector multiplet moduli space.
Heterotic Type I S S T S' U U Notice that the axion-dilatons are mapped to one another [13,14]. Furthermore, the complex structure modulus U of T 2 is unchanged and the Kähler modulus T in Heterotic is mapped to another dilaton-like field denoted S . Indeed, the presence of two dilatons in Type I is not surprising since the theory contains D9-and D5-branes whose coupling constants are given by the imaginary parts of two different fields, namely S and S respectively.
In order to identify theΩ deformation in terms of the type I physical string spectrum, I focus on the universal vector multiplet sector, i.e. the so-called STU-model. Namely, the three vector multiplet moduli are S, S' and U. To keep the discussion clear, I do not consider Wilson lines. In addition to the three vector fields associated to each scalar, there is another vector field stemming from the N = 2 gravity multiplet, the graviphoton, so that there are four gauge fields denoted F I with I = G, S, S , U .
Since the question under consideration is the realisation of the Ω-deformation, recall that one can construct it geometrically as a non-trivial T 2 fibration over space-time, in such a way that, when one goes around the cycles of T 2 , the space-time fields are rotated with an arbitrary angle. This geometric picture of a reduction from six dimensions on the metric with line element 3]] are space-time indices, Z is the complexified T 2 coordinate and Ω,Ω are the space-time rotation matrices depending on two complex parameters 1,2 . Hence, the Lorentz group SU (2) L × SU (2) R is explicitly broken. For practical purposes, redefine them as For the present matter, I work in the topological limit + = 0. From the seminal work [6], one can show that the holomorphic part of the Ω-deformation can be described as a constant background for the self-dual part of the graviphoton field strength. The self-duality condition means that − ≡ is sensitive to SU (2) L only. In this convention, for + = 0, SU (2) R remains unbroken. Consequently, in order to describeΩ in type I string theory, one should turn on a constant background for the self-dual field strength of one of the following fields.
The superscript (−) stresses the fact that the field strengths are self-dual. Notice that I have only written the bosonic part of the vertex operators which should be completed by the fermionic and the Ramond-Ramond parts 2 . As opposed to heterotic, the latter arises typically in orientifold compactifications since the supercharges are combinations of left-and right-moving ones. In addition, I have implicitly assumed that the polarisation µ is chosen in such a way to satisfy the self-duality constraint. Recall that the graviphoton vertex operator is Consequently, one is led to choosing F S (−) and F S (−) as natural candidates forΩ. Here, 'natural' means thatΩ is understood, in some sense, as the complex conjugate of Ω.
In order to rigorously make the selection, I first consider the effect of each of these states on the gauge theory degrees of freedom. More precisely, following the ideas pioneered in [15] and further exploited in [8,11,12], I analyse the possible couplings of the field-strengths with the gauge theory degrees of freedom (including instantons) realised as the massless excitations of the open strings in Dp-D(p+4) branes systems. This is naturally realised in the dual type I string theory [8] studied in the following section.

Vertex Operators
In the present section, I use the D9-branes to realise the gauge theory. By T-duality, this is equivalent to the D3-branes realisation [15], see also [16] for a review and complete analysis. In this setup, gauge theory instantons are realised in terms of D5-branes called D5-instantons wrapping T 2 × K3. This configuration is summarised in the following table.
In the 9-9 sector, the massless excitations consist of a number of N = 2 vector multiplets, each of which containing a vector field A µ , a complex scalar φ as well as four gaugini (Λ αA , Λα A ) that are in the (2, 1) representation of SU (2) + × SU (2) − . The bosonic degrees of freedom stem from the NS sector, while the fermionic ones from the R sector. These fields, taken separately, realise a sector field  Yang-Mills theory in four dimensions. Their vertex operators are In the 5-5 and NS sector, there are ten bosonic moduli that can be written as a real vector a µ and six scalars χ I . From the perspective of the gauge theory living on the world-volume of the D9-branes, a µ parametrises the position of gauge theory instantons. In the Ramond sector, there are sixteen fermionic moduli M αA , λα A . The vertex operators of these states are 3) Here, g 5 is the D5-brane coupling constant. For a Dp-brane, it is given by Finally, let's consider the mixed moduli. From the NS sector, the fermionic coordinates give rise to two Weyl spinors (ωα,ωα) of SO(4). They have the same chirality due to the specific choice of boundary conditions of the D5-branes. In addition, they describe the size of the instanton. In the R sector, one gets two Weyl fermions (µ A ,μ A ) transforming in the fundamental representation of SO (6). The vertex operators for the mixed sectors contain the twist operators that change the coordinates boundary conditions from Dirichlet to Neumann and vice versa. These are bosonic fields ∆,∆ of conformal dimension 1/4. The vertex operators are This summarises the complete description of the gauge theory states in terms of CFT vertex operators. In order to study theΩ-deformation, one must include the closed string operators describing it. As mentioned above, the preserved supercharges are a combination of the left-and right-moving supercharges due to the orientifold action. This implies that the vertex operators for the vector fields have, generically, an NS-NS and a R-R part each. They can be derived explicitly by spectral flow from the universal scalars' vertex operators. In [7], the self-dual graviphoton and U-vector field operators were shown to take the form . Also note that Ψ is the worldsheet fermion in the T 2 direction. It is important to notice that the difference between the two states is merely the sign between the NS-NS and R-R parts. Similarly, the vertex operators for the candidate states, i.e.

Deformed Yang-Mills action
I now calculate the deformations to the Yang-Mills effective action due to the candidates for thē Ω-deformation. In order to achieve this, I calculate all possible tree-level couplings between the selfdual S-and S'-vectors to leading order in α . In the type I theory under study, these are simply disc diagrams whose boundary lies entirely on a D9-brane. In addition, I insert at least one self-dual Sor S'-vector vertex operator in the bulk of the disc diagram (and possibly some graviphotons) while including an arbitrary number of open string insertions at the boundary from the 9-9 sector. In fact, the number of such insertions is highly restricted by the fact that I am only interested in the gauge theory action.
From a practical point of view, to avoid repeating the calculations twice, I split the vertex operator for theΩ-deformation into an NS-NS part V 1 and a R-R one V 2 . It turns out that only few disc amplitudes are potentially non-vanishing in the field theory limit and some of those are summarised in Fig. 1. Figure 1: Disc diagrams in the 9-9 sector. Diagram (a) involves two bosonic boundary insertions and one bulk insertion ofF , diagram (b) two fermionic insertions and oneF , whereas diagram (c) has two bosonic boundary insertions and two bulk insertions of a graviphoton and anF .
I start by evaluating the amplitude with the gaugini. Notice that, due to the specific self-duality of the closed string vertex, the only possible non-trivial couplings are with Λ αA . Furthermore, had one used the graviphoton vertex instead, the amplitude would have vanished trivially by the nonconservation of the SO(2) T 2 -charge. Using the doubling trick, I convert the disc into the full plane with a Z 2 involution. I then split the closed string vertices into their left-and right-moving parts which can be considered as independent. As such, the tree-point disc amplitude becomes effectively a four-point amplitude. In addition, I soak up c-ghost zero-modes on the sphere by attaching c to three dimension-one operators in such a way that the resulting vertex operator is BRST closed. In order to avoid subtleties due to the use of unintegrated vertex operators in different ghost pictures as discussed, for instance, in [16], I don't attach the c-ghost to operators in the zero-picture. The contribution from the NS-NS part ofF is Notice that I have denoted the left and right parts of the NS-NS vertex by two different labels. The double brackets notation means that one should integrate over n − 3 free positions with n being the total number of inserted points. In (3.10), one must integrate over z 4 only. Performing the contractions between the various operators using the standard results summarised in Appendix A, I find (3.11) I have used the notation z ij ≡ z i − z j . The integral over z 4 is straightforward. I identify¯ using a particular normalisation forF µν :F with η c µν being the 't Hooft symbols. Thus, I obtain I now turn to the contribution of the R-R piece of the closed string operator which is After making all possible contractions, the result is (3.15) Therefore, one gets exactly the same result as for the NS-NS contribution: This is to be expected based on supersymmetry arguments. One now puts both results together and notices a clear difference between the S-and the S -vectors. Namely, for the F S one has In fact, to the leading order I am concerned by, there are no further couplings between the closed string vertices and the fermionic fields.
This is already a clear indication thatΩ is realised by the S -vector as I now confirm from the following calculations. Note that this type of cancellations was already observed in a similar setup in [8] where the problem of the refinement is addressed. Now consider the possible couplings with the bosonic open string degrees of freedom. The first non-trivial coupling corresponds to diagram (a) in Figure 1. As before, I split the closed string vertex into its NS-NS and R-R parts. The setup for the vertex operators is as follows.
Hence, I choose to integrate over z 2 only. Notice that only the fermion bilinear term of the gauge field can lead to a non-vanishing amplitude. In addition, one can already set the momenta of the exponentials to zero since this amplitude turns out not to stem from contact terms. The correlator is thus where I have introduced the notation F αβ ≡ (σ µν ) αβ F µν . Using the results of Appendix A, one can calculate the CFT correlator and find Integrating over the disc, one obtains Including the NS-NS part, the total contribution to the action becomes Finally, I analyse the possible quadratic terms in the Ω-deformation. I first consider possible mass terms for the gauge fields, namely I calculate the coupling between two gauge fields, one graviphoton and one S -vector, and choose for convenience to fix the positions z 1 , z 2 and z 4 to −∞, 1 and −ix with x being real (so that z 3 = ix). This is done by attaching a c-ghost to the corresponding vertex operators. In order to keep the calculation simple, I take the gauge fields' vertex operators in the 0-ghost picture.
However, by BRST invariance, this requires including an additional contribution to the vertex (3.1) as discussed in [8,16]. Hence, the vertex operators are explicitly given by Here, I have only written the R-R parts of the closed string vertices whose contributions are considered first. Furthermore, recall that momentum conservation along the Neumann directions implies that p 1 + p 2 + P +P = 0. Even though I compute a coupling at a fixed momentum order, I keep the momenta generic in order to consistently regularise the worldsheet integrals. Equivalently, the contribution of interest arises as a contact term of the form (p i · p j )/(p i · p j ) such that it is crucial to send the momenta to zero only at the end.
There are, in principle, several terms stemming from whether one takes the bosonic or fermionic parts of the gauge fields. First of all, one can show that the terms needed to reinforce BRST invariance of the gauge fields vertex operators do not lead to any non-trivial contributions. There are thus four separate CFT correlators all of which are multiplied by Here, I have defined p 3 = p 4 = P/2 and p 5 = p 6 =P /2. If one takes both gauge fields to give their bosonic term, one obtains the term (3.32) Notice that I have only kept the terms that would be non vanishing once I take into account the polarisation vectors in A 0 . However, in the gauge chosen for the worldsheet positions, this term is zero by the transversality condition p µ 1 A µ = 0. The second term obtained when only the gauge field at z 2 gives its bosonic piece is [ βγ (σ µν ) αδ z 14 z 15 z 36 + αδ (σ µν ) βγ z 13 z 16 z 45 ] Similarly to A 1 , when the gauge field at z 1 only gives its bosonic piece then the correlator vanishes.
Finally, when all vertices give their fermionic terms, one finds   The full correlator is thus A 0 (A 2 + A 4 ) which I now integrate over the worldsheet positions x, z 5 and z 6 . For generic space-time momenta, this is a well defined integral over R × C. Instead of performing directly the calculation using the analytic structure of the integrand, I use the beautiful results of [18] in which this type of complex integrals is mapped to real integrals appearing in colour ordered amplitudes in gauge theory. More precisely, it was proven that Once the leading contributions from the worldsheet integrals is extracted, one evaluates the total amplitude A 0 (A 2 + A 4 ) in the limit of vanishing momenta. The identities for the traces of Pauli matrices derived in Appendix A turn out to be very useful. The result takes the following simple form: One can perform the same analysis for all other possible terms, for instance when including the NS-NS parts of the vertex operators, and I find Before concluding this session, one should check that the full deformation does not completely break supersymmetry, even in the presence ofΩ. More precisely, this boils down to showing that the scalar component of the gauge multiplet remains massless. In order to achieve that, one should calculate all possible couplings between two scalars φ,φ and the Ω,Ω deformation in the field theory limit. Instead of performing the disc amplitude calculations as before, I simply note that this precise analysis has already been done in [19] where it was shown that the mass deformation realised at the level of the worldsheet does not lead to a mass term for the Higgs scalar. The situation here is exactly the same by a mere exchange of the space-time C 2 and the C 2 of the internal space used in [19]. It is important to note that the role of the quadratic deformation is crucial to ensure this property. Namely, as extensively studied in [1], the full description of the Ω-deformation requires the introduction of a background for the scalar field S at second order in momenta and proportional to ¯ . As reviewed in Section 4, in heterotic string theory, this additional ingredient implies an exact decoupling ofΩ from the topological amplitudes F g as a cancellation between bosonic and fermionic degrees of freedom. In the present case, it ensures that supersymmetry is not broken while it does not change the form of the remaining couplings.
To summarise our findings, I write down the full Ω-deformed Yang-Mills action derived from the realisation of the deformation as a constant background of particular string states. The action is which shows that the realisation through the graviphoton and the S -vector is consistent. In the following section, I prove that this statement holds non-perturbatively.

Deformed ADHM action
I now focus on the deformation of the ADHM action stemming from the closed string vertices. As I now show, this confirms the choice of the vertex made previously and leads to the explicit expression of theΩ-deformed instanton effective action as a check. As in the Yang-Mills case, there is a limited number of couplings surviving the field theory limit and self-duality projections. Some of these are summarised in Figure 2.
In principle, there could as well be couplings with mixed D5/D9 boundary conditions. However, due to the self-duality constraint, such couplings are zero. In the case of a general Ω-background, this is of course not any longer true [8].
I first consider the coupling involving fermionic ADHM moduli. That is, for the NS-NS part, has two bosonic boundary insertions and two bulk insertions of a graviphoton and anF .
the g 2 6 factor from the vertices (3.4) cancels with the normalisation of the D5 amplitude. Therefore, I can immediately state the result. Namely, for the S one has , and all the (−1)-picture vertices to be of dimension zero (such that their positions remain unintegrated) so that one must insert one PCO. As I show below, the amplitude takes the form of contact terms in the momenta of the form p i .p j /p i .p j . These contact terms survive in the limit p i → 0. To be able to compute them in a well-defined manner, the momenta p i must be kept generic as they also act as regulators of the worldsheet integrals. The limit of vanishing momenta is only taken at the end of the calculation. Notice, however, that due to the nature of our vertex insertions, none of the ADHM moduli can carry momenta along X µ since the four-dimensional space-time corresponds to directions with Dirichlet boundary conditions for the D5-instantons. Similarly, the S -vector insertions cannot carry momenta along T 2 because of BRST invariance. As a way out, I turn on complex momenta along the K3 directions for all the vertices to make all integrals well-defined. Technically, this means that I first replace K3 by R 4 and compute the amplitude on the D-instanton world-volume T 2 × R 4 . Since the fields appearing in this amplitude survive the orbifold projection 3 , the coupling of interest also exists in the case where R 4 is replaced by K3.
As the role of space-time and internal momenta is crucial, and for the sake of clarity, I write the relevant vertex operators below.
V a (z 1 ) = g 6 a µ (∂X µ − 2ip 1 · χ ψ µ )e 2ip 1 ·Y (z 1 ) , Here, Y i ∈ {X 6 , X 7 , X 8 , X 9 } are coordinates of the internal R 4 . The momenta p i are along these directions, while the momentum of V FS is written as (P µ , P ), where P µ is the space-time part and P is along the Y i directions. Note that after using the doubling trick, the Neumann directions (Z,Z, Y i ) are mapped onto themselves, whereas the Dirichlet ones pick an additional minus sign X µ → −X µ . This is consistent with the fact that the momenta along Neumann directions are conserved. On the other hand, integrating over the zero modes of the Dirichlet directions X µ does not give rise to any conservation law for the momentum P µ .
The three open string vertices contain Chan-Paton labels that should be suitably ordered. For instance, in order to obtain the term Tr(a µ a ν χ), the range of integration is For the other nonequivalent ordering Tr(a µ χ a ν ), the range of integration in z 3 is opposite. It is easy to show that the sum of these two orderings gives Tr(a µ [χ, a ν ]).
For definiteness, I first focus on the term Tr(a µ a ν χ). The contractions of the ghosts, superghosts and the exponentials in momenta yield to the amplitude. This can only contract with ∂Z(y) in V PCO while Ψ(y) contracts withΨ(z). Hence, ψ λ (z) must contract with ψ ν (z 2 ) and from z 1 only ∂X µ (z 1 ) can contribute. The result is (3.54) Next consider the contribution of the second term in (3.48) where there are several contributions.
First, if p 3 · χ(z 3 ) contracts with p 1 · χ(z 1 ), then ψ µ (z 1 ), ψ ν (z 2 ), ψ λ (z) and a space-time fermion ψ σ (y) from the PCO must contract, leaving ∂X σ (y) to contract with the momentum parts of the operators at z andz. This gives a term proportional to P σ . Secondly, notice that the term arising from contracting ψ µ with ψ ν is killed by the transversality condition. The total result is .
On the other hand, if the term p 3 · χ(z 3 ) contracts with χ(y) in V PCO , ∂Y (y) must contract with momentum dependent parts of the vertices. Thus, ψ λ (z) contracts with ψ ν (z 2 ) and only ∂X µ at z 1 contributes so that one obtains . (3.56) The total correlation function is thus given by which I now integrate over z 1 and z 3 . Note that all the terms in A 1 , A 2 and A 3 come with a single power of space-time momentum P µ which is exactly what is required to obtain a coupling to the field strength of the closed string state. However, both A 2 and A 3 are quadratic in the momenta along the Y i directions. Hence, they can only contribute to the amplitude in the zero-momentum limit if the integration over z 1 and z 3 gives a pole 1/(p a · p b ). Clearly, A 0 · A 3 cannot give such a pole 4 . On the other hand, the integral over z 3 for A 0 · A 2 gives a 1/(p 1 · p 3 ) pole. Performing the z 3 integral in the regions (3.52) yields the same result. Consequently, the z 1 integral over the entire real line reads Tr [a µ a ν χ] where I have set all the momenta along the Y i directions to zero given that there are no singularities in the remaining integral. Note that the integral A 0 · A 2 is not gauge invariant. As for the A 0 A 1 term the z 3 and z 1 integrals contain no singularities so that the momenta along the Y i directions can be set to zero. The resulting z 3 integral for both regions (3.52) gives the same result: Tr Adding the two terms (3.58) and (3.59), the total result becomes gauge invariant and the z 1 integration yields 5 : Finally, let us consider the R-R contributions. The vertex operators are the same as above, except for the closed string vertex: Since the total superghost charge is −2, there is no need for a PCO. The total T 2 charge implies that only p 3 · χ Ψ(x 3 ) in (3.48) contributes and, thus, only p 1 · χ ψ µ (x 1 ) in (3.46) contributes. This term is proportional to p 1 · p 3 . As before, the integral over z 3 gives a pole 1/(p 1 · p 3 ) in the channel where z 3 → z 1 . Performing the integrals over z 1 and z 3 gives the same result: Finally, summing over the nonequivalent orderings of the open vertex operators gives we proposed a deformation of the latter to capture alsoΩ as a constant field strength for the vector partner of the Kähler modulus of T 2 denoted F T 6 . Recall that W ij µν is the supergravity multiplet carrying (anti-symmetrised) indices i, j = 1, 2 for the SU (2) R R-symmetry group. It contains the graviphoton field-strength F G , the field strength tensor B i µν of an SU (2) doublet of gravitini and the Riemann tensor.
In order to include additional insertions of F T , I deform the coupling I g as where K T is the descendent superfield whose lowest component is F T . Similarly to the previous type I analysis, it turns out that, in order to obtain a good description ofΩ, one should also include an arbitrary number of the scalar field T at two derivatives. At the level of the worldsheet sigma model, this translates into a quadratic deformation proportional to ¯ as before.

Amplitude calculation
In order to calculate the coupling F g,n , recall that the new couplings while for the T -vectors these are For simplicity, I choose the term in (4.2) in which there are 2g graviphotons and 2n+2 field strengths V T so that the space-time zero-modes are soaked up by two vertices V T . Hence, the amplitude to calculate is which calculates the second derivative of F g,n with respect to T . Here, V φ is the vertex operators of the scalar field T . As explained in [1], after soaking up the space-time zero-modes, one finds that the coupling is integrable in the sense that one can pull out the two derivatives with respect to T . Furthermore, by summing over g, n and m, one can define a generating function in which all the amplitudes with arbitrary number of fields reduce to a Gaussian deformation of the worldsheet sigma-model! That is, the generating function calculates at once all the couplings F g,n which in turn can be recovered by picking a particular power of and¯ . After a careful analysis of (4.6), one shows that the generating function is independent of¯ . Indeed, it is given by The details of the notation and the derivation can be found in [1]. In particular, expanding in the deformation parameters, one finds that with F g denoting here the heterotic one-loop topological amplitude.

Conclusions
In the present paper, I have realised theΩ-deformation, the complex conjugate of the Ω-deformation, in string theory in terms of a physical field in the string spectrum. In type I string theory compactified on T 2 × K3, that is the vector partner of the S scalar which describes the coupling of the D5-branes.
By coupling this field to the graviphoton as well as to all the massless degrees of freedom of the D5-D9 system, I have derived the deformed Yang-Mills and ADHM effective actions. As already noticed in [1], it is crucial to include a quadratic deformation which corresponds to giving a background to the S scalar at quadratic order in the momenta. This proves that the combination of graviphoton and S -vector, together with the quadratic background, is a correct description of the full Ω-background.
Furthermore, as already shown in [1], one can go beyond the pure field theory analysis by calculating, in the dual heterotic theory, the topological amplitudes F g deformed by theΩ-deformation.
Surprisingly, not only one recovers the perturbative part of the Nekrasov partition function, but also the full string result is independent ofΩ. It would be interesting to understand this statement purely at the string level as a Q-exactness of some operator, as in the gauge theory.
From the conceptual point of view, the present analysis shows that the graviphoton differs from the Ω-deformation. Indeed, it is clear that the quadratic deformation is essential to give a correct description of the full non-holomorphic Ω-deformation. Yet, this additional piece corresponds to a different field in the string spectrum. However, Ω and the graviphoton agree at linear order and this is why, practically, one can neglect this subtlety.
As a natural application, it would be interesting to generalise my study beyond the topological limit. This is a priori tedious since, generically, the full Ω-deformation in string theory breaks topological invariance. Yet, simplifications must occur once the completeΩ-deformation is included since the field theory limit is purely holomorphic. It is not clear, however, that the decoupling at the string level would still hold. Indeed, F (±) µν = ∓ 1 2 µνρσ (F (±) ) ρσ . Also, note the following useful identity: