Based on [S. Sasaki, MY arXiv:1608.01436] Introduction
In string theory, extended objects like D-branes have been studied in various contexts. They are BPS solutions in supergravity theories and sometimes called as geometric branes. In addition to these branes, there are other extended objects known as non-geometric branes or . [ J. de Boer, M. Shigemori, Phys. Rev. Lett. 104 (2010), Phys. Rept. 532 (2013) ] It is obtained by U-duality transformations of geometric branes. It has non-trivial monodromy which is due to the U-duality transformation.
The exotic branes which exist in heterotic string theory are less known It is due to the existence of non-Abelian gauge fields Introduction
α’ Heterotic T-duality
Heterotic exotic branes
Summary 5-brane solutions
Exotic branes(Q-, or R-branes)
When we go around the branes in the transverse two directions, the background geometry of the branes non-trivially changes. This is because that the metric and other background fields are governed by multi-valued functions. 5-brane solutions
The 522-brane in type II theory The 522-brane is obtained by performing the T-duality transformations along the transverse directions to the NS5-brane.
the number of isometries in the transverse directions to the brane.
the power of string coupling constant gs-2.
The 522-brane is the solution which has two isometries and its tension is proportional to gs-2. 5-brane solutions
[C. G. Callan, Jr., J. A. Harvey and A. Strominger, Nucl. Phys. B 359 (1991) , Nucl.Phys. B 367 (1991) 60.] Heterotic NS5-brane solution 5-brane solutions
[C. G. Callan, Jr., J. A. Harvey and A. Strominger, Nucl. Phys. B 359 (1991) , Three kinds of Heterotic NS5-brane Nucl.Phys. B 367 (1991) 60.] 5-brane solutions
[C. G. Callan, Jr., J. A. Harvey and A. Strominger, Nucl. Phys. B 359 (1991) , Nucl.Phys. B 367 (1991) 60.] Heterotic Action:
There are the non-abelian gauge field and R2 term and the H-flux is extended by the Chern-Simons terms. The Heterotic NS5-solutions satisfy the eqm derived from the action. Heterotic T-duality
Reduced Heterotic Action [O. Hohm, A. Sen and B. Zwiebach, JHEP 1502 (2015)] Heterotic T-duality
[E. Bergshoeff, B. Janssen and T. Ortin, Class. Quant. Grav. 13 (1996)] Heterotic Buscher rule [O. Hohm, A. Sen and B. Zwiebach, JHEP 1502 (2015)]
When the standard embedding condition is applied, the heterotic Buscher rule reduces to the ordinal one of type II supergravity. Heterotic T-duality
Heterotic Brane web Heterotic T-duality
Generalized metric in heterotic theory;
y and y’ are isometry directions The generalized metric takes the same form in type II supergravities. The second term in the metric G is characteristic to heterotic theories.
θ : 0 → 2π , Heterotic exotic branes
Symmetric 522-brane:
If we choose the coordinate as The supergravity part of symmetric 522-brane solution is the same with the one in type II theory. The omega is proportional to θ. Since the omega is equal to theta, the background fields in the solution are not single-valued anymore. Additionally, the gauge field exists in heterotic theory… Heterotic exotic branes
Symmetric 522-brane:
The gauge generators satisfy the SU(2) algebra; The modified spin connection calculated by the symmetric 522-geometry does not agree with the gauge fields, however, they are gauge-equivalent. Heterotic exotic branes
Symmetric 522-brane:
The generalized metric reduces to the one in type II theory. Even if there is the non-trivial gauge field, The monodromy structure of symmetric 522-brane is same with the type II 522-brane Heterotic exotic branes
Neutral 522-brane:
It is same as the 522-solution in type II theory. Since the α’-term in the generalized metric can be ignored, the monodromy structure is the same with the symmetric 522-brane case. Heterotic exotic branes
Gauge 522-brane:
Note that the Θ is constant parameter. The bulk gauge field satisfied the self-dual equation. Heterotic exotic branes
Gauge 522-brane: For the gauge 522-brane, the gauge field contributes to the generalized metric:
However, since the Θ is not an angular parameter but a constant parameter,
the monodromy becomes trivial. Summary
Summary
There are the three different half BPS five-brane solutions at α’- order are known. From the solutions, we derived following three 522-brane solutions: Summary
Future work
Type I exotic brane.
The world-volume effective action for the non-geometric branes in heterotic theory. Sigma model analysis of the solution.
The double field theory for heterotic supergravity.
BACK YARD
The smearing procedure ; The symmetric solution is governed by the harmonic function H.
When we arrange the NS5-brane as a periodic array, the periodic array of the symmetric NS5-brane solution is governed by the following harmonic function on R3 S1:
Taking the compactification radius small R4 →0, the sum is approximated by the integral over s. We call this procedure as smearing. The result is BACK YARD
Exotic -brane: supergravuty solution J. de Boer , M. Shigemori, [arXiv:1004.2521 [hep-th]] BACK YARD
Symmetric KKM-brane: BACK YARD
Defect gauge NS5-brane: BACK YARD
Smeared gauge KKM-brane: BACK YARD
The structure of gauge generators ;