Superstrings in Graviphoton Background and N= 1/2+ 3/2 Supersymmetry

Home , Graviphoton

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Motivated by the work of Ooguri and Vafa [1] we continue the analysis of [2] of superstrings in flat IR4 with N = 2 supersymmetry with constant self-dual graviphoton field strength. Only in Euclidean space can we turn on the self-dual part of the two form field strength ˙ F αβ while setting the anti-self-dual part F α˙ β to zero. Therefore we limit the discussion to Euclidean space, but we will use Lorentzian signature notation1. The special property of a purely self-dual field strength F αβ is that it does not con- tribute to the energy momentum tensor, and therefore does not lead to a source in the gravity equations of motion. Also, since the kinetic term of the graviphoton does not depend on the dilaton, the background F αβ does not lead to a source in the dilaton equation. Therefore, this background is a solution of the equations of motion. Before turning on the graviphoton our system has N = 2 supersymmetry. What αβ βj happens to it as a result of the nonzero F ? The local N =2 d = 4 chiral gravitini ξµ transformation laws are βj αj˙ ′ αβ βj δξµ = σµαα˙ ε α F + ∇µε (1.1) ˙ where εβj and εβj are the local supersymmetry parameters. The dilatini and antichiral gravitini do not transform into the self-dual graviphoton field strength. The variation (1.1) clearly vanishes for constant εβj and εαj˙ = 0. This shows that the background F αβ does

not aﬀect the four supercharges with one chirality Qαi. The variation (1.1) is not zero for constant εαj˙ showing that the four other standard supercharges are broken by F αβ. However, one can easily make the variation of the chiral gravitini (1.1) vanish if one chooses

βj 1 ′ αβ αj˙ µ ε = α F σ ˙ ε x (1.2) 2 µαα where εαj˙ is constant. So in the presence of this background, one has eight global fermionic symmetries which generate a deformed N =2 d = 4 supersymmetry. One can easily read oﬀ the algebra of the deformed N =2 d = 4 supersymmetry from the above transformations

α α˙ αα˙ µ {Qj , Qk } =2ǫjkσµ P

[Pµ,Qαj]=[Pµ,Pν] = {Qαj,Qβk} =0 (1.3) ′ αβ [Pµ, Qαj˙ ]=2σµβα˙ α F Qαj ′ αβ {Qαj˙ , Qβk˙ } =4ǫjkǫα˙ β˙ α F Mαβ

1 We will use the conventions of Wess and Bagger [3].

1 where Mαβ is the self-dual Lorentz generator which is normalized to transform

γ γ γ [Mαβ,θ ] = θαδβ + θβδα. (1.4)

Note that if one had turned on both a self-dual field strength F αβ and an anti-self-dual ˙ field strength F α˙ β, there would have been a backreaction which would warp the spacetime 2 to (Euclidean) AdS2 × S . The isometries of this space form a PSU(2|2) algebra, and the algebra of (1.3) can be understood as a certain contraction of PSU(2|2). In [4], the 2 superstring was studied in this AdS2 × S background and the structure of the worldsheet 5 action closely resembles the pure spinor version of the AdS5 × S worldsheet action. Since the worldsheet action becomes quadratic in the limit where the anti-self-dual field strength α˙ β˙ d F goes to zero, this limit might be useful for studying AdSd×S backgrounds in a manner analogous to the Penrose limit. In the next section we will extend this supergravity discussion to superstring theory. Using the hybrid formalism for the superstring, the worldsheet action remains quadratic in this graviphoton background, so the theory is trivial to analyze. We will explicitly show that it preserves eight supercharges with the new algebra (1.3). In section 3 we will study D-branes in this background, and will examine the symmetries on the branes. In section 4 we will consider the low energy field theory on the D-brane and will examine its supersymmetries.

2. Superstrings in graviphoton background

In this section and the next one we review and extend the discussion in [1,2] of type II superstrings in IR4 with N = 2 supersymmetry deformed by a self-dual graviphoton (see also [5]). Since we are interested in superstrings in Ramond-Ramond background, the standard NSR formalism cannot be used. Instead, we will use the hybrid formalism (for a review see [6]). The relevant part of the worldsheet Lagrangian is

α˙ 1 1 µ α α˙ α L = ∂x ∂xµ + pα∂θ + p ∂θ + pα∂θ + p ∂θ , (2.1) α′ 2 α˙ α˙ e e e e e e e where µ =0 ... 3, (α, α˙ )=1, 2, and we ignore the worldsheet ﬁelds ρ and the Calabi-Yau sector. Since we use a bar to denote the space time chirality, we use a tilde to denote the worldsheet chirality. Therefore when the worldsheet has Euclidean signature it is

2 parametrized by z and z which are complex conjugate of each other. p, p, p and p are canonically conjugate to θ, θ θ and θ; they are the worldsheet versions of − ∂ etc.. The e e∂θ x e reason x is held ﬁxed in thesee derivativese is that x appears as another independent ﬁeld in (2.1). It is convenient to change variables to

α˙ µ µ α µ α˙ α µ y = x + iθ σαα˙ θ + iθ σαα˙ θ

α µ e 1 d ˙ = p − iθ σ ∂x −eθθ∂θ ˙ + θ ˙ ∂(θθ) α α˙ αα˙ µ α 2 α α˙ 1 3 q = −p − iσµ θ ∂x + θθ∂θ − ∂(θ θθ) (2.2) α α αα˙ µ 2 α 2 α α µ 1 d ˙ = p − iθ σ ∂x − θθ∂θ ˙ + θ ˙ ∂(θθ) α α˙ αα˙ µ α 2 α e e e eα˙ ee1ee e 3e ee q = −p − iσµ θ ∂x + θθ∂θ − ∂(θ θθ) α α αα˙ µ 2 α 2 α e ee ee and to derive e e e ee e e α˙ 1 1 µ α α˙ α L = ∂y ∂yµ − qα∂θ + dα˙ ∂θ − qα∂θ + dα˙ ∂θ + total derivative . (2.3) α′ 2 e e e e e e e ∂ ∂ The new variables in (2.2) are the worldsheet versions of Dα˙ = − α˙ , Qα = α , ∂θ y ∂θ y etc.. The reason y is held fixed in these derivatives is that it appears as an independent field in (2.3). Our definitions of q and q differ from the integrand of the supercharges in [6] by total derivatives which do not affect the charges, but are important for our purpose. e We will also need the worldsheet versions of Dα, Dα and the other two supercharges

µ α˙ 1 e µ α˙ d = −p + iσ θ ∂x − θθ∂θ + θ ∂(θθ) = q +2iσ θ ∂y − 4θθ∂θ α α αα˙ µ α 2 α α αα˙ µ α α˙ µ α˙ 1 µ dα = −pα + iσαα˙ θ ∂xµ − θθ∂θα + θα∂(θθ) = qα +2iσαα˙ θ ∂yµ − 4θθ∂θα 2 (2.4) e α µ e 1 eeee 3 e e ee e e eeee q = p e+ iθ σ ∂x + θθ∂θ ˙ − ∂(θ ˙ θθ) e α˙ α˙ αα˙ µ 2 α 2 α α µ 1 3 q = p + iθ σ ∂x + θθ∂θ ˙ − ∂(θ ˙ θθ). α˙ α˙ αα˙ µ 2 α 2 α e e Sincee thee vertexe operatore foreee a constante self-dualee graviphoton ﬁeld strength is

2 αβ d zqαqβF (2.5) Z in the hybrid formalism, the action remains quadratice in this background and there is no backreaction. The Lagrangian is

α˙ 1 1 µ α α˙ α ′ αβ L = ∂y ∂yµ − qα∂θ + dα˙ ∂θ − qα∂θ + dα˙ ∂θ + α F qαqβ . (2.6) α′ 2 e e e e e e e e 3 The ﬁelds q and q can easily be integrated out using their equations of motion

e α ′ αβ ∂θ = α F qβ (2.7) α ′ αβ ∂eθ = −α F e qβ . e Let us examine the supersymmetries of the Lagrangian (2.6). In addition to being invariant under the Qα and Qα supersymmetries generated by e δθα = εα, δθα = εα, (2.8)

e e where εα and εα are constants, the above Lagrangian is also invariant (up to total derivatives) under thee Qα˙ and Qα˙ supersymmetries generated by e α˙ δθ = −εα˙

α˙ α˙ δθ = −ε eµ µ α˙ α α˙ α δy =2iσe αα˙ (ε θ + ε θ ) α˙ µ . δqα =2iε σµαα˙ ∂y e e (2 9) α˙ µ δqα =2iε σµαα˙ ∂y α ′ αβ α˙ µ δθe =2iαe F εey σµβα˙ α ′ αβ α˙ µ δθ = −2iα F e ε y σµβα˙ . e These transformations are the stringy version of the supergravity variation (1.2). The last two transformations in (2.9) represent the deformation, and are needed because of the de- αβ formation term F qαqβ in the Lagrangian. Unlike the other transformations in (2.9), the µ last two depend explicitlye on y . This fact is similar to the explicit x dependence in (1.2).

Therefore, the corresponding currents qα˙ and qα˙ are not holomorphic and antiholomorphic respectively. They are similar to the currentse of the Lorentz symmetries in the worldsheet theory. In order to establish that (2.9) are not only symmetries of the worldsheet Lagrangian but are also symmetries of the string, we should check that they commute with the BRST

operators. To do that, we use dα and dα of (2.4) to show that under (2.9)

e α˙ α˙ ′ γβ ′ γβ δdα = 8(εα˙ θ )ǫαγα F dβ, δdα = −8(εα˙ θ )ǫαγ α F dβ, (2.10) e e e e 4 where we have used the equations of motion (2.7). Using (2.10) it is clear that the BRST operators α ρ α˙ −ρ G = dαd e , G = dα˙ d e α˙ (2.11) α ρ −ρ G = dαd e , G = dα˙ d e , are invariant. This establishese thate (2.9)e e are symmetriese e e oef the theory. One can easily read oﬀ the algebra of the deformed N = 2 d = 4 supersymmetry

from the above transformations. Identifying (Qα, Qα) with (Qα1,Qα2) and (Qα˙ , Qα˙ ) with

(Qα˙ 2, −Qα˙ 1), one obtains the algebra (1.3). e e e

3. D-branes

Consider first the system without the background graviphoton. If the worldsheet ends on a D-brane, the boundary conditions are easily found by imposing that there is no surface term in the equations of motion. For a boundary at z = z, we can use the boundary conditions θ(z = z) = θ(z = z), q(z = z) = q(z = z), etc. Then the e solutions of the equations of motion are such that θ(z) = θ(z), q(z) = q(z), etc.; i.e. the e e e e e e fields extend to holomorphic fields beyond the boundary. The boundary breaks half the e e e e supersymmetries preserving only qdz + qdz and qdz + qdz. From a spacetime point of view, these unbroken supersymmetriesH H are realizedH linearlyH on the brane, while the other e e e e supersymmetries, which are broken by the boundary conditions, are realized nonlinearly. Now, let us consider the system with nonzero F αβ. After integrating out q and q the relevant part of the worldsheet Lagrangian is e

1 α β Leff = ′2 ∂θ ∂θ . (3.1) α F αβ e e The appropriate boundary conditions are

θa(z = z) = θa(z = z), ∂θa(z = z) = −∂θa(z = z). (3.2)

e e e e e e e The ﬁrst condition states that the superspace has half the number of θs. The second

condition guarantees, using (2.7) that qα(z = z) = qα(z = z). qdz qdz It is clear that the supercharges + e aree preservede by the boundary conditions and are still realized linearly on theH brane.H However, the supercharges qdz + qdz are e e broken and are realized nonlinearly. H H e e 5 The propagator of θ is found to be

α′2F αβ z − w hθα(z, z)θβ(w, w)i = log (3.3) 2πi z − w e e e with the branch cut of the logarithm outside the worldsheet. Thereforee for two points on the boundary z = z = τ and w = w = τ ′

′2 αβ e e ′ α F ′ hθα(τ)θβ(τ )i = sign(τ − τ ). (3.4) 2 Using standard arguments about open string coupling, this leads to

′ {θα,θβ} = α 2F αβ = Cαβ 6=0; (3.5) i.e. to a deformation of the anticommutator of the θs. It is important that since the coordinates θ and y were not aﬀected by the background coupling (2.5), they remain commuting. In particular we derive that [yµ,yν]=[yµ,θα] = 0, and therefore [xµ,xν] 6=0 and [xµ,θα] 6= 0 [2]. For a partial list of other references on noncommuting θs see [7-19].

4. The zero slope limit

The theory on the branes is simpliﬁed in the zero slope limit α′ → 0. If we want to preserve the nontrivial anticommutator of the θs (3.5), we should scale

′ ′ α → 0, F αβ →∞, α 2F αβ = Cαβ = ﬁxed. (4.1)

For C = 0 the low energy ﬁeld theory on the branes is super-Yang-Mills theory. For nonzero C it is [2]

4 µν 1 4 µν 2 S = S0 − i d xC Tr (vµν λλ) + d xC Cµν Tr (λλ) (4.2) Z 4 Z

4 1 µν m 2 µν where S0 = d xTr[− 2 v vµν − 2iλσ λ + D ] is the usual super-Yang-Mills action, v α˙ µν µν αβ is the Yang-MillsR ﬁeld strength, λ is the antichiral gaugino, and C =(σ )αβC . We will now examine the supersymmetries of the action (4.2). 1 α Under the N = 2 supersymmetries generated by Qα + Qα, the chiral superﬁeld W transforms as e ∂ δW β = ηα W β (4.3) + ∂θα

6 α where η+ is the constant supersymmetry parameter. In component ﬁelds, these transformations are [2] µν i δλ = iη+D + σ η+(v + C λλ) µν 2 µν δv = iη+(σ ∇ − σ ∇ )λ µν ν µ µ ν (4.4) µ δD = −η+σ ∇µλ δλ =0.

The action of (4.2) is invariant under these transformations after including the Cµν - dependent terms in the transformations and in the action. 3 However, under the N = 2 supersymmetries corresponding to (Qα − Qα, Qα˙ , Qα˙ ), the component fields transform nonlinearly since these transformations doe not preservee ˙ α˙ α the D-brane boundary conditions θα = θα and θ = θ .2 To determine the nonlinear transformation of these component fields,e it is useful toe recall that in a background with α˙ Abelian chiral and antichiral superfields W α and W , the D-brane boundary conditions α˙ α˙ α α α˙ α α 1 ′ α α˙ 1 ′ α˙ θ − θ = 0 and θ − θ = 0 are modified to θ − θ = 4 α W and θ − θ = 4 α W to ′ leadinge order in α [20,21,22].e Just as the bosonic De-brane boundary conditionse ∂σxµ =0 ′ ν is modified to ∂σxµ = α (∂µAν − ∂νAµ)∂τ x in the presence of the Maxwell vertex op- µ erator dτAµ(x)∂τ x [23,24], the superstring D-brane boundary conditions are modified α˙ α R α 1 ′ α α˙ 1 ′ α˙ to θ − θ = 4 α W and θ − θ = 4 α W in the presence of the super-Maxwell ver- α α˙ tex operatore which includes thee term dτ(W dα + W dα˙ ) [20]. So by comparing with the supersymmetry transformations ofR (θ, θ, θ, θ) in (2.8)(2.9), one learns the nonlinear ˙ α α ′ transformations of the Abelian gauginos λ ande eλ to leading order in α . Note that the nonlinear transformations of all other super-Yang-Mills fields are higher order in α′.

For example, under Qα − Qα, δ(θ − θ) = ε − ε in (2.9) implies that the chiral Abelian

gaugino transforms inhomogeneouslye as e e

α 4i α α α δλ = (ε − ε ) ≡ η−. (4.5) α′ e This transformation clearly leaves invariant the action of (4.2). In the zero slope limit we

should take ε − ε → 0 with ﬁxed η− in order to have a ﬁnite answer.

2 Actually, sincee these supersymmetries are spontaneously broken on the D-brane, the corre-

sponding supercharges (Qα − Qα, Qα˙ , Qα˙ ) do not exist. We will denote by (Qα − Qα, Qα˙ , Qα˙ ) the transformations of the supersymmetries.e e e e 7 Under Qα˙ − Qα˙ , the antichiral Abelian gaugino transforms inhomogeneously as

e α˙ 4i α˙ δλ = (εα˙ − ε ) ≡ ηα˙ , (4.6) α′ − e and therefore in the zero slope limit we should take ε − ε → 0 with fixed η− in order to α α 1 α˙ αβ µ have a finite answer. In addition to (4.6), since δ(θ + θ e) = − 2 η−C yµσβα˙ is finite in γ the zero slope limit, the chiral superfield W transformse as

γ 1 α˙ αβ µ ∂ γ δW = − η−C y σ W . (4.7) 2 µ βα˙ ∂θα

Therefore, to leading order in α′, the component ﬁelds transform as

µν i δλ = iη(y, η−)D + σ η(y, η−)(v + C λλ) µν 2 µν δv = iη(y, η−)(σ ∇ − σ ∇ )λ µν ν µ µ ν (4.8) µ δD = −η(y, η−)σ ∇µλ

δλ = η−.

α 1 α˙ αβ µ where η(y, η−) ≡ − 2 η−C yµσβα˙ . To explicitly check that (4.2) is invariant under the transformation of (4.8), ﬁrst note that under the inhomogeneous transformation of λ,

4 µν 4 µν δinhomogeneousS = −2i d xC Tr [vµν η−λ] + d xC Cµν Tr [(η−λ)(λλ)]. (4.9) Z Z

And under the y-dependent transformation parameterized by η(y, η−) in (4.8),

4 α ∂ δhomogeneousS = d xJµ η(y, η−)α (4.10) Z ∂yµ

α where Jµ is the conserved supersymmetry current associated with the invariance of (4.2) under the global supersymmetry transformation of (4.4). One can easily compute that

ρτ β γ˙ ν α˙ Jµ α =Tr [−2iv (σρτ )α σµ βγ˙ λ +2Cµν σαα˙ λ (λλ)] (4.11) and ∂ 1 βα˙ η(y, η−) = − (η−) ˙ C σ . (4.12) ∂yµ α 2 α αβ µ Using (4.11) and (4.12) in (4.10), one ﬁnds that (4.10) cancels (4.9) so the action is invariant.

8 Lastly, under Qα˙ + Qα˙ , the chiral Abelian gaugino transforms inhomogeneously as e α α˙ α˙ ′ −2 αβ µ α˙ αβ µ δλ = −8(ε + ε )(α ) C yµσβα˙ ≡ −8η+C yµσβα˙ . (4.13)

e α˙ Note that in this case we need εα˙ + ε ∼ α′2 in order to have a ﬁnite transformation in the α˙ ˙ α˙ α˙ α ′2 zero slope limit. Since δ(θ + θ )=(e ε + ε ) ∼ α , we should set it to zero in the zero slope limit. Therefore, undere this transformatione (4.13) is the only variation. To check invariance of (4.2) under (4.13), note that the inhomogeneous transformation of λα leaves µ ∂ α the action invariant since σαα˙ ∂yµ δλ = 0. 1 We conclude that the action (4.2) is invariant under the linear N = 2 supersymmetry 3 transformations (4.3)(4.4) with parameter η+ and under the nonlinear N = 2 supersymmetry transformations (4.5)(4.8)(4.13) with parameters η−, η−, η+. It would be interesting to extend our analysis to higher-order terms in α′. This amounts to ﬁnding a noncommutative superspace generalization to the d = 4 supersymmetric Born-Infeld action. Just as the standard supersymmetric Born-Infeld action realizes N = 1 + 1 supersymmetry (four supercharges are realized linearly and four supercharges are realized nonlinearly) [21,22], the noncommutative superspace Born-Infeld theory should 1 3 realize N = 2 + 2 supersymmetry.

Acknowledgements It is a pleasure to thank H. Ooguri, C. Vafa and E. Witten for helpful discussions. The work of NS was supported in part by DOE grant #DE-FG02-90ER40542 and NSF grant PHY-0070928 to IAS. NB would like to thank the IAS for its hospitality where part of this work was done, and FAPESP grant 99/12763-0, CNPq grant 300256/94-9 and Pronex grant 66.2002/1998-9 for partial ﬁnancial support.

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