Rational Conformal Field Theories in Physics
Explore the concepts of Rational Conformal Field Theories (CFT) in physics, focusing on the conformal group, transformations, applications in statistical mechanics, primary fields, and spin SU(2) fusion rules. Learn how CFT plays a crucial role in modeling sub-atomic particles and solving various physical problems efficiently.
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Presentation Transcript
Handle Squashing in Non- Current Algebra Based Rational Conformal Field Theories Anthony Bennett, YSU Dr. Michael Crescimanno, YSU Project ID: 109
Conformal Field Theory (CFT) Overview By definition, a conformal field theory is a quantum field theory that is invariant under the conformal group. ref. [1] QFT Classical FT + Special Relativity + Quantum Mechanics Classical FT Electricity and Magnetism + Gravitation For modeling sub-atomic particles and quasiparticles
Conformal Field Theory (CFT) Overview conformal group is the set of transformations of spacetime that preserve angles (but not necessarily distances). ref. [1] Ref [7]
What Transformations? Scale Transformation Length changed Angle preserved Ref [7]
Transformations Cont. Flat Space-Time Symmetry Poincar transforms EQUATION Lorentz - EQN EQUATION Translation -
What is it good for? Statistical Mechanics Critical Points E-field at the corner of two conductors Transforms into a more easily solvable problem Ref [7]
Rational Conformal Field Theory 2-D and finite number of primary fields. QFT on Euclidean space invariant under local transformations. Primary field - operator annihilated by the lowering operator. ref. [6] i.e. Quantum Harmonic Oscillator Additional Structure State operator correspondence
Spin SU(2) Fusion Rules Matrix representation of the fusion algebras 0 is the identity Representing using the Hilbert Space |0>=(1,0,0)t , |1/2>=(0,1,0)t , |1>=(0,0,1)t
On Conformal Field Theories with Low Number of Primary Fields Dovgard/Gepner Two relations considered Modular Transformations Fusion Rules EQN Connected through Verlinde formula - Symmetric Affine Variety (SAV) Algorithm works out all SAVs with small number of primary fields
New Fusion Rings Tested up to 8 primaries 4 new fusion rings Don t correspond with current algebra based RCFTs Prior to this all were believed to be current algebra based Dovgard/Gepner ended with hope of future developments
Handle Squashing Operators Extending a previous result to these new rings Handle operator (HO) Expansion of the basis K-matrix Handle squashing operator (HSO) Represents a norm, no obvious reason to always exist in the fusion ring Trace(HSO)=1, always rational coefficients Shown that all HSO are a linear combination of the fields ref[5]
HSO + 4 New Fusion Rings Does this form of the HSO, found to hold for all known current algebra based RCFTs, hold for these four fusion rings? A : B : A : B :
Results The HSO is a linear combination of the fusion rings as expected from previous results ref[5] Further suggests these are indeed non-current algebra based RCFTs A new way to find future RCFTs with possibly a less demanding algorithm? Much more to explore in RCFT
References [1] Qualls, J. (2016, May 19) Lectures on conformal field theory. Prepared for submission to Journal of High Energy Physics. Retrieved January 2019, from the arxiv database, https://arxiv.org/abs/1511.04074. [2] Dovagard, R. and Gepner, D. (2009, Mar 11) On conformal field theories with low number of primary fields. Journal of Physics A, 42(30), 304009. Retrieved January 2019, from the arxiv database, https://arxiv.org/abs/0811.1904 [3] Gepner, D. (2007, October 11) Galois Groups in Rational Conformal Field Theory II. The Discriminant. Physics Letters B, 654(3-4), 113- 120. Retrieved January 2019, from the arxiv database, https://arxiv.org/abs/hep-th/0608140
References Cont. [4] Crescimanno, M. (1993, December 15) Handle Operators of Coset Models. Modern Physics Letters A, 8(20) 1877-1889. Retrieved January 2019, from the arxiv database, https://arxiv.org/abs/hep-th/9312134 [5] Crescimanno, M (1993, March 22) Fusion potentials for G and handle squashing. Nuclear Physics B, 393(1-2), 361-376. Retrieved January 2019, from the arxiv database, https://arxiv.org/pdf/hep- th/9110063.pdf [6] (1991) An introduction to rational conformal field theories. Comments Nuclear Particle Physics, 20(1-2), 23-67. March 2019, http://inspirehep.net/record/324385/files/v20-n1+2-p23.pdf
References Cont. [7] Blumenhagen, R., Plauschinn, E. (2009) Basics in Conformal Field Theory: Lecture Notes, 779, 5-86
