Momentum Sections in Hamiltonian Mechanics and Sigma Models
We show that a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic manifold by considering gauge...
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| Цитувати: | Momentum Sections in Hamiltonian Mechanics and Sigma Models / N. Ikeda // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 27 назв. — англ. |
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| citation_txt | Momentum Sections in Hamiltonian Mechanics and Sigma Models / N. Ikeda // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 27 назв. — англ. |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We show that a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic manifold by considering gauged sigma models on higher-dimensional manifolds.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 076, 16 pages
Momentum Sections in Hamiltonian Mechanics
and Sigma Models
Noriaki IKEDA
Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
E-mail: nikeda@se.ritsumei.ac.jp
Received May 24, 2019, in final form September 29, 2019; Published online October 03, 2019
https://doi.org/10.3842/SIGMA.2019.076
Abstract. We show a constrained Hamiltonian system and a gauged sigma model have
a structure of a momentum section and a Hamiltonian Lie algebroid theory recently intro-
duced by Blohmann and Weinstein. We propose a generalization of a momentum section on
a pre-multisymplectic manifold by considering gauged sigma models on higher-dimensional
manifolds.
Key words: symplectic geometry; Lie algebroid; Hamiltonian mechanics; nonlinear sigma
model
2010 Mathematics Subject Classification: 53D20; 70H33; 70S05
1 Introduction
Recently, relations of physical systems with a Lie algebroid structure and its generalizations have
been found and analyzed in many contexts. For instance, a Lie algebroid [23] and a generalization
such as a Courant algebroid appear in topological sigma models [18], T-duality [9], quantizations,
etc.
Blohmann and Weinstein [2] have proposed a generalization of a momentum map and a Hamil-
tonian G-space on a Lie algebra (a Lie group) to Lie algebroid setting, based on analysis of the
general relativity [1]. It is called a momentum section and a Hamiltonian Lie algebroid. This
structure is also regarded as reinterpretation of compatibility conditions of geometric quantities
such as a metric g and a closed differential form H with a Lie algebroid structure, which was
analyzed by Kotov and Strobl [21].
In this paper, we reinterpret geometric structures of physical theories as a momentum sec-
tion theory, and discuss how momentum sections naturally appear in physical theories. More-
over, from this analysis, we will find a proper definition of a momentum section on a pre-
multisymplectic manifold.
We analyze a constrained Hamiltonian mechanics system with a Lie algebroid structure
discussed in the paper [19], and a two-dimensional gauged sigma models [17] with a 2-form
B-field and one-dimensional boundary. In a constrained Hamiltonian mechanics system, we
consider a Hamiltonian and constraint functions inhomogeneous with respect to the order of
momenta. Then, a zero-th order term in constraints is essentially a momentum section. In
a two-dimensional gauged sigma model, a pre-symplectic form is a B-field, and a one-dimensional
boundary term is a momentum section. Two examples are very natural physical systems, thus,
we can conclude that a momentum section is an important geometric structure in physical
theories.
Recently, a two-dimensional gauged sigma model with a 2-form B-field with three-dimensional
Wess–Zumino term [17] is analyzed related to T-duality in string theory [5, 6, 10, 11, 12, 13]. For
such an application, it is interesting to generalize a momentum section in a pre-multisymplectic
mailto:nikeda@se.ritsumei.ac.jp
https://doi.org/10.3842/SIGMA.2019.076
2 N. Ikeda
manifold. The string theory sigma model with an NS 3-form flux H is the pre-2-symplectic case
in our theory.
In this paper, we consider an n-dimensional gauged sigma model with (n + 1)-dimensional
Wess–Zumino term. The Wess–Zumino term is constructed from a closed (n+1)-form H, which
defines a pre-n-plectic structure on a target manifold M . For gauging, we introduce a vector
bundle E over M , a connection A on a world volume Σ and a Lie algebroid connection Γ on
a target vector bundle E. Consistency conditions of gauging give geometric conditions on a series
of extra geometric quantities η(k) ∈ Ωk
(
M,∧n−kE∗
)
, k = 0, . . . , n − 1. From this analysis, we
propose a definition of a momentum section on a pre-multisymplectic manifold. This definition
comes from a natural physical example, a gauged sigma model. We see that our definition of
a momentum section on a pre-multisymplectic manifold is a generalization of a momentum map
on a multisymplectic manifold [8, 14].
This paper is organized as follows. In Section 2, we explain definitions of a momentum
section and a Hamiltonian Lie algebroid. In Section 3, we show a constrained Hamiltonian
system has a momentum section. In Section 4, we discuss a two-dimensional gauged sigma
model with boundary and show a boundary term gives a momentum section. In Section 5,
we consider gauging conditions of an n-dimensional gauged sigma model with a WZ term and
propose a generalization of a momentum section on a pre-multisymplectic manifold. Section 6
is devoted to discussion and outlook.
2 Momentum section and Hamiltonian Lie algebroid
In this section, we review a momentum section and a Hamiltonian Lie algebroid introduced
in [2].
2.1 Lie algebroid
A Lie algebroid is a unified structure of a Lie algebra, a Lie algebra action and vector fields on
a manifold.
Definition 2.1. Let E be a vector bundle over a smooth manifold M . A Lie algebroid
(E, ρ, [−,−]) is a vector bundle E with a bundle map ρ : E → TM and a Lie bracket [−,−] : Γ(E)
× Γ(E)→ Γ(E) satisfying the Leibniz rule,
[e1, fe2] = f [e1, e2] + ρ(e1)f · e2,
where ei ∈ Γ(E) and f ∈ C∞(M).
A bundle map ρ is called an anchor map.
Example 2.2. Let a manifold M be one point M = {pt}. Then a Lie algebroid is a Lie
algebra g.
Example 2.3. If a vector bundle E is a tangent bundle TM and ρ = id, then a bracket [−,−]
is a normal Lie bracket of vector fields and (TM, id, [−,−]) is a Lie algebroid.
Example 2.4. Let g be a Lie algebra and assume an infinitesimal action of g on a manifold M .
The infinitesimal action g ×M → M determines a map ρ : M × g → TM . The consistency of
a Lie bracket requires a Lie algebroid structure on (E = M × g, ρ, [−,−]). This Lie algebroid is
called an action Lie algebroid.
Momentum Sections in Hamiltonian Mechanics and Sigma Models 3
Example 2.5. An important nontrivial Lie algebroid is a Lie algebroid induced from a Poisson
structure. A bivector field π ∈ Γ
(
∧2TM
)
is called a Poisson structure if [π, π]S = 0, where
[−,−]S is a Schouten bracket on Γ(∧•TM).
Let (M,π) be a Poisson manifold. Then, we can define a bundle map, π] : T ∗M → TM by
π](α)(β) = π(α, β) for all β ∈ Ω1(M). A Lie bracket on Ω1(M) is defined by the so called
Koszul bracket,
[α, β]π = Lπ](α)β − Lπ](β)α− d(π(α, β)),
where α, β ∈ Ω1(M). Then,
(
T ∗M,π], [−,−]π
)
is a Lie algebroid.
One can refer to many other examples, for instance, in [23].
2.2 Lie algebroid differential
We consider a space of exterior products of sections, Γ(∧•E∗) on a Lie algebroid E. Its element
is called an E-differential form. We can define a Lie algebroid differential Ed: Γ
(
∧mE∗
)
→
Γ
(
∧m+1E∗
)
such that
(
Ed
)2
= 0.
Definition 2.6. A Lie algebroid differential Ed: Γ
(
∧mE∗
)
→ Γ
(
∧m+1E∗
)
is defined by
Edα(e1, . . . , em+1) =
m+1∑
i=1
(−1)i−1ρ(ei)α(e1, . . . , ěi, . . . , em+1)
+
∑
i,j
(−1)i+jα([ei, ej ], e1, . . . , ěi, . . . , ěj , . . . , em+1),
where α ∈ Γ
(
∧mE∗
)
and ei ∈ Γ(E).
It is useful to describe Lie algebroids by means of Z-graded geometry [26]. A graded mani-
foldM with local coordinates xi, i = 1, . . . ,dimM , and qa, a = 1, . . . , rankE, of degree zero and
one, respectively, are denoted by M = E[1] for some rank r vector bundle E, where the degree
one basis qa is identified by a section in E∗, i.e., we identify C∞(E[1]) ' Γ(∧•E). A product for
homogeneous elements f, g ∈ C∞(M) has a property, fg = (−1)|f ||g|gf , where |f | is degree of f .
Especially, qaqb = −qbqa. We introduce a derivation of degree −1, ∂
∂qa satisfying ∂
∂qa q
b = δba.
Here, a derivation is a linear operator on a space of functions satisfying the Leibniz rule.
The most general degree plus one vector field on M has the form
Q = ρia(x)qa
∂
∂xi
− 1
2
Ccab(x)qaqb
∂
∂qc
,
where ρia(x) and Ccab(x) are local functions of x. Since qa is of degree 1, Ccab = −Ccba.
Let ea be a local basis in E dual to the basis corresponding to the coordinates qa. Two
functions ρ and C in Q define a bundle map ρ : E → TM and a bilinear bracket on Γ(E) by
means of ρ(ea) := ρia∂i and [ea, eb] := Ccabec. Then, one can prove that these satisfy the definition
of a Lie algebroid, iff
Q2 = 0.
Identifying functions on C∞(E[1]) ' Γ(∧•E∗), Q corresponds to a Lie algebroid differential Ed.
In the remaining of the paper, we identify C∞(E[1]) ' Γ(∧•E∗), and Q to Ed.
4 N. Ikeda
2.3 Momentum section
In this section, a momentum section on a Lie algebroid E is defined [2]. For definition, we
suppose a pre-symplectic form B ∈ Ω2(M) on a base manifold M , i.e., a closed 2-form which
is not necessarily nondegenerate. A Lie algebroid (E, ρ, [−,−]) is one over a pre-symplectic
manifold (M,B).
We introduce a connection (a linear connection) on E, i.e., a covariant derivative D : Γ(E)→
Γ(E⊗T ∗M), satisfying D(fe) = fDe+df⊗e for a section e ∈ Γ(E) and a function f ∈ C∞(M).
A connection is extended to Γ(M,∧•T ∗M ⊗ E) as a degree 1 operator.
In order to define a momentum section, we consider an E∗-valued 1-form γ ∈ Ω1(M,E∗)
defined by
〈γ(v), e〉 = −B(v, ρ(e)),
where e ∈ Γ(E) and v ∈ X(M) is a vector field. Here 〈−,−〉 is a natural pairing of E and E∗.
We introduce the following three conditions for a Lie algebroid E on a pre-symplectic mani-
fold (M,B).
(H1) E is presymplectically anchored with respect to D if
Dγ = 0, (2.1)
where D is a dual connection on E∗ defined by
d〈µ, e〉 = 〈Dµ, e〉+ 〈µ,De〉,
for all sections µ ∈ Γ(E∗) and e ∈ Γ(E). The dual connection extends to a degree 1
operator on Ωk(M,E∗).
(H2) A section µ ∈ Γ(E∗) is a D-momentum section if
Dµ = γ. (2.2)
(H3) A D-momentum section µ is bracket-compatible if
Edµ(e1, e2) = −〈γ(ρ(e1)), e2〉, (2.3)
for all sections e1, e2 ∈ Γ(E). We note these conditions have already appeared in [21] as
compatibility conditions of geometric quantities as a metric and a closed differential form
with a Lie algebroid structure. The equation (H1) is the same as equation (6) and (H2) is
equation (7) in [21].
A Hamiltonian Lie algebroid is defined as follows.
Definition 2.7. A Lie algebroid E with a connection D and a section µ ∈ Γ(E∗) is called weakly
Hamiltonian if (H1) and (H2) are satisfied. If the condition is satisfied on a neighborhood of
every point in M , it is called locally weakly Hamiltonian.
Definition 2.8. A Lie algebroid E with a connectionD and a section µ ∈ Γ(E∗) is called is called
Hamiltonian if (H1), (H2) and (H3) are satisfied. If the condition is satisfied on a neighborhood
of every point in M , it is called locally Hamiltonian.
A bracket-compatible D-momentum section, i.e., conditions (H2) and (H3) are sufficient in
our examples in later section. Thus, in this paper, we mainly consider a bracket compatible
momentum section µ ∈ Γ(E∗) satisfying (H2) and (H3), and do not necessarily require that E
is presymplectically anchored, i.e., (H1). In this case, D2µ is not necessarily zero.
Momentum Sections in Hamiltonian Mechanics and Sigma Models 5
2.4 Lie algebra case: momentum map
A momentum section is a generalization of a momentum map on a symplectic manifold with
a Lie group action. The definition of a momentum section (H1), (H2) and (H3) reduces to the
definition of a momentum map if a Lie algebroid E is an action Lie algebroid.
Suppose B is nondegenerate, i.e., B is a symplectic form. Consider an action Lie algebroid
on E = M × g. It means that an infinitesimal Lie algebra action is given by a bundle map
ρ : g×M → TM , such that
[ρ(e1), ρ(e2)] = ρ([e1, e2]).
The bracket in left hand side is a Lie bracket of vector fields. In this case, we can take a zero
connection, D = d. Then, three axioms of a momentum section reduce to the following equations.
(H1)
dγ(e) = d(ιρ(e)B) = Lρ(e)B = 0. (2.4)
This means that ρ(e) is a symplectic vector field.
(H2) A section µ ∈ Γ(M ×g∗) is regarded as a map µ : M → g∗. Equation (2.2) is that a map µ
is a Hamiltonian for the vector field ρ(e),
dµ(e) = ιρ(e)B. (2.5)
Equation (2.5) leads equation (2.4).
(H3) dµ = γ, i.e., dµ = −B(ρ,−).
Equation (2.3) is equivalent to
ad∗e1 µ(e2) = µ([e1, e2]). (2.6)
for e1, e2 ∈ g. This means that µ is g-equivariant.
Independent conditions are (2.5) and (2.6), which are the definition of an infinitesimally
equivariant momentum map.
Many examples of momentum sections which are not momentum maps have been discussed
in [2]. One can refer to more examples.
3 Constrained Hamiltonian system
We discuss examples of physical systems which have momentum sections and Hamiltonian Lie
algebroid structures. In this section, we consider a constrained Hamiltonian mechanics system
in 1 + 0 dimension analyzed in [19].
Let (N = T ∗M,ωcan) be a symplectic manifold over a smooth manifold M , where ωcan is
a canonical symplectic form on N . We take Darboux coordinates
(
xi, pi
)
such that ωcan =
dxi∧dpi. On this symplectic manifold, we consider a dynamical system. Assume a Hamiltonian
H ∈ C∞(N), and r constraint functions Φa = Φa(x, p), satisfying the following compatibility
condition:
There exist local matrix functions λba = λba(x, p) such that
{H,Φa} = λbaΦb, (3.1)
where {−,−} is the Poisson bracket induced by the symplectic form ωcan. This ensures that H
is preserved by the Hamiltonian flow of the constraints on the constraint surface C := {Φa = 0}.
6 N. Ikeda
Moreover, suppose constraint functions are of the first class, i.e., they satisfy
{Φa,Φb} = CcabΦc, (3.2)
for some functions Ccab = Ccab(x, p) on N .
We assume that constraints Φa, a = 1, . . . , r, are irreducible, i.e., ϕ∗C(dΦ1 ∧ · · · ∧ dΦr) is
everywhere non-zero, where ϕC : C → N is the canonical embedding map of the constraint
surface into the original phase space. Moreover, two sets of irreducible constraints Φa, a =
1, . . . , r, and Φ̃a, a = 1, . . . , r, are equivalent if there exist local matrix functions M b
a = M b
a(x, p)
such that
Φ̃a = M b
aΦb (3.3)
holds true and the matrix
(
M b
a
)r
a,b=1
is invertible when restricted to C.
We take setting of the paper [19]. We require the canonical symplectic form ωcan = dxi ∧
dpi. Then, there is a natural grading of functions with respect to the monomial degree in the
momenta pi. The space of order i or less than i functions is denoted by C∞≤i(T
∗M).
As a typical example which appears in physical applications, we consider the case of Φa ∈
C∞≤1(T ∗M) and H ∈ C∞≤2(T ∗M). These imply
Φa = ρia(x)pi + αa(x), (3.4)
and
H =
1
2
gij(x)pipj + βi(x)pi + V (x). (3.5)
Here ρia(x), αa(x), gij(x), βi(x) and V (x) are local functions of x.
We show that this Hamiltonian mechanics system has a momentum section and a Hamiltonian
Lie algebroid structure.
3.1 Lie algebroid structure on constraints
First, we see equation (3.2) with (3.4). As explained in [19], this equation requires an (anchored
almost) Lie algebroid structure. Counting an order of pi in the equivalence condition (3.3),
matrix functions Ma
b are functions of x. Then, a global structure is a rank r vector bundle E
over M with transition functions (Ma
b )ra,b=1.
The Poisson bracket reduces the order by one or less than one since {pi, xj} = δji and
{pi, pj} = 0. Thus, the equality (3.2) implies Ccab ∈ C∞0 (T ∗M) ∼= C∞(M), which is uniquely
determined due to the irreducibility condition. The 1st order of p of equation (3.2) takes the
form, [ρa, ρb]
i = Ccab(x)ρic, i.e., globally,
[ρ(e1), ρ(e2)] = ρ([e1, e2]), (3.6)
for e1, e2 ∈ Γ(E).
Next we apply (3.2) to the Jacobi identity {{Φa,Φb},Φc}+cycl(abc) = 0. The 1st order of pi
gives (
CeabC
d
ce + ∂jC
d
abρ
j
c + cycl(abc)
)
ρid = 0. (3.7)
From the irreducibility condition on the constraints and the above identity, we may deduce
CeabC
d
ce + ρja∂jC
d
bc + cycl(abc) = σdabc,
Momentum Sections in Hamiltonian Mechanics and Sigma Models 7
for some functions σdabc skewsymmetric in the lower indices and σdabcρ
i
d = 0. If the anchor map ρ
is assumed injective, we have σdabc = 0 and
CeabC
d
ce + ρja∂jC
d
bc + cycl(abc) = 0. (3.8)
It is now straightforward to verify that equations (3.6) and (3.8) yield Lie algebroid axioms,
where the anchor map ρ : E → TM is defined by ρ(ea) = ρia(x)∂i and the Lie bracket is defined
by [ea, eb] = Ccab(x)ec for a basis ea of the fiber of E. We remark that the equivalence (3.3) takes
care of the equivalence of the two sides to not depend on the choice of a chosen frame.
A vector bundle with a bundle map ρ : E → TM and a bilinear bracket [−,−] is an anchored
almost Lie algebroid (E, ρ, [−,−]) if ρ and [−,−] satisfy
[ρ(e1), ρ(e2)] = ρ([e1, e2]),
[e1, fe2] = f [e1, e2] + ρ(e1)f · e2.
If ρ is not injective, we can take an anchored almost Lie algebroid since it satisfies equation (3.7).
We do not discuss these cases in this paper.
We can take a more general algebroid satisfying σdabcρ
i
d = 0 such as a Courant algebroid. We
leave such cases to other analysis.
The second term αa in Φa is considered as components of an E-1-form, α = αa(x)ea ∈ Γ(E∗),
where ea is a basis on E∗. The Poisson bracket (3.2) is equivalent to the condition on α,
Edα = 0. (3.9)
On the other hand, α is determined by (3.4) only up to additions of the form αa 7→ αa +
ρia(x)∂if(x), for a function f on M , which does not modify the symplectic form. Since such
additions to α are the Ed-exact ones, we see that 0th order deformations of p in first class
constraints (3.4) are parametrized by the Q-cohomology of the Lie algebroid at degree one,
[α] ∈ H1
Q(E[1]).
Equation (3.2) and injective assumption for ρ gives a Lie algebroid structure on E and
equation (3.9).
3.2 Hamiltonian, metric and connection
In this section, we explain geometric structures induced from the Hamiltonian (3.5) and the
Poisson bracket (3.1) discussed in [19]. Suppose that in (3.5) the symmetric matrix gij has an
inverse. Then a symmetric tensor gij corresponds to an inverse of a metric g on M . β := βi∂i is
a vector field and V (x) is a global potential function on M . Counting order of p in equation (3.1),
λba is a 1st order function of p, thus it is assumed that λba = gij(x)Γbaj(x)pi + τ ba(x). From
consistency of equation (3.1) with transition functions Ma
b given by equivalence of Φa, we obtain
Γ′ = MΓM−1 + dMM−1,
τ ′ = MτM−1 +
(
β,dMM−1
)
, (3.10)
where Γba = Γbajdx
j . Γba transforms as a connection 1-form on E. τ ba is a local matrix transforming
in equation (3.10).1
We can absorb the term linear in the momenta in the Hamiltonian, βi 7→ 0, at the expense
of redefining the potential V and the E-1-forms α and simultaneously twisting the symplectic
form ωcan by a magnetic field B = dA ∈ Ω2(M) as
ω = ωcan +B,
1An interesting τ is τ = (Γ, β), which case is analyzed later in this section.
8 N. Ikeda
where Ai = gijβ
j and A = Ai(x)dxi. The globally defined 2-form B = dA is obviously regarded
as a pre-symplectic form since dB = 0.
By the above redefinition, constraints and the Hamiltonian become
Φ′a = ρia(x)pi + α′a(x),
H =
1
2
gij(x)pipj + V ′(x).
Here, α′ is an E-1-form defined by 〈α′, e〉 = 〈α, e〉 − ιρ(e)A for all e ∈ Γ(E), and V ′ is defined by
V ′(x) = V (x)− 1
2g(β, β). Equations (3.2) and (3.1) change but are similar equations,
{Φ′a,Φ′b} = CcabΦ
′
c, (3.11)
{H,Φ′a} = λ′baΦ′b, (3.12)
where τ ′ = τ − g−1(Γ, A) and λ′ = λ− g−1(Γ, A) = g−1(Γ, p) + τ ′.
After the above redefinition, we show that geometric structure described by equations (3.11)
and (3.12) have a structure of a momentum section.
The 1st order term of p in equation (3.11) gives the same conditions as (3.2), i.e., (3.11)
requires a Lie algebroid structure on the vector bundle E with the same anchor map ρ and Lie
bracket [−,−] before the redefinition. In the 0th order term of p in equation (3.11), the affine
constraints α changes to
Edα′ = −ρ∗(B), (3.13)
since the new symplectic form ω gives the Poisson bracket {pi, pj} = Bij . Here ρ∗ is the induced
map of the anchor to Ω•(M), mapping ordinary differential forms to E-differential forms. In
particular, ρ∗(B) = 1
2Bijρ
i
aρ
j
bq
aqb ∈ Γ
(
∧2E∗
)
. Equation (3.13) is the same as equation (2.3) in
the condition (H3) by identifying µ = α′.
Let us analyze equation (3.12). As already pointed, the transformation property of Γabi under
the transition function Ma
b shows Γabi is a connection 1-form, thus this defines a Lie algebroid
connection D : Γ(E) → Γ(E ⊗ T ∗M). D and ρ can be combined to define an E-connection
E∇ : Γ(TM)→ Γ(TM ⊗ E∗) on TM :
E∇ev := Lρ(e)v + ρ(Dve), (3.14)
where v ∈ X(M) and e ∈ Γ(E). An E-connection is extended to a tensor product space of TM
and T ∗M .
Equation (3.12) then gives three conditions by considering it to 2nd, 1st, and 0th order in
the momenta. To 2nd order, we obtain the geometrical compatibility equation,
E∇g = 0,
on the metric g.
To 1st order, we get another condition on the system of constraints, It relates the exterior
covariant derivative of α′ induced by D, Dα′ ∈ Γ(E∗⊗T ∗M), to the anchor map ρ, now regarded
as a section of E∗ ⊗ TM :
Dα′ = γ + (τ ′t ⊗ g[)ρ, (3.15)
where γ ∈ Ω1(M,E∗) is a 1-form taking a value on E∗ appeared in the definition of a momentum
section, τ ′t : E∗ → E∗, the transposed of τ ′, and g[ : TM → T ∗M , v 7→ ιvg, as maps on the
corresponding sections. To 0th order one finds that the potential V ′ has to satisfy
EdV ′ = τ ′(α′).
Momentum Sections in Hamiltonian Mechanics and Sigma Models 9
If τ ′ = 0, equation (3.15) becomes
Dα′ = γ,
which is the condition (H2), i.e., equation (2.2), since µ = α′. The condition τ ′ = 0 is τ = g(Γ, A).
The remaining condition of a momentum section is the condition (H1), i.e., equation (2.1),
Dγ = 0. In general, this constrained Hamiltonian system does not satisfy (2.1).2
Therefore, we obtain the following result:
Theorem 3.1. We consider the constraint Hamiltonian system satisfying equations (3.1) and
(3.2) with constraints (3.4) and a Hamiltonian (3.5). Then, B = d(g(β,−)) is a pre-symplectic
form. If ρ is injective and τ ′ = τ − g(Γ, A) = τ − (Γ, β) = 0, α′ = α − ιρA is a bracket
compatible D-momentum section on a Lie algebroid E with respect to a connection D defined by
a connection 1-form Γba. Moreover, if Dγ = D(−ιρB) = 0, it is pre-symplectically anchored.
In τ ′ 6= 0 case, this constrained Hamiltonian system gives a generalization of a momentum
section. It is interesting to see this generalization in a future work.
4 Two-dimensional sigma model with boundary
In this section, we analyze a momentum section and a Hamiltonian Lie algebroid structure in
a two-dimensional sigma model. If a base manifold is in two dimensions and with boundary,
a momentum section naturally appears.
Let Σ be a two-dimensional manifold and M be a d-dimensional target manifold. X : Σ→M
is a smooth map from Σ to M . We start at the following sigma model action with a 2-form
B-field,
S =
1
2
∫
Σ
gij(X)dXi ∧ ∗dXj + bij(X)dXi ∧ dXj , (4.1)
where g is a metric and b ∈ Ω2(M) is a closed 2-form on M . gij(X) and bij(X) are their pullbacks
to Σ. This action is invariant under diffeomorphisms on a worldsheet Σ and on a target space M .
We analyze a general condition that the action S is invariant under other symmetries on M .
In a general setting, an element of a vector space V , or more generally, a section of the vec-
tor bundle E on M , e ∈ Γ(E) acts on M as an infinitesimal transformation generated by
a vector field. A transformation is determined by defining a bundle map to a tangent bundle,
ρ : E → TM . Suppose that ρ define an infinitesimal gauge transformation of X as
δXi = ρ(ε)i = ρia(X)εa, (4.2)
where i = 1, 2, . . . , d are indices of local coordinates on M , ε ∈ Γ(X∗E) is a parameter (a gauge
parameter), and ρ(ea) = ρia(X)∂i by taking a basis of E, ea.
By straight computations, the action (4.1) is in invariant under the transformation (4.2), iff
Lρ(ea)g = 0, (4.3)
Lρ(ea)b = dβa, (4.4)
[ρ(ea), ρ(eb)] = ρ([ea, eb]), (4.5)
where L is a Lie derivative and βa ∈ Ω1(M,E∗) is a 1-form taking a value on E∗. A vector
field ρ(ea) satisfying equation (4.3) is called a Killing vector field. From equation (4.5), a vector
bundle is an anchored almost Lie algebroid.
In this paper, E is a Lie algebroid. In this case, the action S is invariant if equations (4.3)
and (4.4) are satisfied.
2Using an E-connection (3.14) and its extension to Ω2(M), equation (2.1) is equivalent to E∇B = 0, where
γia = −Bijρ
j
a and E∇aBij = ρka∂kBij + ∂iρ
k
aBkj + ∂jρ
k
aBik + ρkbΓb
ajBik + ρkbΓb
aiBkj = 0.
10 N. Ikeda
4.1 Gauged sigma model
We can generalize the above theories by gauging the action (4.1). ‘Gauging’ is a deformation of
the action using a connection 1-form A ∈ Ω1(Σ, X∗E).
A pullback of a basis of a 1-form on M , dXi, is ‘gauged’ using a covariant derivative with
respect to a connection A as
F i = DXi = dXi − ρia(X)Aa.
We can assume Aa has a genuine infinitesimal gauge transformation,
δAa = dεa + [A, ε]a = dεa + CabcA
bεc,
however, Cabc = Cabc(X) is not necessarily constant but a local function on M . We consider
a target space covariant version of the gauge transformation by introducing (a pullback of)
a connection on M , Γabi(X):3
δAa = dεa + Cabc(X)Abεc + Γabi(X)εbDXi,
where the gauge transformation is covariant under the target space diffeomorphism. In summary,
we choose gauge transformations,
δXi = ρia(X)εa, (4.6)
δAa = dεa + Cabc(X)Abεc + Γabi(X)εbDXi. (4.7)
The action (4.1) is generalized to a gauged sigma model action by ‘gauging’ the symmetry to
infinitesimal transformations (4.6) and (4.7). Since the manifold Σ has boundary, we take the
following ansatz for a gauged sigma model action:
S =
1
2
∫
Σ
gij(X)DXi ∧ ∗DXj + bij(X)dXi ∧ dXj +
∫
∂Σ
ηi(X)dXi + µa(X)Aa, (4.8)
where the last two terms are the most general possible boundary terms with some arbitrary local
functions ηi(X) and µa(X). ηi(X)dXi is a pullback of a 1-form on a target space M and µa(X)
is a pullback of an element Γ(E∗) on a target space M . Requiring (4.8) is invariant under gauge
transformations (4.6) and (4.7), we obtain geometric conditions for a metric g, a 2-form b and ρ
and a bracket [−,−]. We obtain the following conditions for the metric, ρ and a bracket,
Lρ(ea)g = Γba ∨ ιρ(eb)g, (4.9)
[ρ(ea), ρ(eb)] = ρ([ea, eb]), (4.10)
where ∨ is a symmetric product of 1-forms. Equation (4.9) is equivalent to E∇g = 0. Though
equation (4.10) is satisfied if (E, ρ, [−,−]) is an anchored almost Lie algebroid, we suppose
(E, ρ, [−,−]) is a true Lie algebroid now.
Next we analyze a condition for a 2-form B-field b. Using db = 0, the gauge transformation
for Sb = 1
2
∫
Σ bij(X)dXi ∧ dXj is
δSb =
∫
Σ
Lρ(ε)b =
∫
Σ
dιρ(ε)b =
∫
∂Σ
ιρ(ε)b.
3We can consider a more general ansatz of a gauge transformation as δAa = Dεa+[A, ε]a = dεa+Ca
bcA
bεc+∆Aa,
where ∆Aa is a 1-form taking a value on a pullback of E, which is linear with respect to the infinitesimal
parameter εa [11, 12, 13].
Momentum Sections in Hamiltonian Mechanics and Sigma Models 11
Thus, requirement of gauge invariance of the total action δS = 0 gives the conditions including
quantities of boundary terms. In local coordinates, straight computations give three equations,
µa = −ηiρia, (4.11)
ρjabji + ρja∂jηi + ηj∂iρ
j
a + Γbaiµb = 0, (4.12)
ρia∂iµb − Ccabµc − ρibΓcaiµc = 0. (4.13)
The first condition (4.11) is µ(e) = −ιρ(e)η for e ∈ Γ(E), the second and third conditions (4.12)
and (4.13) are equivalent to (H2) and (H3), where we identify B = b+ dη. Thus, we obtain the
following result.
Theorem 4.1. We consider a gauged sigma model with boundary, (4.8). Let µ(e) = −ιρ(e)η ∈
Γ(E∗) and B = b + dη ∈ Ω2(M). Let D be a connection defined by Γba. Then, µ is a bracket
compatible D-momentum section with respect to the connection D with a pre-symplectic form B.
If B satisfies (H1), it is pre-symplectically anchored.
Finally, we comment a gauge algebra generated by gauge transformations (4.6) and (4.7).
Gauge transformations must consist of a closed algebra at least on an orbit of equations of
motion. Closure conditions of a gauge algebra generated [δ1, δ2] ∼ δ3 by equations (4.6) and (4.7)
impose extra equations for Xi and Aa, which are topological conditions.
This gives topological condition on a external gauge field Aa and geometry of M . One can
refer to [3] and [27] for more analysis related T-duality. See also [16].
5 Momentum section on pre-multisymplectic manifold
In this section, we propose a generalization of a momentum section to a pre-multisymplectic
manifold. Our strategy is to generalize a gauged sigma model in Section 4. We generalize
a 2-form B-field b to a higher (n + 1)-form h and a two-dimensional manifold Σ to a higher-
dimensional manifold. We naturally obtain a generalization of a momentum section from con-
sistency of these gauged sigma models.
5.1 Gauged sigma model in n dimensions with Wess–Zumino term
We introduce a pre-n-plectic manifold.
Definition 5.1. A pre-n-plectic manifold is (M,h), where M is a smooth manifold and h is
a closed (n+ 1)-form on M .
A pre-n-plectic manifold is also called a pre-multisympletic manifold for n ≥ 2. A pre-n-
plectic manifold is called an n-plectic manifold if h is nondegenerate, i.e., if ιvh = 0 for a vector
field v ∈ X(M) is equivalent to X = 0.
We also introduce a metric g on M . Moreover, by introducing an (n+ 1)-dimensional mani-
fold Ξ with boundary Σ = ∂Ξ, and a map X : Σ → M , we consider a nonlinear sigma model
with a Wess–Zumino term. Compatibility conditions of h and g are determined from gauge
invariance of the following sigma model action with a Wess–Zumino term:
S =
∫
Σ
1
2
gij(X)dXi ∧ ∗dXj +
∫
Ξ
1
(n+ 1)!
hi1...in+1(X)dXi1 ∧ · · · ∧ dXin+1 , (5.1)
where g(X) is a pullback of a metric g and h(X) = 1
(n+1)!hi1...in+1(X)dXi1 ∧ · · · ∧ dXin+1 in the
second term is a pullback of a closed (n + 1)-form h on M . The n = 2 case is most important
since this is string sigma model with an NS-flux 3-form H.
12 N. Ikeda
Invariance conditions of S under the transformation (4.2) of X as in Section 4 gives a similar
condition,
Lρ(ea)g = 0,
Lρ(ea)h = dβa,
[ρ(ea), ρ(eb)] = ρ([ea, eb]), (5.2)
where β is an n-form taking a value on E∗. Equation (5.2) require an anchored almost Lie
algebroid structure on a target vector bundle E.
We consider gauging of an n-dimensional sigma model (5.1) by introducing a connection
A ∈ Ω1(Σ, X∗E) and gauge transformations (4.6) and (4.7). Here, we consider the case that
a vector bundle E is a Lie algebroid for (5.2) again. We take a Hull–Spence type ansatz [17] for
a gauged action, but in our case a gauge structure is not a Lie algebra but a Lie algebroid. The
ansatz is
S = Sg + Sh + Sη, (5.3)
where
Sg =
∫
Σ
1
2
gijDX
i ∧ ∗DXj ,
Sh =
∫
Ξ
1
(n+ 1)!
hi1...in+1(X)dXi1 ∧ · · · ∧ dXin+1 ,
Sη =
∫
Σ
n∑
k=0
1
k!(n− k)!
η
(k)
i1...ikak+1...an
(X)dXi1 ∧ · · · ∧ dXik ∧Aak+1 ∧ · · · ∧Aan ,
where η(k) is a pullback of a k-form onM taking a value on ∧n−kE∗, i.e., η(k)∈X∗Ωk(M,∧n−kE∗).
We require gauge invariance of the above gauged action under the gauge transformations (4.6)
and (4.7), which are the same ones as in two-dimensional case Section 4. Requirement of gauge
invariance imposes conditions for pullbacks of coefficient functions g ∈ Γ
(
S2T ∗M
)
, h ∈ Ωn+1(M)
and η(k) ∈ Ωk
(
M,∧n−kE∗
)
. These identities gives geometric identities of a metric g, H and η(k)
on the vector bundle E on M before pullbacks.4
From concrete computations, the condition of g is
Lρ(ea)g = Γba ∨ ιρ(eb)g,
as in the case of the two-dimensional sigma model. For h and η(k) on M , we obtain the following
conditions on M :5 two algebraic conditions,
η(k−1)(ek, . . . , en) = (−1)kιρ(ek)η
(k)(ek+1, . . . , en) + cycl(ek, . . . , en), (5.4)
ιρ(ek)η
(k)(ek+1, . . . , ek+m, . . . , en) + ιρ(ek+m)η
(k)(ek+1, . . . , ek, . . . , en) = 0,
k = 1, . . . , n− 1, m = 1, . . . , n− k, (5.5)
and three differential equations,
Dη(n−1)(e) = ιρ(e)h̃, k = n, (5.6)
4We use the same notation for geometric quantities on M and their pullbacks. We propose that this structure
gives a momentum section in a pre-n-plectic manifold.
5Note that we obtain identities on h ∈ Ωn+1(M) and η(k) ∈ Ωk
(
M,∧n−kE∗) from conditions for their pullbacks
in the gauged sigma model (5.3).
Momentum Sections in Hamiltonian Mechanics and Sigma Models 13
Lρ(e)η(k)(ek+1, . . . , en) +
n−k∑
i=1
(−1)iη(k)([e, ek+i], ek+1, . . . , ěk+i, . . . , en)
+
n−k∑
i=1
(−1)i〈Γ, ρ(e)〉 ∧ η(k)(ek+1, . . . , en)
−
n−k∑
i=1
(−1)iΓ(e) ∧ ιρ(ek+i)η
(k)(ek+1, . . . , ěk+i, . . . , en)
+
n−k∑
i=1
(−1)i〈ιρ(ek+i)Γ(e) ∧, η(k)(ek+1, . . . , ěk+i, . . . , en)〉 = 0, k = 1, . . . , n− 1, (5.7)
Lρ(e)η(0)(e1, . . . , en) +
n∑
i=1
(−1)iη(0)([e, ek+i], ek+1, . . . , ěk+i, . . . , en)
+
n∑
i=1
(−1)i〈ιρ(ei)Γ(e) ∧, η(0)(e1, . . . , ěi, . . . , en)〉 = 0, k = 0, (5.8)
where h̃ = h + dη(n), e, ei ∈ Γ(E), i = k, . . . , n, Γ is a connection 1-form on E, and 〈−,−〉 is
a natural pairing of E∗ and E. Notation ∧, means both a wedge product on Ωk(M) and a pairing
of E and E∗. Note that δSh =
∫
Ξ Lρ(ε)h =
∫
Σ ιρ(ε)h since dh = 0. For k = n− 1, equation (5.7)
is also written as
Edη(n−1)(e1, e2)−Dη(n−2)(e1, e2) = 0.
In n = 1, equations (5.4)–(5.8) reduce to conditions of a momentum section (H2) and (H3)
by setting µ = η(0), γ = η(1) and B = h̃. In n = 2, equations (5.4)–(5.8) give gauging conditions
of target geometry in [11].
It is natural to impose the following condition corresponding to the condition (H1),
Dιρh̃ = 0. (5.9)
However, this condition is not needed for gauge invariance of a gauged sigma model. As a result,
we need not impose this condition on the definition of a momentum section.
Finally, we obtain the following definition of a multimomentum section on a pre-mutlisym-
plectic manifold.
Definition 5.2. Let (M, h̃) be a pre-n-plectic manifold, where h̃ is a closed (n + 1)-form, and
(E, ρ, [−,−]) be a Lie algebroid over M . We define the following three conditions corresponding
to (H1), (H2) and (H3).
(HM1) E is a pre-n-plectically anchored with respect to D if
Dγ = 0,
where γ = ιρh̃ ∈ Ωn(M,E∗).
(HM2) η(n−1) ∈ Ωn−1(M,E∗) is a D-multimomentum (D-momentum) section if it satisfies
equation (5.6),
Dη(n−1)(e) = ιρ(e)h̃.
(HM3) We define a descent set of multimomentum sections
(
η(k)
)n−2
k=0
by equations (5.4) and
(5.5), where η(k) ∈ Ωk
(
M,∧n−kE∗
)
. A D-multimomentum section and its descents(
η(k)
)n−1
k=0
are bracket-compatible if (5.7) and (5.8) are satisfied.
14 N. Ikeda
Under this definition, we have the same definition of a weakly Hamiltonian Lie algebroid,
Definition 2.7, and a Hamiltonian Lie algebroid, Definition 2.8, but a momentum section is a set
of multimomentum sections η(k) on a pre-multisymplectic manifold
(
M, h̃
)
. A Hamiltonian Lie
algebroid on a pre-multisymplectic manifold is defined as follows.
Definition 5.3. A Lie algebroid E with a connection D and a section η(n−1) ∈ Ωn−1(M,E∗)
is called weakly Hamiltonian if (HM1) and (HM2) are satisfied. If the condition is satisfied on
a neighborhood of every point in M , it is called locally weakly Hamiltonian.
Definition 5.4. A Lie algebroid E with a connection D and a section η(k) ∈ Ωk
(
M,∧n−kE∗
)
,
k = 0, . . . , n−1 is called Hamiltonian if (HM1), (HM2) and (HM3) are satisfied. If the condition
is satisfied on a neighborhood of every point in M , it is called locally Hamiltonian.
We summarize a geometric structure of a gauge sigma model with a (n+1)-form flux h using
the terminology of multimomentum sections.
Theorem 5.5. We consider an n-dimensional gauged sigma model with WZ term (5.3). Then,
η(k) ∈ Ωk
(
M,∧n−kE∗
)
, k = 0, . . . , n − 1, are a bracket compatible D-multimomentum section
and descents with a pre-n-plectic form h̃ = h+ dη(n). If h̃ satisfies (HM1), it is pre-n-plectically
anchored.
5.2 Momentum map on multisymplectic manifold: Lie algebra case
Let a Lie algebroid be an action Lie algebroid E = M×g. Then, we can take a trivial connection
d = D, and a momentum section on a pre-n-plectic manifold reduces to a (multi)momentum
map on a pre-symplectic manifold.
Conditions (5.4)–(5.8) reduce to
η(k−1)(ek, . . . , en) = (−1)k ad∗ek η
(k)(ek+1, . . . , en) + cycl(ek, . . . , en),
ad∗ek η
(k)(ek+m, . . . , ek+m, . . . , en) + ad∗ek+1
η(k)(ek+1, . . . , ek, . . . , en) = 0,
k = 1, . . . , n− 1, m = 1, . . . , n− k,
dη(n−1) = ιρa h̃, k = n, (5.10)
dη(k−1)(e, ek+1, . . . , en) = ad∗e η
(k)(ek+1, . . . , en)
−
n∑
i=k
(−1)i−1η(k)([e, ei], ek+1, . . . , ěi, . . . , en), k = 1, . . . , n− 1,
ad∗e η
(0)(e1, . . . , en) =
n∑
i=1
(−1)i−1η(0)([e, ei], e1, . . . , ěi, . . . , en), k = 0.
A pre-n-plectically anchored condition equation (5.9) is trivially satisfied from equation (5.10),
dιρh̃ = 0.
This condition already appeared in [21].
The above conditions are a direct generalization of a momentum map (multimomentum
map) on a multisymplectic manifold with a Lie group action [8, 14] by setting η(k) = 0 for
k = 0, . . . , n− 2. In this case, η(n−1) is a multimomentum map.
Momentum Sections in Hamiltonian Mechanics and Sigma Models 15
6 Discussion and outlook
We have shown that a simple constrained Hamiltonian mechanics and a two-dimensional gauged
sigma model with boundary have a momentum section and a Hamiltonian Lie algebroid struc-
ture. By generalizing a gauged sigma model to a higher-dimensional gauged sigma model with
WZ term, we have proposed a theory of a multimomentum section on a pre-multisymplectic
manifold.
It is important to compare other generalizations of a moment map theory to a multisymplectic
manifold such as Madsen–Swann’s multimoment map on the n-th Lie kernel [24, 25], a homotopy
moment map [7], and a weak moment map [15].
Though we proposed a momentum section on a pre-multisymplectic manifold (5.4) and (5.8)
from consistency conditions of a higher-dimensional gauged nonlinear sigma model, their geo-
metrical structures should be analyzed more. These structure are described by a Lie algebroid
differential Ed and a covariant derivative D.
In all examples in our paper, the pre-symplectically anchored condition (H1) is not necessary
for consistency of structures. Conditions (H2) and (H3) are essential for physical applications.
More examples are needed for deeper understanding of a momentum section theory.
We have assumed an anchor map ρ is injective in this paper. However we should relax this
condition. If an anchor map ρ is not necessarily injective, we can consider more general algebroid
such as a Courant algebroid [22], a Lie 3-algebroid [20], and higher algebroids, as a symmetry
of a gauged sigma model. This direction is related to a Lie group action on a Courant algebroid
and the reduction [4]. These generalizations are left for future analysis.
We considered an infinitesimal version, i.e., an action of a Lie algebroid on a pre-(multi)
symplectic manifold. A globalization to a Lie groupoid corresponding to a generalization of
a Lie group action is a next problem.
A next step of physical systems in this paper is quantization. One possible quantization is
an equivariant localization using the Duistermaat–Heckman formula, which has been already
discussed in [2]. In this paper, we have obtained more concrete physical models for applications
of the localization. For this purpose, the condition (H1) looks like essential since we need an
equivariant differential such that D2 = 0.
Since a momentum section and a Hamiltonian Lie algebroid structure is a natural structure on
a gauged sigma model, we can hope to obtain new physical results from analysis of a Hamiltonian
Lie algebroid.
Acknowledgments
The author would like to thank Yuji Hirota, Kohei Miura, Satoshi Watamura and Alan Weinstein
for useful comments. He thanks the referees for their careful reading of the manuscript and
especially for their helpful comments.
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1 Introduction
2 Momentum section and Hamiltonian Lie algebroid
2.1 Lie algebroid
2.2 Lie algebroid differential
2.3 Momentum section
2.4 Lie algebra case: momentum map
3 Constrained Hamiltonian system
3.1 Lie algebroid structure on constraints
3.2 Hamiltonian, metric and connection
4 Two-dimensional sigma model with boundary
4.1 Gauged sigma model
5 Momentum section on pre-multisymplectic manifold
5.1 Gauged sigma model in n dimensions with Wess–Zumino term
5.2 Momentum map on multisymplectic manifold: Lie algebra case
6 Discussion and outlook
References
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| id | nasplib_isofts_kiev_ua-123456789-210312 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:06Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ikeda, N. 2025-12-05T09:33:21Z 2019 Momentum Sections in Hamiltonian Mechanics and Sigma Models / N. Ikeda // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D20; 70H33; 70S05 arXiv: 1905.02434 https://nasplib.isofts.kiev.ua/handle/123456789/210312 https://doi.org/10.3842/SIGMA.2019.076 We show that a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic manifold by considering gauged sigma models on higher-dimensional manifolds. The author would like to thank Yuji Hirota, Kohei Miura, Satoshi Watamura, and Alan Weinstein for their useful comments. He thanks the referees for their careful reading of the manuscript and especially for their helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Momentum Sections in Hamiltonian Mechanics and Sigma Models Article published earlier |
| spellingShingle | Momentum Sections in Hamiltonian Mechanics and Sigma Models Ikeda, N. |
| title | Momentum Sections in Hamiltonian Mechanics and Sigma Models |
| title_full | Momentum Sections in Hamiltonian Mechanics and Sigma Models |
| title_fullStr | Momentum Sections in Hamiltonian Mechanics and Sigma Models |
| title_full_unstemmed | Momentum Sections in Hamiltonian Mechanics and Sigma Models |
| title_short | Momentum Sections in Hamiltonian Mechanics and Sigma Models |
| title_sort | momentum sections in hamiltonian mechanics and sigma models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210312 |
| work_keys_str_mv | AT ikedan momentumsectionsinhamiltonianmechanicsandsigmamodels |