Next, we construct the supersymmetric world volume action of the M-theory 5-brane in a flat eleven-dimensional background. One of the major obstacles was that the world volume action of the M-theory 5-brane was not available. In Chapter 7, we present six-dimensional world volume action describing the dynamics of the M-theory five-brane in a flat eleven-dimensional space-time background[146].
Finally, in Chapter 8, double super-Dp brane actions are constructed by performing a duality transformation of the U(l) gauge field with world volume[118].
Chapter 2 Prologue on String Dualities
- Overview of String Dualities
- Worldsheet Properties of Closed String
- Open String, T-dualities and D-branes
- Superstring
- Open Superstring
- Closed Superstring
- D-branes in Type II Theory
- Orientifolds
- F-theory
- Super D-brane and Super M-brane
The only change is that the zero state spectrum in the new variable is the a'/R theory. One can further argue that T-duality is an exact symmetry of the closed string theory[4]. Note that the motion of the D-brane corresponds to the Wilson line in the underlying gauge theory.
The world volume theory of the M-5 brane is a six-dimensional theory whose massless content is N =2 tensor multiples.
Gravitational Anomalies and the Orientifold Group
A gravitational multiplet contains 5 self-dual two-forms, while tensor multiplets contain one anti-self-dual two-form each. Type IIB theory on this orbifold has 5 self-dual and 5 anti-self-dual two-forms coming from the untwisted sector. In a twisted sector, there are 16 anti-self-dual forms from 16 fixed points of R.
Thus, of the anti-self-dual two-forms coming from the twisted sector, 8 are even below S, and 8 are odd.
Open String Sector
- Tadpoles
- Gauge Group and Spectrum
- Anomaly Cancellation
The Klein bottle contains contributions from both the untwisted sector and the sector twisted by R from the original orbifold. Note that the rank of the gauge group is different in the two cases corresponding to two branches of the moduli spaces that are connected. Now the kinetic term for bis of the form A2 which can be thought of as the mass term for the massive gauge field A.
In contrast, the vertex operators for the two tensors B0 and B1 coming from the untwisted sector of the orbifold do not present branch cuts.
Discussion
For example, if y1 and y2 are two fixed points of R that are not related to S, then we can place eight 5-branes on y1 and eight on y2. Now the anti-self-dualistic tensors coming from twisted sectors on both y1 and y2 will be needed for anomaly cancellation. In this theory, the vector multiplets arise from the untwisted sector, while the tensor multiplets arise from the addition of 5-branes from M-theory by reasoning similar to [68, 63].
This is complementary to our construction, where the tensor multiplets arise from the untwisted sector (on smooth K3) and the vector multiplets from the 5-brane addition.
Chapter 4 Strings on Orientifolds
- Introduction
- Some Generalities about Orientifolds
- Orientifolds of Toroidally Compactified Type IIB theory
- An Example in Nine Dimensions
- Type I Theory in Eight Dimensions
- Orientifolds of Type IIB Theory on K3
- General Remarks
- Z 2 0rbifold
- Z3 Orbifold
- Z 4 Orbifold
- Z 6 Orbifold
We can then obtain the orientifold projection (1 + fl/3) /2, where fl is the orientation reversal on the world sheet and f3 is a Z2 involution of the orbifold. The closed string sector of the orientifold is obtained by projecting the spectrum of the original orbifold into states that are invariant under orientifold symmetry. The Klein bottle includes contributions from both the untwisted(U) and the twisted(T) sectors of the original orbifold.
The orbifold symmetry corresponds to an exchange of the two EB factors accompanied by a half-shift on the circle. In the Ramond sector, the null modes WM correspond to the rM matrices of the spacetime Clifford algebra. Recall that the massless representations in D = 6 are labeled by the representations of the small group that is Spin(4) r v SU(2) x SU(2).
Thus, we can immediately read the closed string spectrum from the geometric data of the orientation fold. Considering the orbifold as a limit of a smooth K3, then, except in the case where a is a Z2 rotation, we get a non-trivial action on the cohomology indicated by T. This information is sufficient to determine the spectrum of the to work out orbifold. in the closed string sector.
Of course, the spectrum consists of the closed string sector found in [32] which gives us nr = 8, nH = 12 and the gravity multiplication. One non-trivial check is that the tadpoles of the R-R fields of the twisted sector must now self-cancel for the Klein bottle without any contribution from the open-string sector.
Appendix. Tadpole Calculation
Chapter 5 F-theory
A Note on Orientifold and
Chapter 6 Orientifold and F-theory Duals of CHL Strings
In the strong coupling limit, this configuration is lifted to M-theory compacted on the Mobius band, yielding the 9-dimensional dual of the CHL series. The F-theory dual of the CHL model in 6 dimensions forms a part of the F-theory vacuums in 6 dimensions with N = 2 supersymmetry that has not been explored so far. If the Z2 action used works freely, the adiabatic argument assures us of the duality between the Z2 orbifolds.
The ry8ry9 action preserves all harmonic shapes on the torus, so the supersymmetry is not reduced. From the initial Type I' theory, we obtain the model whose gauge group rank has been reduced by eight. Since there is an assumed duality in 8 dimensions between the F-theory at K3 and the heterotic string theory at T, we expect that one can construct the F-theory dually by working out the Z2-symmetry corresponding to the Z2-symmetry of the CHL. string.
This M-theory orbifold is a special case of the M-theory on (I<3 x T2)/Z2 which is dual to the CHL model in 5 dimensions as argued in [29]. One can consider the models corresponding to other points in the F-theory module space of (I<3 x T2)/Z2. The improved gauge group is the monodromy-invariant part of the apparent local gauge group.
But we have already established the duality between F-theory in (I<3 x T2)/Z2 and the CHL model at a particular point of modulus space, so the above model is necessarily dual to the CHL model in six dimensions. Thus we expect that the theory F in (K3 x T2)/G is dual to the CHL compactification of the heterotic string theory in T4 / G where G and G act the same way on f(2o,4.
Chapter 7 World Volume Theory of l\tI-theory Five-brane
7 .1 Introduction
- Review of the Bosonic Theory
- Formulation Without Manifest Covariance
- The PST Formulation
- Supersymmetrization
- General Coordinate Invariance
- Proof of Kappa Symmetry
- Formulation Without Manifest Covariance
The main one we adopt is based on a formulation in which the general invariance of the coordinates is manifested in only five of the six dimensions. The bosonic part of the five-brane theory, constructed by this method, was recently presented [109]. Another approach to the chiral boson problem uses an infinite number of auxiliary fields.
Besides general coordinate invariance, the other essential symmetry of the world volume theory of any super p-brane is the fermionic symmetry called kappa symmetry. This action corresponds to a partially fixed version of the corresponding action in the PST formulation. When interpreting the 4-form dB'ljJ2dB and the 7-form dB'ljJsde, it should be understood that one of the derivatives must be in the O's direction.
We now have to check whether the general invariance of the coordinates of the bosonic theory in Sect. 3This expression is equal to bµv5, where bµ,op is the covariant extension of the expression given in Eq. The important point is that the Z3 term in 8e H has no counterpart in bosonic theory, so the general coordinate invariance of supersymmetric theory is not an immediate consequence of the corresponding symmetry of bosonic theory.
The rest of the calculation is identical to that for the bosonic theory given in ref. This implies that k(l - 1) is a projection operator, and that half of the components of e can be measured out.
Double-Dimensional Reduction
As is already known, when one of the ten spatial dimensions of M-theory is a small circle of radius R, the theory can be reinterpreted as type IIA string theory in ten dimensions with constant string coupling proportional to R. of M-theory can be then either a five-brane theory or a type-IIA four-brane theory arises depending on whether or not it wraps around the circular dimension. This case is called "two-dimensional reduction", because the dimension of the brane and the dimension of the ambient space-time are reduced by one at the same time.
The first example of this type to be studied was the two-dimensional reduction of M-theory with two branes, which gives the fundamental string of Type IIA [87].). The well-known 4-brane of type IIA string theory is, in fact, a D-brane, which implies that its world volume theory contains an abelian vector gauge field. However, the five-brane theory we have constructed contains an antisymmetric tensor gauge field, which remains one even after the reduction.
However, as we will show elsewhere [118], the D4-brane and 4-brane actions with antisymmetric tensor gauge field obtained below are related to the double world volume. The covariant action for the double brane D4 in ten dimensions can be obtained from the action of the five branes of M-theory by setting. An interesting fact is that after the two-dimensional reduction 1i is no longer invariant with respect to supersymmetry.
After the two-dimensional reduction, the parameter of the induced general coordinate transformation e appears only in quantities (see Eqs. 7.109). The supersymmetry variations of C and H are fully given by the induced general transformation of the o-5 coordinates. Therefore, the supersymmetry of the theory after the two-dimensional reduction is a consequence of both the supersymmetry and the general coordinate invariance of the original 6d theory.
7. 7 Discussion
Chapter 8 Dual D-brane Action
- Introduction
- Dual Born-Infeld Actions
- The Dl-brane
- The D2-brane
- The D3-brane
- The D4-brane
- Discussion
In the case of the D2-brane, we show that the dual action describes the one-dimensional M2-brane of the compact target space. Also, the Bianchi identity of the Maxwell field provides the field equation for field B. The term H /\ b2 in SD will be identified as part of the Wess-Zumino term of the double D-brane.
The induced world volume metric Gµv is the supersymmetric retraction of the lOd string metric 17mn·. The D2 brane action was the first of the super D brane actions to be worked out. Eliminating the U(l) measure field in favor of the dual scalar B, one finds that the dual of the action in eq.
Thus, as expected, we identify the M2 brane action (with a circular 11th dimension) as the dual action of the D2 brane action. As in the case of the Dl brane, a constant shift of C0 is a trivial classical symmetry of the action. Also, we can check the transformation of the dilaton and the axion under the duality transformation.
This dual action of the D4-brane is identical to the action obtained from the two-dimensional reduction of the M5-brane, which was given in Sect. Thus we can conclude that the two-dimensional reduction of the M5-brane with these rescaled variables gives the same action as the double 4-brane action with a constant dilaton in Eq.
Appendix - Duality Transformation of D4-brane
Bibliography
