8
Structures Relevant to Physics
The last chapter the book is devoted to the mathematical interpretation of physical
objects discussed in Part I. The standard references are [1, 4, 9] and [12].
8.1
BV Algebras and the Master Equation
Generalizing Barannikov’s [1], we prove that an odd dg-modular operad morphism
α : F (C ) → T as in Proposition 7.2 can be, in the case when C is a componentwise linear dual of a modular operad M of finite type as in Proposition 7.1,
conveniently described via a solution of a certain master equation in a shifted dgLie algebra succinctly defined in terms of M and T . Recall that [n] for n ≥ 0
denotes the set {1, . . . , n}, with [0] interpreted as the empty set. We denote, as usual,
M (n; g) := M
[n]; g
and T (n; g) := T
[n]; g
, g ∈ A.
Definition 8.1 Let M = (M , d M ) be a dg-modular operad with structure
operations a ◦ b and ◦ uv , and T = (T , d T ) an odd dg-modular operad with
structure operations a • b and • uv . Define
MT(n; g) :=
M (n; g) ⊗ T (n; g)
Σ n
to be, for n ∈ N, g ∈ A, the space of invariants under the diagonal action of the
symmetric group Σ n on the tensor product M (n; g) ⊗ T (n; g).
Let us introduce the following three operations. For e ∈ MT(n; g) put
d(e) :=
d M ⊗ 1 T (n;g) − 1 M (n;g) ⊗ d T
(e) ∈ MT(n; g).
(8.1)
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_8
185
Structures Relevant to Physics
The last chapter the book is devoted to the mathematical interpretation of physical
objects discussed in Part I. The standard references are [1, 4, 9] and [12].
8.1
BV Algebras and the Master Equation
Generalizing Barannikov’s [1], we prove that an odd dg-modular operad morphism
α : F (C ) → T as in Proposition 7.2 can be, in the case when C is a componentwise linear dual of a modular operad M of finite type as in Proposition 7.1,
conveniently described via a solution of a certain master equation in a shifted dgLie algebra succinctly defined in terms of M and T . Recall that [n] for n ≥ 0
denotes the set {1, . . . , n}, with [0] interpreted as the empty set. We denote, as usual,
M (n; g) := M
[n]; g
and T (n; g) := T
[n]; g
, g ∈ A.
Definition 8.1 Let M = (M , d M ) be a dg-modular operad with structure
operations a ◦ b and ◦ uv , and T = (T , d T ) an odd dg-modular operad with
structure operations a • b and • uv . Define
MT(n; g) :=
M (n; g) ⊗ T (n; g)
Σ n
to be, for n ∈ N, g ∈ A, the space of invariants under the diagonal action of the
symmetric group Σ n on the tensor product M (n; g) ⊗ T (n; g).
Let us introduce the following three operations. For e ∈ MT(n; g) put
d(e) :=
d M ⊗ 1 T (n;g) − 1 M (n;g) ⊗ d T
(e) ∈ MT(n; g).
(8.1)
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_8
185