Comments on Worldsheet Description of the Omega Background

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− K X g AT6-87 IPMU11-0094 CALT-68-2837, = ∞ =0 n ,i a rpsdta the that proposed was it 3, G 2 i g g − s 2 S s g ( − ǫ )gaihtn,and graviphotons, 2) b + ∞ → 2 − F N .I eto 3, section In 0. = g b ( − ǫ b supersym- 2 = + ti related is It . ) 1 ! ). 0. 6= , F g ( b have ) (3) S , 2 tions corresponding to the Omega background and the where (z, z¯) are coordinates on T 2. In this case, ǫ± given topological string computation when ǫ+ 6= 0. In section by dΩ as in (5) are complexified. It turns out that, by 5, we discuss interpretation of these backgrounds in the supersymmetry, Nekrasov’s partition function depends context of type II superstring theory. holomorphically on these parameters [7, 14]. The corresponding Kaluza-Klein ansatz is 2 2 ¯ µ µ 2. REVIEW OF THE OMEGA BACKGROUND ds = dx + (dz +2Aµdx )(dz¯ +2Aµdx ). (11) To preserve supersymmetry, the partition function (1) 2.1 Melvin-type geometry is twisted by the R symmetry. Correspondingly, we should turn on the flat gauge field coupled to the SU(2) The Omega background is a Melvin-type geometry R charge, with the metric, i i ¯ i Aj = Aj dz + Aj dz,¯ (12) 2 µ µ 2 2 ds = (dx +Ω dθ) + dθ , (4) i where i = 1, 2 is the SU(2) R symmetry index and Aj µ 4 1 4 where x and θ are coordinates on R and S respectively, has eigenvalues ±ǫ+. In the heterotic string on R × 2 and Ωµ is characterized by the equation, T × K3, the R symmetry is the isometry of K3 along i 1 2 3 4 the Reeb vector. In this case, Aj is given by the Kaluza- F = dΩ= ǫ1dx ∧ dx + ǫ2dx ∧ dx . (5) Klein gauge field associated to this isometry. 2 The parameters ǫ± in Nekrasov’s partition function (1) In the heterotic string theory on T × K3, there are are defined by, three universal vector multiplets: S (dilaton), T (complex structure moduli of T 2), and U (Kähler moduli of 1 2 ǫ± = (ǫ1 ± ǫ2). (6) T ). Combining them with the graviphoton G, there 2 are four gauge fields, AG, AS, AT , AU , which correspond A more invariant way to define ǫ± is to express the field to the off-diagonal elements of the metric (gµz,gµz¯) and strength in the spinor notation (Fαβ, Fα˙ β˙ ) (α, β = 1,2; the NS-NS anti-symmetric tensor (Bµz,Bµz¯), where µ = R4 2 α,˙ β˙ =1, 2) and write 1, 2, 3, 4 are for and (z, z¯) are coordinates on T . By inspecting their vertex operators, their precise relations 2 2 ǫ− = −detFαβ, ǫ+ = −detFα˙ β˙ . (7) are found to be, G G¯ Namely, ǫ− and ǫ+ are related to self-dual and anti-self- F− = (∂[µ,gν]z + ∂[µ,Bν]z)−, F+ = (∂[µ,gν]¯z + ∂[µ,Bν]¯z)+, dual components of the field strength. T T¯ F− = (∂[µ,gν]z − ∂[µ,Bν]z)−, F+ = (∂[µ,gν]¯z − ∂[µ,Bν]¯z)+, When ǫ+ 6= 0, we need to perform the R twist simul- U U¯ taneously in order to preserve some fraction of super- F− = (∂[µ,gν]¯z − ∂[µ,Bν]¯z)−, F+ = (∂[µ,gν]z − ∂[µ,Bν]z)+, symmetry. This requires, in particular, that the internal S S¯ F− = (∂ g + ∂ B )−, F = (∂ g + ∂ B )+, Calabi-Yau 3-fold be non-compact and the Newton con- [µ, ν]¯z [µ, ν]¯z + [µ, ν]z [µ, ν]z (13) stant in four dimensions is zero, resulting in a rigid N =2 theory. where F− and F+ refer to self-dual and anti-self-dual The Melvin-type geometry (4) is locally flat and should components of the gauge field strength. We see, for exam- be contrasted with the Kaluza-Klein ansatz, ple, that the self-dual component of (∂[µ,gν]¯z − ∂[µ,Bν]¯z) T U¯ 2 2 µ 2 is F− , but its anti-self-dual component is F+ . ds = dx + (dθ + Aµdx ) . (8) If we set A¯µ = 0 in (11), anticipating that Nekrasov’s If we choose the Kaluza-Klein gauge field Aµ and the partition function depends holomorphically on ǫ±, corresponding Ωµ for the Melvin-type geometry so that 2 2 µ 2 µ ds = dx + dz(dz¯+2Aµdx )= dx +2Aµdx dz + dzdz.¯ β β˙ (14) Aαα˙ =Ωαα˙ = Fαβ x α˙ + Fα˙ β˙ xα , (9) The field strength components for Aµ are identified with the two metrics (4) and (8) become equal at the linear those for the graviphoton and the STU gauge fields as, 2 order in x. However, they start to differ at O(x ) [6]. In G T Fαβ = Fαβ + Fαβ , the self-dual case ǫ+ = 0, the relation between computa- U¯ S¯ (15) tions in the two backgrounds was pointed out in [3]. In Fα˙ β˙ = Fα˙ β˙ + Fα˙ β˙ . this paper, we propose a generalization of this relation Moreover, if we require that the NS-NS B field is zero, for non-self-dual field strengths. we have F G = F T and F U¯ = F S¯, and thus, We can also start with a (1, 0) theory in six dimensions (for example, the heterotic string theory on K3) G T 1 2 Fαβ = Fαβ = Fαβ , and compactify it on a two-torus T with the metric, 2 (16) ¯ ¯ 1 2 µ µ ¯ µ 2 F U = F S = F . ds = (dx +Ω dz + Ω dz¯) + dzdz,¯ (10) α˙ β˙ α˙ β˙ 2 α˙ β˙ 3

To the leading order in ǫ±, we can identify this as the In the heterotic string theory, elementary BPS parti- Melvin-type geometry. Corrections in higher order in cles are in ground states in the left-moving (supersym- ǫ± will be discussed in section 3. For the Omega back- metric) sector and in the tower of bosonic oscillator exci- ground, we need to perform the R twist to preserve tations in the right-moving (bosonic) sector. From (18), fraction of supersymmetry. As we will discuss in sec- one can deduce that each of such BPS multiplets con- tions 3 and 4, it is realized by turning on the Fayet- tributes to the right-hand side of (18) as, Iliopoulos (FI) terms in the corresponding supergravity ∞ ∞ − 3 backgrounds. 4tǫ−J− 2g−2 2j dt tr e −tµ In the self-dual case (ǫ2 = −detF = 0), the one- ǫ− Fg = (−1) e , (20) + α˙ β˙ t (sinh ǫ−t)2 g=0 0 loop partition function in the Melvin background (10) X Z was shown to reproduce the all genus topological string amplitude in the limit of S → ∞. Let us discuss this in where j is the total spin of the BPS particle and µ is more detail now. its central charge for the N = 2 supersymmetry [3]. Here, the trace is taken only over the SU(2)− representation of the one-particle BPS states. An impor- 2.2 Relation to the topological string tant observation was that the right hand side of (20) is exactly the contribution of each of the BPS multiplets 3 2j −2ǫJ− −βH The genus-g contribution to the topological string par- to Tr(−1) e e from the one-loop computation tition function on a Calabi-Yau manifold X is related to [3, 5]. This was the first evidence for the identification, a scattering amplitude Ag with 2 gravitons and (2g − 2) ∞ 3 graviphotons in type II superstring theory compactified 2g−2 F −2ǫ−J− −βH ǫ− F = log Tr(−1) e e . (21) on X, all at zero momentum and with a particular self- g g=0 dual polarization [1, 2], as X h i For toric Calabi-Yau 3-folds, the topological vertex con- A = (g!)2F . (17) g g structed in [16] can be used to show this identify without The combinatorial factor (g!)2 comes from the superspace relying on the heterotic dual or the S → ∞ limit. Note that the right-hand side of this equation is the integral to relate the F-terms computed by Fg to the low energy effective action of type II superstring theory vacuum amplitude in the N = 2 theory computed in together with the particular choice of the polarization. the self-dual (ǫ+ = 0) Omega background. The left-hand In the heterotic string theory on T 2 × K3, the dilaton side, however, is not a vacuum amplitude, but it is a series S belongs to a vector multiplet. When type II theory on of F-terms evaluated on the self-dual graviphoton back- X has a heterotic string dual, and in the limit S → ∞, ground before performing the superspace integral. If we the topological string partition function can be computed were computing the vacuum amplitude in the left-hand by the heterotic string one-loop as [15] side, we should have included the additional combinatorial factor of (g!)2, among other possible contributions. ∞ ∞ 2g−2 1 2g−2 ǫ− F = ǫ− A g (g!)2 g g=0 g=0 X X (18) 3. SUPERSYMMETRIC CONFIGURATIONS 1 2 −∆Seff 4 2 WITH FIELD STRENGTHS IN FOUR = 2 dτ he iR ×T × ZK3 ǫ− DIMENSIONS Z with When ǫ+ = 0, Nekrasov’s partition function is the vac- 1 1 2 uum amplitude on the self-dual Omega background, and ∆Seff =ǫ− (X ∂Z − χ ψZ )∂X¯ (19) it is related to the F-terms for the self-dual graviphoton Z 1 1 2 background. To find their generalizations for ǫ 6= 0, it + (X¯ ∂Z − χ¯ ψZ )∂¯X¯ . + is natural to look for configurations with both self-dual In (18), τ is the moduli of the worldsheet torus and the and anti-self-dual field strengths turned on. As we will integral is over the fundamental domain. In (19), the see in this section, even the self-dual Omega background integral is over the worldsheet torus, X1,X2 are holo- involves a gauge field other than the graviphoton when in- morphic coordinates on R4 (with their fermionic partner terpreted as a supergravity configuration. The case with χ1,χ2), Z is a holomorphic coordinate on T 2 (with its ǫ+ 6= 0 will involve even more gauge fields. fermionic partner ΨZ ), and ZK3 represents contributions We assume massless hypermultiplets are all uncharged from the K3 part of the target space. We note that the and their scalar components are constant. With nV vec- effective action (19) is invariant under the target space tor multiplets, the supersymmetry transformation laws i Ai supersymmetry we will discuss in section 3.1.1. for the gravitini ψααβ˙ and gaugini λα (i = 1, 2; A = 4

G A¯ 1, ..., nV ) are [17, 18], 3.1.2 Fαβ, Fα˙ β˙ =6 0 and all other = 0

i i G i δψααβ˙ = ∇αα˙ ζβ + Fαβ ζα˙ , This is the configuration proposed in [11] for a world- i i G¯ i sheet description of Nekrasov’s partition function with δψαα˙ β˙ = ∇αα˙ ζβ˙ + Fα˙ β˙ ζα, Ai A αi˙ A i Ai βj (22) ǫ+ 6= 0. The relevant supersymmetry transformation δλα = ∂αα˙ t ζ + (Fαβ δj + ǫαβPj )ζ , laws are, ¯ ¯ ¯ ¯ ˙ δλAi = ∂ tAζαi + (F A δi + ǫ P Ai)ζβj , α˙ αα˙ α˙ β˙ j α˙ β˙ j i i G i δψααβ˙ = ∂αα˙ ζβ + Fαβ ζα˙ , i i i i where (ζα, ζα˙ ) are supersymmetry transformation param- δψ ˙ = ∂αα˙ ζ ˙ , (26) G A αα˙ β β eters, F is the graviphoton field strength, t is a vec- ¯ ¯ ˙ A Ai A βi tor multiplet scalar, F is the corresponding gauge field δλα˙ = Fα˙ β˙ ζ . strength, and ǫαβ,ǫα˙ β˙ are antisymmetric bispinors nor- malized as ǫ = ǫ = 1. We also allowed FI parameters Assuming det F A¯ 6= 0 for some gauge fields, the remain- 12 1˙2˙ α˙ β˙ A P for the vector multiplets. Our task is to find config- i (0)i i ing supersymmetry is parametrized as ζα = ζα , ζα˙ = 0. urations that are annihilated by some of these transfor- i This configuration preserves all of the Qα supercharges, mations. i but breaks the Qα˙ supersymmetry.

3.1. Configurations without FI terms G A A 3.1.3 Fαβ , Fαβ,∂t =6 0 and all other = 0 Let us start with configurations without FI terms. We assume that the graviphoton is self-dual and vector mul- In this case, the relevant supersymmetry transforma- tiplet fields are either self-dual or anti-self-dual. tion laws are, i i G i δψααβ˙ = ∂αα˙ ζβ + Fαβ ζα˙ , G i i 3.1.1 Fαβ =6 0 and all other = 0 δψαα˙ β˙ = ∂αα˙ ζβ˙ , (27) δλAi = ∂ tAζαi˙ + F A ζβi. This is the background for the original topological α αα˙ αβ string theory studied in [1, 2]. In this case, non-trivial Assuming that F G is constant, we can set δψi = 0 parts of the supersymmetry transformation laws are αβ ααβ˙ and δψi = 0 by parametrizing ζi , ζi as, αα˙ β˙ α α˙ i i G i δψααβ˙ = ∂αα˙ ζβ + Fαβζα˙ , i i (23) i G ββ˙ (0)i δψ = ∂αα˙ ζ . ζα = −Fαβx ζ ˙ , αα˙ β˙ β˙ β (28) i (0)i G ζα˙ = ζα˙ . If the self-dual graviphoton field strength Fαβ is constant, the right-hand sides of these equations can be set to zero If [F G, F A]β = 0, we can also solve δλAi = 0 by requiring by choosing, α α that the vector multiplet moduli tA is position dependent i (0)i G ββ˙ (0)i as, ζα = ζα − Fαβ x ζ ˙ , β (24) i (0)i A A 1 G αα˙ A ζα˙ = ζα˙ , t = t + (F x) (F x) . (29) 0 2 αα˙ (0)i (0)i where (ζα , ζα˙ ) are constant spinors. This configuration, however, does not solve the super- Interestingly, this background preserves the same gravity equations of motion. The field strength F A is amount of supersymmetry as the flat Minkowski space defined as, does [19, 20]. In particular, the Hamiltonian is still ex- I pressed as an anti-commutator of the supercharges. The FA = ∂AX FI , (A =1, ..., nV ), (30) supersymmetry algebra is modified as [20], I where X (I = 0, 1, ..., nV ) are projective coordinates {Qi ,Qj } =2ǫijP , α β˙ αβ˙ of the vector multiplet moduli space, and FI ’s are the i G βi field strengths that satisfy the standard Bianchi identi- [Pαα˙ ,Qβ˙ ]=2ǫα˙ β˙ FαβQ , (25) A ties, dFI = 0. Therefore, if the moduli scalar fields t {Qi ,Qj } =4ǫ ǫij F G M αβ, A α˙ β˙ α˙ β˙ αβ are position dependent as in (29), F = const. is not compatible with the Bianchi identities. αβ where Pαα˙ and M are the momentum and the angular Fortunately, we know how to modify the field configu- momentum generators. ration to solve the equations of motion for a case relevant 5 to our problem. Recall that the Kaluza-Klein ansatz (11) The twisting by the R symmetry is closely related ¯ Ai with the gauge field Aµ = 0 gives to turning on of the FI parameters Pj , which is in the adjoint representation of the SU(2) R symmetry 2 2 µ µ ds = dx + dz(dz¯ +2A dx ) (i, j = 1, 2). When the internal space is non-compact 2 µ (31) = dx +2Aµdx dz + dzdz.¯ and allows the R symmetry in four dimensions, one can construct vertex operators on the string worldsheet which From (16), we see that the self-dual F = dA corresponds correspond to turning on the FI terms. In the context of G T 1 to turning on Fαβ = Fαβ = 2 Fαβ . the heterotic string theory, these vertex operators will be On the other hand, the Melvin-type geometry (4) has discussed in detail in the next section. There, it will be the metric, shown why the R twist can be realized by turning on the FI parameters. ds2 = (dxµ +Ωµdz)2 + dzdz¯ It is known that FI parameters are quantized in a 2 µ µ (32) = dx + 2Ωµdx dz +ΩµΩ dzdz + dzdz.¯ supergravity theory with finite Newton constant. How- ever, continuous FI parameters are possible if the internal Comparing the two metrics, we see that the complex space is non-compact and the Newton constant in four structure moduli field tT for the Melvin-type geometry, dimensions is zero. appearing in the coefficient of dzdz in the metric, has In the perturbative heterotic string theory, only the position dependence in four dimensions as U(1) part of the R symmetry is manifest while the full T T 1 µ SU(2) R symmetry is an accidental symmetry in the su- t = t0 + Ω Ωµ, (33) 2 pergravity limit. Although we will use the SU(2)R notation in this section, our supergravity analysis is appli- where tT is the complex moduli of the original T 2. If we 0 cable even if only a U(1) subgroup is manifest since we choose the gauge, R only need to turn on the R twist (and therefore FI terms) G T β for a particular U(1)R direction. A ˙ = A ˙ =Ωαβ˙ = Fαβ x ˙ , (34) αβ αβ β Before describing backgrounds with FI terms, we the moduli field tT (33) shows the same position depen- would like to comment on analytic continuations of field dence as in (29). configurations. In string theory, we often allow moduli A A¯ Therefore, to the leading nontrivial order, the Melvin- fields t and their complex conjugates t as independent type background agrees with the configuration of parameters. For example, in extracting the Gromov- G T T Witten invariants from the topological string theory, one Fαβ, Fαβ ,∂t 6= 0 discussed in this section. The Melvin background with (34) is an exact solution to the su- takes the limit of t¯ → ∞ for the Kähler moduli while pergravity equations of motion since it is locally trivial. keeping t finite. This is allowed since there are sepa- A A¯ Moreover, the supersymmetry (27) is exactly the same rate vertex operators for t and t . Similarly, turning as the one preserved by the Melvin-type background on self-dual field strength Fαβ while keeping Fα˙ β˙ = 0 is [7]. Thus, we conclude that the Melvin-type background allowed since there are separate vertex operators for Fαβ gives a completion of the configuration discussed here, and Fα˙ β˙ on the worldsheet. This is not the case for the when the self-dual field strength is turned on the T di- FI terms. For them, there is only one vertex operator for A A¯ rection (the complex moduli of T 2 for the heterotic string each vector multiplet and we cannot treat P and P A A¯ on T 2 × K3). as independent. If we turn on P , we must turn on P simultaneously. This will become important in some of the cases below. 3.2. Configurations with FI terms

3.2.1. F G , F A , P A 6 and all other The definition of Nekrasov’s partition function (1) in- αβ αβ = 0 = 0 volves twisting by the SU(2) R symmetry. Type II superstring theory compactified on a Calabi-Yau manifold Let us start with a warm-up exercise with only self- X is R symmetric only if X has a special isometry gen- dual field strengths and FI terms turned on. The relevant erated by the Reeb vector [21]. In particular, X must be supersymmetry transformation laws in this case are, non-compact. This is expected from the fact that there i i G i δψ = ∇ ζ + F ζ , is no global symmetry in a quantum gravity theory with ααβ˙ αα˙ β αβ α˙ i i finite Newton constant. Similarly, for the heterotic string δψαα˙ β˙ = ∇αα˙ ζβ˙ , (35) on T 2 ×K3, the R symmetry requires K3 is non-compact Ai A i Ai βj δλ = (F δ + ǫαβP )ζ . and has an isometry generated by the Reeb vector, which α αβ j j rotates the covariantly constant spinor on K3. For ex- When the FI terms are non-zero, we need to take into C2 i A i ample, an isolated ADE singularity (e.g. orbifold of ) account the SU(2)R connection Vµj = Aµ PAj in the co- has such an isometry. variant derivative ∇αα˙ in the first and second lines. In 6 addition, we will need to turn on the spin connection in fields ∂t¯A¯. The relevant supersymmetry transformation four dimensions. laws are, By definition, the FI terms P Ai are in the adjoint j δψi = ∇ ζi + F G ζi , representation of the SU(2) R symmetry (i, j = 1, 2; ααβ˙ αα˙ β αβ α˙ i i hermitian and traceless) and commute with each other, δψαα˙ β˙ = ∇αα˙ ζβ˙ , A B Aβ [P , P ] = 0. On the other hand, Fα is also in the Ai A αi˙ A Ai j (39) δλα = ∂αα˙ t ζ + (Fαβ + Pj )ζα, adjoint representation of SU(2)− ∈ SO(4) for the four- Ai¯ A¯ i Ai¯ βj˙ dimensional rotation. If we turn on the FI terms so that δλα˙ = (Fα˙ β˙ δj + ǫα˙ β˙ Pj )ζ . det P A = det F A, the eigenvalues of P A and F A match A i Ai G A and the rank of the 4×4 matrix (F δ +ǫαβP ) becomes We assume that the field strengths commute, [F , F ]= αβ j j A B A¯ B¯ A A (0) 0, [F , F ] = 0, and [F F ] = 0. 2. Therefore, there are solutions to (F + P )ζ = 0. A A A¯ A¯ Moreover, if we require the field strengths to commute If we set detF = detP and detF = detP simulta- A B A A B B neously, we can find a configuration which preserves half [F , F ] = 0, we have that [F +P , F +P ] = 0,and i i there are simultaneous solutions to (F A + P A)ζ(0) = 0 of the Qα supercharges and half of the Qα˙ supercharges. (0)i (0)i for all directions of the vector multiplets A =1, ··· ,nV . To see this, choose constant spinors ζα and ζα˙ so that i A A A¯ A¯ The integrability of ∇αα˙ ζβ = 0 requires that we tune they are annihilated by (F +P ) and (F +P ). Using the spin connection to cancel the effect of the SU(2)R i (0)i G ββ˙ (0)βi˙ i A i ζα = ζα − Fαβx ζ , connection Vµj = Aµ PAj . This turns on the self-dual (40) component of the Riemann curvature as, i (0)i ζα˙ = ζα˙ , A Rαβγδ ∼ FαβFAγδ. (36) for the supersymmetry transformation laws (39), we find Ai¯ Ai that δλα˙ = 0 is automatic and δλα = 0 can be satisfied The traceless component of the Ricci tensor is zero, if we set as expected from the Einstein equation since the energy- 1 ˙ ˙ tA = tA + (F Gβx F Aαxγβ − F G xββF A xγα˙ ). (41) momentum tensor vanishes for the self-dual gauge fields. 0 2 α ββ˙ γ αβ β˙γ˙ However, the scalar curvature is non-zero, The integrability of ∇αα˙ requires αγ βδ A αβ Ai j A R ∼ ǫ ǫ Rαβγδ ∼ FαβFA = Pj PAi. (37) Rα˙ β˙γ˙ δ˙ ∼ Fα˙ β˙ FAγ˙ δ˙, (42) A This is expected since the FI terms contribute to the cos- and that the scalar curvature is proportional to P PA, mological constant. To understand why the FI parame- which is the trace part of the Einstein equation. In this ters survive in the Einstein equation when the Newton case, the integrability further requires constant in four dimensions is zero, it is convenient to A restore mass dimensions of various parameters. The field Rαβγ˙ δ˙ ∼ FαβFAγ˙ δ˙. (43) A A strength F and the FI parameter P are of dimension The left-hand side is the traceless part of the Ricci cur- 2, while the ǫ± parameters are of dimension 1. Thus, ǫ± vature and the right-hand side is the energy-momentum A must be of the order of P /MPlanck, where MPlanck is tensor for the gauge field, which is non-zero in this case the Planck mass in four dimensions. The Einstein equa- since both self-dual and anti-self-dual components are tion should then set the Ricci curvature to be related turned on. This is consistent with the Einstein equation. 2 to ǫ± without an extra factor of MPlanck, and the effect Note that the moduli scalar fields tA do not contribute survives in the limit of MPlanck → ∞. This scaling is con- to the energy-momentum tensor even though they are sistent with the supergravity transformation (35) and the position dependent since we assume ∂tA¯ = 0. commutation relations (25). As in the case we saw in section 3.1.3, the position The configuration preserves the supersymmetry dependence of tA means that constant F A does not sat-

i (0)i i isfy the Bianchi identities. Thus, we need modify the field ζα = ζα , ζα˙ =0, (38) configuration order by order in an expansion in powers of the gauge field strengths so that the equations of motions which gives rise to half of the Qi charges. α are solved while preserving the same set of supercharges. In the next section, we will propose that the background

G A A¯ A A appropriate for the refined topological string computa- 3.2.2. Fαβ, Fαβ , F ˙ ˙ , P ,∂t =6 0 and all other = 0 αβ tion for the case with ǫ+ 6= 0 is a configuration of this type. This is the most general case we consider in this paper. We can also relax the conditions, detF A = detP A and We turn on almost all the fields, except for the anti-self- detF A¯ = detP A¯, and find a configuration with less su- dual components of the graviphoton F G and the position α˙ β˙ persymmetry. We will show that the Omega background dependence of the anti-holomorphic parts of the moduli with (ǫ+ 6= 0) is a configuration of this type. 7

3.3. Summary ﬁeld, as

G T 1 2 detF = detF = − ǫ−, Let us summarize the supersymmetric configuration αβ αβ 4 studied in this section. (44) U¯ S¯ 1 2 detFα˙ β˙ = detFα˙ β˙ = − ǫ+. (0)i (0)i 4 fields turned on ζα ζα˙ F G 1 1 TST (ǫ = 0) Since both self-dual and anti-self-dual components of αβ + vector multiplet gauge fields are turned on, in order to F G , F A¯ 1 0 discussed in [11] αβ α˙ β˙ preserve supersymmetry, we also need to introduce FI G A A Fαβ, Fαβ ,∂t 0 1 Omega (ǫ+ = 0) terms. This corresponds to the R twist in Nekrasov’s F G , F A , F A¯ , P A,∂tA 0 1/2 Omega (ǫ 6= 0) αβ αβ α˙ β˙ + partition function computation. Since the amount of the ¯ U¯ S¯ F G , F A , F A , P A,∂tA 1/2 1/2 TST (ǫ 6= 0) R twist is proportional to ǫ+, we must set P and P αβ αβ αβ˙ + ˙ as,

The middle two columns count the number of super- ¯ ¯ 1 detP U = detP S = − ǫ2 , (45) charges preserved. On the right column, TST refers to 4 + the topological string theory and Omega is for the Omega U¯ U¯ S¯ S¯ background. The last two rows are the ones discussed so that (F + P ) and (F + P ) can annihilate some in subsection 3.2.2. In the next section, we will pro- spinors. T pose them as the Omega background and the topological As we saw in (13), the self-dual component F− and the U¯ string background for ǫ+ 6= 0. We did not list the config- anti-self-dual component F+ for the T and U fields come uration discussed in subsection 3.2.1 as it was meant to from the same gauge field. Therefore, their FI terms are U¯ T T 1 2 be a warm-up exercise. related as P = P and we must have detP = − 4 ǫ+. T T 1 2 On the other hand, Fαβ is set as detF = − 4 ǫ− by (44). Therefore (F T + P T ) does not have a zero mode unless 4. OMEGA BACKGROUND AND REFINED ǫ+ = ±ǫ−. i TOPOLOGICAL STRING THEORY This configuration preserves half of the Qα˙ super- i charges but breaks all of Qα. The constant field strengths T U¯ S¯ In this section, we will embed the backgrounds dis- of F , F and F do not satisfy the Bianchi identities cussed in the previous section in the heterotic string the- because of the position dependence of the moduli scalar ory on T 2 × K3 and identify the Omega background field. We can modify the configuration order by order in ǫ±. The non-self-dual Melvin-type background with the with ǫ+ 6= 0. We will also propose the background for the corresponding topological string computation. In the R-twist, namely the non-self-dual Omega background, next section, we will discuss interpretations of these back- gives a completion of this iterative procedure since it is an grounds in the context of type II superstring theory on a exact solution to the equations of motion and preserves Calabi-Yau manifold. the same set of supercharges. It is interesting to note that, when ǫ+ = ±ǫ−, we have T T i zero modes for (F +P ) and half of the Qα supercharges are restored. In fact, in this case, the Omega background 4.1. Omega Background twists only a 2-dimensional subspace in R4 keeping the transverse R2 intact. This is known as the Nekrasov- Identifying the supergravity description of the Omega Shatashvili limit [22]. background in R4 × T 2 × K3 is straightforward since the field configuration is locally trivial. The geometry is a flat bundle of R4 × K3 over T 2 so that, as we go along T 2, 4.2. R twist and FI terms we perform the rotation of R4 by the amount specified by (ǫ−,ǫ+) and the isometry on K3 generated by the Reeb To describe the Omega background on the string vector. worldsheet, we need to understand the relation between When we reduce this 10-dimensional description of the the R twist and the FI terms better. To construct the Omega background down to R4, we find that it is a spe- vertex operators for the FI terms, we assume that the cial case of the configuration discussed in section 3.2.2. K3 surface has a special isometry that generates the R We first note that, to the linear order in ǫ±, Melvin-type symmetry. This means, in particular, that K3 is non- R4 2 3 geometry, where is fibered over T , is the same as the compact. In this case, there is a corresponding current JI Kaluza-Klein geometry, where T 2 is fibered over R4. We on the left-moving (supersymmetric) sector of the world- saw in (16), such a Kaluza-Klein geometry for the het- sheet theory. However, it does not preserve the world- erotic string theory on R4 × T 2 × K3 requires turning on sheet BRST symmetry by itself. To construct a physi- the S, T , and U gauge fields as well as the graviphoton cal vertex operator, we must combine it with one of the 8

3 SU(2) current JN=4 in the N = 4 superconformal alge- we will try to identify what they are. There are several 3 3 bra in the left-mover. The combination (JI + JN=4) is constraints we need to take into account in order to make BRST invariant. the identification. If there is a current J with conformal weight (0, 1) in [1] We require that ǫ− and ǫ are self-dual and anti-self the right-moving (bosonic) sector, there is a correspond- + dual field strengths for the Kaluza-Klein gauge field g . ing vector multiplet in four dimensions. For example, the µz µ Using the field identification (13), we find vertex operator for the gauge field is Aµ∂X J . The FI term for this multiplet is constructed using the BRST in- G T ǫ− = (∂[µ,gν]z)− = F− + F− , variant R symmetry generator as (J 3 + J 3 )J . In fact, (48) I N=4 U¯ S¯ one can verify that it is in the same supermultiplet as ǫ+ = (∂[µ,gν]z)+ = F+ + F+ . the moduli scalar field for J by commuting these vertex To minimize the set of assumptions, we require that no operators with the target space supercharges as, G T U¯ S¯ field strength other than F− , F− , F+ and F+ is turned 1 2 3 3 ϕ on. We also assume that only the FI terms for the T and {Q , [Q , (J + J )J ]} = ǫαβe ψZ J . (46) α β I N=4 U fields are turned on since they are the ones directly The right-hand side is the picture one vertex operator related to the R twist, as we saw in 4.2. ϕ T for the moduli scalar field, where e is the bosonized [2] We will find that F− has to be non-zero. Thus, both superghost. self-dual and anti-self-dual components of vector multi- The vertex operators for the T and U gauge fields are, plet gauge fields are turned on. We require that the relevant configuration preserves the maximally allowed T µ ¯ T¯ µ ¯ ¯ (Aµ )−∂X ∂Z, (Aµ )+∂X ∂Z, amount of supersymmetry under the circumstance. In (47) U µ U¯ µ i (A )−∂X ∂¯Z,¯ (A ) ∂X ∂Z.¯ this case, it means half of the Qα supersymmetry and µ µ + i half of the Qα˙ supersymmetry. T U¯ We note that the general Omega background with This means that (A )− and (A )+ have the same FI µ µ ǫ 6= 0 preserves half of the supersymmetry of the self- term, P T = P U¯ , which couples to the vertex operator + T¯ U dual Omega background with ǫ+ = 0. Our proposal for (J 3 + J 3 )∂Z¯ . Similarly, (A ) and (A )− have the I N=4 µ + µ the refined topological string background preserves half T¯ U 3 same FI term, P = P , with the vertex operator (JI + of the supersymmetry of the self-dual graviphoton back- 3 ¯ ¯ JN=4)∂Z. ground for the ordinary topological string theory. T T¯ The vertex operators for P and P constructed in T U¯ this way can be identified with the vertex operators for [3] The FI term for F− is the same as the FI term for F+ . T U¯ the dz and dz¯ components of the Kaluza-Klein gauge field By the constraint [2], we must have detF− = detF+ . (12) associated to the isometry along the Reeb vector on ¯ [4] Since P S = 0 by the assumption [1], we need to turn K3. In this way, we understand that turning on the FI ¯ off F S. parameters is related to the R twist. This is analogous + to the relation between the twisting parameter in the Assuming these constraints, there is a unique choice: Melvin-type geometry and the introduction of the con- G stant field strength to the four-dimensional Kaluza-Klein F− = ǫ− − ǫ+, gauge field. T U¯ F− = F+ = ǫ+, (49) T U¯ P = P = ǫ+. 4.3. Proposal for Refined Topological String Theory We note that, in the self-dual limit ǫ+ = 0, this configuration reduces to the self-dual graviphoton background As we reviewed in section 2.2, when ǫ+ = 0, Nekrasov’s for the original topological string theory. partition function has the asymptotic expansion which We propose that the refined topological string theory gives the all genus topological string partition function. computes scattering amplitudes of the linear superposi- The former is the partition function for the Omega G T tions of various gauge fields specified by (49). More ex- background, where Fαβ = Fαβ are turned on, whereas n¯G,n¯T ,n¯U¯ plicitly, let AnG,nT ,n ¯ ,nP be the scattering amplitude of the latter computes scattering amplitudes for self-dual U 2 self-dual gravitons, nG +¯nG − 2 self-dual graviphotons graviphotons, with 2 self-dual gravitons. In 4.1, we dis- G T F , nT +n ¯T self-dual vector fields for F , n ¯ +n ¯ ¯ cussed the generalization of the Omega background to αβ αβ U U anti-self-dual vector fields for F U¯ , and n FI terms for the non-self-dual case. It is natural to ask what the cor- α˙ β˙ P responding generalization of the topological string the- P T = P U¯ , all at zero-momentum. Here the numbers with ory is. Following the standard terminology, we call it and without bar are defined by the two different polar- the refined topological string theory. Assuming that this izations used in [1, 2] for the self-dual case and [11, 12] string theory still computes some scattering amplitudes, for the anti-self-dual case. (For a precise specification 9 of the polarizations, see the expression in (51)). In the 4.4. Test of the Proposal unrefined case (18), the worldsheet charge conservation dictates nG =n ¯G, whereas more generic scattering am- Although we have not been able to solve the world- plitudes are non-zero here. The generating function for sheet theory for the background proposed in the above, scattering amplitudes for (49) is then given by we can discuss aspects of the proposed string amplitudes. In particular, we present a modest test of this proposal in ∞ the zero-slope limit of the heterotic string, where the one- 2g−2 F (ǫ−,ǫ+)= gs Fg(b) loop scattering amplitude and the corresponding gener- g=0 ating function (50) can be computed. This is the limit ∞ X nG+¯nG nT +¯nT +nU¯ +¯nU¯ +nP where Nekrasov’s partition function is originally defined (ǫ− − ǫ+) ǫ + n¯G,n¯T ,n¯U¯ and computed from the field theory. = 2 2 AnG,nT ,nU¯ ,nP (ǫ− − ǫ )(nG!¯nG!nT !¯nT !nU¯ !¯nU¯ !nP !) nG,n¯G,nT , + The contribution from an elementary hypermultiplet X n¯T ,nU¯ , is, n¯U¯ ,nP =0 ∞ −µt 1 2 −∆Seff dt e = d τhe iR4×T 2×K3, . (52) ǫ2 − ǫ2 − + Z 0 t sinh(ǫ− + ǫ+)t sinh(ǫ− − ǫ+)t (50) Z In this case, there is no effect of the R twist (i.e., the FI term), and the computation is essentially the same as i − i − 1 − 1 where we recall ǫ+ = 2 gs(b + b ), ǫ = 2 gs(b − b ). the one in [11, 23, 24]. On the other hand, the R twist The deformation of the worldsheet action ∆Seff is given is relevant for an elementary vector multiplet, and its by one-loop contribution is, ∞ dt −2cosh(2ǫ t)e−µt + . (53) 1 1 ¯ 2 t sinh(ǫ− + ǫ )t sinh(ǫ− − ǫ )t ∆Seff =(ǫ− − ǫ+) (X ∂Z − χ ψZ )∂X Z0 + + Z 1 1 2 + (X¯ ∂Z − χ¯ ψZ )∂¯X¯ The extra −2 cosh(2ǫ+t) is precisely due to the FI term that gives the worldsheet R symmetry twist in the left- 1 2 1 2 1 2 1 2 + ǫ+ X ∂X − χ χ + X¯ ∂X¯ − χ¯ χ¯ ∂Z¯ moving ground states. These reproduce the contributions Z of the hyper and vector multiplets to Nekrasov’s partition 1 2 1 2 1 2 1 2 + ǫ+ X¯ ∂X − χ¯ χ + X ∂X¯ − χ χ¯ ∂Z¯ function (1), and agree with the general formula, Z 3 3 − 3 − 3 2J−+2J+ 4tǫ−J− 4tǫ+J+ 3 3 ¯ 2 dt tr(−1) e −µt + ǫ+ (JI + JN=4)∂Z + O(ǫ ), log Z(ǫ+,ǫ−)= e t sinh(ǫ− + ǫ )t sinh(ǫ− − ǫ )t Z + + (51) Z (54) proposed in [9, 25]. Our proposal also reproduces some of generic features which corresponds to the configuration (49). This is a of Nekrasov’s partition function. FI term P T contains generalization of (18) for the non-self-dual background. (J 3 + J 3), and it generates twisting of the worldsheet The particular polarization and the combinatorial factor N=4 I −1 R symmetry. This leads to the twisting of the target (n !¯n !n !¯n !n ¯ !¯n ¯ !n !) in (50) are needed in order G G T T U U P space R symmetry via the conformal field theory for K3. for the effective worldsheet action (51) to preserve the Secondly, in the field theory limit, the amplitude has supersymmetry discussed in section 3.2.2. the manifest reflection symmetry ǫ± →−ǫ± because the As discussed in section 3.2.2, we need to turn on ∂tT U(1) R symmetry is enhanced to the SU(2) R symme- at O(ǫ2). In addition, the four-dimensional space should try. This reflection symmetry is an important feature of be curved appropriately to preserve the supersymmetry. Nekrasov’s partition function (see e.g. [9, 26]). The requirement for the position dependence of tT can also be traced to the worldsheet operator product expansion (OPE) and renormalization. The OPE of the vertex 5. TYPE II INTERPRETATION G T operators F− and F− generates the complex structure moduli ∂Z∂Z¯ for the moduli field tT . 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