Publications

  1. Inclusions and positive cones of von Neumann algebras,
    J. Operator Theory Vol. 64, Issue 2 (2010), 435-452. published final version in JOT
    arXiv:0801.4259 pdf file
    We consider cones in a Hilbert space associated to two von Neumann algebras and determine when one algebra is included in the other. If a cone is assocated to a von Neumann algebra, the Jordan structure is naturally recovered from it and we can characterize projections of the given von Neumann algebra with the structure in some special situations.
  2. Representation theory of the stabilizer subgroup of the point at infinity in Diff(S1),
    Internat. J. Math. Vol. 21, No. 10 (2010), 1297–1335. published final version in IJM
    arXiv:0905.0875, pdf file,
    The group Diff(S1) of the orientation preserving diffeomorphisms of the circle S1 plays an important role in conformal field theory. We consider a subgroup B0 of Diff(S1) whose elements stabilize ``the point at infinity''. This subgroup is of interest for the actual physical theory living on the punctured circle, or the real line.
    We investigate the unique central extension K of the Lie algebra of that group. We determine the first and second cohomologies, its ideal structure and the automorphism group. We define a generalization of Verma modules and determine when these representations are irreducible. Its endomorphism semigroup is investigated and some unitary representations of the group which do not extend to Diff(S1) are constructed.
  3. Ground state representations of loop algebras,
    Ann. Henri Poincaré Vol. 12, No. 4 (2011), 805-827. published final version in AHP
    arXiv:1005.0270 pdf file
    Let g be a simple Lie algebra, Lg be the loop algebra of g. Fixing a point in S1 and identifying the real line with the punctured circle, we consider the subalgebra Sg of Lg of rapidly decreasing elements on R. We classify the translation-invariant 2-cocycles on Sg. We show that the ground state representation of Sg is unique for each cocycle. These ground states correspond precisely to the vacuum representations of Lg.
  4. (with W. Dybalski) Asymptotic completeness in a class of massless relativistic quantum field theories,
    Commun. Math. Phys. Vol. 305, No. 2 (2011), 427-440. published final version in CMP
    arXiv:1006.5430 pdf file
    This paper presents the first examples of massless relativistic quantum field theories which are interacting and asymptotically complete. These two-dimensional models are obtained by an application of a deformation procedure, introduced recently by Grosse and Lechner, to chiral conformal quantum field theories. The resulting models may not be strictly local, but they contain observables localized in spacelike wedges.
  5. (with P. Camassa, R. Longo and M. Weiner) Thermal States in Conformal QFT. I,
    to appear in Commun. Math. Phys.
    arXiv:1101.2865 pdf file
    We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A of von Neumann algebras on R. In this first part, we focus on completely rational net A. Our main result here states that, if A is completely rational, there exists exactly one locally normal KMS state φ. Moreover, φ is canonically constructed by a geometric procedure. A crucial role is played by the analysis of the "thermal completion net" associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.
  6. (with W. Dybalski) Infraparticles with superselected direction of motion in two-dimensional conformal field theory,
    to appear in Commun. Math. Phys.
    arXiv:1101.5700 pdf file
    Particle aspects of two-dimensional conformal field theories are investigated, using methods from algebraic quantum field theory. The results include asymptotic completeness in terms of (counterparts of) Wigner particles in any vacuum representation and the existence of (counterparts of) infraparticles in any charged representation of a given chiral conformal field theory. Moreover, an interesting interplay between infraparticle's direction of motion and the superselection structure is demonstrated in a large class of examples. This phenomenon resembles electron's momentum superselection expected in quantum electrodynamics.
  7. Noninteraction of waves in two-dimensional conformal field theory,
    to appear in Commun. Math. Phys.
    arXiv:1107.2662 pdf file
    In higher dimensional quantum field theory, irreducible representations of the Poincaré group are associated with particles. Their counterpart in two-dimensional massless models are "waves" introduced by Buchholz. In this paper we show that waves do not interact in two-dimensional Möbius covariant theories and in- and out-asymptotic fields coincide. We identify the set of the collision states of waves with the subspace generated by the chiral components of the Möbius covariant net from the vacuum. It is also shown that Bisognano-Wichmann property, dilation covariance and asymptotic completeness imply Möbius symmetry.
    Under natural assumptions, we observe that asymptotic fields in Poincaré covariant theory are conditional expectations between appropriate algebras. We show that a two-dimensional massless theory is asymptotically complete and noninteracting if and only if it is a chiral Möbius covariant theory.
  8. Construction of wedge-local nets of observables through Longo-Witten endomorphisms,
    to appear in Commun. Math. Phys.
    arXiv:1107.2629 pdf file
    A convenient framework to treat massless two-dimensional scattering theories has been established by Buchholz. In this framework, we show that the asymptotic algebra and the scattering matrix completely characterize the given theory under asymptotic completeness and standard assumptions.
    Then we obtain several families of interacting wedge-local nets by a purely von Neumann algebraic procedure. One particular case of them coincides with the deformation of chiral CFT by Buchholz-Lechner-Summers. In another case, we manage to determine completely the strictly local elements. Finally, using Longo-Witten endomorphisms on the U(1)-current net and the free fermion net, a large family of wedge-local nets is constructed.
  9. (with P. Camassa, R. Longo and M. Weiner) Thermal States in Conformal QFT. II,
    arXiv:1109.2064 pdf file
    We continue the analysis of the set of locally normal KMS states w.r.t. the translation group for a local conformal net A of von Neumann algebras on the real line. In the first part we have proved the uniqueness of KMS state on every completely rational net. In this second part, we exhibit several (non-rational) conformal nets which admit continuously many primary KMS states. We give a complete classification of the KMS states on the U(1)-current net and on the Virasoro net Vir1 with the central charge c=1, whilst for the Virasoro net Virc with c>1 we exhibit a (possibly incomplete) list of continuously many primary KMS states. To this end, we provide a variation of the Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework: if there is an inclusion of split nets A in B and A is the fixed point of B w.r.t. a compact gauge group, then any locally normal, primary KMS state on A extends to a locally normal, primary state on B, KMS w.r.t. a perturbed translation. Concerning the non-local case, we show that the free Fermi model admits a unique KMS state.

Ph.D. Thesis

Low dimensional quantum field theory and operator algebras, pdf file, October 2011. Supervisor: Roberto Longo

Talks

  1. A new construction of causal nets of operator algebras (after Lechner),
    at the Tokyo operator algebras seminar, University of Tokyo, June 2007.
  2. An introduction to Algebraic QFT,
    at the Functional analysis junior, Kyoto seminar house, September 2007.
  3. Positive cones and half-sided modular inclusions,
    at the RIMS conference "Research on operator algebras and mathematical physics", Kyoto University, RIMS, January 2008.
  4. Another analogue of the Borel-Weil theory on loop groups,
    at the Tokyo operator algebras seminar, University of Tokyo, June 2008.
  5. Representations of the stabilizer subgroup at the point of infinity in Diff(S1), slide
    at the conference "Noncommutative Geometry and Quantum Physics", Vietri sul Mare, August 2009.
  6. Symmetric representations of Diff(R),
    at the Tokyo operator algebras seminar, University of Tokyo, December 2009.
  7. Covariant representations of Diff(R), slide
    at the workshop "Foundations and Constructive aspects of QFT", University of Goettingen, January 2010.
  8. KMS states on conformal nets, slide
    at the mathematical phyics group seminar, University of Goettingen, August 2010.
  9. KMS states on conformal nets,
    at the workshop "Foundations and Constructive aspects of QFT", University of Leipzig, November 2010.
  10. Thermal states in conformal CFT (invited), slide
    at the EU-NCG 4th Annual Meeting, Institute of Mathematics of the Romanian Academy, Bucharest, April 2011.
  11. Deformation of chiral conformal field theory,
    at the operator algebra seminar, University of Rome "Tor Vergata", May 2011.
  12. Deformation of chiral conformal field theory, slide
    at GDRE GREFI-GENCO meeting, Istitut Henri Poincaré, Paris, May 2011.
  13. Deformation of chiral CFT through Longo-Witten endomorphisms,
    at the workshop "Foundations and Constructive aspects of QFT", University of Goettingen, July 2011.
  14. Construction of wedge-local nets of observables through Longo-Witten endomorphisms, slide
    at the workshop "Modern Trends in AQFT", University of Pavia, September 2011.
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