Abstract
We consider some aspects of the curved BPS domain walls and their supersymmetric Lorentz invariant vacua of the four dimensional N = 1 supergravity coupled to a chiral multiplet. In particular, the scalar manifold can be viewed as a two dimensional Kähler-Ricci soliton generating a one-parameter family of Kähler manifolds evolved with respect to a real parameter, τ. This implies that all quantities describing the walls and their vacua indeed evolve with respect to τ. Then, the analysis on the eigenvalues of the first order expansion of BPS equations shows that in general the vacua related to the field theory on a curved background do not always exist. In order to verify their existence in the ultraviolet or infrared regions one has to perform the renormalization group analysis. Finally, we discuss in detail a simple model with a linear superpotential and the Kähler-Ricci soliton considered as the Rosenau solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
For a review see, for example: Aharony, O., Gubser, S.S., Maldacena, J., Ooguri, H., Oz, Y.: Large N Field Theories, String Theory and Gravity. Phys. Rep. 323, 183 (2000), and references therein
Cardoso G.L., Dall’Agata G., Lüst D.: Curved BPS domain walls and RG flow in five dimensions. JHEP 0203, 044 (2002)
Cardoso G.L., Lüst D.: The holographic RG flow in a field theory on a curved background. JHEP 0209, 029 (2002)
Freedman D.Z., Gubser S.S., Pilch K., Warner N.P.: Renormalization group flows from holography-supersymmetry and a c-theorem. Adv. Theor. Math. Phys. 3, 363 (1999)
Cvetic M., Griffies S., Rey S.J.: Static domain walls in N = 1 supergravity. Nucl. Phys. B 381, 301 (1992)
Cvetic M., Soleng H.H.: Supergravity domain walls. Phys. Rep. 282, 159 (1997)
Ceresole A., Dall’Agata G., Girvayets A., Kallosh R., Linde A.: Domain walls, near-BPS bubbles and probabilities in the landscape. Phys. Rev. D74, 086010 (2006)
Gunara B.E., Zen F.P.: Kähler-Ricci Flow, Morse Theory, and Vacuum Structure Deformation of N = 1 Supersymmetry in Four Dimensions. Adv. Theor. Math. Phys 13, 217 (2009)
Gunara B.E., Zen F.P., Arianto : BPS Domain Walls and Vacuum Structure of N = 1 Supergravity Coupled To A Chiral Multiplet. J. Math. Phys. 48, 053505 (2007)
Cao, H.-D.: Existence of gradient Kähler-Ricci solitons. In: Elliptic and parabolic methods p.1 in geometry. In: Chow, B., Gulliver, R., Levy, S., Sullivan, J. (eds.), Wellesley, MA: A K Peters, 1996
Cao H.-D.: Limits of solutions to the Kähler-Ricci flow. J. Diff. Geom. 45, 257 (1997)
Higashijima, K., Itou, E.: Wilsonian Renormalization Group Approach to N = 2 Supersymmetric Sigma Models. Prog. Theor. Phys. 108, 737; A New Class of Conformal Field Theories with Anomalous Dimensions. Prog. Theor. Phys. 109, 751 (1997); Three Dimensional Nonlinear Sigma Models in the Wilsonian Renormalization Method, Prog. Theor. Phys. 110, 563 (2003)
Nitta M.: Conformal Sigma Models with Anomalous Dimensions and Ricci Solitons. Mod. Phys. Lett. A20, 577 (2005)
Cremmer, E., Julia, B., Scherk, J., Ferrara, S., Girardello, L., van Nieuwenhuizen, P.: Spontaneous Symmetry Breaking and Higgs Effect in Supergravity Without Cosmological Constant. Nucl. Phys. B147, 105 (1979); Witten, E., Bagger, J.: Quantization of Newton’s Constant in Certain Supergravity Theories. Phys. Lett. B115, 202 (1982)
Wess J., Bagger J.: Supersymmetry and Supergravity 2nd ed. Princeton University Press, Princeton, NJ (1992)
D’Auria, R., Ferrara, S.: On Fermion Masses, Gradient Flows and Potential in Supersymmetric Theories. JHEP 0105, 034 (2001); Andrianopoli, L., D’Auria, R., Ferrara, S.: Supersymmetry reduction of N-extended supergravities in four dimensions. JHEP 0203, 025 (2002)
Carstea S.A., Visinescu M.: Special solutions for Ricci flow equation in 2D using the linearization approach. Mod. Phys. Lett. A20, 2993 (2005)
Topping, P.: Lectures on Ricci Flow. London Mathematical Society Lecture Note Series 325, Cambridge: Cambridge University Press, 2006, and references therein
Cao, H.-D., Zhu, X.-P.: Hamilton-Perelman’s Proof of the Poincaré and the Geometrization Conjecture. A revised version of the article which originally appeared in Asian J. Math. 10, 165, 2006. available at http://arxiv.org/list/math.DG/0612069, 2006 and references therein
Rosenau P.: Fast and Superfast Diffusion Processes. Phys. Rev. Lett. 74, 1056 (1995)
De Wit, B., Herger, I.: Anti-de Sitter Supersymmetry. Lect. Notes Phys. 541, Berlin-Heidelberg-NewYork: Springer, 2000, pp. 79–100
Ceresole A., Dall’Agata G., Kallosh R., van Proeyen A.: Hypermultiplets, Domain Walls and Supersymmetric Attractor. Phys. Rev. D64, 104006 (2001)
Kuznetsov Y.A.: Elements of Applied Bifurcation Theory. Springer-Verlag, New York (1998)
Chow, B.: Lectures on Ricci Flow, Lectures given at Clay Mathematics Institutes Summer School, Berkeley, 2005, and references therein, available at http://www.claymath.org/programs/summer_school/2005/program.php#riccil-3, 2005
Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions 9th ed. Dover Publications, New York (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. W. Gibbons
Rights and permissions
About this article
Cite this article
Gunara, B.E., Zen, F.P. Deformation of Curved BPS Domain Walls and Supersymmetric Flows on 2d Kähler-Ricci Soliton. Commun. Math. Phys. 287, 849–866 (2009). https://doi.org/10.1007/s00220-009-0744-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0744-1