N = 2 Strings and the Twistorial Calabi-Yau

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In a beautiful recent paper [1], Witten provided highly non-trivial evidence that the planar amplitudes of N = 4 supersymmetric U(N) Yang-Mills theory in four dimensions, once transformed to the twistor space CP3|4, are supported on holomorphic curves (see also the followup work [2, 3].) This work was based on the spinor formalism for gauge theory amplitudes; see e.g. the early work [4–6] and more recent papers [7, 8]. Since holomorphic curves are traditional hallmarks of string theory, it was attempted in [1] to provide a tentative string theory interpretation of their appearance. This interpretation involved the topological B model on the Calabi-Yau manifold CP3|4, with N D5-branes wrapping the Calabi-Yau. The computation of the amplitude then reduced to a current algebra calcula- tion on the holomorphic curves as suggested long ago in [9]. However, the objects lying on the holomorphic curves were proposed to be D1-instantons, rather than worldsheet instantons. This was somewhat puzzling, as was already noted in [1], as one would have expected that these instantons should correspond to spacetime instantons rather than to perturbative amplitudes of gauge theory. Here we provide an alternative stringy realization which resolves these puzzles as well as extending the identiﬁcation of [1]: We propose that the string theory in question is the A model on CP3|4, with N D-branes wrapped on the Lagrangian subspace RP3|4. Moreover, we rephrase the observations of [1] as the statement that the amplitudes of N = 4 supersymmetric Yang-Mills on spacetime with signature + + −−, when twistor-transformed to functions on RP3|4, are supported on the real boundaries of holomorphic curves in twistor space CP3|4. We identify the holomorphic curves in question as A model worldsheet instantons ending on Lagrangian D-branes; these instantons correct the Chern-Simons amplitudes, as in the stringy realization of Chern-Simons theories on Calabi-Yau manifolds [10]. This observation extends the suggestion in [1] beyond the planar limit: Indeed, the diagrams of N = 4 Yang-Mills, which can be organized a la ’t Hooft [11] into worldsheets with boundaries, now correspond to A model worldsheets with the same topology and with boundaries on the D-branes. Our proposal also sharpens the statement of what the signature of the spacetime for the N = 4 supersymmetric Yang-Mills theory is: It is the + + −− signature, which is precisely

1 the one arising naturally for N = 2 strings [12]. A stringy realization a la ’t Hooft of N =4 Yang-Mills was expected from the N = 2 string perspective. As we will explain in the next subsection, this is not an accident; we are giving evidence that our A model on CP3|4 is equivalent to an N = 2 string theory. Moreover, the gravity version of this theory, which as suggested in [1] should be N = 4 conformal supergravity, is equivalent to the corresponding N = 2 closed string theory, with signature + + −−. This also explains the fact that the propagators of the gravity theory for N = 2 strings are expected to be 1/(k2)2 [12] as is the case for conformal gravity. We actually make one more conjecture: We conjecture that mirror symmetry maps the A model on CP3|4 to the B model on the Calabi-Yau which is a quadric in CP3|3 ×CP3|3. In the B model, string theory receives no corrections from worldsheet instantons, and therefore the perturbative diagrams of the theory in twistor space should give rise directly to N =4 super Yang-Mills. Indeed it is known that this identification works at least classically [13], and the fact that this B model would not require instantons to obtain N = 4 amplitudes was noted in [1]. Taking this statement as the starting point, we see that we can demystify the fact that the twistor transformed N = 4 Yang-Mills amplitudes lie on the real boundaries of holomorphic curves. If our conjecture is correct, this is essentially a consequence of mirror symmetry! The organization of this paper is as follows. At the end of this section we summarize how N = 2 strings enter the picture, as the relevant literature seems to be unfamiliar to most readers, and is one of the main motivations of this paper. In Section 2 we briefly review topological A and B model strings with branes included. In Section 3 we make our proposal in the context of the A model and show that the field theory observations in [1] are compatible with our interpretation. In section 4 we discuss the possible role of mirror symmetry.

1.1 Historical background

The string worldsheet can be viewed as a theory of 2d gravity coupled to matter. The corresponding 2d gravity theories can be classiﬁed by the number of supersymmetries N . Bosonic strings and superstrings correspond to N = 0 and N = 1 worldsheet supersymmetry

2 respectively, and the heterotic string is where left-movers and right-movers have (NL, NR)= (0, 1). The case with 2 worldsheet supersymmetries (known as N = 2 string theory) has been studied far less. It was first discussed in [14,15]. Its target space meaning was clarified in [12], where it was shown that the closed string sector leads to self-dual gravity, and in [16, 17] where it was shown that the heterotic or open string sector corresponds to self-dual Yang- Mills, in four dimensions with signature + + −−. Moreover, it was shown that for this string theory the infinite tower of string oscillations are absent from the physical spectrum (they are BRST trivial). This is what one would ordinarily expect from a non-critical string theory, such as c ≤ 1 bosonic strings. Indeed, in a sense, for N = 2 strings the “non-critical” theories are equivalent to “critical” theories (the analog of the gap 1 bosonic string theory, namely, a twisted N = 2 worldsheet supersymmetry was used to construct a bosonic-string-like BRST complex. This N = 2 topological string has two versions [19, 20], known as the A and B models, exchanged by mirror symmetry. It was shown in [10] that the open topological strings in the A version lead to ordinary Chern- Simons theory on Lagrangian submanifolds of a Calabi-Yau space, while the B version leads to holomorphic Chern-Simons theory on the full Calabi-Yau. The corresponding closed string theory in the B model was studied in [21], leading to a gravity theory quantizing complex structure of Calabi-Yau threefolds. In any case, the target space description always involves only finitely many fields. Since the N = 2 string also involves only finitely many target space fields, it was natural to expect [21] that the N = 2 string is also related to some kind of topological string; this was indeed shown to be the case in [22]. The theory in question is the N = 4 topological string theory, so named because it is connected to a twisted version of N = 4 worldsheet supersymmetry. Moreover, this formulation leads to vanishing theorems [22] for N = 2 string amplitudes for n> 3 points on flat space at all loops. It is also useful computationally, leading in certain cases to computations of amplitudes at all loops [23]. Thus both the N = 2 and the N = 4 topological strings emerged as string theories

3 for which string oscillations are absent from the physical spectrum. It was suggested in [21] (pages 140-141) that these two theories are indeed also related to each other. The motivation for this was that in the open N = 2 topological string one finds holomorphic Chern-Simons theory on a three complex dimensional Calabi-Yau space (topological B model), while on the other hand N = 2 strings, via the twistor transform, map self-dual Yang-Mills geometry to holomorphic bundles in twistor space [24] which is also of complex dimension 3. However, there was an obstacle to realizing this idea: the twistor space is CP3 which is not a Calabi- Yau threefold. A few ideas emerged to resolve this puzzle [25–27]: N = 2 strings naturally lead to a theory in 2 complex dimensions [12], with Lorentz group U(1, 1). Although this is not necessarily a problem, it was suggested in [25] that one could try to recover Lorentz invariance by “integrating” over the choices of U(1, 1) ⊂ SO(2, 2), which leads to bringing in the twistor sphere as part of the physical space. However, it was found that this was not possible unless one also added fermions and supersymmetrized the theory to N = 4 in the target space. This echoes the fact that CP3 is not a Calabi-Yau manifold but CP3|4 is, as was recently exploited in [1]; thus there is a possibility to unify the two topological strings, as also discussed by [27, 28]. We believe that the results of [1] and this note should be viewed as a step toward such a unification. Together with the recent observations in [36] that N = 2 topological strings are equivalent to non-critical bosonic strings and superstrings, we can view the equivalence of N = 2 strings (which can also be viewed as “non-critical”) with N = 2 topological strings as an extension of the principle that all non-critical strings are unified by topological strings1. 1It is amusing to note that the dimension of the critical topological string is correlated with the upper limit of the central charge of non-critical strings. However, depending on how the twisting is done there are two choices for thec ˆ of the corresponding N = 2 topological theory [29]. In particular, the c = 1 non-critical bosonic string gets mapped toc ˆ =3andˆc = −1, which are precisely the cases of most interest in topological strings (the latter being explored here and in [1]).

4 2 Brief review of A and B models

In this section we provide a brief review of A and B model strings on Calabi-Yau manifolds. The reader can consult [30] for a more thorough review. To make the discussion simple we first discuss the case which has been most studied, namely the case where the Calabi-Yau is an ordinary (bosonic) manifold. These topological models are defined by topological twisting of the N = 2 supersymmetry on the worldsheet of the supersymmetric sigma model with Calabi-Yau target space. In the closed string case, this twisting just amounts to shifting the spins of all worldsheet fields by

J → J + α/2, where choosing α to be the U(1)R or U(1)A charge defines the A or B model respectively. After this twist one of the two supercharges becomes scalar and can therefore be used as a BRST operator. Correlation functions of BRST-invariant operators are then independent of worldsheet position, and depend on the background in a very restricted way — namely they depend only on the Kähler moduli (A model) or complex moduli (B model). Most importantly for our purposes, the fermionic BRST symmetry acting on the sigma model field space implies that correlation functions are localized on BRST-invariant field configurations. In the A model these are holomorphic maps of the worldsheet into the target (worldsheet instantons), while in the B model they are simply the constant maps. In this sense the B model reduces to classical geometry of points while the A model admits more complicated corrections from holomorphic curves.

2.1 Open string case

The A and B models also admit BRST-invariant boundary conditions for open strings. In the A model such a boundary condition is obtained by restricting the endpoints of the string to lie on a Lagrangian subspace Y of the Calabi-Yau manifold X. In the B model the endpoints are coupled instead to a holomorphic bundle, supported on some holomorphic cycle inside the Calabi-Yau. The corresponding open string field theories were described in [10]. Let us first discuss the A model. In this case the BRST cohomology on the worldsheet gets identified with the ordinary de Rham cohomology on Y . Therefore the target space fields can be packaged

5 together as

A = A0 + A1 + ··· + Am. (2.1)

Here Ak is a k-form on Y , and m is the real dimension of Y . The target space action consists of two parts. One part is the Chern-Simons theory on Y , with action 1 2 SCS = Tr A ∧ dA + A ∧ A ∧ A . (2.2) gs ZY 3 This part of the action can be considered as coming from disc diagrams in which the worldsheet has degenerated to a Feynman diagram; these conﬁgurations lie at the boundary of ﬁeld space and were called “virtual instantons” in [10]. In this Chern-Simons theory the

one-form part of A, A1, is viewed as a U(N) connection (if we have N branes), while the

other Ak can be viewed as the usual ghost ﬁelds which arise in covariant quantization of Chern-Simons [31, 32]. The second contribution to the action comes from honest holomorphic maps (Σ, ∂Σ) → (X,Y ). Each such instanton modiﬁes the action by adding a Wilson loop term, giving

S = C e−ti Tr Pexp A, (2.3) inst i I Xi γi where ti is the area of the i-th instanton and γi is its boundary inside Y . The coeﬃcients Ci will depend on the details of the map. The full target space ﬁeld theory can thus be viewed as the Chern-Simons theory coupled to a background source of Wilson lines, determined by the worldsheet instantons. Thus the full action is given by

S = SCS + Sinst. (2.4)

Note that in the Chern-Simons action the only nonzero terms come from total cohomology degree m. The worldsheet origin of this rule is the ghost number symmetry. This is a worldsheet U(1) symmetry under which Ak has charge k and the BRST operator Q has charge 1; this symmetry is anomalous, so that on the worldsheet Σ it has a background charge −cχˆ (Σ). In computing an A model amplitude the net contribution from ghosts (integral over worldsheet moduli space) is 3χ(Σ), so the total ghost number before any insertions are included is (3 − cˆ)χ(Σ). One therefore sees thatc ˆ = 3 is special, because

6 in this case the partition function on any Riemann surface can be nonvanishing without including any insertions. The story is generally similar in the B model, with one important difference. In this model, as we described above, the only BRST invariant field configurations are the constant maps. Therefore there are no worldsheet instanton corrections; the full open string theory for any Calabi-Yau target space is reduced to a holomorphic version of Chern-Simons, with action 1 2 S = Ω ∧ Tr dA + A ∧ A ∧ A . (2.5) gs ZX 3

2.2 Supermanifold case

Let us now consider the case of the A model on a Calabi-Yau supermanifold X, of complex dimension (m|n). Once again the physical open string ﬁelds make up the de Rham cohomology of the Lagrangian submanifold Y , which has real dimension (m|n); therefore A will have components of all form degrees (m′|n′) where 0 ≤ m′ ≤ m and 0 ≤ n′ ≤ n. The Chern-Simons action is then written exactly as in the bosonic case,

1 2 SCS = Tr A ∧ dA + A ∧ A ∧ A . (2.6) gs ZY 3 The requirement that the integral make sense over Y means that the only nonzero terms will involve ﬁelds with total bosonic degree m as well as total fermionic degree n. For a detailed discussion of forms in the fermionic directions see [33]. Counting fermions as degree −1 we then have the condition that the total degree should be m − n. Just as in the bosonic case, this rule can be understood as a consequence of the worldsheet ghost number symmetry; the target space fermions contribute negatively to the anomaly, so the background charge on the worldsheet Σ is now (n − m)χ(Σ), or simply n − m for the disc. In this supermanifold setting it turns out that the “physical” target space ﬁelds corre-

spond to the A1|∗ components, i.e. choosing the degree of the bosonic form to be 1, without any restriction on the fermionic degree. Of course all fields contribute to the action, but the rest should be viewed as ghosts in the target space. A special case of interest for us will be when (m|n) = (3|4). In this case we have five physical fields on the 3-manifold, which we

7 write A1|0, χ1|1,φ1|2, χ˜1|3,G1|4. Plugging into the action we get schematically 1 SCS = Tr (G ∧ (dA + A ∧ A)+˜χ ∧ DAχ + φDAφ + χ ∧ χ ∧ φ) . (2.7) gs ZY The full eﬀective action also has contributions from honest worldsheet instantons, i.e. holomorphic maps (Σ, ∂Σ) → (X,Y ), which modify the above action by sourcing Wilson lines as in the bosonic case. These worldsheet instantons will play the starring role in our proposal.

3 Our proposal

Now we want to explain our proposal for a string theory which computes the perturbative scattering amplitudes of N = 4 super Yang-Mills coupled to N = 4 conformal supergravity. The string theory we use is the A model on CP3|4, with N branes wrapped on the Lagrangian submanifold RP3|4 (this is a Lagrangian cycle just because it is the fixed locus of an an- tiholomorphic involution.) As discussed above, the string field theory of this model is real Chern-Simons theory on RP3|4, enriched by extra contributions coming from nondegenerate worldsheet instantons with boundary on RP3|4. Then we propose that the on-shell perturbative amplitudes of N = 4 super Yang-Mills coupled to N = 4 conformal supergravity in signature + + −− can be computed by the following procedure. First Fourier transform the wavefunctions of the external particles to the real twistor space RP3|4; this is the real version of the twistor transform [34] as was used in [1, 3]. Then treat the transformed wavefunctions as physical states (vertex operators) in the A model and compute the scattering amplitude for these physical states using standard string perturbation theory. Moreover, we identify the ‘t Hooft diagrams of the gauge theory 2 with string worldsheets of the same topology. In particular, gYM = gs as usual. The dictionary relating Yang-Mills particles to A model vertex operators will work simi- larly to [1]; namely, in the string field theory, the fields with fermionic degree k correspond to Yang-Mills fields of helicity 1 − k/2. But unlike [1] the relevant fields here are coming from differential forms in the fermionic directions, rather than from expanding the field in the fermionic directions of the supermanifold. In particular, even though we obtain from degenerate instantons the Chern-Simons action (2.7), which is the same as in [1], the logic

8 of its derivation is somewhat different. Furthermore, on top of the degenerate instantons leading to the Chern-Simons action, we also end up with Wilson lines on loops coming from honest holomorphic instantons ending on RP3|4. The evidence in favor of this proposal from the perturbative Yang-Mills side can be taken directly from [1]. Namely, Witten observed that certain tree level and one-loop planar amplitudes of N = 4 super Yang-Mills perturbation theory obey certain novel differential equations; these differential equations are equivalent to the claim that when Fourier transformed to RP3|4, the amplitudes are supported only on special configurations of points. Specifically, the amplitudes vanish unless all insertions on RP3|4 are restricted to lie on an algebraic curve in RP3|4. This algebraic curve has to have degree d given by the formula

d = q + l − 1. (3.1)

Here q is the number of ingoing negative helicity gluons in the Yang-Mills scattering process, and l is the number of loops in Yang-Mills. It is easy to derive this rule from our string theory from considerations of ghost charge and counting of fermionic zero modes, as we will show below. First we make one remark: in [1] the possibly disconnected curves are actually connected by the Chern-Simons terms; our setup allows a similar mechanism, but the Chern- Simons terms have their usual interpretation as degenerate worldsheet instantons, so that all contributions to the amplitude can be interpreted as coming from a single connected instanton. In principle we can consider a worldsheet instanton with an arbitrary number of handles h, and b boundaries lying on RP3|4. To compare with the results of [1], we will restrict to diagrams with h = 0 and single-trace contributions, though one of the nice features of our setup is that we should be able to address the subleading contributions as well. For planar diagrams the number of Yang-Mills loops l is related to b by

l = b − 1. (3.2)

Let (r|s) denote the real bosonic/fermionic (virtual) dimension of the moduli space of worldsheet instantons of degree d with h handles and b boundaries. Then the counting of worldsheet ghost numbers in the open A model implies that

r − s = (ˆc − 3)χ(Σ) = (D − 3)(2 − 2h − b), (3.3)

9 where D is the complex superdimension of the Calabi-Yau. For the case at hand D =3−4= −1, so we have r − s = 4(2h + b − 2). (3.4)

On the other hand, if we consider degree d curves in CP3, the bosonic virtual dimension of the moduli space is given by index theory as 4d. This is 4d complex dimensions, but we want to consider only curves which are cut in half by RP3; this amounts to taking a real section of the moduli space, and gives for the real dimension

r =4d. (3.5)

Combining these results and taking h = 0, we ﬁnd the fermionic dimension of the moduli space, s =4d − 4b +8. (3.6)

For the integral over moduli space to be non-vanishing we need to choose an observable with fermion number s. Each negative helicity gluon is a fermionic form of degree 4 on RP3|4, and through the vertex operators this gets related to a fermionic form of degree 4 on the instanton moduli space. So if we have q of them for a non-vanishing amplitude we must have

4q = s =4d − 4b +8 ⇒ q = d − b +2= d − l +1, (3.7) which is Witten’s prediction (3.1). Note that the instanton conﬁugrations considered in [1] are the doubles of the open string instantons considered here. Moreover, the genus g of the doubled surface is the same as the number of boundaries of our worldsheet instanton minus one (as we are restricting here to planar diagrams h = 0). The strict rule g = l rather than g ≤ l of [1] follows because the propagators of the Chern-Simons theory are also part of the worldsheet instanton for us. To ﬁnish the comparison to Yang-Mills theory we should also account for the factor e−A, which is the classical action of an instanton of area A, and would therefore appear in the A model amplitude. Writing a for the area of a degree 1 curve, we have A = ad for a holomorphic curve of degree d. We can absorb the factor e−ad in two rescalings: we rescale

a gs → e gs, (3.8)

10 and also rescale the vertex operators V− for all negative helicity gluons by

a V− → e V−. (3.9)

Using the fermion number counting rule (3.1) (and its simple generalization when h =6 0) we can verify that this rescaling indeed eliminates dependence on the K¨ahler parameter a, which therefore seems to play only a trivial role in this model. This is reminiscent of [33], where it was shown that the K¨ahler form is indeed decoupled from the BRST invariant observables in certain A models on supermanifolds.

4 Mirror symmetry

So far we have proposed a stringy twistorial formulation of N = 4 supersymmetric U(N) Yang-Mills coupled to N = 4 conformal supergravity as the A model topological string on CP3|4 coupled to N branes wrapping RP3|4. We have seen that worldsheet instantons do contribute to various amplitudes in the twistor space. As is well known, the main point of mirror symmetry is to compute precisely such worldsheet instanton contributions. Namely, in the mirror description the instantons disappear; the field theory in the B model captures the worldsheet instantons of the A model. This strategy has been successfully applied to Calabi- Yau threefolds in the presence of Lagrangian branes, starting with [35] and culminating in a recent work [36]. Thus we would expect this strategy to work here as well. How does mirror symmetry work for Calabi-Yau supermanifolds? In fact Calabi-Yau supermanifolds were first studied precisely for a better understanding of mirror symmetry [33]! Namely, the puzzle that rigid Calabi-Yaus cannot have ordinary Calabi-Yau mirrors (as the mirror would not have any Kähler classes) was resolved in [33] by extending the space of bosonic Calabi-Yaus to include Calabi-Yau supermanifolds. In particular, examples were found where the mirror of a rigid Calabi-Yau is a Calabi-Yau supermanifold, realized as a

ni|mi complete intersection in a product of projective supermanifolds i CP . In such cases Q the superdimension of the manifold and the mirror agree (which is required because the superdimension determines thec ˆ in the U(1) current algebra), but the individual bosonic and fermionic dimenions need not be equal. In particular it was found in [33] that a Calabi- Yau of dimension (n|0) can be mirror to a Calabi-Yau with dimension (n + d|d).

11 The techniques to derive mirror geometry from the linear sigma model description have been found in [37] in the context of bosonic Calabi-Yaus. To treat CP3|4 one would need to extend these techniques to the case of Calabi-Yau supermanifolds. This should be possible, given that CP3|4 does admit a simple linear sigma model realization. Here we would like to make a conjecture about what the mirror is. Whatever it is, it will carry the holomorphic version of Chern-Simons. Moreover, via a twistor transform, it should realize Yang-Mills perturbation theory directly from holomorphic Chern-Simons, since that theory has no corrections from worldsheet instantons. Therefore the full classical Yang-Mills should be realized completely in a classical way after the twistor transform. There is only one such formulation presently known: classical solutions of the full Yang-Mills can be identiﬁed with holomorphic data on the quadric in CP3|3 × CP3|3 [13]. We thus conjecture that the mirror of CP3|4 is a quadric in CP3|3 × CP3|3. First of all, note that this quadric is a Calabi-Yau supermanifold. This is already a non-trivial test. In fact, it was already conjectured in [1] that the B model on this quadric could reproduce N = 4 super Yang-Mills. Furthermore, both manifolds have the same superdimension, −1, as required by mirror symmetry. We are currently investigating the validity of this mirror conjecture. If the conjecture is proven it will explain, more or less from ﬁrst principles, the fact that the twistor transformed amplitudes lie on holomorphic curves; it is because the A model on CP3|4 is the mirror of a B model on twistor space whose perturbation theory is the same as that of ordinary N =4 super Yang-Mills.

Acknowledgements

We thank Nathan Berkovits, Sergei Gukov, LuboˇsMotl, Hirosi Ooguri, Warren Siegel and Edward Witten for several helpful conversations. This research is supported in part by NSF grants PHY-0244821 and DMS-0244464.

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