Supersymmetric interactions of a six-dimensional self-dual tensor and fixed-shape second quantized strings

"Curvepole (2,0)-theory"is a deformation of the (2,0)-theory with nonlocal interactions. A"curvepole"is defined as a two-dimensional generalization of a dipole. It is an object of fixed two-dimensional shape whose boundary is a charged curve that interacts with a two-form gauge field. Curvepole theory was previously only defined indirectly via M-theory. Here we propose a supersymmetric Lagrangian, constructed explicitly up to quartic terms, for an"abelian"curvepole theory, which is an interacting deformation of the free (2,0) tensor mutliplet. This theory contains fields whose quanta are curvepoles (i.e., fixed-shape strings). Supersymmetry is preserved (at least up to quartic terms) if the shape of the curvepoles is (2d) planar. This nonlocal 6d QFT may also serve as a UV completion for certain (local) 5d gauge theories.


I. INTRODUCTION
Gaining a better understanding of the 6d (2, 0)-theory, originally discovered in [1], has become a major objective in contemporary theoretical high energy physics. The (2, 0)-theory's relation to the low-energy limit of M5branes [2], as well as its power to elucidate the strong coupling dynamics of 4d and lower dimensional gauge theories, has been realized at the outset [1], and further developed in [3][4][5], with a remarkably unified picture emerging more recently in [6][7][8][9][10][11] and other works. These developments generate a strong motivation for finding a fundamental description of the (2, 0)-theory. Several promising approaches have been proposed over the years (see for example [12][13][14][15][16][17][18][19][20][21][22]), but the problem is still open. Part of the difficulty stems from the problem that (at least on the Coulomb branch) the theory describes an interacting anti-self-dual two-form gauge field, and the charged objects that couple to two-forms are strings rather than point particles.
It is therefore interesting to explore the various possibilities that arise from interacting two-form field theories, and in this letter we take first steps to construct a supersymmetric field theory describing a deformation of a free (2, 0) tensor multiplet (containing a two-form gauge field) where some of the scalars and spinors take the form of extended string-like objects whose boundary is charged under the two-form field.
The sort of theory we are looking for was put forward in [23] (referred to by another moniker) as an M-theoretic construction. In order to better explain what kind of 6d field theory we seek to construct, it is useful to recall the "dipole-theories" in 4d. In [24][25][26], motivated by the construction of Yang-Mills theory on a noncommutative space [27][28][29], a theory of "fundamental" dipoles inter- * ganor@berkeley.edu acting with a gauge field was constructed and embedded in string theory. The term "fundamental" here means that the dipoles of dipole-theory are quanta of a fundamental field, rather than composites of other fields. For example, a scalar fundamental dipole field φ couples to a gauge field A µ dx µ via the covariant derivative where l is a constant vector from the (−q)-charged endpoint to the (+q)-charged endpoint of the dipole, and x is taken at the center of the dipole. Such theories arise naturally in string theory and in the context of noncommutative geometry, as explained in [24,25]. The goal of this paper is to construct a 6d counterpart of the dipole-theories where the role of the gauge field A µ dx µ is played by a gauge 2-form B with anti-self-dual field strength H = − * H (and at 0 th order H = dB). The role of the dipole fields will then be played by "curvepole" fields. Let us define a curvepole to be a non-pointlike particle that has the shape of a fixed oriented closed curve C ⊂ R 6 and that interacts with a given two-form field B = 1 2 B µν dx µ ∧ dx ν (µ, ν, · · · = 1, . . . , 6 label the coordinates on Euclidean R 6 ) via the action i (x+C) B, where (x + C) denotes the result of translating C by the spacetime vector x. We would also like the role of the covariant derivative (1) to be played by [30]: However, (2) is incompatible with anti-self-duality and will require a small modification, to be explained in III.
We also wish to preserve some amount of supersymmetry, and in fact, we will require (1, 0)-SUSY. The twoform B will be part of a tensor multiplet (containing also a real scalar, and two Weyl spinors), and the curvepole field φ will be part of a hyper-multiplet (containing two complex scalars and a complex Weyl spinor).
The interaction also modifies the anti-self-duality relation dB = − * dB, as we shall see.
Before proceeding to the details, let us discuss another reason to be interested in the curvepole deformation of the (2, 0)-theory. As was suggested in [23,24,31], such theories arise naturally in M-theory. Following [23], we probe with an M2-brane a flat M-theory background of the form R 1,2 × T 2 × W Ω , where W Ω is the 6dimensional flat space formed by fibering R 5 [parameterized by y = (x 6 , . . . , x 10 )] over S 1 (of radius r and parameterized by x 3 ) with a twist Ω ∈ Spin(5). (I.e., we identify x 3 ≃ x 3 + 2πr together with y ≃ Ωy.) For simplicity, we also take T 2 to be a product of two circles of radius r (parameterized by x 4 , x 5 ). We then set Ω = exp(ih) [for h a fixed element of the Lie algebra so(5)], insert an M2brane probe at x 3 = · · · = x 10 = 0, and take the limit h → 0, (Planck length) ℓ p → 0, r/ℓ p → 0, with hℓ 4 p /r 2 kept finite. Note that at the location of the M2-brane, directions 3, 4, 5 form a small T 3 of volume (2πr) 3 → 0. If also h = 0, we can apply U-duality to turn the M2brane into an M5-brane wrapped on a large T 3 , which in turn is described at low-energy by a free tensor multiplet. It was argued in [23,31] that for nonzero h the dynamics has features of a curvepole theory. We will now proceed with a purely field theoretic construction, which we conjecture is applicable to the case that h preserves half the supersymmetry, i.e., annihilates a spinor of Spin(5).

A. Spinor conventions
The 6d spin group is Spin(6) ∼ = SU (4), and we use lowercase Roman indices (a, b, c, · · · = 1, . . . , 4) to denote components of a chiral spinor, with lower indices for the fundamental representation 4 of SU (4) (e.g., ψ a ), and upper indices for the dual representation 4 (e.g., ψ a ). A vector in the representation 6 with components V µ will be represented by an antisymmetric V ab = −V ba , normalized so that the scalar product of two vectors V and U is Components of a 2-form B µν are written in spinor notation as traceless B b a (with B a a = 0), and normalized so that c . An anti-self-dual 3-form with tensor components H (−) µνσ is expressed in spinor notation as a symmetric H ab = H ba , while a self-dual 3-form is expressed as H ab = H ba . They are normalized so that H ab H ab = c .

B. Curvepole integrals
The action presented below depends on the choice of a fixed (two-dimensional) open surface Σ ⊂ R 6 with boundary C = ∂Σ. For a (either fundamental or composite) field Φ(x), we define the curvepole integral: and the anti-curvepole integral: We call Σ the curvepole surface, and say that it is balanced if for arbitrary Φ and Ψ, It can also be expressed as Φ γδ αβ = Φ αβ γδ . We will call Σ n-planar if it can be embedded in some R n subspace of R 6 . It is not hard to see that a 2-planar curvepole is balanced [in which case the integrand of (5) vanishes], but the reverse is not necessarily true. In fact, any parity invariant Σ is balanced.

III. THE CURVEPOLE DEFORMATION
We will start with the action I 0 of the free 6d tensor theory, and add interaction terms I 1 + I 2 + · · · , where I n is a polynomial of order (n + 2) in the fundamental fields, and each of its terms contains a total of n curvepole integrals. At each order we will modify the supersymmetry variation δ = δ 0 + δ 1 + δ 2 + · · · , where δ 0 is the free SUSY variation that is linear in the fields, δ 1 is quadratic in the fields with terms that have one curvepole integral, etc. The calculations have been performed up to quartic order, so the results presented below satisfy The curvepole deformation preserves a U (1)×SU (2) subgroup of the Sp(2) R-symmetry of the free tensor multiplet. SU (2) doublets will carry indices i, j, k, · · · = 1, 2. They will be lowered with the antisymmetric ǫ ij . The surviving anticommuting (1, 0) SUSY parameters, which are U (1) neutral SU (2) doublets and chiral spinors, will be denoted by η ai . The U (1) symmetry will be further discussed in subsection III B.

A. Quadratic terms
The fields of the (2, 0) tensor multiplet are: a 2-form gauge field B b a , an SU (2) singlet scalar ϕ, an SU (2) doublet anticommuting spinor χ i a , an SU (2) doublet scalar φ i and its complex conjugate φ i , and an SU (2) singlet anticommuting spinor ψ a and its complex conjugate ψ a . The first three fields form a (1, 0) tensor multiplet, and the last four form a hypermultiplet. The SUSY transformations are

The action is
Note that the coupling constant of the 3-form field strength has the self-dual value √ 2π. The self-dual part of the 3-form field strength H (+) µνσ decouples, and we will make sure below that interaction terms are always expressed in terms of the anti-self-dual H (−) µνσ only.

B. Cubic terms
At 0 th order, the U (1) ⊂ Sp(2) current that the curvepole deformation preserves is given by The fields that will become curvepoles are those that are charged under this U (1). However, the covariant derivative (2) is not satisfactory, because interaction terms in our action are allowed to depend on B only through H (−) .
To overcome this, we recall that for dipole-theories there is a way to rewrite (1) so that only the field strength F µν = 2∂ [µ A ν] appears [24]. One performs a field re- where P is some fixed path from − 1 2 l to 1 2 l, and x + P is the translation of the path by x. Then e iq (x+P ) Aµdx µ D For curvepole interactions, guided by the above example, we replace q P F µν (x + y)dy ν with ± H (−) µνσ νσ . We thus add to I 0 the interaction term − 1 2 J µ H (−) µνσ νσ and look for a supersymmetric completion. The results are described below. We define a vector field which will play the role of an effective gauge field. We also define an effective gluino field ρ ia and a composite field S i a by Then, the cubic interactions in the action are given by To be consistent with the equations of motion, the antiself-duality condition is modified to To write the correction to the SUSY transformation at this order, we define λ≡η a i χ i c c a , and set δ 1 ϕ = 0,

C. Quartic terms
We define an SU (2) triplet "meson" composite field We then find that satisfies (6) provided the balanced curvepole condition (5) holds. (We will not present δ 2 here, but we claim that δ 0 I 2 + δ 1 I 1 vanishes up to 0 th order equations of motion, which is equivalent to the existence of a δ 2 such that δ 0 I 2 +δ 1 I 1 +δ 2 I 0 = 0.) The balanced curvepole condition is needed to cancel terms of the form φ∂ψ ψψ .
The first term V µ V µ φ i φ i in (13) combines with −J µ V µ of (11) and the kinetic term of (7) to give the action of a scalar that is minimally coupled to the effective gauge field V µ . The second term of (13) is part of an effective magnetic coupling. It can also be derived by formally dualizing an electrically coupled H µνσ using the standard technique of treating H as the independent field and introducing a Lagrange multiplier for the Bianchi identity dH = 0, then integrating over H. (See [32] for a thorough explanation on how to couple a theory of anti-self-dual fields to a 3-form source, and see [33][34][35] for related calculations in the context of other nonlocal gauge theories.) The terms in I 2 were determined by requiring supersymmetry up to (and including) quartic terms. But SUSY alone is not sufficient to uniquely determine I 2 , because it turns out that there is a term Thus, I 2 → I 2 + cI ′ 2 , together with δ 2 → δ 2 + cδ ′ 2 , will also be a solution to (6) for any arbitrary constant c. Nevertheless, (13) is uniquely determined by imposing the "hollowness" requirement to be introduced next.

IV. HOLLOW CURVEPOLES
We would like the curvepole theory defined by I = I 0 + I 1 + I 2 + · · · to depend only on the curve C = ∂Σ, and not on the way we fill it with the curvepole surface Σ. We will refer to terms that only depend on ∂Σ as hollow.
As it stands, however, (11) and (13) appear to depend on the bulk of Σ. Upon closer inspection, we see that the last term of (13) and the last term of (11) in fact do depend only on C. This is because ∂ ν Φ , and for fixed d, this is antisymmetric in a, b, and hence a vector that is proportional to ∂ µ χ i d µν , which depends only on C. The terms in the action that are not explicitly hollow are . (14) The first term of (14) could have been hollow if H (−) were a closed 3-form. But this is only true at 0 th order. The equations of motion that follow from I 0 + I 1 are However, it turns out that the total action is hollow up to a field redefinition, provided Σ is 3-planar. To see this, consider a deformation of Σ given by y µ → y µ + ξ µ , where ξ : Σ → R 6 is an infinitesimal vector that vanishes on C. Then, for any field Φ, the variation of the curvepole integral is where ξ σ (· · · ) is defined as the integral (3) with ξ σ (y) inserted in the integrand, and similarly ξ σ (· · · ) is given by (4) with ξ σ (−y). Next, we augment (16) with the field redefinition δB γδ = πi 4 ǫ αµνσγδ ξ α J µ νσ (note that i will be absent in Minkowski signature), as well as . Then a straightforward calculation shows that The RHS of (17) vanishes for a 3-planar Σ, because in a suitable coordinate system there are only three possible values for the indices β, γ, ν, σ for which the curvepole integral does not vanish. Thus, if C is 2-planar, we can deform Σ in the direction of an arbitrary 3-plane without affecting the theory. This demonstrates that the bulk of Σ is "incorporeal." Recall that a 2-planar curvepole Σ is balanced. We now see that even if only C is 2-planar, as long as Σ is 3-planar the theory is supersymmetric (up to quartic order), although perhaps not manifestly so.

V. DISCUSSION
We have constructed, up to quartic order, a supersymmetric Lagrangian that describes the interactions of a curvepole field with a tensor gauge field B µν whose field strength is anti-self-dual. Requiring, in addition to supersymmetry, that the interactions be only through the boundary C = ∂Σ, in the sense of section IV, determines the quartic terms uniquely.
The question of quintic and higher terms is open. Terms of order n (within I n−2 ) have a total of (n − 2) · 's and · 's. Basic dimensional analysis shows that no such terms are explicitly hollow for n ≥ 5 (since each · would require an additional ∂ in order to be expressible as an integral over C). Thus, to preserve "hollowness", we would expect the only modification to be that J µ in (13) should be corrected from its 0 th order form (8) to include the contribution of interactions. It is easy to derive it order by order as the Noether current of the U (1) symmetry. At order n = 6 possible anomalies should also be considered. The SUSY transformations (12) involve anomalous chiral rotations of fermions. However, since V is not a true gauge field, this is not expected to be a problem. In fact, without the chiral rotation the SUSY algebra closes only up to an anomalous U (1) gauge transformation, and hence supersymmetry would be anomalous itself (see [36]). The chiral rotation ensures that the anticommutator of two SUSY transformations is a translation, and hence is expected to be anomaly free. The fermionic field redefinitions of section IV also have an anomaly proportional to ε(dV) 3 , and it is possible that the "hollowness" condition suffers from anomalies at n ≥ 6. An anomaly would manifest itself as a nonzero δ ξ variation of a 4-point correlator of tensor fields, of the form HHHH . This variation, however, also receives a 1-loop contribution from the field redefinition , introduced in IV. The complete result, and in particular the question of whether a hollowness-anomaly exists or not, would require further analysis.
We expect the action constructed here to describe the low-energy degrees of freedom of the M-theory setup described at the end of section I when the twist h has a nonzero kernel in the spinor representation (and hence preserves SUSY). It would be interesting to determine Σ from the twist. That setup has a formal dual with an M5-brane probing a background with strong 4-form flux [25,31]. The limit of small Σ might therefore be related to the order-by-order expansion of the interaction of an M5-brane with 4-form flux (see also [21,[37][38][39][40][41]). It would also be interesting to connect the M-theory construction of [24,31] with the field theory construction of [42] (using BLG theory [43,44]), and to generalize to other 6d theories [45][46][47][48][49].
Curvepole theories can be viewed as higher dimensional analogs of dipole-theories, and the latter have interesting supergravity duals [25] that have found applications in the study of rotating black holes [50,51]. We would expect a curvepole deformation to exist for any number of M5-branes, and it would be interesting to identify deformations of AdS 7 × S 4 [52] that depend on Σ as a parameter.
Curvepole theories can also serve as novel kinds of UV completion of 5d theories. Consider curvepole theory on R 5 × S 1 , with Σ = I × S 1 , where I ⊂ R 5 is a straight segment. Σ is 2-planar and hence balanced. We can take one end of the segment to ∞ and the other end at the origin. According to section IV, it does not matter in which direction we take the segment. The theory is therefore local on R 5 , and at low energy it becomes a U (1) gauge field interacting with a charged hyper-multiplet. (See [53] for a recent analysis of UV completions of supersymmetric 5d theories.) ACKNOWLEDGMENTS I am grateful to Kevin Schaeffer for a discussion on [23,24,31] in 2012, in which he explicitly wrote down the covariant derivative form (2), and which subsequently prompted me to search for a supersymmetric extension. I would also like to thank Keshav Dasgupta for helpful comments, Kuo-Wei Huang for helpful correspondence, Neil Lambert for sharing with me the yet unpublished results of [41], and the Aspen Center for Theoretical Physics and the organizers of the 2017 Aspen Winter Conference "Superconformal Field Theories in d ≥ 4", for allowing me to present preliminary results of this work.