CALT-68-2837, IPMU11-0094 Comments on Worldsheet Description of the Omega Background Yu Nakayama1 and Hirosi Ooguri1, 2 1California Institute of Technology, 452-48, Pasadena, California 91125, USA 2Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan Nekrasov’s partition function is defined on a flat bundle of R4 over S1 called the Omega back- ground. When the fibration is self-dual, the partition function is known to be equal to the topological string partition function, which computes scattering amplitudes of self-dual gravitons and gravipho- tons in type II superstring compactified on a Calabi-Yau manifold. We propose a generalization of this correspondence when the fibration is not necessarily self-dual. PACS numbers: 1. INTRODUCTION [3]. When ǫ+ 6= 0, Nekrasov’s partition function can still be expanded as, The topological string theory has been used to com- ∞ pute a variety of physical observables in type II super- 2g−2 Z(ǫ+,ǫ−) = exp gs Fg(b) , (3) string theory and M theory compactified on a Calabi-Yau g=0 ! manifold. The genus-g contribution Fg to the partition X i − i − function is related to the scattering amplitude of 2 self- 1 − 1 where ǫ+ = 2 gs(b + b ), ǫ = 2 gs(b − b ). dual gravitons and (2g − 2) self-dual graviphotons in the It is natural to ask if there is a worldsheet description zero momentum limit and computes the corresponding F- of the coefficients Fg(b). For toric Calabi-Yau manifolds, term in the low energy effective action in four dimensions generalizations of the topological vertex for Fg(b) have [1, 2]. The all genus partition function can be used to been proposed [8, 9]. For Calabi-Yau manifolds realized count the number of BPS particles in M theory compact- by fibrations over Riemann surfaces, deformed matrix ified to five dimensions [3]. Its absolute value squared is models have been proposed [10]. related to the generating function for BPS states in type Recently, Antoniadis, et al. proposed a worldsheet def- II string theory compactified to four dimensions [4]. inition of Fg(b) [11]. It involves scattering amplitude For a five-dimensional rigid N = 2 supersymmetric AnG,nF among 2 gravitons, (2nG − 2) graviphotons, and R4 S1 theory on × , which can be realized, for exam- 2nF gauge fields in vector multiplets. When there is a ple, by M theory on a non-compact Calabi-Yau 3-fold heterotic string dual on T 2×K3, it was proposed that the X, Nekrasov’s partition function is defined by 2nF gauge fields belong to the linear dilaton multiplet S, which is a vector multiplet in the heterotic string theory. − 3 − 3 3 − F 2ǫ−J− 2ǫ+(J++JR) βH Z(ǫ+,ǫ−) = Tr (−1) e e e , (1) The scattering amplitude in the heterotic dual has been computed in [12] in the limit of S → ∞. It is related where the trace is taken over the Hilbert space of the to certain higher derivative terms in N = 2 supersym- R4 3 theory on , J± are the Cartan generators of SU(2)± metric effective action whose superconformal structure R4 3 subgroups of the SO(4) rotation of , JR is the Cartan was studied in [13]. In this paper, we propose a different arXiv:1106.5503v1 [hep-th] 27 Jun 2011 generator of the SU(2) R symmetry, and H is the trans- interpretation of the Omega background. S1 lation generator for whose radius is set by β [5–7]. It As we will review in the next section, even when can be regarded as a vacuum amplitude on a Melvin-type ǫ = 0, the Omega background for Nekrasov’s partition R4 S1 + geometry, where the is fibered over the (with addi- function (1) and the self-dual graviphoton background tional twist by the R symmetry). This geometry is called used to compute the topological string partition function the Omega background. Fg are similar but not identical. In this paper, we will When ǫ+ = 0 and when the N = 2 theory arises from generalize each of these backgrounds for ǫ 6= 0. R4 S1 + M theory on × × X for some Calabi-Yau 3-fold This paper is organized as follows. In section 2, we re- X, this BPS state counting problem is related to the view the geometry of the Omega background and explain topological string partition function as, why Nekrasov’s partition function is equal to the topolog- ∞ ical string partition function when ǫ+ = 0. In section 3, 2g−2 we study configurations with non-zero field strengths for Z(ǫ+ =0,ǫ− = gs) = exp gs Fg(X) , (2) g=0 ! the graviphoton and vector multiplet gauge fields and ex- X amine the amounts of supersymmetry they preserve. In where gs is the genus counting parameter, and Fg(X) is section 4, we discuss their realizations in the heterotic the genus-g topological string partition function for X string on T 2 × K3 and propose string theory configura- 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 cor- responding 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 (com- plex structure moduli of T 2), and U (K¨ahler 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˙ .