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Zero-energy states in supersymmetric matrix models
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-3456-5846
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The work of this Ph.D. thesis in mathematics concerns the problem of determining existence, uniqueness, and structure of zero-energy states in supersymmetric matrix models, which arise from a quantum mechanical description of the physics of relativistic membranes, reduced Yang-Mills gauge theory, and of nonperturbative features of string theory, respectively M-theory. Several new approaches to this problem are introduced and considered in the course of seven scientific papers, including: construction by recursive methods (Papers A and D), deformations and alternative models (Papers B and C), averaging with respect to symmetries (Paper E), and weighted supersymmetry and index theory (Papers F and G). The mathematical tools used and developed for these approaches include Clifford algebras and associated representation theory, structure of supersymmetric quantum mechanics, as well as spectral theory of (matrix-) Schrödinger operators.

Place, publisher, year, edition, pages
Stockholm: KTH , 2010. , 88 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 10:06
Keyword [en]
supermembrane matrix models, supersymmetric quantum mechanics, zero-energy states, Clifford algebra, matrix-valued Schrödinger operator, spectral theory, bounds for negative eigenvalues
National Category
Mathematics Other Physics Topics
Identifiers
URN: urn:nbn:se:kth:diva-12846ISBN: 978-91-7415-662-1 (print)OAI: oai:DiVA.org:kth-12846DiVA: diva2:319330
Public defence
2010-06-04, Sal F3, KTH, Lindstedtsvägen 26, Stockholm, 14:00 (English)
Opponent
Supervisors
Note
QC20100629Available from: 2010-05-19 Created: 2010-05-17 Last updated: 2010-06-29Bibliographically approved
List of papers
1. Dynamical Symmetries in Supersymmetric Matrix
Open this publication in new window or tab >>Dynamical Symmetries in Supersymmetric Matrix
2008 (English)In: Documenta Mathematica, ISSN 1431-0643, E-ISSN 1431-0635, Vol. 13, 103-116 p.Article in journal (Refereed) Published
Abstract [en]

We reveal a dynamical SU(2) symmetry in the asymptotic description of supersymmetric matrix models. We also consider a recursive approach for determining the ground state, and point out some additional properties of the model(s).

Keyword
supersymmetric matrix models, asymptotic analysis, dynamical symmetry, recursive approach
National Category
Other Physics Topics Mathematics
Identifiers
urn:nbn:se:kth:diva-12837 (URN)000257322900002 ()2-s2.0-58149263155 (Scopus ID)
Note

QC20100629

Available from: 2010-05-17 Created: 2010-05-17 Last updated: 2014-10-15Bibliographically approved
2. On the geometry of supersymmetric quantum mechanical systems
Open this publication in new window or tab >>On the geometry of supersymmetric quantum mechanical systems
2008 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 49, no 6, 062101-1-062101-15 p.Article in journal (Refereed) Published
Abstract [en]

We consider some simple examples of supersymmetric quantum mechanical systems and explore their possible geometric interpretation with the help of geometric aspects of real Clifford algebras. This leads to natural extensions of the considered systems to higher dimensions and more complicated potentials.

Place, publisher, year, edition, pages
American Institute of Physics, 2008
Keyword
supersymmetric quantum mechanics, geometric algebra, Clifford algebra, supermembrane toy model, harmonic oscillator
National Category
Other Physics Topics Mathematics
Identifiers
urn:nbn:se:kth:diva-12835 (URN)10.1063/1.2937096 (DOI)000257284400001 ()2-s2.0-46449109389 (Scopus ID)
Note
QC20100629Available from: 2010-05-16 Created: 2010-05-16 Last updated: 2010-06-29Bibliographically approved
3. Octonionic twists for supermembrane matrix models
Open this publication in new window or tab >>Octonionic twists for supermembrane matrix models
2009 (English)In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 10, no 2, 339-356 p.Article in journal (Refereed) Published
Abstract [en]

A certain G(2) x U(1) invariant Hamiltonian arising from the standard membrane matrix model via conjugating any of the supercharges by a cubic, octonionic, exponential is proven to have a spectrum covering the whole half-axis R-vertical bar. The model could be useful in determining a normalizable zero-energy state in the original SO(9) invariant SU(N) matrix model.

Place, publisher, year, edition, pages
Basel: Birkhäuser Verlag, 2009
Keyword
supersymmetric matrix models, octonionic structure, deformation, essential spectrum
National Category
Other Physics Topics Mathematics
Identifiers
urn:nbn:se:kth:diva-12839 (URN)10.1007/s00023-009-0411-2 (DOI)000266371900003 ()2-s2.0-67650251464 (Scopus ID)
Note
QC20100629Available from: 2010-05-17 Created: 2010-05-17 Last updated: 2011-03-18Bibliographically approved
4. Construction of the zero-energy state of SU(2)-matrix theory: Near the origin
Open this publication in new window or tab >>Construction of the zero-energy state of SU(2)-matrix theory: Near the origin
2009 (English)In: Nuclear Physics B, ISSN 0550-3213, Vol. 817, no 3, 155-166 p.Article in journal (Refereed) Published
Abstract [en]

We explicitly construct a (unique) Spin(9) x SU(2) singlet state, phi, involving only the fermionic degrees of freedom of the supersymmetric matrix-model corresponding to reduced 10-dimensional super-Yang Mills theory, respectively supermembranes in 11-dimensional Minkowski space. Any non-singular wave-function annihilated by the 16 supercharges of SU(2) matrix theory must, at the origin (where it is assumed to be non-vanishing) reduce to phi.

Place, publisher, year, edition, pages
Elsevier B.V., 2009
Keyword
supersymmetric matrix models, singlet state, representation theory, recursive approach
National Category
Other Physics Topics Mathematics
Identifiers
urn:nbn:se:kth:diva-12840 (URN)10.1016/j.nuclphysb.2009.01.020 (DOI)000272811100001 ()2-s2.0-67349266205 (Scopus ID)
Funder
Swedish Research Council
Note
QC20100629Available from: 2010-05-17 Created: 2010-05-17 Last updated: 2011-03-18Bibliographically approved
5. Spin(9) average of SU(N) matrix models
Open this publication in new window or tab >>Spin(9) average of SU(N) matrix models
2009 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 50, no 4, 043510-1-043510-7 p.Article in journal (Refereed) Published
Abstract [en]

We apply a method of group averaging to states and operators appearing in (truncations of) the Spin(9)xSU(N) invariant matrix models. We find that there is an exact correspondence between the standard supersymmetric Hamiltonian and the Spin (9) average of a relatively simple lower-dimensional model.

Place, publisher, year, edition, pages
American Institute of Physics, 2009
Keyword
supersymmetric matrix models, zero-energy states, group averaging, harmonic oscillator, matrix-Schrödinger operator
National Category
Other Physics Topics Mathematics
Identifiers
urn:nbn:se:kth:diva-12841 (URN)10.1063/1.3105925 (DOI)000266596800036 ()2-s2.0-65549137712 (Scopus ID)
Note
QC20100629Available from: 2010-05-17 Created: 2010-05-17 Last updated: 2011-03-18Bibliographically approved
6. Weighted Supermembrane Toy Model
Open this publication in new window or tab >>Weighted Supermembrane Toy Model
2010 (English)In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 92, no 2, 125-141 p.Article in journal (Refereed) Published
Abstract [en]

A weighted Hilbert space approach to the study of zero-energy states of supersymmetric matrix models is introduced. Applied to a related but technically simpler model, it is shown that the spectrum of the corresponding weighted Hamiltonian simplifies to become purely discrete for sufficient weights. This follows from a bound for the number of negative eigenvalues of an associated matrix-valued Schrodinger operator.

Place, publisher, year, edition, pages
Springer Netherlands, 2010
Keyword
supersymmetric quantum mechanics, matrix models, zero-energy states, weighted Hilbert spaces, matrix-valued Schrödinger operator, Cwikel-Lieb-Rozenblum inequality
National Category
Other Physics Topics Mathematics
Identifiers
urn:nbn:se:kth:diva-12842 (URN)10.1007/s11005-010-0383-7 (DOI)000277243800003 ()2-s2.0-77952092107 (Scopus ID)
Note
QC20100629Available from: 2010-05-17 Created: 2010-05-17 Last updated: 2011-01-03Bibliographically approved
7. Some spectral bounds for Schrödinger operators with Hardy-type potentials
Open this publication in new window or tab >>Some spectral bounds for Schrödinger operators with Hardy-type potentials
(English)Manuscript (preprint) (Other academic)
Keyword
Schrödinger operator, Hardy inequality, ground state representation, transformation of quadratic form, bounds for negative spectrum
National Category
Mathematical Analysis Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-12843 (URN)
Note
QC20100629Available from: 2010-05-17 Created: 2010-05-17 Last updated: 2010-06-29Bibliographically approved

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