Natural Science Forum / Physics / Research / November 2003

I was fortunate enough to have had Barton Zwiebach as my PhD thesis
supervisor between 1993-1997 at MIT (though he was probably somewhat
less fortunate to have had me as his student!).

Anyway, it was certainly true back then, and judging from his and his
colleagues' recent work, it seems just as true now, that the driving
motivation for his research has been the desire to develop a manifestly
background independent formulation of string theory. Such a formulation
promises to provide a unified description of all string backgrounds
(i.e. the entire string 'theory space') which would make it possible to
ask (and perhaps even answer) questions about the selection of
particular string vacua (such as the one, assuming there is one, which
describes our universe), to greatly simplify our understanding of
string dualities, and in general to give us a more complete (including
non-perturbative) understanding of what string theory really is. In
this note, I would like to explicitly put forward what I believe this
manifest background independent formulation will look like.

[I had actually proposed what the solution back in 1997 (after which I
changed fields - though admittedly string theory has always for me been
a hobby rather than a way of life). But a slightly better understanding
of D-branes in the context of string field theory (I had no clue about
D-branes back then), which really only struck me after reading by
chance the recent review article by Taylor & Zwiebach, and subsequently
re-reading Zwiebach's older paper on open-closed string field theory,
have compelled me to update and slightly revise my earlier thoughts on
the matter:

http://arxiv.org/abs/hep-th/0311017
http://arxiv.org/abs/hep-th/9705241
http://arxiv.org/abs/hep-th/9709215
]

Zwiebach not only had the tremendous insight to realise that this
ultimate goal may best be achieved by combining the language of the
powerful Batalin-Vilkovisky quantisation method with the operator
formalism of conformal field theory, but also the patience and
perseverance to turn this into a fully functional formulation of
(bosonic) string field theory:

http://arxiv.org/abs/hep-th/9206084

The fact that the BV formalism has been applied (and indeed required)
to its full extent has meant that any result that results which are
developed in the context of string theory will in general have
consequences for field theory as a whole. The fact that this may well
eventually lead to a unified formalism which can at once describe the
entire space of physical theories is motivation enough that such
research is worthwhile.

By the end of 1997, it was clear that the closed bosonic string action
could be described in terms of the complete set of string vertices,
which are the non-negative-dimensional moduli spaces of punctured
Riemann surfaces as follows:

S = Q + f(B)

where f is a function which maps vertices onto functions of the
fields in the theory Q is the kinetic term, and where B is the sum of
string vertices,

B = Sum_{g,n,m} B^m_{g,n}

where the action satisfies the quantum BV master equation,

{S,S} + 1/2 Delta S = 0

(For open-closed strings, the vertices must be generalised to bordered
Riemann surfaces). To consider string theories around non-conformal
backgrounds, surfaces having both symmetric and antisymmetric punctures
were introduced. An important point to note is that the string vertices
themselves are completely background independent, and satisfy a set of
non-trivial recursion relations which are somewhat analogous to the
master equation for the action S, and which in fact guarantee that
the action satisfies the master equation:

partial B = V'_{0,3} + T^2_{0,1} + KB - IB - 1/2 {B,B} - Delta B

Here, partial K, B, and I are operators acting on surfaces (the
antibracket {B,B} and the Delta are standard BV operators which
sew together punctures on surfaces), and V' and T are special
surfaces (refer to Zwiebach's earlier papers for more details):

http://arxiv.org/abs/hep-th/9606153 and references therein

Note that the dimension of the moduli space of a decorated Riemann
surface of genus g with n symmetric punctures and m antisymmetric
punctures is 6g-6+2n+3m. Thus, the vertices appearing in the string
action were those with (i) g > 0 (note that the g=1, n=m=0,
corresponding to the one-loop vacuum graph is a special case having
dimension 2 rather than 0), (ii) g = 0, 2n+3m>5 (with suitable
generalisations for the open-closed theory).

It turned out that this was not the end of the story.  Zwiebach showed
that certain moduli spaces of apparently negative dimension, despite
not contributing to the action, could be interpreted as operators which
acted on other vertices (via antibracket sewing) - for example, the
sphere with one symmetric and one antisymmetric puncture (with g=0 and
n=m=1), on sewing to a punctured surface changes an symmetric puncture
into an antisymmetric one. During my PhD research, I was able to extend
these results to identify the remaining operators and special surfaces
(partial, V', T, K, I and Q) with some of the remaining
'negative-dimensional' vertices. A remarkable simplification resulted
where the theory now took the following form:

S = f(B)

B = Sum_{g,n,m} B^m_{g,n}

{B,B} + 1/2 Delta B = 0

where the summation now includes all vertices except the unpunctured
sphere B^0_{0,0} and the once-punctured sphere B^0_{0,1}.
Remarkably, the vertices themselves now satisfy a full quantum BV
master equation of their own, guaranteeing that the action satisfies
the same equation. Note that this strongly suggests that the BV master
equation (and hence BV quantisation) has a significance at the level of
Feynman diagrams in field theory, even when the actual fields (i.e. the
specific background) has not been specified.

http://arxiv.org/abs/hep-th/9706128
http://arxiv.org/abs/hep-th/9709126

In this form, the only background dependency which remains in the
theory is in the string field | Psi >, and a slight mysterious ghost
number three state |F> associated with the choice of string
background (and which therefore presumably also determines the string
field), which together can be used to define the mapping f from the
string vertices B to the action S.

Now, Zwiebach's work on string theory around non-conformal backgrounds
suggested strongly that |F>, and hence the choice of string
background should be associated with the once-punctured sphere
B^0_{0,1} (the possible forms of |F> are stated explicitly in his
OCSFT paper). So in my final 'paper' (if it can even be called that -
it was more a bunch of ideas thrown together before graduating), I
speculated that this association was correct. No harm would be done to
the action as the term corresponds to the tree-level tadpole graph
which is taken to vanish anyway. I am still fairly certain that this
association is correct.

http://arxiv.org/abs/hep-th/9709215

At the time I had also made the speculation that the sewing of this
once-punctured sphere to a string vertex was equivalent to applying the
BV Delta operator to that vertex (I recall my supervisor being
rather upset by this postulated association - though admittedly he was
upset by some of the earlier other associations I had made too). I
finally added the unpunctured sphere B^0_{0,0} to complete the full
set of string vertices. This also seemed harmless as the vertex could
at most contribute a harmless constant to the action, and having no
punctures, could not be sewn to other vertices. The result was the
entire theory could now be summarised by the *classical* master
equation for the sum of string vertices:

{B,B}=0

where B is summed over all vertices with g,n,m >= 0. These formed a
manifest background-independent basis from which the action could in
principle be derived. With hindsight, I still believe that this is a
correct form for the manifest background independent formulation of the
theory. However, I think the interpretations I proposed (as mentioned
in the previous paragraph), need to be revised as follows:
It was only after re-reading Zwiebach & Taylor's recent papers that I
realised that the closed string tadpole graph was actually represented
by a closed string being absorbed into a D-brane with a contribution to
the action of the form < B | Psi >, where B is the surface state
corresponding to the D-brane. I had previously interpreted the
unpunctured half of the once-punctured sphere as a semi-infinite tube
which had to end on another puncture, thus adding a handle to the
surface to which the once-punctured sphere was sewn (i.e. the action of
the BV Delta sewing operator). It now appears that this
interpretation was not correct - rather the sewing operation should
presumably be related to the sewing of a D-brane to a puncture on the
target surface. It seems that this should have the effect of simply
removing the sewn puncture from the surface, though no such operator
seems to appear in the operator formalism of the theory to date.

Ideally it will be found/can be postulated that (i) the tadpole term in
the action vanishes, and (ii) the sewing of the once-punctured sphere
annihilates the sewn surface, so that we can simply include and then
just forget about this term.

Turning my attention to the unpunctured sphere again, I noted that the
minimal area metric condition basically turns it into a length of tube
- presumably ending on D-branes if we are to take a lesson from the
tadpole term. The contribution to the action should probably be
something like <B|B> (with possibly some ghostly insertions) which is
field independent, and hence can only contribute a harmless constant to
the action. Now although the unpunctured sphere cannot naively be sewn
as it contains no punctures, its interpretation as a tube with D-branes
sitting on both ends, and given the D-brane's known predilection for
gobbling up closed strings, makes it irresistable to associate the
sewing of the unpunctured sphere with the action of the Delta
operator - the D-branes on each end each absorb a puncture (i.e. closed
string) on the target surface, and thereby attach both ends of the tube
to the surface and increasing the genus by one.

And as Forrest Gump says, "...and that's all I have to say about that".
I am sure Barton and colleagues will fill in the details in due course.

Best wishes,

Sabbir Rahman.

[Moderator's note: there were a lot of dollar signs and backslashes in
this post.  Since this newsgroup is not read by computers, we
discourage use of such LaTeX-isms, and I have deleted them, while
keeping the notations for superscript and subscript, which are
actually useful. - jb]