Kinematic Lie Algebras From Twistor Spaces

We analyze theories with color-kinematics duality from an algebraic perspective and find that any such theory has an underlying BV${}^{\color{gray} \blacksquare}$-algebra structure, extending the ideas of arXiv:1912.03110. Conversely, we show that any theory with a BV${}^{\color{gray} \blacksquare}$-algebra features a kinematic Lie algebra that controls interaction vertices, both on- and off-shell. We explain that the archetypal example of a theory with BV${}^{\color{gray} \blacksquare}$-algebra is Chern-Simons theory, for which the resulting kinematic Lie algebra is isomorphic to the Schouten-Nijenhuis algebra on multivector fields. The BV${}^{\color{gray} \blacksquare}$-algebra implies the known color-kinematics duality of Chern-Simons theory. Similarly, we show that holomorphic and Cauchy-Riemann (CR) Chern-Simons theories come with BV${}^{\color{gray} \blacksquare}$-algebras and that, on the appropriate twistor spaces, these theories organize and identify kinematic Lie algebras for self-dual and full Yang-Mills theories, as well as the currents of any field theory with a twistorial description. We show that this result extends to the loop level under certain assumptions.


INTRODUCTION AND SUMMARY
Color-kinematics (CK) duality [2,3] (see reviews [4][5][6][7][8]) is a surprising property of certain field theories, that allows for their scattering amplitudes to be split into a kinematic component or numerators and gauge Lie algebra or color numerators, such that the kinematic numerators mirror the algebraic properties of the color numerators.This was first observed for the tree-level amplitudes of Yang-Mills theory, but many other theories also exhibit CK-duality.CK-duality is the cornerstone of the double copy prescription [2,3], which constructs gravity scattering amplitudes from a simple combination of pairs of corresponding kinematical numerators for Yang-Mills theory, suggesting deep, illuminating connections between the known fundamental theories of nature and providing cutting-edge predictions in gravitational-wave astronomy, see e.g.[9][10][11].
Concretely, a theory is CK-dual if its n-point amplitudes A n can be written as where Γ n denotes the set of cubic Feynman graphs with n external lines, d γ are the product of 1 p 2 ℓ over all propagator lines ℓ, with p ℓ their momenta, c γ are the color numerators, i.e. products of gauge Lie algebra structure constants, as prescribed by the diagram γ, and n γ are the kinematic numerators, built from momenta and polarization tensors.Furthermore, c γ and n γ obey the same antisymmetry under the interchange of edges in γ, and c γ1 + c γ2 + c γ3 = 0 implies n γ1 + n γ2 + n γ3 = 0. CKduality thus suggests the existence of a kinematic Lie algebra (KLA) from which the n γ are constructed.
Instead of working at the level of amplitudes, we consider CK-duality and the KLA directly at the level of actions [39? -41]: given tree-level CK-duality, one can always render the action CK-dual using infinitely many auxiliary fields, but at the cost of unitarity, which is broken by Jacobians arising from field redefinitions [40? , 41].It therefore remains to identify an organizational principle for the resulting tower of auxiliaries and avoid nonlocal field redefinitions altogether.Following [1], we find this organizational principle in the form of BValgebras, which ensure the existence of a kinematic Lie algebra. 1he prime example of a theory with BV -algebra is Chern-Simons (CS) theory, and many field theories can be equivalently formulated as CS-type theories on twistor spaces.Using this picture, we are able to reproduce and generalize e.g. the results of [26,27,42].We show that the currents of such field theories come with a KLA which we can readily identify.In many cases, this KLA extends to the amplitudes, and for a special class, it implies conventional CK-duality.For theories for which the anomalies discussed in [43] are absent, our arguments extend to the loop level.
Our results significantly improve the understanding of KLAs and comprise concrete and new examples.They highlight the power of the action perspective on CKduality as an organizational principle, and our improved algebraic understanding has the power to streamline the computation of the kinematic numerators important in the double copy construction of gravity scattering amplitudes, cf.[26].

CHERN-SIMONS THEORY
We start with the illustrative example of ordinary nonabelian CS theory, which demonstrates all essential features.For our purposes, it is convenient to work with Batalin-Vilkovisky (BV) quantization [44,45], which introduces unphysical fields called antifields in addition to the physical fields and ghosts, and use differential form notation to hide Lorentz indices.The BV action of CS theory reads as where A is a gauge potential 1-form; c is the ghost field, a Grassmann-odd scalar function; A + and c + are the corresponding antifields (an odd 2-form and an even 3-form); and all fields take values in a color Lie algebra g.This action is the Maurer-Cartan action for the differential graded (dg) Lie algebra of differential forms with values in g, (Ω • ⊗ g, d), whose Lie bracket is the wedge product composed with the Lie bracket of g, whose differential is the exterior derivative d, and whose grading is such that Ω p carries ghost number 1 − p, see e.g.[46].After color-stripping, we are left with the dg commutative algebra (Ω p , d) of ordinary differential forms under wedge product and exterior derivative.
It is well-known that CS scattering amplitudes are trivial.Instead, [42] considers correlators of harmonic differential forms.To compute these, note that where the codifferential d † is defined as d † α = −(−1) p ⋆d⋆α for a p-form α using the Hodge operator with respect to the Minkowski metric, and the d'Alembertian.The propagator is now given by −d † .Using (3), we may decompose the identity operator on differential forms as where Π Harm projects onto the harmonic forms.The operator −d † allows us to introduce a "derived bracket" on Ω for all α ∈ Ω p and β ∈ Ω • .Since −d † is a second-order differential operator, [−, −] defines a so-called Gerstenhaber bracket on Ω • , which is a degree-shifted Lie algebra.Furthermore, the bracket [−, −] maps pairs of physical fields to physical fields and encodes their interactions.Thus, this defines the KLA2 for correlators of harmonic forms, which therefore can be brought into the form (1).One can show [47,48] that this KLA is isomorphic to the Schouten-Nijenhuis algebra of totally antisymmetric tensor fields, the natural Gerstenhaber algebra on three-dimensional Minkowski space.
Truncating K to degree 0 yields the KLA K 0 commonly discussed in the literature, which here is the spacetime diffeomorphism algebra.The above straightforwardly extends to holomorphic CS theory on 3 , with the real p-forms and the KLA replaced by the complex (0, p)-forms and the evident holomorphic version of the Schouten-Nijenhuis algebra, respectively.
More generally, the structure (Ω • , d, ∧, −d † ) is an instance of what is known as a BV -algebra [1,26] with = , which we explore below.

COLOR-KINEMATICS DUALITY ALGEBRAICALLY
Consider a field theory whose tree amplitudes (1) arise from the Feynman diagram expansion of a BV action.
The corresponding vector space of fields L is graded by the ghost number and so, we may write L = p∈ L p , where the elements of L p carry ghost number 1 − p.The free part of the action is captured by a kinematic operator, which is a linear map d on L with d 2 = 0 that decreases ghost number by 1 (mapping antifields to fields and fields to zero).All interaction terms are cubic which are captured by a product on L that conserves ghost number.Gauge invariance implies that this L forms a dg Lie algebra [41,46,49], which generalizes our previous Ω p ⊗ g.
The Feynman expansion (1) implies that a propagator h exists that inverts d on propagating (off-shell) fields.Thus, the identity operator on fields decomposes as where Π on-shell is the projector on on-shell fields, generalizing (4).It is always possible to choose h such that h 2 = 0 [50].Splitting h = b into the denominator and numerator b leads to db + bd = .
Color-stripping now amounts to factorizing L = g ⊗ B into the color Lie algebra g and a dg commutative algebra (B, d, m) with differential d and product m [41].Denoting the color-stripped propagator also by b , we have which generalizes (3), and is a second-order differential operator with respect to m.If also b is a second-order differential operator that squares to zero, 3  for all x, y ∈ B is a Gerstenhaber bracket on B of which ( 5) is a special instance.As in the CS case, the KLA (with all BV fields) is then simply Truncating K to degree 0 yields the usual KLA K 0 .Mathematically, (B, d, m, b) is a BV algebra [51] with Gerstenhaber bracket given by ( 9), and promotes this BV algebra to a BV -algebra [1,26].
Conversely, in a theory with a BV -algebra, the cubic vertices in Feynman diagrams are governed by a KLA and a "color" Lie algebra.When coincides with the d'Alembertian , this implies CK-duality (up to potential anomalies).Otherwise, complications may arise such that it is not possible to write the amplitudes in the form (1).Then, we merely speak of a theory with a KLA.
Apart from the CS theories already discussed, these ideas extend to field theories whose linearized equations of motions are encoded in a differential with a natural codifferential.This is the case for all theories whose solutions can be described in terms of flat connections on twistor spaces.In the following, we discuss two examples in detail.We stress that even in the absence of an action principle, we still have a BV -algebra and, thus, a KLA for the numerators of the corresponding tree-level currents.
The space of color-stripped fields forms a dg commutative algebra, with product given by the wedge product and differential ∂red .It further forms a BV -algebra where ι X denotes the contraction of a differential form with a vector field X.A quick calculation shows that Recall that the actions ( 13) and ( 11) are equivalent, i.e. they share the same tree-level amplitudes.We can compute these by embedding external states on Ê 4 , given by harmonic gauge potentials, into A ∈ Ω 0|0,1|0 red , respecting the gauge condition bA = 0. We then use the trivial Feynman rules derived from (13)  The full KLA K is isomorphic to the Schouten-Nijenhuis-type Lie algebra K of bosonic holomorphic totally antisymmetric tensor fields on Z with Lie bracket This construction generalizes to dimensionally reduced SDYM theory and theories with any amount of supersymmetry, following [58,59] or, e.g.[60].
To match the literature, note that the lowest order in η i of the gauge potential A describes the gluon.Since to this order, ι Ê0 A can be gauged away [53, §5.2],only holomorphic multivector fields spanned by the E α in the Schouten-Nijenhuis-type Lie algebra K contribute to K0 .On spacetime, these parameterize translations along selfdual planes spanned by ), reproducing the KLA identified in [27].

MAXIMALLY-SUPERSYMMETRIC YANG-MILLS THEORY
Similar arguments also hold for full maximallysupersymmetric Yang-Mills (MSYM) theory. 4However, the KLA will be such that CK-duality is not immediate. 4Recall that N = 3 SYM theory is perturbatively equivalent to N = 4 SYM theory.
The relevant action here is again of the form ( 13), with Z replaced by L, Ω 3|4,0|0 replaced by the holomorphic measure identified in [62,63], and the dg commutative algebra (Ω 0|0,•|0 red , ∂red ) replaced by the restricted bosonic CR differential forms Ω 0|0,•|0 CR that depend holomorphically on (η i , θ i ) = (η α i λ α, θ iα µ α ) for i = 1, 2, 3 and with no antiholomorphic fermionic directions, which are endowed with the differential ∂CR = êF ÊF + êL ÊL + êR ÊR .The second-order differential operator Thus, we obtain the BV -algebra While L is not compatible with Wick rotation, B SYM and the contained KLA are: KK-expand the theory along P 1 × P 1 , obtaining a cubic field theory with an infinite tower of KK fields on Ê 4 , on which ( ∂CR , b, ) act as "(∞ × ∞)-matrices of differential operators." The complexified KK fields all carry complex Spin(4)representations5 , and imposing reality conditions suitable for Minkowski space Wick-rotates both the fields and the operators ( ∂CR , b, ). 6n view of CK-duality, our above propagator involving the inverse of , where M, N, . . .label KK modes, has a striking similarity to the YM propagator in R ξ -gauge in that both lead to unphysical singularities (e.g.k µ k ν k 4 for h µν ξ ) in individual Feynman diagrams.These singularities signal the propagation of unphysical longitudinal modes.With all external states physical, their contributions have to cancel, and for R ξ -gauge this follows from Ward identities.It is natural to expect that the same occurs for (potentially Wick-rotated) Cauchy-Riemann CS theory, and there is a KK tower of generalized Ward identities that allows to replace the propagator b with b .If true, then the BV -algebra guarantees CK-dual numerators: the numerators computed with the propagator b are automatically CK-dual.We study this in upcoming work.
The extension to the loop level depends on the assumption that the relevant (ambi)twistor theories correctly describe the loop amplitudes.The equivalences between spacetime field theories and twistorial CS-type theories only extends to the loop level if certain twistor space anomalies vanish [43].Provided that Cauchy-Riemann CS theory on L is anomaly-free with no further problems reducing the path-integral measure from twistor space fields to spacetime fields, then Cauchy-Riemann CS theory on L captures also loop amplitudes, and we obtain a loop-level KLA.
Let us sketch an argument that this is true for the Cauchy-Riemann CS anomaly.A codimension k Leviflat Cauchy-Riemann manifold M foliates into holomorphic leaves M t .If the space of leaves T is a k-dimensional manifold, one easily checks that L satisfies these conditions, then the Cauchy-Riemann CS partition function is log Z(M ) = t∈T ω log Z(M t ), where ω is a volume form (defining the path integral measure) on T , and where Z(M t ) is the partition function of holomorphic CS theory (with the same field content) on M t : the full theory on M is anomaly-free if the corresponding holomorphic theory on M t is anomaly-free.
Thus, it suffices to study holomorphic CS theory on M t , or even (as anomalies are integrals of local objects) on small patch in M t , where global issues (e.g.nonzerodegree bundles) disappear.In the weak-coupling limit, MSYM theory.Perturbation theory with propagator b and gauge bA = 0 for the field A reproduces MSYM amplitudes on four-dimensional Minkowski space: semi-classical equivalence fixes the interaction vertices; the propagator b is the inverse of the kinematic operator almost-everywhere (i.e.modulo measurezero sets) in momentum space.
anomaly contributions from bosons and fermions are equal and opposite [43], since their linearized actions in the presence of an external gauge field coincide up to statistics: supersymmetry ensures anomaly cancellation in holomorphic CS theory and Cauchy-Riemann CS theory.Without supersymmetry, this argument fails.
Indeed, for nonsupersymmetric twistorial holomorphic CS theory (semi-classically equivalent to SDYM theory) there is an anomaly [43] that, via the preceding argument, implies an anomaly for nonsupersymmetric twistorial Cauchy-Riemann CS theory (semi-classically equivalent to YM theory).Even if the KLA algebra does imply tree-level CK-duality, it will be anomalous.This is consistent with, and elucidates, the conclusion that CKduality can be realized as an anomalous symmetry of a semi-classically equivalent YM-BV action [? ], as well as the proof (by exhaustion) that there are no CK-dual four-point two-loop numerators for bosonic YM theory assuming that they can be derived from local Feynman rules [64].

Ê 4 .
together with the propagator h = b Ê 4 .This Feynman diagram ex- pansion manifests the KLA contained in B SDYM with = Hence, MSSDYM theory possesses CK- duality, and the twistor action produces a CK-dualitymanifesting spacetime action for MSSDYM theory after Kaluza-Klein (KK)-expanding along P 1 .Integrating out the KK tower of auxiliary fields reproduces the Siegel action.
Data Management.No additional research data beyond the data presented and cited in this work are needed to validate the research findings in this work.For the purpose of open access, the authors have applied a Creative Commons Attribution (CC-BY) license to any Author Accepted Manuscript version arising.