8
I Physics Preliminaries
V to each puncture. This vector space could be the deRham complex of the target
manifold, as is the case for the topological string, or the quantum mechanical Hilbert
spaces of physical degrees of freedom such as photons or gravitons. The geometric
BV master equation is then transferred to V by means of a two-dimensional
conformal field theory defined on the geometric vertices. This conformal field theory
furthermore endows V with an inner product, more precisely, with a symplectic
form. This provides the data needed to define a BV action on V , that is a field
theory action of closed strings.
It turns out that, at genus zero, the vertices of this BV action satisfy the axioms
of a cyclic L ∞ -algebra, which is a differential graded Lie algebra up to homotopy.
Thus, we find that the mathematical structure of classical closed string field theory
is simply that of a cyclic L ∞ -algebra supported by the vector space V of closed
string states together with a symplectic form. Similarly, at genus zero, any consistent
covariant open string field theory carries a structure of a cyclic A ∞ -algebra, that is,
a differential graded associative algebra up to homotopy.
We already mentioned above that open string field theory is not consistent at the
quantum level without inclusion of closed strings. This raises the question about the
algebraic structure of the combined system of open and closed string field theory. At
genus zero, it is that of an open-closed homotopy algebra which is a realization of an
L ∞ -morphism that maps closed string states to open string vertices. This map turns
out to be a quasi-isomorphism, that is, there is an isomorphism between physical
closed string states and classically consistent open string field theories.
If we include closed string loops, the elementary geometric vertices are realized
by genus g Riemann surfaces with punctures. The quantum master equation then
involves the BV operator Δ. This originates from the corresponding operation acting
on a geometric vertex, which increases the genus of Σ by one by glueing two
punctures with a zero length propagator and thus comes with a power of ¯
h in the
BV equation. If the algebraic vertices are interpreted as functions on the (graded
symmetric) tensor products of the physical Hilbert space (denoted by V above) then
Δ can be implemented as the inverse of the symplectic form ω on V . The algebra
realized by these functions subject to the BV master equation is the loop homotopy
algebra (or quantum L ∞ -algebra). Since closed string field theory is perturbatively
consistent, this is the algebraic structure of the complete quantum field theory of
closed strings.
The quantum open-closed homotopy algebra has an analogous physical interpretation as the classical (genus zero) one. For this the natural framework is that of
IBL ∞ -algebras. The quantum open-closed vertices realize an IBL ∞ -morphism that
maps closed string states to open string vertices of higher genus. We will discuss all
this in the present lecture notes from both mathematics and physics points of view.
I Physics Preliminaries
V to each puncture. This vector space could be the deRham complex of the target
manifold, as is the case for the topological string, or the quantum mechanical Hilbert
spaces of physical degrees of freedom such as photons or gravitons. The geometric
BV master equation is then transferred to V by means of a two-dimensional
conformal field theory defined on the geometric vertices. This conformal field theory
furthermore endows V with an inner product, more precisely, with a symplectic
form. This provides the data needed to define a BV action on V , that is a field
theory action of closed strings.
It turns out that, at genus zero, the vertices of this BV action satisfy the axioms
of a cyclic L ∞ -algebra, which is a differential graded Lie algebra up to homotopy.
Thus, we find that the mathematical structure of classical closed string field theory
is simply that of a cyclic L ∞ -algebra supported by the vector space V of closed
string states together with a symplectic form. Similarly, at genus zero, any consistent
covariant open string field theory carries a structure of a cyclic A ∞ -algebra, that is,
a differential graded associative algebra up to homotopy.
We already mentioned above that open string field theory is not consistent at the
quantum level without inclusion of closed strings. This raises the question about the
algebraic structure of the combined system of open and closed string field theory. At
genus zero, it is that of an open-closed homotopy algebra which is a realization of an
L ∞ -morphism that maps closed string states to open string vertices. This map turns
out to be a quasi-isomorphism, that is, there is an isomorphism between physical
closed string states and classically consistent open string field theories.
If we include closed string loops, the elementary geometric vertices are realized
by genus g Riemann surfaces with punctures. The quantum master equation then
involves the BV operator Δ. This originates from the corresponding operation acting
on a geometric vertex, which increases the genus of Σ by one by glueing two
punctures with a zero length propagator and thus comes with a power of ¯
h in the
BV equation. If the algebraic vertices are interpreted as functions on the (graded
symmetric) tensor products of the physical Hilbert space (denoted by V above) then
Δ can be implemented as the inverse of the symplectic form ω on V . The algebra
realized by these functions subject to the BV master equation is the loop homotopy
algebra (or quantum L ∞ -algebra). Since closed string field theory is perturbatively
consistent, this is the algebraic structure of the complete quantum field theory of
closed strings.
The quantum open-closed homotopy algebra has an analogous physical interpretation as the classical (genus zero) one. For this the natural framework is that of
IBL ∞ -algebras. The quantum open-closed vertices realize an IBL ∞ -morphism that
maps closed string states to open string vertices of higher genus. We will discuss all
this in the present lecture notes from both mathematics and physics points of view.