38
3 String Theory
so that the consistency condition for S is the algebraic BV equation
{S, S} = 0 .
This is the stringy generalization of (2.16).
Before we return to the geometric BV equation, we would like to touch upon
the quantum corrections to the classical BV equation (3.11). The qualitatively new
feature here is the presence of loops, obtained by connecting two punctures of
the same vertex by a propagator. A generic Riemann surface appearing in this
way will not be a fundamental vertex since it is obtained through composition
of vertices with propagators. In the BV quantization of field theory such loops
generically produce what is usually called ultra-violet (UV) divergences arising
when the length of such a propagator shrinks to zero. These divergences have to
be regularized, for instance, by introducing a cut-off. If the theory in question is
renormalizable, there is a suitable redefinition of the vertices such that the cutoff can be removed. In string theory the situation is different. Indeed, due to the
modular invariance of the world-sheet conformal field theory, the part of the moduli
space corresponding to a very short, collapsed handle, is equivalent, by a modular
transformation, to a region where this loop propagator is long. This feature is at
the origin of the UV-finiteness of string theory. One way of implementing this
feature in the decomposition of ˆ
P g,n is to demand that any Jordan curve on the
punctured Riemann surface, not homotopic to a point, has length bigger or equal
2π. The length is measured with respect to the metric of minimal area for a Riemann
surface of genus g with n-punctures. This data together with the coordinate curves
homotopic to the punctures define the moduli space ˆ
P g,n . Note that, according to
this prescription, the punctures are replaced by stubs of finite length.
One possible obstruction to this decomposition of the moduli space is that
connecting stubs by a propagator of a non-negative length does not cover (or
overcovers) the moduli space at genus g + 1 and n − 2 punctures. In this case,
inserting a propagator of zero length introduces an artificial boundary which needs
to be compensated by introducing an elementary vertex of genus g + 1 and n − 2
punctures that covers the missing region in the moduli space such that its boundary
cancels the one introduced by the zero-length propagator. Thus, we have to extend
the geometric BV equation (3.11), introducing an operation Δ that corresponds to
the twist-sewing of two punctures of the same vertex with a zero-length propagator.
We will write the corresponding BV equation as
∂ν g+1,1 + ¯
hΔν g,3 = 0
∂ν g+1,2 + ¯
hΔν g,4 = 0
∂ν g+1,3 + ¯
hΔν g,5 +
1
2
g 1 +g 2 =g+1
{ν g 1 ,2 , ν g 2 ,2 } = 0, etc.,
3 String Theory
so that the consistency condition for S is the algebraic BV equation
{S, S} = 0 .
This is the stringy generalization of (2.16).
Before we return to the geometric BV equation, we would like to touch upon
the quantum corrections to the classical BV equation (3.11). The qualitatively new
feature here is the presence of loops, obtained by connecting two punctures of
the same vertex by a propagator. A generic Riemann surface appearing in this
way will not be a fundamental vertex since it is obtained through composition
of vertices with propagators. In the BV quantization of field theory such loops
generically produce what is usually called ultra-violet (UV) divergences arising
when the length of such a propagator shrinks to zero. These divergences have to
be regularized, for instance, by introducing a cut-off. If the theory in question is
renormalizable, there is a suitable redefinition of the vertices such that the cutoff can be removed. In string theory the situation is different. Indeed, due to the
modular invariance of the world-sheet conformal field theory, the part of the moduli
space corresponding to a very short, collapsed handle, is equivalent, by a modular
transformation, to a region where this loop propagator is long. This feature is at
the origin of the UV-finiteness of string theory. One way of implementing this
feature in the decomposition of ˆ
P g,n is to demand that any Jordan curve on the
punctured Riemann surface, not homotopic to a point, has length bigger or equal
2π. The length is measured with respect to the metric of minimal area for a Riemann
surface of genus g with n-punctures. This data together with the coordinate curves
homotopic to the punctures define the moduli space ˆ
P g,n . Note that, according to
this prescription, the punctures are replaced by stubs of finite length.
One possible obstruction to this decomposition of the moduli space is that
connecting stubs by a propagator of a non-negative length does not cover (or
overcovers) the moduli space at genus g + 1 and n − 2 punctures. In this case,
inserting a propagator of zero length introduces an artificial boundary which needs
to be compensated by introducing an elementary vertex of genus g + 1 and n − 2
punctures that covers the missing region in the moduli space such that its boundary
cancels the one introduced by the zero-length propagator. Thus, we have to extend
the geometric BV equation (3.11), introducing an operation Δ that corresponds to
the twist-sewing of two punctures of the same vertex with a zero-length propagator.
We will write the corresponding BV equation as
∂ν g+1,1 + ¯
hΔν g,3 = 0
∂ν g+1,2 + ¯
hΔν g,4 = 0
∂ν g+1,3 + ¯
hΔν g,5 +
1
2
g 1 +g 2 =g+1
{ν g 1 ,2 , ν g 2 ,2 } = 0, etc.,