A note on Faddeev--Popov action for doubled-yet-gauged particle and graded Poisson geometry

The section condition of Double Field Theory has been argued to mean that doubled coordinates are gauged: a gauge orbit represents a single physical point. In this note, we consider a doubled and at the same time gauged particle action, and show that its BRST formulation including Faddeev--Popov ghosts matches with the graded Poisson geometry that has been recently used to describe the symmetries of Double Field Theory. Besides, by requiring target spacetime diffeomorphisms at the quantum level, we derive quantum corrections to the classical action involving dilaton, which might be comparable with the Fradkin--Tseytlin term on string worldsheet.

Despite the nomenclature, DFT is not truly doubled: a prescription called section condition should be imposed on all the variables appearing in the theory, such as physical fields and local parameters. The section condition has been argued to imply that the doubled coordinates are actually gauged: a gauge orbit or an equivalence class in the doubled coordinate space corresponds to a single physical point [28]. This idea of 'coordinate gauge symmetry' is naturally realized in sigma models where the doubled target spacetime coordinates are dynamical fields and thus can be genuinely gauged [29][30][31][32][33][34][35][36][37][38][39].
It is the purpose of the present note to revisit the coordinate gauge symmetry from the viewpoint of a constrained system, and along the way establish a connection with the aforementioned graded geometric approach. Specifically we shall show that the doubled-yet-gauged particle action constructed in [33] can be formulated as a simple constrained system whose BRST phase space matches with the graded manifold adopted in [62,63]. In particular, Faddeev-Popov ghosts carrying an O(D, D) vector index are mapped to the Grassmann odd coordinates of the graded manifold. On top of that, the Poisson bracket and the BRST charge agree with [62,63].
The organization of the manuscript is as follows. In the remaining of this Introduction we review the section condition, the coordinate gauge symmetry, and certain elements of the graded geometric approach to DFT. Section 2 contains our main results which split into three parts. Firstly, we introduce a constant projector into a section, and using this we reformulate the section condition as well as the coordinate gauge symmetry. Secondly, we consider the BRST formulation of the doubled-yet-gauged particle action [33], and point out its connection to the graded Poisson geometry [62,63]. Thirdly, by requiring target spacetime diffeomorphisms at the quantum level, we derive quantum corrections to the classical action involving DFT-dilaton, which are analogous to the Fradkin-Tseytlin term on string worldsheet. We conclude with comments in section 3.
Note added: After the first version of our note, a preprint [72] appeared on arXiv which discusses the coordinate gauge symmetry within the context of higher geometry.

Section condition
DFT postulates O(D, D) symmetry as the first principle with an invariant metric, It decomposes the doubled coordinates into two parts, where the Greek letters are (usual) D-dimensional vector indices.
In DFT, it is necessary to impose the section condition: Acting on arbitrary functions in the theory, say Φ r , as well as their products like Φ s Φ t , the O(D, D) invariant D'Alembertian should vanish, leading to the notion of weak and strong constraints, Here we are considering a set, {Φ r , Φ s , Φ t , · · · }, formed by all the functions in DFT including physical fields, gauge parameters, and their derivatives (as well as numerical constants). The section condition is easily solved by letting∂ Diffeomorphisms in DFT consist in the transformations of the doubled coordinates, which induce the following transformation rule for tensors (or tensor density with weight ω), The active version of this passive transformation of tensors is the 'generalized Lie derivative' [2,6], Thanks to the section condition, the generalized Lie derivatives are closed under commutator:

Coordinate gauge symmetry
The section condition has been shown to be equivalent to a certain translational invariance [28,31], where the shift parameter ∆ A is 'derivative-index-valued', meaning that its superscript index should be identifiable as that of derivative: The invariance of every function in DFT (1.10) may suggest that the doubled coordinates are actually gauged by the shift [28]: That is to say, each gauge orbit-or equivalence class-corresponds to a single physical point. This idea of 'coordinate gauge symmetry' has been applied and tested in various contexts. The finite transformation of tensorsà la Hohm and Zwiebach [73] is equivalent to the exponentiation of the generalized Lie derivative, only up to the equivalence relation (1.12) [28] (c.f. [74][75][76]). The usual coordinate basis of one-forms, dx A , However, if we literally gauge the one-form by introducing a derivative-index-valued connection, it becomes DFT-diffeomorphism covariant, (1.14) Further, it is invariant under the coordinate gauge symmetry and thus qualifies as a physical quantity, Using this gauged one-form, we can define a gauge invariant and (arguably) physically meaningful proper length in doubled spacetime as a path integral over the gauge connection [77], recover the doubled (and gauged) string action by Hull [29] [31], and extend to Green-Schwarz superstring [34], U-duality covariant exceptional string actions [35,36] as well as point-like particle actions [33,[37][38][39] (see (2.12) later).

Graded geometric approach
Symmetries of DFT, which are encompassed by the generalized Lie derivative (1.8) with C-bracket (1.9), have been revisited using graded geometry in [61] and further studied in [62,63]. The point of this approach is to reproduce (among other things) the generalized Lie derivative, using tools from graded geometry. In the following, we shall introduce a few elements of this approach which are relevant to our work. Of particular interest for us is the appearance of a graded manifold with coordinates, {x A , p B , θ C }, where x A and p B are usual Grassmann even variables and θ A 's are odd (i.e. θ A 's anti-commute). This graded manifold is endowed with the graded Poisson bracket, where deg(F ) is zero or one for even or odd F respectively. 1 The graded bracket is graded anti-symmetric and satisfies the graded Jacobi identity, , can be identified with a function in the graded manifold: The graded Poisson bracket then provides i) an inner product, ii) an expression very similar to an exterior derivative, iii) the generalized Lie derivative, iv) the C-bracket as a derived bracket, The first equality in (1.21) is due to the Jacobi identity (1.17), the second holds from , and the resulting expression is analogous to the well-known (undoubled) "Cartan's magic formula", i ξ d + d i ξ = L ξ . 1 All the derivatives are a priori set to act from left, ∂F

Main Results
Formulation of the section condition by a projector Here we develop some formalism which will make the somewhat colloquial notion, 'derivative-indexvalued' (1.11), more concrete, and enable us to analyze the constrained system of the doubled-yet-gauged sigma model powerfully later. Specifically, we describe the section condition by a constant projection matrix, P A B , along the line of the earlier works [29,30,49] and the more recent para-Hermitian approach [50][51][52][53][54][55][56][57][58], as These relations imply 2) and the section condition is now fulfilled as 2 Its orthogonal complementary projection matrix follows The middle relation in (2.1) implies that the rank of the projection is D as J AB P AB = P A A = P A A = D. In other words, the section is D-dimensional. Specifically,P A B projects the doubled coordinates into a sectionP A B x B , as it satisfies the section condition of the form (2.1): with ∂ A x B = J AB , Accordingly, all the variables in DFT are functions ofP A B x B only, independent of P A B x B , fulfilling the section condition,P A B ∂ B = 0.
Crucially, any derivative-index-valued vector is P-projected, from (2.2), Conversely, any P-projected vector is derivative-index-valued, That is to say, being P-projected is equivalent to being derivative-index-valued. From now on, we shall always make use of the projector whenever it is necessary to consider the notion of derivative-indexvaluedness. First of all, we reformulate the coordinate gauge symmetry (1.12) and the translational invariance (1.10) equivalently as Once again, P A B is the constant projector of (2.1) and V B is an arbitrary variable carrying an O(D, D) index, c.f. [30,58].
Explicitly for the choice of the section as∂ µ ≡ 0 (1.5), or up to O(D, D) duality rotations, we have 3 Doubled-yet-gauged particle action Now we focus on the doubled-yet-gauged particle action constructed in [33], Here m is the particle mass, l s is a fundamental length scale, e is the einbein, and Compared to (1.14), the derivative-index-valued gauge connection is now equivalently set to be P-projected Further, H AB is the DFT-metric satisfying two defining properties, to which the most general solutions and thus all the possible DFT geometries have been classified in [24].
The action is invariant under worldline diffeomorphisms, 15) and the coordinate gauge symmetry, We combine these two local symmetries, with the shift of the parameter, Thanks to the shift, δx A now assumes a covariant form, while δe and δA A are separately given by the time derivative of each gauge parameter.

Hamiltonian action
In order to obtain more insights into the coordinate gauge symmetry from the view point of a constrained system, e.g. [78], we reformulate the doubled-yet-gauged particle action (2.12) into the Hamiltonian form, where the Hamiltonian is given by Now, A A and e are Lagrange multipliers and generate two first-class constraints, which Poisson-commute, [H A H} P.B. = 0, upon imposing the section condition, P C D ∂ C H AB = 0. The dynamics is governed by the total Hamiltonian, H total = A A H A + eH, Integrating out the auxiliary momenta, p A , in (2.18) one recovers (2.12). Surely, the two first-class constraints reflect the underlying two gauge symmetries of the Hamiltonian action (2.18), the coordinate gauge symmetry and the worldline diffeomorphisms: 4 The difference between (2.17) and (2.22) amounts to the so-called trivial gauge symmetry [78]. It is important to remark that the former constraint in (2.20) is projected,

BRST formulation
Finally, let us extend the classical action, S in (2.12), to the Faddeev-Popov gauge-fixed action, where θ A and ϑ α with α = 1, 2 are Grassmann odd variables. Readers might not immediately recognize standard ghost terms, but the above action contains them precisely and we have a (good) reason to spell the action like above, which we explain shortly.
First of all, integrating out the auxiliary variables, k A , k, we are fixing the gauge, In our case, f i jk = 0 as it is Abelian (2.34).
Decomposing the odd variable, θ A , into two parts, we may identify the standard BC ghost term, from (2.4), (2.17), up to total derivative, 5 The ghost number U is defined as which ranges from −D to +D.
Similarly for the worldline diffeomorphisms, we identify 29) and the corresponding bc ghost term, along with the ghost number, u, Intriguingly, an O(1, 1) structure has appeared with the invariant metric given by the second Pauli matrix, σ 2 , which might hint at the 'doubling' of the worldline, c.f. [79]. It is also amusing to observe that the total derivatives in (2.27), (2.30) actually contribute to the action (2.24) through the ghost number changes at boundaries as 1 2ls (U + u) τ =+∞ τ =−∞ . The BRST differential, δ Q , with the nilpotency, δ 2 Q = 0, is given by (2.31) 5 If we fix the section,∂ µ ≡ 0, with (2.10), we note On-shell we haveθ A = 0,θ α = 0, and these make δ Q e, δ Q A A trivial and thus consistent with the gauge fixing (2.25). Now, we proceed to the Hamiltonian formulation of the Faddeev-Popov action (2.24). We denote the canonical momenta of x A , θ B , ϑ i by p A , Π B , π i respectively, and write the Poisson bracket, The dynamics after the gauge fixing (2.25) is governed by the Hamiltonian (2.19), subject to the two firstclass constraints (2.20) and further two additional second-class constraints, (2.33) The first-class constraints form an Abelian algebra, The relevant Dirac bracket reads which satisfies recovering more familiar bracket structure of the ghost system, 6 (2.37) Thus, on the surface of the second-class constraints, the Poisson bracket reduces to Dirac bracket given by (2.38) The first-class constraints (2.20) make up the BRST charge, where the nilpotency is ensured, with C A ∂ A = 0 (the section condition), by The charge, Q, generates the BRST symmetry (2.31) through the Dirac bracket,

Quantum correction
We consider quantizing the Dirac bracket (2.36). We set a vacuum state, |0 , which is annihilated byp A , B A , andb. Any physical state, |Ψ = Ψ(x)|0 , having trivial ghost numbers should satisfŷ where the covariant derivative, ∇ A , was defined in [8] and gives the expression after the first equality. This new dilaton contribution might be comparable with the Fradkin-Tseytlin term on string worldsheet, c.f. [80].
The modification (2.44) amounts to quantum corrections to the Hamiltonian constraint, and accordingly to the classical action,

Conclusion
In this note, we have shown that the BRST formulation of the doubled-yet-gauged particle action (2.24) naturally produces the graded Poisson geometry of [62,63]. The Grassmann odd variable, θ A , of the graded Poisson bracket carries an O(D, D) vector indices. Thus, if it is to be identified as a ghost of any BRST system, the underlying gauge symmetry should be about the doubled spacetime itself, which we have shown to be the 'coordinate gauge symmetry', x A ∼ x A + P A B V B (2.8). One message our work may convey is that, the investigation of "spaces" can be performed by studying not only the functions defined on them but also the coordinate systems adopted for them, such as the doubled-yet-gauged coordinate system.
A few comments are in order. With B A =P A B θ B (2.26), if we set, instead of (1.18), we may realize an exterior derivative precisely, c.f. (1.20), Now, allowing O(D, D) duality rotations as well as coordinate gauge symmetry in addition to the DFTdiffeomorphisms for the gluing of overlapping patches, c.f. [49,81], the notion of de Rham cohomology should differ from the usual undoubled one.
It would be of interest for future work to generalize our analysis to sigma-models where the constant projector (2.1) is promoted to a local object, c.f. [52,58,82]. Another interesting direction to pursue would be to promote the Abelian coordinate gauge symmetry to a non-Abelian version, as well as to extend this research to the doubled-yet-gauged string actions [29,31,[34][35][36].