Superstring theory is the first promising candidate of quantum gravity, per- haps unified with strong and electroweak interactions. Still, our current under-.
I
XX-PUB-5659 September. 0’)
ON STRING
THEORY AND AXIONIC INSTANTONS*
STRINGS
1991
&
Soo- Jong Rey + Stanford Linear Stanford University,
Accelerator Stanford,
Center CA 94309
ABSTRACT Progress in understanding -
dimensional axionic Superstring
exact superconformal
instantons and axionic strings is reported.
theory is the first promising
haps unified with strong and electroweak standing
is largely based on weak coupling
candidate of quantum
interactions.
physics of superstring
tive understandings:
to name. a few, dynamical
Therefore,
just
constant
problem
in this talk,
topics in superstring tons. Recently,
and spacetime
analysis. On the other
seem to require nonperturbasupersymmetry
singularity
theory to the above direction:
considerable
suitably
under-
breaking,
are such examples.
I will discuss what I find one of the most ‘interesting stringy
progress in nonperturbative
strings has been made. Here, I will restrict dimensions,
gravity, per-
Still, our current
perturbative
hand, most of interesting cosmological
field theories of four-
compactified
solitons and instanaspects of noncritical
myself to critical
to four dimensions
superstrings
in ten-
keeping N = 1 spacetime
supersymmetry. At scales well below that may be described
of compactification,
by a low-energy
supergravity
massless string
theory. It is observed that, in
all known string theories, there arises ubiquitously tiplet:
graviton,
excitations .- _ -.* Work supported
dilaton
and Kalb-Ramond
depends largely
a massless gravity
(axion)
in part by the Department
of Energy, contract
supermul-
fields. The other massless
on the way a specific compactification
7perm&ent address: Center for Theoretical Haven CT 06511 USA
Presented Vancouver,
excitations
scheme is
DE-ACO3-76SF00515
Physics, Sloane Laboratory,
at the Particle and BC, Canada, August
Fields 18-22,
Yale University,
‘91, 1991
New
--
chosen. Since I would like to understand superstrings,
..
I will only keep the gravity
intrinsic multiplet
nonperturbative
aspects of
in the low-energy
whose dy-
namics is described by an effective acti0n.l S eff = ~3!~xdi?[ -R( Q) + (higher J Here, Q2, denotes a general affine connection
orders)].
(1)
with nonzero dilaton
and torsion
Cl,iab = L@ + $=Vbl@ + Hpab,
1.
Axionic
Instanton
After Wick rotation
to Euclidean
tions to self-dual Einstein
spacetime, we find axionic instantons
as solu-
equation2
.- - _
7yyb(Q)
= uz,““b(
This turns out to be equivalent to d@ = f”H solution G,,(x)
i-2). and R,,
(2) = 0. We find N-instanton
as = S,,,
IQ01 - xcJ2'
e-‘($) = 5 .,127+
which is stabilized
N &a H,,,A=~T~pvA~(x
by a conserved topological
An exact superconformal be N = 1 supersymmetric
- xa)D,
(x - x,)4
(3)
charge Q = & &,‘aEH = Z.
field theory of the axionic instanton
5214(2) x u;( 1) = u(2) WZNW
Q denotes level of su(2) WZNW theory. The self-duality
a=1
is found’ to
model of ir = 4. Here,
model and 5 a background
charge of u(l)
equation of Eq.(2) b ecomes an algebraic self-duality 7=*&i-
In this case, the supersymmetric
u(2) WZNW
(4) model can be written
Sw=~{JtT(~~t~~)+~ Jtr(E’dx)3}+i J&(2) ln(det where C = exp($X”
$ ia - X)
-possesses a larger symmetry:
as3
C+det Ct)+(ferm;ons)
(5) E U(2). It can be shown”y3 that the instanton
doubly extended N = 4 SUQ-~(~)
2
x
sur(2)
x
ut( 1)
superconformal u(2) currents,
invariance.
Denoting
we find the generators T = - -& K”K” Go=
K” E $8X0, K s J $ i@ A g out of the of doubly extended
N = 4 SCA to be
+ K . K) - (IE”d%I!” •t %‘q)
•t +K”
-(~“K”+JI.K-~.~r\8)+ld~” & 7
(6)
-
u = &x0. If the background
N = 4 SCA with
bly extended central
charge is c = B
i contributes duality
is satisfied,
symmetry
central the total
axionic instantons
gauge connection
tended (4,4)
KM subalgebra
worldsheet
guarantees
is identified
a difficult
charge $. central
SUQ-i(2)
the original5
x sui(2)
U( 1) background
Therefore,
with
The charge
if the algebraic
self-
charge adds up to 6. Wonderfully,
spin connection
u(1) alstring in
possess doubly
ex-
-1,3 This hidden N = 4 superconformal
a nonrenormalization
of the axionic
expansions. Patching
sectors into a modular
dou-
x u(l).
in type II string and in heterotic
supersymmetry.
all orders in worldsheet perturbation tiholomorphic
produces
N = 4 SCA remains to hold even after we twist
extended
gebra! Therefore,
off, Eq.(6)
= 6 - $. In our model,
an additional
Eq.(4)
the doubly which
charge $ is dropped
invariant
step to analyze even though
partition
significant
instanton
solution
holomorphic function
to
and an-
seems to be
progress has been achieved
recently.sv7 In the axionic
instanton
background,
it was shown2 that
there exist only
two chiral fermion
zero modes (as opposed, e.g., to four nonchiral
in Eguchi-Hanson
instantons)
1 expansions.
to the lowest order in worldsheet
The exact SCFT of axionic instanton
orders as well. The spacetime supersymmetry holomorphic
(1,0) primary
fields. Denoting
3
zero modes perturbation
enables us to prove this to all
is generated by contour integrals of zero modes of N = 4 supercurrents
I -
asG~~JT~,G,~$T~,N=l if they annihilate
..
{G;,Gb,}l0~~=
sp acetime supersymmetry
the Ramond
vacuum
SQb(Lo- &)\O>R=
that G0 annihilate
Since the Ramond
a background
only
it is easy to show that
0, in which contribution
model cancels that from u(1) with imposing
10 >R. First,
is guaranteed
from su(2) WZNW
charge. On the other hand,
(O>R, we have
ground state is GSO projected,
I’ii(O >R=
$10 >R, Eq.(7)
implies that (1 f I’,) 10>R= 0. Therefore, there arises only two chiral fermionic zero modes if and only if the algebraic self-duality this is yet another proof of the aforementioned also entails a very significant metry .
breaking’
low-energy
y = kIJ&
is satisfied. Indeed,
nonrenormalization
implications2
theorem. It
to dynamical
since only when there exist precisely
supersym-
two chiral fermion
zero
modes, gravitino
can acquire nonzero mass, thus breaks local supersymmetry.
2.
Strings
Axionic
It is also possible to find a superconformal that
the axionic
instanton
was a source of quantized
field. That is why compact, instanton.
Furthermore,
rectly interpreted
field theory of axionic strings. Recall
sum
WZNW
integer-valued
model was relevant for the axionic
level Q of SUQ(~) WZNW
model was corfield. Funda-
mental string is an axionic string, thus a source of Kalb-Ramond
electric field.
we expect a relevant
model with metric
signature
ous value, and interpreted exact SCFTs. radial direction
(+ + -).
to be noncompact,
1) x u(l)
and au-~(1,
1) x u(l)i
WZNW
models as
modei twisted
by 4 along
and a compact U( 1) model, while the second a su+( u( 1) model with a background
signature
scribes correct dilaton
(-
+ ++)
and (- - ++)
and Kalb-Ramond
order solution ‘” but aparently
1) WZNW
electric charge. Indeed, for axionic
1,l) WZNW
charge $ which may be
as a wrong sign u(1) model with a background
spacetime
su+(l,
In this case, the level Q takes a continu-
The first consists of a su(1, 1) WZNW
model and a Feign-Fuks -- - have
SCFT
as a Kalb-Ramond
string, we find3vg ~%-~,+(l,
interpreted
magnetic
magnetic
charge of Kalb-Ramond
Therefore,
as quantized
Kalb-Ramond
charge t. Thus, they
respectively.
field backgrounds
The first3 de-
of a known lowest
different global geometry. The second is intimately
4
-
related to the axionic instanton and u+(l)
--+ u:(l)
The total
central
critical
through tra.nsformations”:
suck
3 su+(l,
1)
(The reader is warned that this is not equivalent to ~(1, 1)). charge is ?tot = (3 + 4/Q)
six if an algebraic self-duality
+ (1 - 4/y2).
condition
7 = &fl
This equals to the is satisfied.
In fact,
the doubly extended N = 4 algebra Eq.(5) remains to hold in the axionic string as well, and the axionic strings are spacetime supersymmetric worldsheet
perturbation
is that the manifold . tion with nontrivial
expansions!
Furthermore,
is a solution of (+ $ --)
a generalized connection,
to all orders in the
its spacetime interpretation
signature self-dual Einstein
Therefore,
ths axionic string
constitutes
soliton solution of N = 2 gauged string theoryl’ at critical
My motivations
of studying
(1) Recently,
black hole in critical
was shown to be described by su(l, 1)/u(l)
dimension!
connected to one-dimensional
N = 1 supersymmetry
bosonic string
gauged WZNW
can generalize it to an N = 1 supersymmetric by a manifestly
a
the axionic string are two-fold.
two-dimensional
pears to be intimately
equa-
theory
model,12 which ap-
noncritical
strings.
One
black hole, which is described
su(1, 1)/u(l)
gauged WZNW
model.
If-one choose Landau gauge, the black hole SCFT consists of su(1, 1) WZNW model, negative metric free field theory and (b, c) ghosts. They contribute tral charges ir = (3 + 4/Q), 1 and -2 a SCFT of su-o( 1,l) Furthermore,
x u(1) WZNW
respectively.
symmetry
The first two combines to
model with central charge 2 = 4 + 4/Q.
it can be shown that the WZNW
N = 4 superconformal
model possess a doubly extended
as well! This is almost identical
SCFTs of axionic strings. The N = 4 superconformal related to Harmonic
cen-
superspace and twistor
to the above
theories being intimately
theory, the above unexpected
con-
nections between the 2-D black hole and the axionic strings might shed light on uncovering
hidden VV, symmetry
(2) Even though conceptually gate topological
in c = 1 quantum
gravity.
simple, I found it technically
sectors of N = 1 superstrings.
hard to investi-
The gauged N = 2 string theory
is very simple since string field consists only of one scalar field, the Ktihler potential, yet rich enough to allow soliton sectors. Therefore, technical
questions such as collective
coordinates,
I do expect that many
moduli
space, definitions
of
aspects of string theory are -_-. topological charge etc. and other nonperturbative easier- to investigate and to understand in gauged N = 2 string theory. The author is grateful
to C. Callan, J.A. Harvey, G.T. Horowitz,
5
E. Mar-
tinec, A. Stromingeer
and C. Vafa for useful conversations
This work was supported
in part
by the U.S. DOE
on related subjects.
under
Grant
DE-AC-OS-
76SF00515.
References 1. J. Scherk and J.H. Schwarz, Phys. Lett. 52B
(1974) 347.
2. S.-J. Rey, Phys. Rev. D41 (1991); A xionic String Instantons Energy
Implications,
and L. Clavelli,
in ‘Particle
World
Theory
Scientific
tons, SLAC preprint 4. C.G. Callan,
(August,
and Superstrings’,
Pub. (November,
3. S.-J. Rey, Ezact Superconformal
eds. B. Harms
1989).
Field Theory of Azionic
Strings and Instan-
1991).
J.A. Harvey and A. Strominger,
erotic Instantons
and Their Low-
and Solitons,
EFI-91-03,
Worldsheet
PUPT-1233
Approach
preprint
5. A. Sevrin, W. Troost and A. van Proeyen,
Phys. Lett. 208B
6. J.L.
Properties
Petersen
and -A. Taormina,
N = 4 Superconformal Models’,
EFI-90-61
7. H. Ooguri,
Algebras
preprint
Modular and Their
Connection
to Het-
(1991).
(1988) 447.
of Doubly
Extended
to Rational
Torus
(1990).
S.-J. Rey and C. Vafa, unpublished
8. J. Louis, Status of Supersymmetry
Breaking
work (1991). in String
Theory, in this Pro-
ceeding and references therein. 9. S.-J. Rey, unpublished N = 2 String
(March,
Self-Dual (August,
Cosmic Strings 1991).
HUTP-91/01
preprint
(1990)
1389: Geometry
(1991).
On String Theory and Black Holes, IASSNS-HEP-91/12
1991).
in Gauged
and J.A. Harvey, Phys. Rev. Lett. 63 (1989) 719.
and C. Vafa, Mod. Phys. Lett. A5
N = 2 Strings, 12. E. Witten,
(1990);
Theory, SLAC preprint
10. A. Dabholkar 11. H. Ooguri
work
preprint
of