# BFV extensions for mechanical systems with Lie-2 symmetry

Aliaksandr Hancharuk and Thomas Strobl  Institut Camille Jordan, Université
Claude Bernard Lyon 1
43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

###### Abstract

We consider mechanical systems on $T^{*}M$ with possibly irregular and
reducible first class contraints linear in the momenta, which thus correspond
to singular foliations on $M$. According to a recent result, the latter ones
have a Lie-infinity algebroid $({\cal M},Q)$ covering them, where we restrict
to the case of Lie-2 algebroids. We propose to consider $T^{*}{\cal M}$ as a
potential BFV extended phase space of the constrained system, such that the
canonical lift of the nilpotent vector field $Q$ yields automatically a
solution to the BFV master equation. We show that in this case, the BFV
extension of the Hamiltonian, providing a second corner stone of the BFV
formalism, may be obstructed. We identify the corresponding complex governing
this second extension problem explicitly (the first extension problem was
circumvented by means of the lift of the Lie-2 algebroid structure). We
repeatedly come back to the example of angular momenta on
$T^{*}{\mathbb{R}}^{3}$: in this procedure, the standard free Hamiltonian does
not have a BFV extension—while it does so on
$T^{*}({\mathbb{R}}^{3}\backslash\\{0\\})$, with a relatively involved ghost
contribution singular at the origin.

##  I Introduction

One of the main assumptions in textbooks about constrained systems—see, e.g.,
[1]—is that the constraint functions $G_{a}$ are regular on the unconstrained
symplectic manifold. Important existence theorems about BRST-BV [2, 3] or BFV
[4, 5] extensions have then been proven in this setting only—see, however, [6,
7, 8] for a notable exception. Physically realistic systems often do not
satisfy the regularity conditions. In realistic situations, group actions can
have fixed points and there the corresponding constraints are not regular. And
in non-abelian Yang-Mills gauge theories on compact spaces, for example, the
constraints become non-regular at reducible connections.

On the other hand, without irreducibility and, in particular, regularity
assumptions, often the BV and BFV extensions become hard mathematical
problems. Let us illustrate this statement in the finite dimensional setting:
Suppose you are given a set $v_{a}$ of vector fields on a manifold $M$
satisfying

$[v_{a},v_{b}]=C^{c}_{ab}v_{c}$ (1)

for some functions $C^{c}_{ab}$. Then the corresponding ”BRST differential”
$Q$ is readily written down as

$Q=\xi^{a}v_{a}-\tfrac{1}{2}C^{a}_{bc}\,\xi^{b}\xi^{c}\tfrac{\partial}{\partial\xi^{a}}$
(2)

where $\xi^{a}$ are the ”ghosts”, odd coordinates on an extended graded
manifold. At first sight, a short calculation, using (1) together with the
Jacobi identity for the Lie bracket of vector fields, seems to establish
$Q^{2}=0$. However, if the vector fields $v_{a}$ are not linarly independent
at each $x\in M$, there exist functions $t^{a}_{I}$ such that

$t^{a}_{I}\,v_{a}\equiv 0.$ (3)

This happens already for the standard three vector fields $L_{a}$ generating
rotations in ${\mathbb{R}}^{3}$ where $x^{a}L_{a}\equiv 0$. Then the structure
functions $C^{a}_{bc}$ entering (1) are not unique:

$C^{a}_{bc}\sim C^{a}_{bc}+t^{a}_{I}B^{I}_{bc}.$ (4)

The question if, within this equivalence class, there _exists_ a choice of the
structure functions $C^{c}_{ab}$ such that $Q$ as defined in (2) squares to
zero is, by the work of Vaintrob [9], equivalent to the question if the
singular foliation generated by the vector fields $v_{a}$ comes from a Lie
algebroid covering it.111In the case of the angular momenta $L_{a}$, there is
a preferred choice of structure functions, namely the constant ones. They can
be identified with the structure constants of the Lie algebra $so(3)$. Indeed
this example of rotations comes from a simple Lie algebroid, namely the action
Lie algebroid $so(3)\times{\mathbb{R}}^{3}$. And this question is, as of this
day, an unsolved mathematical problem.

In the present paper, we want to consider a Hamiltonian constrained system
defined on a cotangent bundle $T^{*}M$, with first class [11] constraints
$G_{a}$ linear in the momenta and a Hamiltonian quadratic in them. Linear
constraints are of the form

$G_{a}(x,p)\equiv v_{a}^{i}(x)p_{i}\,.$ (5)

So we can identify them with the vector fields $v_{a}$ on $M$ above, since the
first class property of the constraints $G_{a}$ is tantamount to precisely
(1). We do not restrict to merely regular constraints, as mostly done in
[12]—otherwise one deals with regular foliations (on $M$ as well as on
$T^{*}M$)—but we permit leaves of different dimensions a priori. A quadratic
Hamiltonian has the form

$\mathrm{Ham}=\tfrac{1}{2}g^{ij}(x)p_{i}p_{j}.$ (6)

We assume the coefficient matrix $g^{ij}$ to be non-degenerate, thus
${\mathrm{Ham}}$ is the standard Hamiltonian corresponding to a metric $g$ on
$M$. We require the usual compatibility of the constraints with the dynamics,

$\\{G_{a},\mathrm{Ham}\\}=\omega_{a}^{b}\,G_{b}$ (7)

for some functions $\omega_{a}^{b}$ on $T^{*}M$. This equips the base manifold
$M$ with the structure of a singular Riemannian foliation [13, 12, 14].

We assume here that (3) captures all dependencies of the vector fields and
there are no dependencies between the functions $t_{I}^{a}$ on $M$. In more
mathematical terms, this means that one has a sequence of vector bundles

$\quad 0\to F\stackrel{{\scriptstyle
t}}{{\longrightarrow}}E\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}TM\,,$
(8)

such that $0\to\Gamma(F)\to\Gamma(E)\to{\cal F}\to 0$ is exact. Here ${\cal
F}$ denotes the subset of vector fields obtained as the image of $\rho$, a
generating set of which corresponds to the vector fields $v_{a}=\rho(e_{a})$.
Likewise, the map $t$ gives rise to the functions $t^{a}_{I}$ after the choice
of local bases in $e_{a}$ and $b_{I}$ in $E$ and $F$, respectively.

And now the situation is different: Under these assumptions, it has been
proven recently [15] that there always exists a homological vector field $Q$,
an extension of (2), which describes a Lie 2-algebroid covering the singular
foliation generated by the vector fields $v_{a}$.

Note that while (3) yields the reducibility conditions

$t_{I}^{a}G_{a}\equiv 0,$ (9)

the above does not imply automatically that there are no further
reducibilities between the constraints (5). This is because to find all
reducibilities, we need to find all independent functions $t_{I}^{a}$
satisfying (9) on $T^{*}M$—and not just on $M$.

Consider, e.g., $G_{a}=\varepsilon_{abc}x_{b}p_{c}$, angular momenta. Then
besides $x^{a}G_{a}\equiv 0$, we have the additional dependency
$p_{a}G_{a}\equiv 0$. But furthermore, there are dependencies between these
dependencies and in the end one finds an infinite set of reducibility
conditions leading to an _infinite_ tower of ghosts for ghosts in its BFV
description. We will illustrate the beginning of this procedure at the end of
this article and provide a complete description elsewhere. In principle,
having such a complete description of the tower of dependencies between the
constraints (5) on $T^{*}M$, one can construct a Koszul-Tate resolution, which
then guarantees the existence of a BFV extension also in the singular context.
However, this description can be very cumbersome for practical purposes
apparently and one may be interested in potential shortcuts.

As such the following procedure proposes itself: Let us return to the setting
described by (8). As mentioned above, there is a homological vector field $Q$
associated naturally to (8). If we consider its Hamiltonian lift to the
cotangent bundle of the graded space underlying $Q$, we get a function ${\cal
Q}_{BFV}$ of degree one. Due to $Q^{2}\equiv\tfrac{1}{2}[Q,Q]=0$, this
function satisfies automatically the BFV master equation

$({\cal Q}_{BFV},{\cal Q}_{BFV})_{\scriptscriptstyle{BFV}}=0.$ (10)

Also it satisfies some other necessary requirements posed on the BFV charge,
like the correct appearance of the constraints (5) and the redundancies (3) at
the lowest orders in the ghosts. We may thus consider regarding ${\cal
Q}_{BFV}$ as the BFV charge of our constrained system.

A price to pay is that now, in general, it is no more guaranteed that the
${\cal Q}_{BFV}$-cohomology describes the functions on the (possibly singular)
reduced phase space correctly. There is another famous example, however, where
such a deficiency is normally disregarded: The AKSZ formalism [16, 17, 18]
yields a BV extension of the Poisson sigma model or the Chern-Simons gauge
theory likewise in a much faster way than when following the usual extension
procedure. If one compares its cohomology at degree zero with gauge invariant
functionals modulo the Euler-Lagrange equations, however, the cohomology one
finds in the AKSZ framework is in general too large for non-trivial topologies
of the source manifold. To see this explicitly, consider an abelian Chern-
Simons theory on a 3-manifold $\Sigma$ where the space of solutions to the
Euler-Lagrange equations consist of closed 1-forms and gauge symmetries
correspond to adding exact 1-forms to them, thus leading to simply
$H^{1}_{dR}(\Sigma)$, and compare this to the results about the corresponding
AKSZ-cohomology at degree zero in [19].

We describe the construction of the BFV phase space obtained in this way in
more detail in the subsequent section. However, there is—in contrast to the
Lagrangian BV formalism—one more essential ingredient of the cohomological
approach on the Hamiltonian level: One needs an extension ${\cal H}_{BFV}$ of
the Hamiltonian (6) such that

$({\cal Q}_{BFV},{\cal H}_{BFV})_{\scriptscriptstyle{BFV}}=0\,.$ (11)

Equations (10) and (11) are the two indispensable corner stones of the BFV
formalism. Under the given assumptions underlying the shortcut for the
construction of ${\cal Q}_{BFV}$, the existence of the extension of the
Hamiltonian is in general not guaranteed. Its potential obstructions is the
main subject of the present article.

##  II BFV phase space and charge

Much as (2) defines a Lie algebroid on $E$ [9], every homological vector field
$Q$ of degree 1 on

${\cal M}:=E[1]\oplus F[2],$ (12)

which is necessarily of the form

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\xi^{a}\rho_{a}^{i}\tfrac{\partial}{\partial
x^{i}}-\left(\eta^{I}t_{I}^{a}+\tfrac{1}{2}\xi^{b}\xi^{c}C_{bc}^{a}\right)\tfrac{\partial}{\partial\xi^{a}}$
(13)
$\displaystyle+\left(\tfrac{1}{6}\xi^{a}\xi^{b}\xi^{c}h^{I}_{abc}-\Gamma^{I}_{Ja}\eta^{J}\xi^{a}\right)\tfrac{\partial}{\partial\eta^{I}}\,,$

defines a Lie 2-algebroid structure on the vector bundle $E\oplus F$. Here $E$
and $F$ are the vector bundles appearing in (8) and the numbers in the
brackets denote the degrees carried by the local fiber-linear coordinates
$\xi^{a}$ and $\eta^{I}$ on $E$ and $F$, respectively, in the graded
description of the Lie 2-algebroid.

The coefficient functions entering (13) all have some algebraic and/or
geometrical meaning: One way of viewing them is that the terms linear in
$(\xi,\eta)$ give rise to a complex (8)—which in our case we require, in
addition, to be exact on the level of sections—, those quadratic in them are
2-brackets, which are not tensorial but satisfy a Leibniz rule, and the cubic
one, which is tensorial, is a 3-bracket, which, if non-zero, reflects the fact
that the 2-bracket between sections of $E$ then does not satisfy the Jacobi
identity, giving rise to a Lie 2-algebra. For some introduction to Lie
2-algebras see [20], for the more general $L_{\infty}$-algebras see [21, 22].
Alternatively, we may view, e.g., the coefficients $\Gamma^{I}_{Ja}$ as a
local expression for an $E$-covariant derivative on $F$. The tensor
$h\in\Gamma(F\otimes\Lambda^{3}E^{*})$ then satisfies ${}^{E}\mathrm{D}h=0$,
where ${}^{E}\mathrm{D}$ denotes the corresponding exterior covariant
$E$-derivative.

There is a vast literature on the geometry of Lie algebroids, see, e.g., [23,
24, 25]. For further details about the geometry of Lie 2-algebroids see [26,
27]. For a general Lie 2-algebroid, the complex (8) is not exact (also not on
the level of sections). Under the condition of exactness of the sequence on
the level sections, thus providing a resolution of the $C^{\infty}(M)$-module
of the vector fields generated by $v_{a}$, i.e. by the image of $\rho\colon
E\to TM$, the existence of a Lie 2-algebroid structure extending a given
sequence (8) was proven in [15]. As mentioned, this is tantamount to the
existence of a vector field (13) on the corresponding ${\cal M}$ of (12) which
squares to zero.

Given these data, we now construct the BFV phase space as the cotangent bundle
$T^{*}{\cal M}$ of (12). This extends the classical variables
$(x^{i},p_{i})\in T^{*}M$ by a ghost $\xi^{a}$ (odd and of degree one) for
each of the constraints and a ghost-for-ghosts $\eta^{I}$ (even and of degree
two) for each of the dependencies (9). Both of the latter two ghost families
are accompanied by their momenta of opposite degrees. All this comes together
with the BFV symplectic form

$\omega_{{\scriptscriptstyle{BFV}}}=\mathrm{d}x^{i}\wedge\mathrm{d}p_{i}+\mathrm{d}\xi^{a}\wedge\mathrm{d}\pi_{a}+\mathrm{d}\eta^{I}\wedge\mathrm{d}{\cal
P}_{I}\,.$ (14)

It is of degree zero and thus so also the graded Poisson bracket
$(\cdot,\cdot)_{{\scriptscriptstyle{BFV}}}$ it induces, which takes the
following form:

$\\{p_{i},x^{j}\\}=\delta^{j}_{i},\enspace\\{\pi_{a},\xi^{b}\\}=\delta^{b}_{a},\enspace\\{{\cal
P}_{I},\eta^{J}\\}=\delta^{J}_{I}.$

Regarding the vector field (13) as a function on the BFV phase space
$T^{*}{\cal M}$ (by replacing the derivatives simply by the corresponding
momenta), one has

$\displaystyle{\cal Q}_{BFV}$ $\displaystyle=$
$\displaystyle\left(\xi^{a}\rho_{a}^{i}\right)p_{i}-\left(\eta^{I}t_{I}^{a}+\tfrac{1}{2}\xi^{b}\xi^{c}C_{bc}^{a}\right)\pi_{a}$
(15)
$\displaystyle+\left(\tfrac{1}{6}\xi^{a}\xi^{b}\xi^{c}h^{I}_{abc}-\Gamma^{I}_{Ja}\eta^{J}\xi^{a}\right){\cal
P}_{I}\,,$

where we use the notation $\rho_{a}^{i}\equiv v_{a}^{i}$. We notice that, as
required by the standard BFV procedure, it indeed extends $\xi^{a}G_{a}$ by
terms such that (10) holds true.

The classical constrained system on $T^{*}M$ considered here is already
uniquely determined by the underlying singular foliation. This is in contrast
to the coefficient functions entering (15): they are constrained by $Q^{2}=0$,
but, at each step of an extension, they are not unique. This should be also
reflected on the BFV level: Indeed, e.g. a change (4) of the almost Lie
bracket on $E$ can be obtained by lifting the degree preserving diffeomorphism

$\eta^{I}\mapsto\eta^{I}+\tfrac{1}{2}B^{I}_{ab}\xi^{a}\xi^{b}$ (16)

from ${\cal M}$ to $T^{*}{\cal M}$. Such a change of the structure functions
$C^{a}_{bc}$ does not come for free: also other quantities entering ${\cal
Q}_{BFV}$ are then changed correspondingly. E.g., $h$ receives an additive
contribution by the $E$-covariant exterior derivative of
$B\in\Gamma(F\otimes\Lambda^{2}E^{*})$, $h\mapsto h+{}^{E}\mathrm{D}B+\ldots$,
where the dots denote terms quadratic in $B$ (see [26]).

##  III Geometrical interpretation of ${\cal H}_{BFV}$

We first observe that the BFV bracket decreases the polynomial degree
$\mathrm{pol}$ of momenta $(p_{i},\pi_{a},{\cal P}_{I})$ by one. Since, in
addition, both ${\cal Q}_{BFV}$ and ${\mathrm{Ham}}$ are homogeneous with
respect to $\mathrm{pol}$, we may assume always that the extension ${\cal
H}_{BFV}={\mathrm{Ham}}+\ldots$ satisfies $\mathrm{pol}({\cal H}_{BFV})=2$.
Identifying momenta with vector fields, we thus may view ${\cal H}_{BFV}$ as a
graded-symmetric 2-vector field on (12),

$H\equiv H_{BFV}\in\Gamma(S^{2}T{\cal M})$ (17)

and re-interpret the second master equation (11) as

${\cal L}_{Q}H=0$ (18)

where $Q=Q_{Lie2}\in\Gamma(T{\cal M})$. For later use, we remark that a basis
of $\Gamma(S^{2}T{\cal M})$ is spanned by

$\tfrac{\partial}{\partial\eta^{I}}\tfrac{\partial}{\partial\eta^{J}},\>\tfrac{\partial}{\partial\eta^{I}}\tfrac{\partial}{\partial\xi^{a}},\>\tfrac{\partial}{\partial\eta^{I}}\tfrac{\partial}{\partial
x^{i}},\>\tfrac{\partial}{\partial\xi^{a}}\tfrac{\partial}{\partial\xi^{b}},\>\tfrac{\partial}{\partial\xi^{a}}\tfrac{\partial}{\partial
x^{i}},\>\tfrac{\partial}{\partial x^{i}}\tfrac{\partial}{\partial x^{j}}$
(19)

where a graded symmetrization is understood and the degrees are $-4$, $-3$,
$-2$, $-2$, $-1$, and $0$, respectively. $H$ being of degree zero, one needs
coefficient functions on ${\cal M}$ of the respective opposite degrees when
using this basis. A graded manifold equipped with a degree 1 vector field is
called a Q-manifold [28]; if there is, in addition, (17) of degree 0
satisfying (18), we call it an $HQ$-manifold.

Already an $H$-structure alone, that is a degree zero tensor (17) without any
compatibility conditions, contains interesting geometrical data, at least if
the coefficients of the very last term in (19) form a non-degenerate matrix—in
which case we then call the $H$-structure non-degenerate. It then defines a
metric $g$ on $M$ (subject to no further conditions). But this is not all:
Consider the terms linear in $p_{i}\sim\partial_{i}$. Due to the non-
degeneracy of $g^{ij}$, we may absorb all such terms by completing the square
with a redefinition of the momenta $p_{i}\to p_{i}^{\nabla}$, where

$p_{i}^{\nabla}:=p_{i}-\Gamma^{a}_{bi}\xi^{b}\pi_{a}-(\Gamma^{I}_{Ji}\eta^{J}-\tfrac{1}{2}\gamma_{abi}^{I}\xi^{a}\xi^{b}){\cal
P}_{I}\,.$ (20)

Then the 2-tensor (17), rewritten as a function ${\cal H}$ in
$C^{\infty}(T^{*}{\cal M})$, takes the form

${\cal
H}=\tfrac{1}{2}g^{ij}p_{i}^{\nabla}p_{j}^{\nabla}+\tfrac{1}{2}\Sigma_{(2)}^{ab}\pi_{a}\pi_{b}+\Sigma_{(3)}^{aI}\pi_{a}{\cal
P}_{I}+\tfrac{1}{2}\Sigma_{(4)}^{IJ}{\cal P}_{I}{\cal P}_{J}$ (21)

where $\Sigma_{(k)}^{\cdot\cdot}$ are functions on ${\cal M}$ of degree $k$.
Thus, for example,

$\Sigma_{(2)}^{ab}=\tfrac{1}{2}\Sigma_{cd}^{ab}\xi^{c}\xi^{d}+\Sigma_{I}^{ab}\eta^{I}$
(22)

for appropriate tensors $\Sigma_{cd}^{ab}$ and $\Sigma_{I}^{ab}$. The
coefficients in (20) have a geometrical interpretation: Consider a coordinate
transformation $\xi^{a}\mapsto M^{a}_{b}(x)\xi^{b}$ (all other coordinates
unchanged). Lifting it to $T^{*}{\cal M}$ so as to leave (14) invariant—or,
equivalently, considering its tangent map—yields
$\pi_{a}\mapsto(M^{-1})^{b}_{a}\pi_{b}$ together with $p_{i}\mapsto
p_{i}+(M^{-1})^{a}_{c}M^{c}_{b,i}\xi^{b}\pi_{a}$ (all other momenta
unchanged). Requiring (20) to remain form-invariant under such transformations
shows that $\Gamma^{a}_{b}=\Gamma^{a}_{bi}\mathrm{d}x^{i}$ are the local
1-forms representing a connection $\nabla$ on $E$. Similarly,
$\Gamma^{I}_{J}=\Gamma^{I}_{Ji}\mathrm{d}x^{i}$ corresponds to a covariant
derivative $\nabla$ on $F$. $\gamma\in\Gamma(F\otimes\Lambda^{2}E^{*}\otimes
T^{*}M)$ is needed, finally, to also establish form covariance of (20) w.r.t.
(16); it is part of a connection in a graded sense.

An almost Q-structure (a degree one vector field $Q$ which does not
necessarily square to zero) equips the graded manifold (12) with the structure
of a (not necessarily exact) sequence of vector bundles as in (8) together
with 2-brackets and one 3-bracket. A non-degenerate H-structure equips it in
the lowest order with a metric $g$ on the base and in the next order—the terms
linear in the momenta $p_{i}$ (see also (20))—with a connection defined on all
of (8) (note that we can always equip $TM$ with the canonical Levi-Civita
connection of $g$, moreover).

Given an almost HQ-structure, where we have both, an almost Q-structure $Q$
and an $H$ as above, (18) determines compatibility conditions. To obtain
tensorial formulas for these on the nose, it is useful to re-express (15) in
terms of the variables (20). Then, e.g., the new coefficient of the term
quadratic in $\xi$ is
$-C^{a}_{bc}+\rho^{i}_{b}\Gamma^{a}_{ci}-\rho^{i}_{c}\Gamma^{a}_{bi}=:{}^{E}T^{a}_{bc}$
and has the geometric interpretation of what is called an $E$-torsion
${}^{E}T\in\Gamma(E\otimes\Lambda^{2}E^{*})$ [13], generalizing the ordinary
torsion on a tangent bundle to $E$. The bracket of two covariant vector fields
(20) yields the curvatures of $\nabla$, both on $E$ and $F$ (as well as a
contribution proportional to $\mathrm{D}\gamma$). To lowest orders we find:

$\displaystyle{\cal L}_{Q}H$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\xi^{a}\left({}^{E}\nabla_{\\!e_{a}}\,g\right)^{ij}\partial_{i}^{\nabla}\partial_{j}^{\nabla}$
$\displaystyle $
$\displaystyle-\tfrac{1}{2}\xi^{b}\xi^{c}\left(S^{a}_{bci}-t^{a}_{I}\gamma^{I}_{bci}-\Sigma^{da}_{bc}\rho^{k}_{d}g_{ki}\right)g^{ij}\partial_{j}^{\nabla}\tfrac{\partial}{\partial\xi^{a}}$
(23)
$\displaystyle+b^{I}\left(t^{a}_{I;j}g^{ji}+\Sigma^{ba}_{I}\rho_{b}^{i}\right)\partial_{i}^{\nabla}\tfrac{\partial}{\partial\xi^{a}}+\ldots\,.$

Here $\partial_{i}^{\nabla}$ is the vector field corresponding to (20) and
${}^{E}\nabla$ denotes the $E$-covariant derivative acting on $TM$ according
to ${}^{E}\nabla_{s}v=[\rho(s),v]+\rho(\nabla_{v}s)$, where $s\in\Gamma(E)$
and $v\in\Gamma(TM)$. $\Sigma_{(2)}^{ab}$ has been decomposed according to
(22), a semicolon denotes a covariant derivative, and we introduced the
abbreviation

$S^{a}_{bci}={}^{E}T^{a}_{bc;i}+\rho_{b}^{j}R^{a}_{cji}-\rho_{c}^{j}R^{a}_{bji}.$
(24)

The tensor $S\in\Gamma(E\otimes\Lambda^{2}E^{*}\otimes T^{*}M)$ measures the
compatibility of the 2-bracket and the connection $\nabla$ on $E$, see [13].
As for Jacobi identities of the brackets in a higher Lie algebroid, also here
we do not expect (18) to ensure $S$ to vanish on the nose, but to instead do
so up to some appropriate boundary term only; what this means precisely is
subject of the next section.

Let us mention that the current method is efficient in obtaining Bianchi type
of identites for quantities such as $S$. $Q^{2}=0$, which we have when $Q$
comes from a Lie 2-algebroid, implies that the application of another ${\cal
L}_{Q}$ to (23) vanishes identically. From this one can read off, for example,
that (24) satisfies

${}^{E}\mathrm{D}S-\nabla\langle t,h\rangle_{\\!\scriptscriptstyle{F}}=0\,.$
(25)

Here ${}^{E}\mathrm{D}$ denotes the $E$-exterior covariant derivative
associated naturally to ${}^{E}\nabla$: on $TM$ it acts as specified above, on
$E$ according to
${}^{E}\nabla_{s}s^{\prime}=[s,s^{\prime}]_{E}+\nabla_{\rho(s^{\prime})}s$ for
every $s,s^{\prime}\in\Gamma(E)$, and on elements in $\Lambda^{\bullet}E^{*}$
by a straightforward generalization of the Cartan formula for the de Rahm
differential ([26, 13]). The identity (25)—specialized to a Lie algebroid,
where $h=0$—was of essential importance in the construction of the BV-
extension [29] and the above derivation constitutes a significant
simplification of the one provided there.

##  IV Complex governing the extension

In the construction of (15), the sequence of vector bundles (8) and its
exactness on the level of sections plays a crucial role. There is a similar
sequence which governs the extension problem (6),(7),(11), but, despite being
constructed out of the previous one, this one is in general no longer exact on
the level of sections.

Let us denote the complex (8) by ${\cal E}^{\bullet}$ (so, in particular,
${\cal E}^{0}=TM$, ${\cal E}^{-1}=E$, and ${\cal E}^{-2}=F$) and tensor it
with itself shifted to the right, ${\cal F}^{\bullet}:={\cal
E}^{\bullet}\otimes{\cal E}^{\bullet}[-1]$, where, by definition, ${\cal
E}^{i}[-1]={\cal E}^{i-1}$. At degree zero, e.g., one has ${\cal
F}^{0}=TM\otimes E\oplus E\otimes TM$, where in the first term both factors
carry degree 0, while in the second one, elements in $E$ enter with degree -1
and those in $TM$ with degree +1, again adding up to 0.

By a standard construction (see, e.g., [30]), ${\cal F}^{\bullet}$ is again a
complex. It is, however, not yet the sequence ${\cal G}^{\bullet}$ we are
interested in:

${\cal
G}^{\bullet}:=0\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}S^{2}F\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}F\otimes
E\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}F\otimes
TM\oplus\Lambda^{2}E\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}E\otimes
TM\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}S^{2}TM\,,$ (26)

where the degrees are such that ${\cal G}^{0}=S^{2}TM$, ${\cal
G}^{-1}=E\otimes TM$, etc. Now one observes that ${\cal
G}^{\bullet}\xhookrightarrow{}{\cal F}^{\bullet}[1]$ (essentially it is
embedded as a graded symmetrization). For example, typical elements
$\Phi\in{\cal F}^{0}$ and $\varphi\in{\cal G}^{-1}$ are of the form
$\Phi=\Phi^{ia}\,\partial_{i}\otimes
e_{a}+\bar{\Phi}^{ai}\,e_{a}\otimes\partial_{i}$ and
$\varphi=\varphi^{ai}\,e_{a}\otimes\partial_{i}$, respectively, and then
${\cal G}^{-1}$ is embedded into ${\cal F}^{0}$ by the diagonal map,
$\Phi^{ia}:=\varphi^{ai}$, $\bar{\Phi}^{ai}:=\varphi^{ai}$. The codifferential
$\delta$ is easily identified with

$\delta:=\rho_{a}^{i}p_{i}\tfrac{\partial}{\partial\pi_{a}}+t_{I}^{a}\pi_{a}\tfrac{\partial}{\partial{\cal
P}_{I}}$ (27)

when replacing the vector fields (19), which provide a basis for ${\cal
G}^{\bullet}$, by the corresponding momenta.

In general, for the cohomology of a tensor product of two complexes there is a
Künneth formula. It says that even if the cohomology of each of the complexes
is trivial, which is the case here, there can still be a non-zero contribution
called “torsion”. Below we will provide an example that, in general, ${\cal
G}^{\bullet}$ is not exact, even not on the level of sections.

It is remarkable that (27) also generates the coboundary operator of the
complex (8). In fact, if all the independent relations between the constraints
$G_{a}=\rho^{i}_{a}p_{i}$ on $T^{*}M$ are provided by the functions
$t^{a}_{I}(x)$ (these are all such dependencies that depend on $x$ only, but
there could be further ones that also depend on momenta $p$ in principle),
then the operator (27) acting on $C^{\infty}(T^{*}{\cal M})$ would be the
Koszul-Tate differential [1]. And in that case, it would have no cohomology on
$C^{\infty}(T^{*}{\cal M})$ and neither so when acting on (26), which is the
restriction of $(C^{\infty}(T^{*}{\cal M}),\delta)$ to functions quadratic in
the momenta—such as the complex (8) can be identified with its restriction to
the subcomplex linear in the momenta. But in general, the true Koszul-Tate
complex is much bigger, as we will illustrate by means of an example at the
end of this article.

The main point here is, however, that it is precisely the cohomology of (26)
that governs the extendability problem ${\cal H}_{BFV}={\mathrm{Ham}}+\ldots$
satisfying (11) for ${\cal Q}_{BFV}$ as in (15).

_If_ $\Gamma({\cal G}^{\bullet})$ is exact, the existence of the BFV extension
$H_{BFV}$ is guaranteed. This follows from a standard consideration: $({\cal
Q}_{BFV},\cdot)=\delta+X_{rest}$. At each step when adding terms from the
right to the left in (26), one finds an expression that is already
$\delta$-closed. Now, $\delta$ having no cohomology, one can always add a
$\delta$-exact contribution from the next level so as to cancel it. In
general, the resulting expression for $H_{BFV}$ will contain all possible
terms compatible with degrees, as is also the case for (15); together they
then define an HQ-structure on (12).

The converse is certainly not true: The complex $\Gamma({\cal G}^{\bullet})$
can have non-trivial cohomology, but the extension problem for a particular
Hamiltonian (6) may still lead to trivial cohomology classes and thus be
unobstructed.

The absence of a cohomology also leads to remarkable, purely geometrical
formulas. To illustrate this, assume that

$H^{-1}(\Gamma({\cal G}^{\bullet}),\delta)=0.$ (28)

Then for every $\varphi\in\Gamma({\cal G}^{-1})$ such that $\delta\varphi=0$,
there is some
$\psi\equiv\Psi^{Ii}b_{I}\otimes\partial_{i}+\tfrac{1}{2}M^{ab}e_{a}\wedge
e_{b}\in\Gamma({\cal G}^{-2})$ such that $\varphi=\delta\psi$. Concretely,

$\rho_{a}^{i}\,\varphi^{aj}+\rho_{a}^{j}\,\varphi^{ai}=0\quad\Rightarrow\quad\varphi^{ai}=t^{a}_{I}\Psi^{Ii}+\rho_{b}^{i}M^{ab}\,,$
(29)

where $M^{ab}(x)=-M^{ba}(x)$. The exactness at a given degree is preserved if
one tensors a complex with a fixed $C^{\infty}(M)$-module; this permits adding
spectator indices to the quantities in (29).

As a first application, let us return to the initial compatibility condition
(7). It can be shown [14] that (7) holds true in arbitrary local patches iff
there exists a connection $\nabla$ on $E$ such that its induced $E$-connection
annihilates the metric, ${}^{E}\nabla g=0$. If $\Gamma^{b}_{ai}$ denote the
connection coefficients of $\nabla$, we can choose
$\omega_{a}^{b}:=\Gamma_{ai}^{b}(x)g^{ij}p_{j}$. The difference between two
connections is a tensor field and then (7) implies that this tensor is
$\delta$-closed. This provides the following equivalence for the choice of the
connection coefficients

$\omega^{ai}_{b}\sim\omega^{ai}_{b}+t^{a}_{I}\Psi^{Ii}_{b}+\rho_{c}^{i}M^{ac}_{b},$
(30)

where $M^{ac}_{b}=-M^{ca}_{b}$, corresponding to $\delta$-exact contributions.
If (28) holds true, these are _all_ the ambiguities in the choice of the
connection on $E$ such that ${}^{E}\nabla g=0$.

Even more remarkable is the following decomposition that one finds for the
tensor $S$ defined in (24). Using a connection $\nabla$ such that
${}^{E}\nabla g=0$, the geometrical quantity $S$ is always $\delta$-closed and
therefore, assuming (28), it can be decomposed into some
$\gamma\in\Gamma(F\otimes\Lambda^{2}E^{*}\otimes T^{*}M)$ and
$\Sigma\in\Gamma(\Lambda^{2}E\otimes\Lambda^{2}E^{*})$ as follows:

$S^{a}_{bci}=t^{a}_{I}\gamma^{I}_{bci}+\rho_{d}^{j}g_{ij}\Sigma^{da}_{bc}\,.$
(31)

There are, in general, no explicit expressions for these tensors. In fact,
they are even not uniquely determined: We can change $\gamma$ and $\Sigma$
simultaneously by

$\displaystyle\Sigma^{ab}_{cd}$ $\displaystyle\mapsto$
$\displaystyle\Sigma^{ab}_{cd}+t^{a}_{I}\,V^{Ib}_{cd}-t^{b}_{I}\,V^{Ia}_{cd}$
$\displaystyle\gamma^{I}_{abi}$ $\displaystyle\mapsto$
$\displaystyle\gamma^{I}_{abi}+g_{ij}\rho_{c}^{j}\,V^{Ic}_{ab}$ (32)

for any $V\in\Gamma(F\otimes E\otimes\Lambda^{2}E^{*})$ without changing
$S$—and these are all such ambiguities, if also $H_{\delta}^{-2}(\Gamma({\cal
G}))=0$.

Finally, the identity $\\{t^{a}_{I}G_{a},H\\}$ yields

$t^{a}_{I;i}g^{ij}\\!\rho_{a}^{k}\,p_{j}p_{k}=0\,.$ (33)

Thus, $t^{a;i}_{I}\equiv t^{a}_{I;j}g^{ij}$ satisfies the condition of (29)
and, if (28) holds true, there exist tensors $\tilde{\Psi}$ and $\tilde{M}$
such that

$t^{a;i}_{I}=t^{a}_{J}\tilde{\Psi}^{Ji}_{I}+\rho^{i}_{b}\tilde{M}_{I}^{ab}\,.$
(34)

These considerations can be considered as holding true on the purely
geometrical level, without any application to physical models.

On the other hand, as explained above, the $\delta$-cohomology governs the
physical extension problem here. For example, the second line in (23) says
that $S$ needs to be $\delta$-exact (and, by a general feature of the
procedure, also shows implicitly that it is already $\delta$-closed),
$S=\delta(\gamma+\Sigma)$, where $\gamma$ and $\Sigma$ are the quantities
entering ${\cal H}$, see (20), (21), and (22). In deriving (31) we only
demanded $S$ to be $\delta$-exact with respect to some tensors, but in a
slight abuse of notation we denoted them already by $\gamma$ and $\Sigma$, the
quantities we need to choose for the expansion (21). We have not done so in
(34), which can be rewritten as $\nabla t=\delta(\tilde{\Psi}+\tilde{M})$ and
one may wonder where the analogue of the first term on its right hand side is
in comparison to the last line in (23), while evidently $\tilde{M}_{I}^{ab}$
can be identified with $\Sigma_{I}^{ab}$. In fact, this depends on the
connection on $F$: if one uses a different connection than the one appearing
in (20), then their difference is a tensor, which can be identified with
$\tilde{\Psi}^{Ji}_{I}$. So also the last line in (23) says that $\nabla t$
should be $\delta$-exact.

##  V Examples with $t=0$

The prototype of a singular constraint
$T^{*}{\mathbb{R}}\cong{\mathbb{R}}^{2}$ is $G=xp$. It is singular at the
origin, the constraint surface has the shape of a cross. The corresponding
singular foliation on $M={\mathbb{R}}$ is generated by $v=x\partial_{x}$ or,
by putting $E={\mathbb{R}}\times M$ with an anchor such that $\rho(1)=v$. The
bundle $F$ then has rank zero and the exact sequence is of length one only.
The corresponding BFV charge is simply ${\cal Q}_{BFV}=\xi xp$.

There is no compatible non-degenerate Hamiltonian coming from a metric $g$ in
this case, since ${\mathbb{R}}$ with a non-regular leaf structure does not
admit a singular Riemannian foliation. However, if we start with a Hamiltonian
${\mathrm{Ham}}=\tfrac{1}{2}h(x)p^{2}$, dropping the condition that $h=1/g$,
then a BFV extension exists as long as $h$ vanishes at least quadratically
near the origin; it then takes the form: ${\cal
H}_{BFV}=\tfrac{1}{2}h(x)p^{2}+x^{2}\left(\tfrac{h}{x^{2}}\right)^{\prime}\xi\pi
p$.

More generally, if $t=0$, the anchor map $\rho\colon E\to TM$ needs to be
injective on a dense open subset of $M$: this follows from injectivity of the
induced map from $\Gamma(E)$ to $\Gamma(TM)$ when $t=0$ (see the exactness
condition expressed in the sentence containing (8)). And if one insists on a
non-degenerate metric $g$ satisfying (7), then $t=0$ implies injectivity of
the anchor map $\rho\colon E\to TM$ everywhere. This in turn implies that the
foliation on $M$ and the constraints (5) on $T^{*}M$ are regular.

Regular foliations can still yield geometrically interesting examples. To
illustrate this, consider even $E=TM$, $\rho=\mathrm{id}$. While the BFV
charge is very simple in this case, ${\cal Q}_{BFV}=\xi^{i}p_{i}$, non-trivial
geometry remains in the BFV extension of the Hamiltonian: To satisfy the
compatibility (7), we need a connection $\nabla$ on $TM$ such that

$g_{ij;k}+T_{ijk}+T_{jik}=0,$ (35)

where $T_{ijk}=g_{il}T^{l}_{jk}$; here $T^{l}_{jk}$ denote the components of
the torsion tensor. The Levi-Civita connection satisfies this condition
evidently. The ambiguity (30) translates into the freedom of the choice of the
torsion of the connection, but maintaining (35) instead of metricity. One then
finds

${\cal
H}_{BFV}=\tfrac{1}{2}g^{ij}p_{i}^{\nabla}p_{j}^{\nabla}+\tfrac{1}{2}R^{k}{}_{ij}{}^{l}\xi^{i}\xi^{j}\pi_{k}\pi_{l}+\tfrac{1}{4}T^{k}{}_{ij}{}^{;l}\xi^{i}\xi^{j}\pi_{k}\pi_{l}$
(36)

where $R^{i}{}_{jkl}$ denote the components of the Riemann tensor. If we
choose the Levi-Civita connection for $\Gamma^{i}{}_{jk}$ in
$p_{i}^{\nabla}=p_{i}+\Gamma^{j}{}_{ki}\xi^{k}\pi_{j}$, then (35) is satisfied
identically and the last term in (36) disappears, but one is still left with
the curvature term.

Inspection of (24) shows that the last two terms in (36) combine into the
tensor $S$ for $E=TM$. In [12], such a BFV extension was provided under the
assumption that $S$ vanishes. The extension (36) of the Hamiltonian
generalizes this result, with the interesting contributions of curvature and
torsion which combe into the geometrical quantity $S$, which, for $E=TM$, in
turn can be identified also with the curvature of a dual connection, see [13].

##  VI Angular Momentum I:
Example of an obstruction

Let us now consider the phase space $T^{*}{\mathbb{R}}^{3}$ with constrained
angular momentum, $G_{a}=\varepsilon_{abc}x^{b}p^{c}$ or
$\vec{G}=\vec{x}\times\vec{p}$. Then one has

${\cal
Q}_{BFV}=\vec{\xi}\cdot(\vec{x}\times\vec{p})+\tfrac{1}{2}(\vec{\xi}\times\vec{\xi})\cdot\vec{\pi}-\eta\,\vec{x}\cdot\vec{\pi}\,.$
(37)

The second term is the (cotangent lift of the) Chevalley-Eilenberg
differential of the Lie algebra $so(3)$, the first two terms the Chevalley-
Eilenberg differential for $so(3)$ acting on ${\mathbb{R}}^{3}$, which then
corresponds to the BRST charge (2) of the action Lie algebroid
$E=so(3)\times{\mathbb{R}}^{3}$. The last term takes care of the dependency
$\vec{x}\cdot\vec{G}\equiv 0$ of the constraints. It is easy to verify that
the expression (37) satisfies the master equation (10).

In this example, $M={\mathbb{R}}^{3}$, $E={\mathbb{R}}^{3}\times M$,
$F={\mathbb{R}}\times M$, and the maps $\rho$ and $t$ in (8) can be identified
with the sections
$\rho=\varepsilon_{abc}x^{b}e^{a}\otimes\tfrac{\partial}{\partial x^{c}}$ and

$t=x^{a}e_{a}\otimes b^{*}\,,$ (38)

respectively; here $(e^{a})_{a=1}^{3}$ denotes a basis in $E^{*}$ and $b^{*}$
a basis in $F^{*}$. The kernel of $\rho$ is one-dimensional outside the
origin, while it is all of $E_{\vec{x}=0}={\mathbb{R}}^{3}$ at the origin
$0\in M$. The map $t$ spans the radial vectors in $\ker\rho$ for all
$\vec{x}\neq 0$, but, for continuity reasons, vanishes at $0$. Thus, the
complex (8) has no cohomology outside the origin, but
$H_{\vec{x}=0}^{-1}({\cal E},\delta)={\mathbb{R}}^{3}$. And still—again for
continuity reasons (every radial vector field has to vanish at the origin)—it
is exact on the level of sections: $H^{\bullet}(\Gamma({\cal
E}^{\bullet}),\delta)=0$. Correspondingly, the extension problem for the BFV
charge (37) when taking into account (only) the $x$-dependent dependencies of
the constraints, has not been obstructed.

Let us now turn to the extension problem of the Hamiltonian (6), the main
subject of this paper, for the standard metric on $M={\mathbb{R}}^{3}$,

${\mathrm{Ham}}=\tfrac{1}{2}\vec{p}\cdot\vec{p}\,.$ (39)

For this purpose we first equip the bundles $E$ and $F$ with their canonical
flat connections (for what concerns $E$, this is also motivated by the fact
that (7) is satisfied with $\omega_{a}^{b}=0$). Then one has

$\nabla t=\mathrm{d}x^{a}\otimes e_{a}\otimes
b^{*}\stackrel{{\scriptstyle\sim}}{{=}}\tfrac{\partial}{\partial x^{a}}\otimes
e_{a}\otimes b^{*}\,,$ (40)

where in the second equality we used the standard metric of $M$ for the
identification, then yielding an element in $\Gamma(E\otimes
TM)\otimes_{C^{\infty}(M)}\Gamma(F^{*})$. It is easy to see that
$\delta(\nabla t)=0$ which is equivalent to (33): applying $\rho$ to $e_{a}$
and symmetrizing over the two ensuing entries in $TM$ gives zero due to the
contraction with the $\varepsilon$-tensor.

Now the main observation of this short section: $\nabla t$ cannot be $\delta$
exact, i.e. it cannot be of the form (34), since both $t$ and $\rho$ vanish at
the origin. This shows, on the one hand, that here

$H^{-1}(\Gamma({\cal G}^{\bullet}),\delta)\neq 0,$ (41)

and, on the other hand, that the BFV extension of (39) is indeed obstructed:
As we learn from the last line in (23), we _need_ $\nabla t$ to be exact for
the BFV-extension of the Hamiltonian within the present framework.

Note also that any other choice of connections on $E$ and $F$ would not help,
since their contribution to $\nabla t$ vanish at $x=0$ as well. In fact, the
cohomology class of $\nabla t$ does not depend on such choices.

##  VII Examples with Lie-2 gauge symmetry

As a technically more involved example, we truncate the expansion for the BFV
extension of the Hamiltonian (while keeping the general form for ${\cal
Q}_{BFV}$ as in (15)): Putting $\Sigma_{3}$, $\Sigma_{4}$, and
$\Sigma^{ab}_{I}$ to zero in (21), one obtains222This example is an
improvement of a result of [12] obtained in the context of Cartan-Lie
algebroids, where the $\Sigma$-term in (42) is absent. During the preparation
of the present work, we were informed by N. Ikeda that he found a similar
extension in the context of Courant algebroids, see [33].

${\cal
H}_{BFV}=\tfrac{1}{2}g^{ij}p_{i}^{\nabla}p_{j}^{\nabla}+\tfrac{1}{4}\Sigma_{cd}^{ab}\xi^{c}\xi^{d}\pi^{\nabla}_{a}\pi^{\nabla}_{b}\,,$
(42)

where $p_{i}^{\nabla}$ is given by (20) and
$\pi_{a}^{\nabla}=\pi_{a}-\lambda_{ab}^{I}\xi^{b}{\cal P}_{I}$ for some
$\lambda\in\Gamma(\Lambda^{2}E^{*}\otimes F)$. This extension of $\pi_{a}$ to
$\pi_{a}^{\nabla}$ has the effect of making (42) covariant with respect to
both, changes of frames as well as (16), where then $\lambda\mapsto\lambda+B$.
Now we learn from (23) that, under these assumptions, _necessary_ conditions
for the validity of (11) are

${}^{E}\nabla g=0\>,\quad\nabla t=0\>,\quad\,S=\delta(\gamma+\Sigma).$ (43)

Note that the second condition implies that $t$ needs to have constant rank.
On the other hand, the last condition in (43) is not overly restrictive, since
always $\delta S=0$.

A priori, there are many more conditions to be satisfied for (11) to hold, but
with the following trick, one may show that they can be all reduced to one
only: For this purpose, we first remark that within (43) we can always change
the two quantities $\gamma$ and $\Sigma$—which are then to enter the extension
(42)—according to (32). Since $t$ has a constant rank, we can choose some
$C\subset E$ such that $E=C\oplus t(F)$. One may now see that the
transformations (32) can be used to assure
$\Sigma\in\Gamma(\Lambda^{2}C\otimes\Lambda^{2}E^{*})$. Then the only
remaining condition is:

${}^{E}\mathrm{D}_{\lambda}\Sigma=0\,.$ (44)

Here ${}^{E}\mathrm{D}_{\lambda}$ is defined as ${}^{E}\mathrm{D}$ but
replacing the 2-bracket on $E$ by
$[s,s^{\prime}]_{\lambda}=[s,s^{\prime}]+t(\lambda(s,s^{\prime}))$, where
$\lambda$ is the tensor entering $\pi^{\nabla}_{a}$.

We finally remark that from (25) one can conclude an equation similar to (44):
One first observes that the operators ${}^{E}\mathrm{D}$ and (27) commute,
$[{}^{E}\mathrm{D},\delta]=0$. Using (25), one then finds that
$\delta({}^{E}\mathrm{D}(\gamma+\Sigma))=\langle t,\nabla h\rangle_{F}$, where
the $h$-contribution only adds to ${}^{E}\mathrm{D}\gamma$. If
$H^{-2}(\Gamma({\cal G}^{\bullet}),\delta)=0$ holds, moreover, one finds that
${}^{E}\mathrm{D}\Sigma$ is part of a coboundary for some
$\tilde{V}\in\Gamma(\Lambda^{3}E^{*}\otimes E\otimes F)$. (44) then translates
into the condition that $\tilde{V}$ needs to be a contraction of $\lambda$
with $\Sigma$.

##  VIII Angular Momentum II:
Outside of the spatial origin

We want to illustrate the above formulas by means of the example of the
angular momentum we discussed already before, but this time excluding the
origin $\vec{x}=0$.

On the parts of $T^{*}{\mathbb{R}}^{3}$ where $\vec{x}\neq 0$, the dependency
$\vec{x}\cdot\vec{G}\equiv 0$ is in fact already sufficient to describe all
the dependencies of the constraints: this is the case since, assuming
$\vec{x}\neq 0$, the constraint surface $\vec{G}=\vec{x}\times\vec{p}\approx
0$ implies $\vec{p}\approx\lambda\vec{x}$ for some $\lambda\in{\mathbb{R}}$.
Thus the dependency $\vec{p}\cdot\vec{G}\equiv 0$ is automatically implemented
when $\vec{x}\cdot\vec{G}\equiv 0$ is taken care of.

Therefore now, even if one follows the Koszul-Tate procedure, one does not
need to introduce an additional ghost of degree two to take into account all
the dependencies of the constraints: the BFV charge (37) is the one that gives
the correct cohomology of observables when the $\vec{x}$-origin is excluded.
This is, on the other hand, not the case, if the last term in (37) is
supressed—as one finds it sometimes in the literature in the treatment of
mechanical models with rotational invariance in the BRST-B(F)V formalism.

Excluding the $\vec{x}$-origin, there are now no more obstructions to BFV-
extend the classical Hamiltonian (39) to an ${\cal H}_{BFV}$ satisfying the
second master equation (11). And, as it turns out, it can even be put into the
special form (42). As we learn from (43), we need for this that the tensor
(38) is covariantly constant. Let us choose for this purpose the connections
on $E$ and $F$ by means of

$\nabla e_{a}=-\frac{x_{a}}{r^{2}}\,\mathrm{d}x^{b}\otimes e_{b}$ (45)

and $\nabla b=0$, respectively. Note that the first choice implies that the
radial section $x^{a}e_{a}$ is covariantly constant, $\nabla(x^{a}e_{a})=0$.
Together these choices indeed yield $\nabla t=0$. It was this condition that
we could not satisfy previously; here it is possible also only due to the
singularity of the connection $\nabla$ on $E$ when approaching the origin.

One also verifies easily, that this choice of the connection on $E$ guarantees
${}^{E}\nabla g=0$, a necessary condition for the BFV extension even without
the restricting ${\cal H}_{BFV}$ to be of the form (42), see the first line in
(23). The tensor $S$ looks as follows with the above choices for the
connection:

$S=\frac{\epsilon_{bce}x^{e}}{r^{2}}e_{a}\otimes e^{b}\otimes
e^{c}\otimes\mathrm{d}x^{a}\,.$ (46)

$S$ now has a contribution coming from the tensor $\gamma$ in its
decomposition (31), which enters the covariantized momentum (20):

$\vec{p}^{\nabla}=\vec{p}-\frac{1}{r^{2}}\vec{\pi}(\vec{\xi}\cdot\vec{x})+\frac{\vec{x}}{2r^{4}}\vec{\xi}\cdot(\vec{\xi}\times\vec{x}){\cal
P}\,.$ (47)

The complete BFV extension of the Hamiltonian then takes the form

${\cal
H}_{BFV}=\tfrac{1}{2}\,\vec{p}^{\nabla}\\!\\!\cdot\\!\vec{p}^{\nabla}-\frac{1}{4r^{4}}\bigl{[}\vec{\xi}\cdot(\vec{\xi}\times\vec{x})\bigr{]}\,\bigl{[}\vec{\pi}\cdot(\vec{\pi}\times\vec{x})\bigr{]}.$
(48)

This corresponds to the following tensor $\Sigma$ in (42),

$\Sigma^{ab}_{cd}=-\frac{\varepsilon_{abe}\varepsilon_{cdg}x^{e}x^{g}}{r^{4}}\,,$
(49)

which satisfies the consistency condition (44) for $\lambda=0$. Together with
the above choice for $\gamma$, this completes the decomposition (31) of (46),
which then satisfies the last condition in (43). According to the discussion
following (42), this then proves the validity of (11), which certainly one can
also verify directly to hold true for (37) and (48).

The extension of (39) to ${\cal H}_{BFV}$ is not unique, also not for a fixed
choice of the BFV charge (37). One can, at each stage, change contributions to
${\cal H}={\cal H}_{BFV}$ in (21) by $\delta$-exact terms (once more
underlining the importance of the complex ${\cal G}^{\bullet}$ for the given
extension problem). For a given Hamiltonian, the first such an ambiguity is
provided by the choice of the connection on $E$, which we found in (30). It is
now not difficult to see that with the choice
$\Psi^{i}_{b}=\tfrac{1}{r^{2}}\delta^{i}_{b}$,
$M^{ac}_{b}=\tfrac{1}{r^{2}}\epsilon_{acb}$, the connection coefficients
corresponding to (45) can be made to vanish: $\nabla e_{a}=0$ for this
modified, equivalent connection. Now the connection on $F$, if taken as the
corresponding coefficient in (20), is fixed by requiring the vanishing of the
last line in (23), i.e. by requiring $\nabla
t=\rho^{i}_{b}\Sigma^{ab}e_{a}\otimes\mathrm{d}x^{i}$: One finds

$\nabla b=\frac{x_{a}\mathrm{d}x^{a}}{r^{2}}\otimes b\,,$ (50)

which, in particular, has the property that the section $\frac{1}{r}b$ is
covariantly constant, $\nabla(\tfrac{1}{r}b)=0$. After the dust clears, one
then finds the following, relatively short BFV-Hamiltonian

${\cal
H}_{BFV}=\tfrac{1}{2}\vec{p}^{\nabla}\cdot\vec{p}^{\nabla}+\frac{\vec{\pi}\cdot(\vec{\pi}\times\vec{x})}{2r^{2}}\eta\,$
(51)

where

$\vec{p}^{\nabla}=\vec{p}-\frac{\vec{x}}{r^{2}}\eta{\cal P}\,.$ (52)

In the extension (48), the last term in the decomposition (22) vanishes. In
(51), on the other hand, it is the first term that does so, with the
$\eta$-coefficient $\Sigma_{I}^{ab}\equiv\Sigma^{ab}$ taking the form

$\Sigma_{ab}=\varepsilon_{abc}\frac{x^{c}}{r^{2}}\,.$ (53)

##  IX Angular Momentum III:
An infinite Koszul-Tate resolution

Before ending this article, we want to address briefly what happens if one
follows the usual BFV procedure in the singular context, not taking the
shortcut by the resolution of the singular foliation as advocated here. We
want to sketch this at the example of the angular momentum when not excluding
the spatial origin $\vec{x}=0$. As mentioned above, in this case, one needs to
take both dependencies $\vec{x}\cdot\vec{G}=0$ and $\vec{p}\cdot\vec{G}=0$
into account separately. As a preparation for the construction of the BFV
charge and the BFV Hamiltonian, one now determines a Koszul-Tate (KT)
resolution of the ideal generated by the constraints $G_{a}$. The graded
coordinates introduced for this purpose, called the anti-ghosts in [1],
subsequentely serve as the momenta to the ghosts that one needs for the BFV
charge. For the angular momentum, we have three anti-ghosts ${\pi}_{a}$ of
degree -$1$, which provide the generators of the ideal after application of
the KT differential: $\delta_{KT}\left({\pi}_{a}\right)=G_{a}\,.$ For the two
dependencies between these constraints, we introduce two anti-ghosts of degree
-$2$, ${\cal P}$ and $\bar{\cal P}$, such that:

$\delta_{KT}{\cal P}=x^{a}\pi^{a}\>,\quad\delta_{KT}\bar{\cal
P}=p^{a}\pi^{a}\,.$ (54)

Note that $\delta_{KT}$ coincides with $\delta$ on the ghost momenta we
introduced before, see (27), but now we have more of them. And it does not
stop with the additional $\bar{\cal P}$. The reducibility functions $x^{a}$
and $p^{a}$ are not independent on-shell:
$x^{a}p^{b}-x^{b}p^{a}=\epsilon^{abc}G_{c}\approx 0$. This implies that one
needs additional anti-ghots ${\cal P}^{a}_{3}$ of degree -$3$, which then give

$\delta_{KT}{\cal P}^{a}_{3}=x^{a}\bar{\cal P}-p^{a}{\cal
P}+\tfrac{1}{2}\varepsilon_{abc}\pi^{b}\pi^{c}\,.$ (55)

Here the last term is needed to ensure $(\delta_{KT})^{2}=0$. However, on the
space generated by $(x^{a},p_{a},\pi_{a},{\cal P},\bar{\cal P},{\cal
P}^{a}_{3})$ one now finds non-trivial $\delta_{KT}$-cohomology classes at
degree -$3$. This in turn requires the introduction of six anti-ghosts of
degree -$4$ such that

$\delta{\cal P}^{a}_{4}=\varepsilon_{abc}x^{b}{\cal P}_{3}^{c}+\pi^{a}{\cal
P},\qquad\delta\bar{\cal P}^{a}_{4}=\varepsilon_{abc}p^{b}{\cal
P}_{3}^{c}+\pi^{a}\bar{\cal P}\,.$ (56)

And this procedure does not stop. 333We will come back to a complete
description of this resolution elsewhere. By construction, there is no
cohomology of $\delta_{KT}$ except at degree zero. Therefore, with this
starting point, ${\cal Q}_{BFV}$ and the extension ${\cal H}_{BFV}$ of (6)
always exist in principle. But there is a price to be paid: the underlying
space of ghosts and anti-ghosts consists of an infinite tower of them.

##  X Summary and Outlook

In this article we considered mechanical models of constrained systems where,
as is often the case in physics, at least if the topology of spacetime is non-
trivial, the constraints are not necessarily everywhere regular and where they
are permitted to have reducibilities of the first level. We provided a
shortcut for the construction of a BFV-type charge ${\cal Q}_{BFV}$, bypassing
a potentially intricate Koszul-Tate procedure. We then addressed the
conditions under which an extension ${\cal H}_{BFV}$ of the Hamiltonian
exists. In particular, we identified the complex (26) governing this extension
problem, providing sufficient conditions for the exitence of ${\cal H}_{BFV}$
in this way.

There are several challenges for future work. One is to extend the analysis to
higher level reducibilities, bringing Lie $\infty$-algebroids into play.
Another is to derive the honest BFV extension using the Koszul-Tate algorithm
for irregular constraints, even if at the cost of possibly introducing an
infinite tower of ghosts. Finally, one may want to see how all this plays out
at the quantum level. We intend to come back to some of these points
elsewhere.

##  XI Acknowledgements

We thank Camille Laurent-Gengoux and Hadi Nahari for valuable discussions and
Noriaki Ikeda for making us aware of [33]. T.S. furthermore wants to thank all
of them for collaborations on related subjects, which were essential for the
success of the current project. We also thank Jim Stasheff and Marc Henneaux
for remarks on the manuscript. A.H. is grateful to the ESI in Vienna for
support to join the program “Higher structures and Field Theory” as a Junior
Fellow.

##  XII Appendix: Angular Momentum IV

In this appendix we want to return once more to the example of the angular
momentum in ${\mathbb{R}}^{3}$. While the coordinates and momenta enter the
constraint surface $\vec{x}\times\vec{p}\approx 0$ symmetrically, this is not
the case for the Hamiltonian, see (39). Now, in a region where $\vec{p}\neq
0$, excluding the origin in momentum space, one may use the BFV charge

$\bar{\cal
Q}_{BFV}=\vec{\xi}\cdot(\vec{x}\times\vec{p})+\tfrac{1}{2}(\vec{\xi}\times\vec{\xi})\cdot\vec{\pi}-\bar{\eta}\,\vec{p}\cdot\vec{\pi}\,,$
(57)

where we introduced a ghost-for-ghost pair $(\bar{\eta},\bar{\cal P})$ to
implement the dependency $\vec{p}\cdot(\vec{x}\times\vec{p})\equiv 0$. Like
when excluding the spatial origin $\vec{x}\neq 0$, there is no obstruction for
the BFV extension of the Hamiltonian, but in this case it even agrees with the
classical one, $\bar{\cal H}_{BFV}=\tfrac{1}{2}\vec{p}\cdot\vec{p}$.

One may wonder, if and how one might obtain the much more involved BFV
extensions (48) and (51) from this simple solution to the extension problem in
regions where both $\vec{x}$ and $\vec{p}$ are non-vanishing.

We only have a partial, semi-rigorous answer to this question: Introduce a BFV
charge ${\cal Q}^{\prime}$ that incorporates the two dependencies given by
$\vec{x}$ and $\vec{p}$ in a symmetrical fashion,

${\cal
Q}^{\prime}=\vec{\xi}\cdot(\vec{x}\times\vec{p})+\tfrac{1}{2}(\vec{\xi}\times\vec{\xi})\cdot\vec{\pi}-\eta\,\vec{x}\cdot\vec{\pi}-\bar{\eta}\,\vec{p}\cdot\vec{\pi},$
(58)

with now two conjugate ghost-for-ghost pairs $(\eta,{\cal P})$ and
$(\bar{\eta},\bar{\cal P})$. The charge (58) squares to zero in the obvious
canonical bracket, $({\cal Q}^{\prime},{\cal Q}^{\prime})^{\prime}=0$. It also
agrees with a truncation of the honestly constructed BFV charge in an infinite
tower expansion when following the Koszul-Tate procedure mentioned at the end
of the main text above. But being such a truncation, where one drops all
higher ghost-for-ghost contributions, it does not have the correct cohomology,
however—therefore, the argument is only semi-rigorous at this stage.

There now is a ${\cal Q}^{\prime}$-closed extension of the classic Hamiltonian
on this extended phase space and it is even globally defined: The simple

${\cal H}^{\prime}=\tfrac{1}{2}\vec{p}\cdot\vec{p}-\eta\bar{\cal P}$ (59)

is readily seen to satisy $({\cal Q}^{\prime},{\cal H}^{\prime})^{\prime}=0$.

We may obtain the two sought-for BFV formulations from the above one by two
different reductions in the extended phase space, where one has the canonical
coordinates $(\vec{x},\vec{p},\vec{\xi},\vec{\pi},\eta,{\cal
P},\bar{\eta},\bar{\cal P})$. The first one follows from the evident choice

$\eta:=0\,,\;{\cal P}:=0.$ (60)

It reproduces the BFV formulation $(\bar{\cal Q}_{BFV},\bar{\cal H}_{BFV})$
valid on regions with $\vec{p}\neq 0$.

Note that the elimination of a canonical pair, here $(\eta,{\cal P})$, does
not affect the brackets between the remaining variables. But one still needs
to ensure that the new quantities satisfy the two master equations after
constraining to (60). One way of doing this is to see if they remain to have
vanishing brackets when replacing the original BFV bracket
$(\cdot,\cdot)^{\prime}$ by a Dirac bracket (more precisely, by an adaptation
of the Dirac bracket, introduced originally for second class constraints [11],
to the extended phase space): Restraining a graded symplectic space by putting
two even functions $\alpha$ and $\beta$ which satisfy
$(\alpha,\beta)^{\prime}=1$ to zero, one replaces the original bracket
$(f,g)^{\prime}$ between two functions $f$ and $g$ by

$(f,g)_{D}:=(f,g)^{\prime}-(f,\alpha)^{\prime}\,(\beta,g)^{\prime}+(f,\beta)^{\prime}\,(\alpha,g)^{\prime}\,.$
(61)

Restriction to the subspace given by $\alpha=\beta=0$ and inverting the
restricted graded symplectic form, in the end is reproduced by this bracket
directly. Since both, (58) and (59) commute with $\eta$ in the
$(\cdot,\cdot)^{\prime}$ bracket, in view of (61), it is clear that the two
master equations still hold for the induced bracket when implementing (60).

The situation changes, however, if one wants to proceed in the same way by
replacing (60) with the corresponding barred equations. The reason is that now
$(\bar{\eta},{\cal H}^{\prime})^{\prime}$ does not vanish, nor does $({\cal
Q}^{\prime},\bar{\cal P})^{\prime}$, this then would impede the vanishing of
$({\cal Q}^{\prime},{\cal H}^{\prime})_{D}$. To take care of this, one may
search for deforming the condition $\bar{\cal P}=0$ by $\bar{\cal P}=F$ for
some function $F$ of the unbarred variables. This procedure permits to find
the function $F$ by an expansion in the ghosts, which can be chosen such that
the deformed conditions become

$\bar{\eta}:=0\,,\;\bar{\cal
P}:=\frac{1}{2r^{2}}\left(2\vec{p}\cdot\vec{x}\,{\cal P}-\eta{\cal
P}^{2}-\vec{\pi}\cdot(\vec{\pi}\times\vec{x})\right).$ (62)

With this ”gauge” the two master equations remain valid and one obtains
precisely the BFV data ${\cal Q}_{BFV}$ and ${\cal H}_{BFV})$ as given by (37)
and (51), respectively. The data corresponding to (37) and (48), on the other
hand, result from these by an additional canonical transformation.

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