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## Über dieses Buch

Traditionally, Lie theory is a tool to build mathematical models for physical systems. Recently, the trend is towards geometrization of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry which is very helpful in understanding its structure. Geometrization and symmetries are meant in their widest sense, i.e., representation theory, algebraic geometry, infinite-dimensional Lie algebras and groups, superalgebras and supergroups, groups and quantum groups, noncommutative geometry, symmetries of linear and nonlinear PDE, special functions, and others. Furthermore, the necessary tools from functional analysis and number theory are included. This is a big interdisciplinary and interrelated field.

Samples of these fresh trends are presented in this volume, based on contributions from the Workshop "Lie Theory and Its Applications in Physics" held near Varna (Bulgaria) in June 2013.

This book is suitable for a broad audience of mathematicians, mathematical physicists, and theoretical physicists and researchers in the field of Lie Theory.

## Inhaltsverzeichnis

### Revisiting Trace Anomalies in Chiral Theories

This is a report on work in progress about gravitational trace anomalies. We review the problem of trace anomalies in chiral theories in view of the possibility that such anomalies may contain not yet considered CP violating terms. The research consists of various stages. In the first stage we examine chiral theories at one-loop with external gravity and show that a (CP violating) Pontryagin term appears in the trace anomaly in the presence of an unbalance of left and right chirality. However the imaginary coupling of such term implies a breakdown of unitarity, putting a severe constraint on such type of models. In a second stage we consider the compatibility of the presence of the Pontryagin density in the trace anomaly with (local) supersymmetry, coming to an essentially negative conclusion.

Loriano Bonora, Stefano Giaccari, Bruno Lima De Souza

### Complete T-Dualization of a String in a Weakly Curved Background

We apply the generalized Buscher procedure, to a subset of the initial coordinates of the bosonic string moving in the weakly curved background, composed of a constant metric and a linearly coordinate dependent Kalb-Ramond field with the infinitesimal strength. In this way we obtain the partially T-dualized action. Applying the procedure to the rest of the original coordinates we obtain the totally T-dualized action. This derivation allows the investigation of the relations between the Poisson structures of the original, the partially T-dualized and the totally T-dualized theory.

Lj. Davidović, B. Nikolić, B. Sazdović

### Modular Double of the Quantum Group S L q ( 2 , ℝ ) $$SL_{q}(2, \mathbb{R})$$

The term “quantum group”, introduced by V. Drinfeld (Proceedings of ICM-86, Berkeley, vol. 1, p. 798. AMS, Providence, 1987), applies in fact to two dual objects:

q

-deformation of the algebra

??

$$\mathcal{A}$$

of functions on the Lie group and that for the universal enveloping algebra

??

$$\mathcal{U}$$

of the corresponding Lie algebra. See Faddeev [1] for the short history. It is instructive to stress, that the construction of

q

-deformation originates in the theory of the quantum integrable models and conformal field theory [see Faddeev [2]]. In this lecture I plan to survey some new developments on a representative example of the rang 1

SL

(2) case.

### Physical Ageing and New Representations of Some Lie Algebras of Local Scale-Invariance

Indecomposable but reducible representations of several Lie algebras of local scale-transformations, including the Schrödinger and conformal Galilean algebras, and their applications in physical ageing are reviewed. The physical requirement of the decay of co-variant two-point functions for large distances is related to analyticity properties in the coordinates dual to the physical masses or rapidities.

Malte Henkel, Stoimen Stoimenov

### New Type of ?? = 4 $$\mathcal{N} = 4$$ Supersymmetric Mechanics

We give a short account of the superfield approach based on deformed analogs of the standard

??

=

4

,

d

=

1

$$\mathcal{N}=4,d=1$$

superspace and present a few models of supersymmetric quantum mechanics constructed within this new framework. The relevant superspaces are the proper cosets of the supergroup

SU

(2 | 1). As instructive examples we consider the models associated with the worldline

SU

(2 | 1) supermultiplets (

1,4,3

) and (

2,4,2

). An essential ingredient of these models is the mass parameter

m

which deforms the standard

??

=

4

,

d

=

1

$$\mathcal{N}=4,d=1$$

supersymmetry to

SU

(2 | 1) supersymmetry.

Evgeny Ivanov, Stepan Sidorov

### Vector-Valued Covariant Differential Operators for the Möbius Transformation

We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials. These differential operators are symmetry breaking for the pair of Lie groups

(

S

L

(

2

,

)

,

S

L

(

2

,

)

)

$$(SL(2, \mathbb{C}),SL(2, \mathbb{R}))$$

that arise from conformal geometry.

Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner

### Semi-classical Scalar Products in the Generalised SU(2) Model

In these notes we review the field-theoretical approach to the computation of the scalar product of multi-magnon states in the Sutherland limit where the magnon rapidities condense into one or several macroscopic arrays. We formulate a systematic procedure for computing the 1∕

M

expansion of the on-shell/off-shell scalar product of

M

-magnon states in the generalised integrable model with

SU

(2)-invariant rational

R

-matrix. The coefficients of the expansion are obtained as multiple contour integrals in the rapidity plane.

Ivan Kostov

### Weak Poisson Structures on Infinite Dimensional Manifolds and Hamiltonian Actions

We introduce a notion of a weak Poisson structure on a manifold

M

modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra

??

C

(

M

)

$$\mathcal{A}\subseteq C^{\infty }(M)$$

which has to satisfy a non-degeneracy condition (the differentials of elements of

??

$$\mathcal{A}$$

separate tangent vectors) and we postulate the existence of smooth Hamiltonian vector fields. Motivated by applications to Hamiltonian actions, we focus on affine Poisson spaces which include in particular the linear and affine Poisson structures on duals of locally convex Lie algebras. As an interesting byproduct of our approach, we can associate to an invariant symmetric bilinear form

κ

on a Lie algebra

??

$$\mathfrak{g}$$

and a

κ

-skew-symmetric derivation

D

a weak affine Poisson structure on

??

$$\mathfrak{g}$$

itself. This leads naturally to a concept of a Hamiltonian

G

-action on a weak Poisson manifold with a

??

$$\mathfrak{g}$$

-valued momentum map and hence to a generalization of quasi-hamiltonian group actions.

K.-H. Neeb, H. Sahlmann, T. Thiemann

### Bethe Vectors of gl(3)-Invariant Integrable Models, Their Scalar Products and Form Factors

This short note corresponds to a talk given at

Lie Theory and Its Applications in Physics

(Varna, Bulgaria, June 2013) and is based on joint works with S. Belliard, S. Pakuliak and N. Slavnov, see arXiv:1206.4931, arXiv:1207.0956, arXiv:1210.0768, arXiv:1211.3968 and arXiv:1312.1488.

Eric Ragoucy

### Polylogarithms and Multizeta Values in Massless Feynman Amplitudes

The last two decades have seen a remarkable development of analytic methods in the study of Feynman amplitudes in perturbative quantum field theory. The present lecture offers a physicists’ oriented survey of Francis Brown’s work on singlevalued multiple polylogarithms, the associated multizeta periods and their application to Schnetz’s graphical functions and to

x

-space renormalization. To keep the discussion concrete we restrict attention to explicit examples of primitively divergent graphs in a massless scalar QFT.

Ivan Todorov

### Reduction of Couplings in Quantum Field Theories with Applications in Finite Theories and the MSSM

We apply the method of reduction of couplings in a Finite Unified Theory and in the MSSM. The method consists on searching for renormalization group invariant relations among couplings of a renormalizable theory holding to all orders in perturbation theory. It has a remarkable predictive power, since it leads to relations between gauge and Yukawa couplings in the dimensionless sectors and relations involving the trilinear terms and the Yukawa couplings, as well as a sum rule among scalar masses in the soft breaking sector, at the GUT scale. In both the MSSM and the FUT model we predict the masses of the top and bottom quarks and the light Higgs in remarkable agreement with the experiment. Furthermore we also predict the masses of the other Higgses, as well as the supersymmetric spectrum, the latter being in very comfortable agreement with the LHC bounds on supersymmetric particles.

S. Heinemeyer, M. Mondragón, N. Tracas, G. Zoupanos

### A SUSY Double-Well Matrix Model as 2D Type IIA Superstring

We discuss correspondence between a simple supersymmetric matrix model with a double-well potential and two-dimensional type IIA superstrings on a nontrivial Ramond–Ramond background. In particular, we can see direct correspondence between single trace operators in the matrix model and vertex operators in the type IIA theory by computing scattering amplitudes and comparing the results in both sides.

Fumihiko Sugino

### f(R)-Gravity: “Einstein Frame” Lagrangian Formulation, Non-standard Black Holes and QCD-Like Confinement/Deconfinement

We consider

f

(

R

)

=

R

+

R

2

$$f(R) = R + R^{2}$$

gravity interacting with a dilaton and a special non-standard form of nonlinear electrodynamics containing a square-root of ordinary Maxwell Lagrangian. In flat spacetime the latter arises due to a spontaneous breakdown of scale symmetry and produces an effective charge-confining potential. In the

R

+

R

2

gravity case, upon deriving the explicit form of the equivalent

local

“Einstein frame” Lagrangian action, we find several physically relevant features due to the combined effect of the gauge field and gravity nonlinearities such as: appearance of dynamical effective gauge couplings and

confinement-deconfinement transition effect

as functions of the dilaton vacuum expectation value; new mechanism for dynamical generation of cosmological constant; deriving non-standard black hole solutions carrying additional constant vacuum radial electric field and with non-asymptotically flat “hedge-hog”-type spacetime asymptotics.

E. Guendelman, A. Kaganovich, E. Nissimov, S. Pacheva

### The D-Brane Charges of C 3/ ℤ 2 $$\mathbb{Z}_{2}$$

The charges of WZW D-branes form a finite abelian group called the charge group. One approach to finding these groups is to use the conformal field theory description of D-branes, i.e. the charge equation. Using this approach, we work out the charge groups for the non-simply connected group

C

3

2

$$C_{3}/\mathbb{Z}_{2}$$

, which requires knowing the NIM-rep of the underlying conformal field theory.

Elaine Beltaos

### On Robertson Walker Solutions in Noncommutative Gauge Gravity

Robertson–Walker solution is presented in terms of gauge fields in a de Sitter gauge theory of gravity (Chamseddine and Mukhanov, J High Energy Phys 3:033, 2010). For a vanishing torsion analogous (Zet et al., Int J Mod Phys C15(7):1031, 2004) we present the field strength tensor and the scalar analogous of the Ricci scalar. Following the noncommutative generalization (Chamseddine, Phys Lett B504:33, 2001) for the de Sitter gauge theory of gravity we study how the noncommutativity of space-time deform, through noncommutative parameters, the homogeneous isotropic solution of the commutative gauge theory of gravity. The study is realized with special conceived analytical procedures under GRTensorII for Maple that we designed for the specific quantities of the gauge theory of gravity (Babeti (Pretorian), Rom J Phys 57(5–6):785, 2012). Noncommutative deformations are obtained using a star product deformation of space time and the Seiberg–Witten map to express the deformed fields in terms of undeformed ones and noncommutative parameter. We analyze a space-time (Fabi et al., Phys Rev D78:065037, 2008) and a space-space noncommutativity. The gauge fields, the field strength tensor and the noncommutative analogue of the metric tensor, the noncommutative scalar analog to Ricci scalar are followed until second order in noncommutative parameter.

Simona Babeti

### Some Power-Law Cosmological Solutions in Nonlocal Modified Gravity

Modified gravity with nonlocal term

R

1

(

)

R

,

$$R^{-1}\mathcal{F}(\square )R,$$

and without matter, is considered from the cosmological point of view. Equations of motion are derived. Cosmological solutions of the form

a

(

t

)

=

a

0

|

t

t

0

|

α

,

$$a(t) = a_{0}\vert t - t_{0}\vert ^{\alpha },$$

for the FLRW metric and

k

= 0, ±1, are found.

Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, Zoran Rakic

### On Nonlocal Modified Gravity and Cosmology

Despite many nice properties and numerous achievements, general relativity is not a complete theory. One of actual approaches towards more complete theory of gravity is its nonlocal modification. We present here a brief review of nonlocal gravity with its cosmological solutions. In particular, we pay special attention to two nonlocal models and their nonsingular bounce solutions for the cosmic scale factor.

Branko Dragovich

### Vertex Operator Approach to Semi-infiniteSpin Chain: Recent Progress

Vertex operator approach is a powerful method to study exactly solvable models. We review recent progress of vertex operator approach to semi-infinite spin chain. (1) The first progress is a generalization of boundary condition. We study

U

q

(

s

l

̂

(

2

)

)

$$U_{q}(\widehat{sl}(2))$$

spin chain with a triangular boundary, which gives a generalization of diagonal boundary (Baseilhac and Belliard, Nucl Phys B873:550–583, 2013; Baseilhac and Kojima, Nucl Phys B880:378–413, 2014). We give a bosonization of the boundary vacuum state. As an application, we derive a summation formulae of boundary magnetization. (2) The second progress is a generalization of hidden symmetry. We study supersymmetry

U

q

(

s

l

̂

(

M

|

N

)

)

$$U_{q}(\widehat{sl}(M\vert N))$$

spin chain with a diagonal boundary (Kojima, J Math Phys 54(043507):40 pp., 2013). By now we have studied spin chain with a boundary, associated with symmetry

U

q

(

s

l

̂

(

N

)

)

$$U_{q}(\widehat{sl}(N))$$

,

U

q

(

A

2

(

2

)

)

$$U_{q}(A_{2}^{(2)})$$

and

U

q

,

p

(

s

l

̂

(

N

)

)

$$U_{q,p}(\widehat{sl}(N))$$

(Furutsu and Kojima, J Math Phys 41:4413–4436, 2000; Yang and Zhang, Nucl Phys B596:495–512, 2001; Kojima, Int J Mod Phys A26:1973–1989, 2011; Miwa and Weston, Nucl Phys B486:517–545, 1997; Kojima, J Math Phys 52(01351):26 pp., 2011), where bosonizations of vertex operators are realized by “monomial”. However the vertex operator for

U

q

(

s

l

̂

(

M

|

N

)

)

$$U_{q}(\widehat{sl}(M\vert N))$$

is realized by “sum”, a bosonization of boundary vacuum state is realized by “monomial”.

Takeo Kojima

### Thermopower in the Coulomb Blockade Regime for Laughlin Quantum Dots

Using the conformal field theory partition function of a Coulomb-blockaded quantum dot, constructed by two quantum point contacts in a Laughlin quantum Hall bar, we derive the finite-temperature thermodynamic expression for the thermopower in the linear-response regime. The low-temperature results for the thermopower are compared to those for the conductance and their capability to reveal the structure of the single-electron spectrum in the quantum dot is analyzed.

Lachezar S. Georgiev

### On a Pair of Difference Equations for the 4 F 3 Type Orthogonal Polynomials and Related Exactly-Solvable Quantum Systems

We introduce a pair of novel difference equations, whose solutions are expressed in terms of Racah or Wilson polynomials depending on the nature of the finite-difference step. A number of special cases and limit relations are also examined, which allow to introduce similar difference equations for the orthogonal polynomials of the

3

F

2

and

2

F

1

types. It is shown that the introduced equations allow to construct new models of exactly-solvable quantum dynamical systems, such as spin chains with a nearest-neighbour interaction and fermionic quantum oscillator models.

E. I. Jafarov, N. I. Stoilova, J. Van der Jeugt

### Spin Chain Models of Free Fermions

We consider the integrable open spin chain models formulated through the generators of the Hecke algebras which are realized in terms of free fermions.

Č. Burdík, A. P. Isaev, S. O. Krivonos, O. Navrátil

### Group Analysis of Generalized Fifth-Order Korteweg–de Vries Equations with Time-Dependent Coefficients

We perform enhanced Lie symmetry analysis of generalized fifth-order Korteweg–de Vries equations with time-dependent coefficients. The corresponding similarity reductions are classified and some exact solutions are constructed.

Oksana Kuriksha, Severin Pošta, Olena Vaneeva

### A Construction of Generalized Lotka–Volterra Systems Connected with ?? ?? n ( ℂ ) $$\mathfrak{s}\mathfrak{l}_{n}(\mathbb{C})$$

We construct a large family of Hamiltonian systems which are connected with root systems of complex simple Lie algebras. These systems are generalizations of the KM system. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a system to a quadratic Lotka–Volterra system. We classify all possible Lotka–Volterra systems that arise via this algorithm in the

A

n

case.

S. A. Charalambides, P. A. Damianou, C. A. Evripidou

### Systems of First-Order Ordinary Differential Equations Invariant with Respect to Linear Realizations of Two- and Three-Dimensional Lie Algebras

The complete group classification of systems of two first-order ordinary differential equations with respect to point transformations linear in dependent variables is carried out.

Oksana Kuriksha

### On Principal Finite W-Algebras for Certain Orthosymplectic Lie Superalgebras and F(4)

We study finite

W

-algebras associated to even regular (principal) nilpotent elements for basic classical Lie superalgebras. We describe the principal finite

W

-algebras for Lie superalgebras

??

??

??

(

1

|

2

)

,

??

??

??

(

1

|

4

)

,

??

??

??

(

2

|

2

)

$$\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\vert 2),\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\vert 4),\mathfrak{o}\mathfrak{s}\mathfrak{p}(2\vert 2)$$

,

??

??

??

(

3

|

2

)

$$\mathfrak{o}\mathfrak{s}\mathfrak{p}(3\vert 2)$$

, and obtain partial results for the exceptional classical Lie superalgebra

F

(4).

Elena Poletaeva

### Super-de Sitter and Alternative Super-Poincaré Symmetries

It is well-known that de Sitter Lie algebra

??

(

1

,

4

)

$$\mathfrak{o}(1,4)$$

contrary to anti-de Sitter one

??

(

2

,

3

)

$$\mathfrak{o}(2,3)$$

does not have a standard

2

$$\mathbb{Z}_{2}$$

-graded superextension. We show here that the Lie algebra

??

(

1

,

4

)

$$\mathfrak{o}(1,4)$$

has a superextension based on the

2

×

2

$$\mathbb{Z}_{2} \times \mathbb{Z}_{2}$$

-grading. Using the standard contraction procedure for this superextension we obtain an

alternative

super-Poincaré algebra with the

2

×

2

$$\mathbb{Z}_{2} \times \mathbb{Z}_{2}$$

V. N. Tolstoy

### Localizations of U q ( ?? ?? ( 2 ) ) $$U_{q}(\mathfrak{s}\mathfrak{l}(2))$$ and U q ( ?? ?? ?? ( 1 | 2 ) ) $$U_{q}(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\vert 2))$$ Associated with Euclidean and Super Euclidean Algebras

We construct homomorphisms from the Euclidean and super Euclidean algebras,

??

??

??

(

2

)

$$\mathfrak{i}\mathfrak{s}\mathfrak{o}(2)$$

and

U

(

??

??

??

̃

(

2

)

)

$$U(\widetilde{\mathfrak{i}\mathfrak{s}\mathfrak{o}}(2))$$

, onto their images in localizations of

U

q

(

??

??

(

2

)

)

$$U_{q}(\mathfrak{s}\mathfrak{l}(2))$$

and

U

q

(

??

??

??

(

1

|

2

)

)

$$U_{q}(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\vert 2))$$

, respectively, and, conversely, we describe homomorphisms of

U

q

(

??

??

(

2

)

)

$$U_{q}(\mathfrak{s}\mathfrak{l}(2))$$

and

U

q

(

??

??

??

(

1

|

2

)

)

$$U_{q}(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\vert 2))$$

into localizations of

U

(

??

??

??

(

2

)

)

$$U(\mathfrak{i}\mathfrak{s}\mathfrak{o}(2))$$

and

U

(

??

??

??

̃

(

2

)

)

$$U(\widetilde{\mathfrak{i}\mathfrak{s}\mathfrak{o}}(2))$$

. These homomorphisms give results on the relationship between the representation theory of the respective algebras, and, in particular, lead to new representations of

U

q

(

??

??

??

(

1

|

2

)

)

$$U_{q}(\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\vert 2))$$

.

Patrick Moylan

### On the 2D Zero Modes’ Algebra of the SU(n) WZNW Model

A quantum group covariant extension of the chiral parts of the Wess-Zumino-Novikov-Witten (WZNW) model on a compact Lie group

G

gives rise to two matrix algebras with non-commutative entries. These are generated by “chiral zero modes”

a

α

i

,

ā

j

β

$$a_{\alpha }^{i}\,,\bar{a}_{j}^{\beta }$$

which combine, in the 2

D

model, into

Q

j

i

=

a

α

i

a

̄

j

α

$$Q_{j}^{i} = a_{\alpha }^{i} \otimes \bar{ a}_{j}^{\alpha }$$

. The

Q

-operators provide important information about the internal symmetry and the fusion ring. Here we review earlier results about the

SU

(

n

) WZNW

Q

-algebra and its Fock representation for

n

= 2 and make the first steps towards their generalization to

n

≥ 3.

### Breaking s o ( 4 ) $$so(4)$$ Symmetry Without Degeneracy Lift

We consider on

S

R

3

the quantum motion of a scalar particle of mass

m

, perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter,

, the 3D orbital angular momentum, and another, an external scale introducing continuous parameter,

B

. We show that a loss of the geometric hyper-spherical

so

(4) symmetry of the free motion can occur that leaves intact the unperturbed

??

2

$$\mathcal{N}^{2}$$

-fold degeneracy patterns, with

??

=

(

+

n

+

1

)

$$\mathcal{N} = (\ell+n + 1)$$

and

n

denoting the nodes of the wave function. Our point is that although the number of degenerate states for any

??

$$\mathcal{N}$$

matches dimensionality of an irreducible

so

(4) representation space, the corresponding set of wave functions do not transform irreducibly under any

so

(4). Indeed, in expanding the Scarf I wave functions in the basis of properly identified

so

(4) representation functions, we find power series in the perturbation parameter,

B

, where 4D angular momenta

K

[

,

??

1

]

$$K \in [\ell,\mathcal{N}- 1]$$

contribute up to the order

??

2

m

R

2

B

2

??

1

K

$$\mathcal{O}\left (\frac{2mR^{2}B} {\hslash ^{2}} \right )^{\mathcal{N}-1-K}$$

. In this fashion, we work out an explicit example on a symmetry breakdown by external scales that retains the degeneracy. The scheme extends to

so

(

d

+ 2) for any

d

.

M. Kirchbach, A. Pallares Rivera, F. de J. Rosales Aldape

### On the Relation Between an ?? = 1 $$\mathcal{N} = 1$$ Supersymmetric Liouville Field Theory and a Pair of Non-SUSY Liouville Fields

We discuss a relation between the tensor product of the

??

=

1

$$\mathcal{N} = 1$$

super-Liouville field theory with the imaginary free fermion and a certain projected tensor product of the real and the imaginary Liouville field theories. Using techniques of two dimensional, conformal field theory we give a complete proof of their equivalence in the NS sector.

### Multi-Point Virtual Structure Constants and Mirror Computation of CP 2-Model

CP

2

-model. Communications in Number Theory and Physics,

7

(3), 411–468 (2013)) with the same title, which is a joint work with Dr. M. Shimizu.

Masao Jinzenji

### N-Conformal Galilean Group as a Maximal Symmetry Group of Higher-Derivative Free Theory

It is shown that for

N

odd the

N

-conformal Galilean algebra is the algebra of maximal Noether symmetry group, both on the classical and quantum level, of free higher derivative dynamics.

Krzysztof Andrzejewski, Joanna Gonera

### Virasoro Structures in the Twisted Vertex Algebra of the Particle Correspondence of Type C

In this paper we study the existence of Virasoro structures in the twisted vertex algebra describing the particle correspondence of type C. We show that this twisted vertex algebra has at least two distinct Virasoro structures: one with central charge 1, and a second with central charge − 1.

Iana I. Anguelova

### On the Correspondence Between Mirror-Twisted Sectors for N = 2 Supersymmetric Vertex Operator Superalgebras of the Form V ⊗ V and N = 1 Ramond Sectors of V

Using recent results of the author along with Vander Werf, we present the classification and construction of mirror-twisted modules for N = 2 supersymmetric vertex operator superalgebras of the form

V

V

$$V \otimes V$$

for the signed transposition mirror map automorphism. In particular, we show that the category of such mirror-twisted sectors for

V

V

$$V \otimes V$$

is isomorphic to the category of N = 1 Ramond sectors for

V

.

Katrina Barron

### Operadic Bridge Between Renormalization Theory and Vertex Algebras

A construction is presented that provides a correspondence between renormalization groups in models of perturbative massless Quantum Field Theory and models of vertex algebras.

Nikolay M. Nikolov

### Superfields and Vertex Algebras in Four Dimensions

This contribution is short presentation of the work (Nedanovski, D, Superconformal vertex algebras in four dimensions. arXiv:1401.0884v1 [hep-th]) in which the vertex algebra techniques in four dimensions are used for developing a superfield formalism for quantum fields with extended superconformal symmetry.

Dimitar Nedanovski

### Special Reduced Multiplets and Minimal Representations for SO(p,q)

Using our previous results on the systematic construction of invariant differential operators for non-compact semisimple Lie groups we classify the special reduced multiplets and minimal representations in the case of SO(p,q).

### On the Structure of Green’s Ansatz

It is well known that the symmetric group has an important role (via Young tableaux formalism) both in labelling of the representations of the unitary group and in construction of the corresponding basis vectors (in the tensor product of the defining representations). We show that orthogonal group has a very similar role in the context of positive energy representations of

o

s

p

(

1

|

2

n

,

)

$$osp(1\vert 2n, \mathbb{R})$$

. In the language of parabose algebra, we essentially solve, in the parabosonic case, the long standing problem of reducibility of Green’s Ansatz representations.

Igor Salom

### Parafermionic Algebras, Their Modules and Cohomologies

We explore the Fock spaces of the parafermionic algebra introduced by H.S. Green. Each parafermionic Fock space allows for a free minimal resolution by graded modules of the graded two-step nilpotent subalgebra of the parafermionic creation operators. Such a free resolution is constructed with the help of a classical Kostant’s theorem computing Lie algebra cohomologies of the nilpotent subalgebra with values in the parafermionic Fock space. The Euler-Poincaré characteristic of the parafermionic Fock space free resolution yields some interesting identities between Schur polynomials. Finally we briefly comment on parabosonic and general parastatistics Fock spaces.

Todor Popov

### On Non-local Representations of the Ageing Algebra in d ≥ 1 Dimensions

Non-local representations of the ageing algebra for generic dynamical exponents

z

and for any space dimension

d

≥ 1 are constructed. The mechanism for the closure of the Lie algebra is explained. The Lie algebra generators contain higher-order differential operators or the Riesz fractional derivative. Covariant two-time response functions are derived. An application to phase-separation in the conserved spherical model is described.

Stoimen Stoimenov, Malte Henkel

### The Quantum Closet

The equivalence postulate approach to quantum mechanics entails a derivation of quantum mechanics from a fundamental geometrical principle. Underlying the formalism there exists a basic cocycle condition, which is invariant under

D

-dimensional finite Möbius transformations. The invariance of the cocycle condition under finite Möbius transformations implies that space is compact. Additionally, it implies energy quantisation and the undefinability of quantum trajectories. I argue that the decompactification limit coincides with the classical limit. Evidence for the compactness of the universe may exist in the Cosmic Microwave Background Radiation.

Alon E. Faraggi

### Shape-Invariant Orbits and Their Laplace-Runge-Lenz Vectors for a Class of “Double Potentials”

We derive exact

E

= 0 classical solutions for the following class of Hamiltonians with “

double potentials

H

D

:

=

p

2

2

m

+

V

D

(

r

)

,

$$\displaystyle{H_{D}:= \frac{\mathbf{p}^{2}} {2m} + V _{D}(\,r),}$$

where

V

D

:

=

γ

r

2

+

2

μ

+

λ

r

2

+

4

μ

,

0

μ

.

$$\displaystyle{V _{D}:= - \frac{\gamma } {r^{2+2\mu }} + \frac{\lambda } {r^{2+4\mu }}\;,\ \ \ \forall \ \ 0\neq \mu \in \mathbb{R}.}$$

For

μ

=

1

2

$$\mu = -1/2$$

and

μ

=

1

$$\mu = -1$$

the

H

D

yields the Kepler and oscillator systems for

E

≠ 0, respectively. The classical orbits of

H

D

are

shape invariant

for a wide range of

γ

and

λ

, in the sense that each maximum of their orbits

r

(

φ

)

$$r(\varphi )$$

is followed by a minimum after an angular shift of

Δ

φ

=

π

2

μ

$$\varDelta \varphi =\pi /2\mu$$

. We map the LRL vector

M

:

=

(

M

1

,

M

2

)

$$\mathbf{M}:= (M_{1},M_{2})$$

of the Kepler problem to a complex expression

M

μ

$$M_{\mu } \in \mathbb{C}$$

, which is conserved for every

μ

. We use

M

μ

to derive a general expression for the orbit

r

(

φ

,

μ

;

γ

,

λ

)

$$r(\varphi,\mu;\gamma,\lambda )$$

for all

μ

≠ 0. We also contrast the limit of the above orbits as

λ

0

$$\lambda \rightarrow 0$$

with those considered by Daboul and Nieto for the power-law potentials

V

P

:

=

γ

r

2

+

2

μ

$$V _{P}:= -\gamma /r^{2+2\mu }$$

.

Jamil Daboul

### Quantization on Co-adjoint Group Orbits and Second Class Constraints

We make a comparison between two schemes for quantization of dynamical systems with non-trivial phase space—the geometric quantization based on co-adjoint group orbits and second class constraints method. It is shown that the Hilbert space of a system with second class constraints always has, contrary to the geometric quantization, infinite dimension.

Michail Stoilov

### Some Kind of Stabilities and Instabilities of Energies of Maps Between Kähler Manifolds

We treat the variational problem of the energy of the map between two Riemannian manifolds. It is known that any holomorphic or anti-holomorphic map

f

:

M

→

N

between compact Kähler manifolds is stable for the variation

f

t

of

f

with fixed Kähler metrics compatible with the holomorphic structures. Is this also stable for the variation

g

t

of the metric

g

of

M

with fixed volume of

M

and fixed isometric map

f

? In this paper, we show that the answer is no if the dimension of

M

is no less than 3. This paper is a expositary note of Taniguchi and Udagawa (Characterizations of Ricci flat metrics and Lagrangian submanifolds in terms of the variational problem. To appear in Glasgow Math. J).

Tetsuya Taniguchi, Seiichi Udagawa
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