ygor.geurts@gmail.com
Jun30-08, 05:00 AM
Hi,
I'm confused about the following string theory related isssue. Closed
superstring theory is defined by a non-linear sigma model and has
local N=(1,1) worldsheet supersymmetry. This sigma model corresponds
to a superconformal field theory (SCFT) with local N=(1,1)
supersymmetry. It comprises of a matter and ghost part, with total
vanishing central charge. In particular, the matter part has a central
charge of 15. In addition it turns out that the target spacetime is
ten-dimensional and has local N=(1,1) ten-dimensional supersymmetry.
Now upon CFT compactification (so not just geometrical
compactification) one splits the matter SCFT, which has N=(1,1) local
susy, into two parts: an external part corresponding to the four
dimensions we observe and an internal part, corresponding the the
compactified dimensions. Since the central charge of both should add
up to 15, one takes the external SCFT to have c=6 and the internal one
to have c=9.
Now in many places, including the 'string theory on calabi-yau
manifolds' notes by brian greene, suddenly instead of a local N=(1,1)
c=9 SCFT one considers a N=(2,2) c=9 SCFT for the internal part! Now I
dont understand the why of this (apparently the (2,2) is needed to
ensure four-dimensional local N=2 supersymmetry, but I cant find an
accessible explanation of this). Moreover I wonder if this (2,2) is
local or global susy? And if we want (2,2) why didnt we start with
local (2,2) on the worldsheet in the first place? And what happens
with the original ten-dimensional N=(1,1) susy ? Is there still susy
in the internal part after taking this worldsheet N=(2,2) SCFT? And
does the external SCFT also need to suddenly have N=(2,2) ?
I hope anyone can help me out with the above questions,
Ygor
I'm confused about the following string theory related isssue. Closed
superstring theory is defined by a non-linear sigma model and has
local N=(1,1) worldsheet supersymmetry. This sigma model corresponds
to a superconformal field theory (SCFT) with local N=(1,1)
supersymmetry. It comprises of a matter and ghost part, with total
vanishing central charge. In particular, the matter part has a central
charge of 15. In addition it turns out that the target spacetime is
ten-dimensional and has local N=(1,1) ten-dimensional supersymmetry.
Now upon CFT compactification (so not just geometrical
compactification) one splits the matter SCFT, which has N=(1,1) local
susy, into two parts: an external part corresponding to the four
dimensions we observe and an internal part, corresponding the the
compactified dimensions. Since the central charge of both should add
up to 15, one takes the external SCFT to have c=6 and the internal one
to have c=9.
Now in many places, including the 'string theory on calabi-yau
manifolds' notes by brian greene, suddenly instead of a local N=(1,1)
c=9 SCFT one considers a N=(2,2) c=9 SCFT for the internal part! Now I
dont understand the why of this (apparently the (2,2) is needed to
ensure four-dimensional local N=2 supersymmetry, but I cant find an
accessible explanation of this). Moreover I wonder if this (2,2) is
local or global susy? And if we want (2,2) why didnt we start with
local (2,2) on the worldsheet in the first place? And what happens
with the original ten-dimensional N=(1,1) susy ? Is there still susy
in the internal part after taking this worldsheet N=(2,2) SCFT? And
does the external SCFT also need to suddenly have N=(2,2) ?
I hope anyone can help me out with the above questions,
Ygor