Schrödinger's Equation for Conformal Symmetry
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Volker Schomerus
IGST 2018, Copenhagen
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
0.1 CFT and Conformal Symmetry
0.1 CFT and Conformal Symmetry
our present knowledge of conformal symmetry is incomplete
CFT is everywhere: 2
nd
order phase transitions, IR dynamics
of many interesting QFTs, AdS/CFT correspondence ….
Understanding of perturbative & non-perturbative dynamics
is based on the study of both local and non-local observables
‘t Hooft, Wilson line, surface,
defect, interface operators …
Analyis and construction of their correlators relies on mathe-
matics of conformal symmetry G = SO(1,d+1) … yet …
weights of SO(1,1) & SO(d)
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
0.2 Conformal Partial Waves
0.2 Conformal Partial Waves
… are the CFT-analogues of plane waves in Fourier theory
e.g. 4-pt fcts
CPW
(u
1
,u
2
)
3J symbol
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
0.2 Conformal Partial Waves
0.2 Conformal Partial Waves
… are the CFT-analogues of plane waves in Fourier theory
e.g. 4-pt fcts
CPW
(u
1
,u
2
)
3J symbol
Defect 2-pt fcts
CPW
What kind of functions are the CPWs ?
CPWs are wave functions of integrable N-particle Schrödinger
problem in coordinate space and in weight/momentum space.
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
0.3 Main Results and Plan
0.3 Main Results and Plan
``Euclidean’’ Heckman-Opdam hypergeometric functions
and degenerations of virtual Koornwinder polynomials
1.
Review.
CPWs and the Calogero-Sutherland potential
2. Extension.
Defects blocks and the N-particle CS model
3. Integrability.
Bi-spectral duality: weights ↔ coordinates
Hyperbolic Calogero-Sutherland
model for BC
N
root system in u
i
[Isachenkov, VS] [Isachenkov,Liendo,Linke,VS] …
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
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= sections of a vector bundle on 2-sided coset space K\G/K
with fiber V
SO(d-2)
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
1.1 Conformal Partial Waves
1.1 Conformal Partial Waves
… are G invariants in TP of 4 principal series representations
2 – dimensional
[cross ratios]
Space of
tensor structures
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
1.2 The Casimir Equation
1.2 The Casimir Equation
m is volume of K x K orbit through u
Scalar CPWs:
[M. Isachenkov, VS, E. Sobko]
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1.2 The Casimir Equation (contnd)
1.2 The Casimir Equation (contnd)
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
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u
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radial coordinates
[Hogervorst,Rychkov]
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
1.3 Calogero-Sutherland Models
1.3 Calogero-Sutherland Models
Integrable multi-particle genera-
lization of Poeschl-Teller model
Associated with non-reduced
root system – here with BC
N
Eigenvalue problem ↔ hypergeometrics
[Heckman,Opdam]
[Calogero 71] [Sutherland 72]
The scattering problem for particles in a Weyl chamber is solved
Harish-Chandra functions: single plane plane waves asymptotics
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| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
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| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
2.1 Conformal Defect Operators
2.1 Conformal Defect Operators
D
0
D
d-1
e.g. D
0
: 2d parameters D
d-1
: d+1 parameters
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
2.2 Conformal Partial Waves
2.2 Conformal Partial Waves
Space of CPWs for two scalar defects
D
p
and
D
q
can be realized as
↔ [Gadde]
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
Scalar
CPW
s:
[Isachenkov,Liendo,Linke,VS]
2.3 The Casimir Equation
2.3 The Casimir Equation
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
2.4 Some Applications
2.4 Some Applications
All defect blocks for any value of N were constructed in terms of
multi-variate hypergeometrics.
For N = 2 we found complete set of relations with 4-point blocks
extending results by
[Liendo,Linke,Isachenkov,VS]
[Billo,Goncalves,Lauria,Meineri] [Liendo,Meneghelli]
We found a Lorentzian inversion formula extending
[Caron-Huot]
Computation of bulk-defect OPE coefficients for large spins.
[Alday et al.] [Caron-Huot][Lemos,Liendo,Meineri,Sarkar]
related with
work in progress
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
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| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
3.1 Dependence on weights/momenta
3.1 Dependence on weights/momenta
Dolan & Osborn noticed that scalar blocks obey shift equations
Eq. (5.1) from hep-th/0309180
[Dolan,Osborn]
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
3.2 Ruijsenaars-Schneider model
3.2 Ruijsenaars-Schneider model
2
nd
order difference eq: rational Ruijsenaars-Schneider model
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
3.3 Hyperbolic RS model
3.3 Hyperbolic RS model
Rational Ruijsenaars-Schneider model possesses integrable
hyperbolic/trigonometric deformation
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Wave functions of this bi-spectrally self-dual RS model are
(virtual) Koornwinder polynomials (functions)
Conformal partial waves obtained by degeneration
| Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018
4 Outlook and Conclusions
4 Outlook and Conclusions
Integrable quantum mechanics provides a new approach to CPWs
that is powerful
is flexible
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[VS,Sobko][Buric,VS,Sobko]
Series expansions, recurrence relations, integral formulas
…
↔ [Cornagliotto,Lemos,VS] …
↔ SUSY gauge theory
Many aspects need to be further developed
Volker Schomerus presented on Schrödinger's equation for conformal symmetry at IGST 2018 in Copenhagen. The talk was based on collaborative work with M. Isachenkov, E. Sobko, P. Liendo, Y. Linke, M. Cornagliotto, and M. and explored the implications of conformal symmetry in quantum mechanics.
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Presentation Transcript
Schrdingers equation for Conformal Symmetry Volker Schomerus IGST 2018, Copenhagen Based on work with M. Isachenkov, E. Sobko, P. Liendo, Y. Linke; M. Cornagliotto, M. Lemos, I. Buric, T. Bargheer
0.1 CFT and Conformal Symmetry CFT is everywhere: 2nd order phase transitions, IR dynamics of many interesting QFTs, AdS/CFT correspondence . Understanding of perturbative & non-perturbative dynamics is based on the study of both local and non-local observables Primary fields ??, (?) t Hooft, Wilson line, surface, defect, interface operators weights of SO(1,1) & SO(d) Analyis and construction of their correlators relies on mathe- matics of conformal symmetry G = SO(1,d+1) yet our present knowledge of conformal symmetry is incomplete | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 2
0.2 Conformal Partial Waves are the CFT-analogues of plane waves in Fourier theory e.g. 4-pt fcts (u1,u2) CPW 3J symbol = ~ Zonal spherical functions | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 3
0.2 Conformal Partial Waves are the CFT-analogues of plane waves in Fourier theory e.g. 4-pt fcts (u1,u2) CPW 3J symbol Defect 2-pt fcts q p CPW ? ? p How do they depend on Conformal cross ratios u What kind of functions are the CPWs ? and parameters of field ? | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 4
0.3 Main Results and Plan CPWs are wave functions of integrable N-particle Schr dinger problem in coordinate space and in weight/momentum space. Hyperbolic Calogero-Sutherland Rational Ruiijsenaars - Schneider model in ?, model for BCN root system in ui [Isachenkov, VS] [Isachenkov,Liendo,Linke,VS] ``Euclidean Heckman-Opdam hypergeometric functions and degenerations of virtual Koornwinder polynomials 1. Review. CPWs and the Calogero-Sutherland potential 2. Extension. Defects blocks and the N-particle CS model 3. Integrability. Bi-spectral duality: weights coordinates | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 5
Review Talks at IGST 2016 [VS], IGST 2017 [Sobko] | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 6
1.1 Conformal Partial Waves are G invariants in TP of 4 principal series representations = sections of a vector bundle on 2-sided coset space K\G/K 2 dimensional [cross ratios] with fiber V SO(d-2) Space of tensor structures Principal series reps induced from fd irrep of K = SO(1,1) x SO(d) on ??, ??, - valued functions on the coset space | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 7
1.2 The Casimir Equation Eigenvalue equation for the quadratic Casimir element C2 of the conformal group G on space ? of conformal partial waves [M. Isachenkov, VS, E. Sobko] m is volume of K x K orbit through u Scalar CPWs: Calogero-Sutherland model = 2 Poeschl-Teller particles with interaction | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 8
1.2 The Casimir Equation (contnd) ??,?in Dolan-Osbon conventions CS eigenfunctions ???,?? [Isachenkov,VS] uiradial coordinates [Hogervorst,Rychkov] [ Dolan, Osborn] | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 9
1.3 Calogero-Sutherland Models [Calogero 71] [Sutherland 72] Integrable multi-particle genera- lization of Poeschl-Teller model Associated with non-reduced root system here with BCN Eigenvalue problem hypergeometrics The scattering problem for particles in a Weyl chamber is solved ??? ?? ? [Heckman,Opdam] ?? for Harish-Chandra functions: single plane plane waves asymptotics Much is known: Poles in space of momenta ?, series representations . | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 10
Extensions Spinning blocks Defect blocks Superconformal blocks Thermal blocks Multi-point blocks | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 11
2.1 Conformal Defect Operators Isometries of a p-dimensional conformal defect form subgroup ?? = ?? ?,? + ? ?? ? ? ? p = 0: isometries of pair of points (dilations, rotations); ??= ? Conformal defect possesses dim?/?? = (p+2)(d-p)parameters D0 Dd-1 e.g. D0: 2d parameters Dd-1: d+1 parameters | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 12
2.2 Conformal Partial Waves Space of CPWs for two scalar defects Dpand Dqcan be realized as as functions on the 2-sided coset space ??\?/?? number Nof ``cross ratios ??? ?? ? /??= ??? ? ??? ?? ??? ??+ ??? ? = ? = ???(? ?,? + ?) [Gadde] ? = ?? ? ? ?? ? ? ? ? ??,? | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 13
2.3 The Casimir Equation Eigenvalue equation for the quadratic Casimir element C2 of the conformal group G on space ? of conformal partial waves m is volume of ?? ?? orbit through ? = (??, ,??) [Isachenkov,Liendo,Linke,VS] Scalar CPWs: ? = ???(? ?,? + ?) | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 14
2.4 Some Applications All defect blocks for any value of N were constructed in terms of multi-variate hypergeometrics. [Liendo,Linke,Isachenkov,VS] For N = 2 we found complete set of relations with 4-point blocks [Billo,Goncalves,Lauria,Meineri] [Liendo,Meneghelli] extending results by We found a Lorentzian inversion formula extending [Caron-Huot] Computation of bulk-defect OPE coefficients for large spins. work in progress related with [Alday et al.] [Caron-Huot][Lemos,Liendo,Meineri,Sarkar] | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 15
Integrability | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 16
3.1 Dependence on weights/momenta Dolan & Osborn noticed that scalar blocks obey shift equations Eq. (5.1) from hep-th/0309180 [Dolan,Osborn] | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 17
3.2 Ruijsenaars-Schneider model After multiplication with some factor c = c(??,??) one obtains 2nd order difference eq: rational Ruijsenaars-Schneider model Comments: This generalizes to wave functions ??? of the BCN Calogero-Sutherland model and hence to defect blocks. note??+??= ??? ?? ? Rational RS contains exponential of ??= ???& is rational in ?? ???& exponential of ?? ? Hyperbolic CS is polynomial in ?? = | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 18
3.3 Hyperbolic RS model Rational Ruijsenaars-Schneider model possesses integrable hyperbolic/trigonometric deformation deformation parameter q ?? on coordinates u Dependence of its wave functions ?? determined by dual hyperbolic Ruijsenaars-Schneider model Casimir differential equation obtained by degeneration q 1 Wave functions of this bi-spectrally self-dual RS model are (virtual) Koornwinder polynomials (functions) Conformal partial waves obtained by degeneration | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 19
4 Outlook and Conclusions Integrable quantum mechanics provides a new approach to CPWs that is powerful by embedding into modern theory of multivariate hypergeometric functions SUSY gauge theory Series expansions, recurrence relations, integral formulas is flexible Applies to conformal defects, spinning correlators superconformal symmetry [VS,Sobko][Buric,VS,Sobko] [Cornagliotto,Lemos,VS] Many aspects need to be further developed | Schroedinger's Equation for Conformal Symmetry | Volker Schomerus, 20.8.2018 Page 20
