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{{Short description|Algebra describing 2D conformal symmetry}} {{Group theory sidebar}} In [[mathematics]], the '''Virasoro algebra''' is a complex [[Lie algebra]] and the unique nontrivial [[Lie algebra extension#Central|central extension]] of the [[Witt algebra]]. It is widely used in [[two-dimensional conformal field theory]] and in [[string theory]]. It is named after [[Miguel Ángel Virasoro (physicist)|Miguel Ángel Virasoro]]. == Structure == The '''Virasoro algebra''' is [[linear span|spanned]] by '''generators''' {{math|''L<sub>n</sub>''}} for {{math|''n'' ∈ ℤ}} and the '''[[central charge]]''' {{mvar|c}}. These generators satisfy <math>[c,L_n]=0</math> and {{Equation box 1 |indent =:: |equation = <math>[L_m, L_n] = (m - n) L_{m+n} + \frac{c}{12} (m^3 - m) \delta_{m+n,0}.</math> |border colour = #0073CF |bgcolor=#F9FFF7}} The factor of <math>\frac{1}{12}</math> is merely a matter of convention. For a derivation of the algebra as the unique central extension of the [[Witt algebra]], see [[Lie algebra extension#Virasoro algebra|derivation of the Virasoro algebra]] or Schottenloher,<ref>{{Cite book |date=2008 |title=A Mathematical Introduction to Conformal Field Theory |url=https://link.springer.com/book/10.1007/978-3-540-68628-6 |series=Lecture Notes in Physics |volume=759 |language=en |doi=10.1007/978-3-540-68628-6 |isbn=978-3-540-68625-5 |issn=0075-8450}}</ref> Thm. 5.1, pp. 79. The Virasoro algebra has a [[Presentation of a group|presentation]] in terms of two generators (e.g. {{mvar|L}}<sub>3</sub> and {{mvar|L}}<sub>−2</sub>) and six relations.<ref>{{Cite journal | doi = 10.1007/BF01218387 | title = A presentation for the Virasoro and super-Virasoro algebras | journal = Communications in Mathematical Physics| volume = 117 | issue = 4 | pages = 595 | year = 1988| last1 = Fairlie | first1 = D. B. | last2 = Nuyts | first2 = J. | last3 = Zachos | first3 = C. K. |bibcode = 1988CMaPh.117..595F | s2cid = 119811901 | url = http://projecteuclid.org/euclid.cmp/1104161819 | url-access = subscription }}</ref><ref>{{Cite journal | doi = 10.1007/BF01221412 | title = Redundancy of conditions for a Virasoro algebra | journal = Communications in Mathematical Physics | volume = 122 | issue = 1 | pages = 171–173 | year = 1989 | last1 = Uretsky | first1 = J. L. |bibcode = 1989CMaPh.122..171U | s2cid = 119887710 }}</ref> The generators <math>L_{n>0}</math> are called annihilation modes, while <math>L_{n<0}</math> are creation modes. A basis of creation generators of the Virasoro algebra's [[universal enveloping algebra]] is the set :<math> \mathcal{L} = \Big\{ L_{-n_1} L_{-n_2} \cdots L_{-n_k}\Big\}_{\begin{array}{l} k\in\mathbb{N} \\ 0 < n_1 \leq n_2 \leq \cdots n_k\end{array}} </math> For <math>L\in \mathcal{L}</math>, let <math>|L|= \sum_{i=1}^k n_i</math>, then <math>[L_0,L] = |L|L</math>. ==Representation theory== In any indecomposable representation of the Virasoro algebra, the central generator <math>c</math> of the algebra takes a constant value, also denoted <math>c</math> and called the representation's central charge. A vector <math>v</math> in a representation of the Virasoro algebra has '''conformal dimension''' (or conformal weight) <math>h</math> if it is an eigenvector of <math>L_0</math> with eigenvalue <math>h</math>: : <math> L_0 v = hv</math> An <math>L_0</math>-eigenvector <math>v</math> is called a '''primary state''' (of dimension <math>h</math>) if it is annihilated by the annihilation modes, : <math> L_{n>0} v = 0</math> ===Highest weight representations=== A [[Weight (representation theory)|highest weight representation]] of the Virasoro algebra is a representation generated by a primary state <math>v</math>. A highest weight representation is spanned by the <math>L_0</math>-eigenstates <math>\{Lv\}_{L\in\mathcal{L}}</math>. The conformal dimension of <math>Lv</math> is <math>h+|L|</math>, where <math>|L|\in\mathbb{N}</math> is called the '''level''' of <math>Lv</math>. Any state whose level is not zero is called a '''descendant state''' of <math>v</math>. For any <math>h,c\in\mathbb{C}</math>, the [[Verma module]] <math>\mathcal V_{c,h}</math> of central charge <math>c</math> and conformal dimension <math>h</math> is the representation whose basis is <math>\{Lv\}_{L\in\mathcal{L}}</math>, for <math>v</math> a primary state of dimension <math>h</math>. The Verma module is the largest possible highest weight representation. The Verma module is indecomposable, and for generic values of <math>h,c\in\mathbb{C}</math> it is also irreducible. When it is reducible, there exist other highest weight representations with these values of <math>h,c\in\mathbb{C}</math>, called '''degenerate representations''', which are quotients of the Verma module. In particular, the unique irreducible highest weight representation with these values of <math>h,c\in\mathbb{C}</math> is the quotient of the Verma module by its maximal submodule. A Verma module is irreducible if and only if it has no singular vectors. ===Singular vectors=== A singular vector or null vector of a highest weight representation is a state that is both descendant and primary. A sufficient condition for the Verma module <math>\mathcal V_{c,h}</math> to have a singular vector is <math>h=h_{r,s}(c)</math> for some <math>r,s\in\mathbb{N}^*</math>, where :<math> h_{r,s}(c) = \frac14\Big( (\beta r - \beta^{-1}s)^2-(\beta-\beta^{-1})^2\Big)\ ,\quad \text{where} \quad c=1-6(\beta-\beta^{-1})^2\ . </math> Then the singular vector has level <math>rs</math> and conformal dimension :<math> h_{r,s}+rs = h_{r,-s} </math> Here are the values of <math>h_{r,s}(c)</math> for <math>rs\leq 4</math>, together with the corresponding singular vectors, written as <math>L_{r,s}v</math> for <math>v</math> the primary state of <math>\mathcal{V}_{c,h_{r,s}(c)}</math>: :<math> \begin{array}{|c|c|l|} \hline r,s & h_{r,s} & L_{r,s} \\ \hline \hline 1,1 & 0 & L_{-1} \\ \hline 2,1 & -\frac12 +\frac{3}{4} \beta^2 & L_{-1}^2 -\beta^2 L_{-2} \\ \hline 1,2 & -\frac12 + \frac{3}{4}\beta^{-2} & L_{-1}^2 -\beta^{-2} L_{-2} \\ \hline 3,1 & -1 +2 \beta^2 &L_{-1}^3 -4\beta^2 L_{-1}L_{-2}+2\beta^2(2\beta^2+1)L_{-3} \\ \hline 1,3 & -1 +2 \beta^{-2} &L_{-1}^3 -4\beta^{-2} L_{-1}L_{-2}+2\beta^{-2}(2\beta^{-2}+1)L_{-3} \\ \hline 4,1 & -\frac32 +\frac{15}{4} \beta^2 & \begin{array}{r} L_{-1}^4 -10\beta^2L_{-1}^2L_{-2} +2\beta^2\left(12\beta^2+5\right) L_{-1}L_{-3} \\ +9\beta^4 L_{-2}^2 -6\beta^2\left(6\beta^4+4\beta^2+1\right)L_{-4} \end{array} \\ \hline 2,2 & \frac34\left(\beta-\beta^{-1}\right)^2 & \begin{array}{l} L_{-1}^4-2\left(\beta^2+\beta^{-2}\right)L_{-1}^2L_{-2} +\left(\beta^2-\beta^{-2}\right)^2 L_{-2}^2 \\ +2\left(1+\left(\beta+\beta^{-1}\right)^2\right) L_{-1}L_{-3} -2\left(\beta+\beta^{-1}\right)^2 L_{-4} \end{array} \\ \hline 1,4 & -\frac32 +\frac{15}{4} \beta^{-2} & \begin{array}{r} L_{-1}^4 -10\beta^{-2}L_{-1}^2L_{-2} +2\beta^{-2}\left(12\beta^{-2}+5\right) L_{-1}L_{-3} \\ +9\beta^{-4} L_{-2}^2 -6\beta^{-2}\left(6\beta^{-4}+4\beta^{-2}+1\right)L_{-4} \end{array} \\ \hline \end{array} </math> Singular vectors for arbitrary <math>r,s\in\mathbb{N}^*</math> may be computed using various algorithms,<ref name="ken92"/><ref name="BYB"/> and their explicit expressions are known.<ref name="wat24"/> If <math>\beta^2\notin\mathbb{Q}</math>, then <math>\mathcal V_{c,h}</math> has a singular vector at level <math>N</math> if and only if <math>h=h_{r,s}(c)</math> with <math>N=rs</math>. If <math>\beta^2\in\mathbb{Q}</math>, there can also exist a singular vector at level <math>N</math> if <math>N= rs + r's'</math> with <math> h=h_{r,s}(c)</math> and <math> h+rs = h_{r',s'}(c)</math>. This singular vector is now a descendant of another singular vector at level <math>rs</math>. The integers <math>r,s</math> that appear in <math>h_{r,s}(c)</math> are called '''Kac indices'''. It can be useful to use non-integer Kac indices for parametrizing the conformal dimensions of Verma modules that do not have singular vectors, for example in the critical [[Random_cluster_model#Two-dimensional_case|random cluster model]]. ===Shapovalov form=== For any <math>c,h\in\mathbb{C}</math>, the involution <math>L_n\mapsto L^*=L_{-n}</math> defines an automorphism of the Virasoro algebra and of its universal enveloping algebra. Then the '''Shapovalov form''' is the symmetric bilinear form on the Verma module <math>\mathcal{V}_{c,h}</math> such that <math>(Lv,L'v)= S_{L,L'}(c,h)</math>, where the numbers <math>S_{L,L'}(c,h)</math> are defined by <math>L^*L'v \underset{|L|=|L'|}{=} S_{L,L'}(c,h)v</math> and <math>S_{L,L'}(c,h)\underset{|L|\neq |L'|}{=}0</math>. The inverse Shapovalov form is relevant to computing [[Virasoro conformal block]]s, and can be determined in terms of singular vectors.<ref name="fqs24"/> The determinant of the Shapovalov form at a given level <math>N</math> is given by the '''Kac determinant formula''',<ref name="ff84"/> : <math>\det \left(S_{L,L'}(c,h)\right)_{\begin{array}{l} L,L'\in\mathcal{L} \\ |L|=|L'|=N\end{array}} = A_N \prod_{1\le r,s\le N} \big(h - h_{r,s}(c)\big)^{p(N-rs)},</math> where <math>p(N)</math> is the [[partition function (number theory)|partition function]], and <math>A_N </math> is a positive constant that does not depend on <math>h</math> or <math>c</math>. ===Hermitian form and unitarity=== If <math>c,h\in\mathbb{R}</math>, a highest weight representation with conformal dimension <math>h</math> has a unique [[Hermitian form]] such that the Hermitian adjoint of <math>L_n</math> is <math>L_n^\dagger = L_{-n}</math> and the norm of the primary state <math>v</math> is one. In the basis <math>(Lv)_{L\in\mathcal{L}}</math>, the Hermitian form on the Verma module <math>\mathcal V_{c,h}</math> has the same matrix as the Shapovalov form <math>S_{L,L'}(c,h)</math>, now interpreted as a [[Gram matrix]]. The representation is called '''unitary''' if that Hermitian form is positive definite. Since any singular vector has zero norm, all unitary highest weight representations are irreducible. An irreducible highest weight representation is unitary if and only if * either <math> c\geq 1</math> with <math>h\geq 0</math>, * or <math>c \in \left\{1 - \frac{6}{m(m + 1)}\right\}_{m=2,3,4,\ldots}=\left\{ 0, \frac12, \frac{7}{10}, \frac45, \frac67, \frac{25}{28}, \ldots\right\}</math> with <math>h\in \left\{h_{r,s}(c)=\frac{\big((m + 1) r - ms\big)^2 - 1}{4m(m + 1)}\right\}_{\begin{array}{l} r=1,2,...,m-1 \\ s=1,2,...,m\end{array}}</math> [[Daniel Friedan]], Zongan Qiu, and [[Stephen Shenker]] showed that these conditions are necessary,<ref name="fqs84"/> and [[Peter Goddard (physicist)|Peter Goddard]], [[Adrian Kent]], and [[David Olive]] used the [[coset construction]] or [[GKO construction]] (identifying unitary representations of the Virasoro algebra within tensor products of unitary representations of affine [[Kac–Moody algebra]]s) to show that they are sufficient.<ref name="gko86"/> ===Characters=== The character of a representation <math>\mathcal{R}</math> of the Virasoro algebra is the function :<math> \chi_\mathcal{R}(q) = \operatorname{Tr}_{\mathcal{R}} q^{L_0-\frac{c}{24}}. </math> The character of the Verma module <math> \mathcal{V}_{c,h}</math> is :<math> \chi_{\mathcal{V}_{c,h}}(q) = \frac{q^{h-\frac{c}{24}}}{\prod_{n=1}^\infty (1-q^n)} = \frac{q^{h-\frac{c-1}{24}}}{\eta(q)}=q^{h-\frac{c}{24}}\left(1+q+2q^2+3q^3+5q^4+\cdots\right), </math> where <math>\eta</math> is the [[Dedekind eta function]]. For any <math>c\in\mathbb{C}</math> and for <math>r,s\in \mathbb{N}^*</math>, the Verma module <math>\mathcal{V}_{c,h_{r,s}}</math> is reducible due to the existence of a singular vector at level <math>rs</math>. This singular vector generates a submodule, which is isomorphic to the Verma module <math>\mathcal{V}_{c,h_{r,s}+rs}</math>. The quotient of <math>\mathcal{V}_{c,h_{r,s}}</math> by this submodule is irreducible if <math>\mathcal{V}_{c,h_{r,s}}</math> does not have other singular vectors, and its character is :<math> \chi_{\mathcal{V}_{c,h_{r,s}}/\mathcal{V}_{c,h_{r,s}+rs}} = \chi_{\mathcal{V}_{c,h_{r,s}}} -\chi_{\mathcal{V}_{c,h_{r,s}+rs}} = (1-q^{rs}) \chi_{\mathcal{V}_{c,h_{r,s}}}. </math> Let <math>c=c_{p,p'}</math> with <math>2\leq p<p'</math> and <math>p,p'</math> coprime, and <math>1\leq r \leq p-1</math> and <math>1\leq s\leq p'-1</math>. (Then <math>(r,s)</math> is in the Kac table of the corresponding [[Minimal model (physics)|minimal model]]). The Verma module <math>\mathcal{V}_{c,h_{r,s}}</math> has infinitely many singular vectors, and is therefore reducible with infinitely many submodules. This Verma module has an irreducible quotient by its largest nontrivial submodule. (The spectrums of minimal models are built from such irreducible representations.) The character of the irreducible quotient is :<math>\begin{align} &\chi_{\mathcal{V}_{c,h_{r,s}}/(\mathcal{V}_{c,h_{r,s}+rs}+\mathcal{V}_{c,h_{r,s}+(p-r)(p'-s)}) } \\ &= \sum_{k\in\mathbb{Z}} \left(\chi_{\mathcal{V}_{c,\frac{1}{4pp'}\left((p'r-ps+2kpp')^2-(p-p')^2\right)}}-\chi_{\mathcal{V}_{c,\frac{1}{4pp'}\left((p'r+ps+2kpp')^2-(p-p')^2\right)}}\right). \end{align} </math> This expression is an infinite sum because the submodules <math>\mathcal{V}_{c,h_{r,s}+rs}</math> and <math>\mathcal{V}_{c,h_{r,s}+(p-r)(p'-s)}</math> have a nontrivial intersection, which is itself a complicated submodule. ==Applications== ===Conformal field theory=== In two dimensions, the algebra of local [[conformal map|conformal transformations]] is made of two copies of the [[Witt algebra]]. It follows that the symmetry algebra of [[two-dimensional conformal field theory]] is the Virasoro algebra. Technically, the [[conformal bootstrap]] approach to two-dimensional CFT relies on [[Virasoro conformal block]]s, special functions that include and generalize the characters of representations of the Virasoro algebra. ===String theory=== Since the Virasoro algebra comprises the generators of the conformal group of the [[worldsheet]], the [[Stress–energy tensor|stress tensor]] in [[string theory]] obeys the commutation relations of (two copies of) the Virasoro algebra. This is because the conformal group decomposes into separate diffeomorphisms of the forward and back lightcones. Diffeomorphism invariance of the worldsheet implies additionally that the stress tensor vanishes. This is known as the [[Virasoro constraint]], and in the [[Quantum mechanics|quantum theory]], cannot be applied to all the states in the theory, but rather only on the physical states (compare [[Gupta–Bleuler formalism]]). ==Generalizations== ===Super Virasoro algebras=== {{main|Super Virasoro algebra}} There are two [[Lie superalgebra|supersymmetric ''N'' = 1 extensions]] of the Virasoro algebra, called the [[Neveu–Schwarz algebra]] and the [[Ramond algebra]]. Their theory is similar to that of the Virasoro algebra, now involving [[Grassmann number]]s. There are further extensions of these algebras with more supersymmetry, such as the [[N = 2 superconformal algebra|''N'' = 2 superconformal algebra]]. ===W-algebras=== {{main|W-algebra}} W-algebras are associative algebras which contain the Virasoro algebra, and which play an important role in [[two-dimensional conformal field theory]]. Among W-algebras, the Virasoro algebra has the particularity of being a Lie algebra. ===Affine Lie algebras=== {{main|affine Lie algebra}} The Virasoro algebra is a subalgebra of the universal enveloping algebra of any affine Lie algebra, as shown by the [[Sugawara construction]]. In this sense, affine Lie algebras are extensions of the Virasoro algebra. ===Meromorphic vector fields on Riemann surfaces=== The Virasoro algebra is a central extension of the Lie algebra of meromorphic vector fields with two poles on a genus 0 [[Riemann surface]]. On a higher-genus compact Riemann surface, the Lie algebra of meromorphic vector fields with two poles also has a central extension, which is a generalization of the Virasoro algebra.<ref>{{cite journal | last1 = Krichever | first1 = I. M. | last2 = Novikov | first2 = S.P. | year = 1987 | title = Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons | journal = Funkts. Anal. Appl. | volume = 21 | issue = 2| pages = 46–63 | doi = 10.1007/BF01078026 | s2cid = 55989582 }}</ref> This can be further generalized to supermanifolds.<ref>{{Cite journal | doi = 10.1016/0393-0440(94)00012-S| title = Super elliptic curves| journal = Journal of Geometry and Physics| volume = 15| issue = 3| pages = 252–280| year = 1995| last1 = Rabin | first1 = J. M. |arxiv = hep-th/9302105 |bibcode = 1995JGP....15..252R | s2cid = 10921054}}</ref> ===Vertex algebras and conformal algebras=== The Virasoro algebra also has [[vertex operator algebra|vertex algebraic]] and [[Lie conformal algebra|conformal algebraic]] counterparts, which basically come from arranging all the basis elements into generating series and working with single objects. ==History<!--'Virasoro operator' and 'Virasoro operators' redirect here-->== The Witt algebra (the Virasoro algebra without the central extension) was discovered by [[É. Cartan]]<ref name="car09"/> (1909). Its analogues over finite fields were studied by [[E. Witt]] in about the 1930s. The central extension of the Witt algebra that gives the Virasoro algebra was first found (in characteristic ''p'' > 0) by [[R. E. Block]]<ref name="blo66"/> (1966, page 381) and independently rediscovered (in characteristic 0) by [[I. M. Gelfand]] and [[Dmitry Fuchs]]<ref name="gf69"/> (1969). The physicist [[Miguel Ángel Virasoro (physicist)|Miguel Ángel Virasoro]]<ref name="vir70"/> (1970) wrote down some operators generating the Virasoro algebra (later known as the '''Virasoro operators'''<!--boldface per WP:R#PLA-->) while studying [[dual resonance model]]s, though he did not find the central extension. The central extension giving the Virasoro algebra was rediscovered in physics shortly after by J. H. Weis, according to Brower and Thorn<ref name="bt71"/> (1971, footnote on page 167). ==See also== {{div col}} *[[Conformal field theory]] *[[Goddard–Thorn theorem]] *[[Heisenberg algebra]] *[[Lie conformal algebra]] *[[Pohlmeyer charge]] *[[Super Virasoro algebra]] *[[W-algebra]] *[[Witt algebra]] *[[WZW model]] {{div col end}} ==References== {{reflist|refs = <ref name="wat24">{{cite journal | last=Watts | first=Gérard M T | title=Explicit expressions for Virasoro singular vectors | journal=Journal of High Energy Physics | date=2025 | volume=2025 | issue=4 | page=131 | doi=10.1007/JHEP04(2025)131 | arxiv=2412.07505 | bibcode=2025JHEP...04..131W }}</ref> <ref name="gko86"> {{cite journal |author1=P. Goddard, A. Kent |author2=D. Olive |name-list-style=amp |year=1986 |title=Unitary representations of the Virasoro and super-Virasoro algebras |journal=[[Communications in Mathematical Physics]] |volume=103 |issue=1 |pages=105–119 |mr=0826859 |zbl=0588.17014 |doi=10.1007/BF01464283|bibcode = 1986CMaPh.103..105G |s2cid=91181508 |url=http://projecteuclid.org/euclid.cmp/1104114626 }}</ref> <ref name="fqs84">{{cite journal |author=Friedan, D., Qiu, Z. and Shenker, S. |year=1984 |title=Conformal invariance, unitarity and critical exponents in two dimensions |journal=[[Physical Review Letters]] |volume=52 |pages=1575–1578 |doi=10.1103/PhysRevLett.52.1575 |issue=18|bibcode = 1984PhRvL..52.1575F |s2cid=122320349 }}</ref> <ref name="ff84">{{cite book | last1=Feigin | first1=B. L. | last2=Fuchs | first2=D. B. | title=Topology | chapter=Verma modules over the virasoro algebra | publisher=Springer Berlin Heidelberg | publication-place=Berlin, Heidelberg | volume=1060 | date=1984 | isbn=978-3-540-13337-7 | doi=10.1007/bfb0099939 | pages=230–245}}</ref> <ref name="fqs24">{{cite journal | last1=Fortin | first1=Jean-François | last2=Quintavalle | first2=Lorenzo | last3=Skiba | first3=Witold | title=The Virasoro Completeness Relation and Inverse Shapovalov Form | journal=Physical Review D | date=2025 | volume=111 | issue=8 | page=085010 | doi=10.1103/PhysRevD.111.085010 | arxiv=2409.12224 | bibcode=2025PhRvD.111h5010F }}</ref> <ref name="BYB">P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', 1997, {{ISBN|0-387-94785-X}}.</ref> <ref name="vir70">{{cite journal | last=Virasoro | first=M. A. | title=Subsidiary Conditions and Ghosts in Dual-Resonance Models | journal=Physical Review D | volume=1 | issue=10 | date=1970-05-15 | issn=0556-2821 | doi=10.1103/PhysRevD.1.2933 | pages=2933–2936| bibcode=1970PhRvD...1.2933V }}</ref> <ref name="car09">{{cite journal |author=E. Cartan |year=1909 |title=Les groupes de transformations continus, infinis, simples |journal=Annales Scientifiques de l'École Normale Supérieure |volume=26 |pages=93–161 |jfm=40.0193.02 |doi=10.24033/asens.603|doi-access=free }}</ref> <ref name="blo66"> {{cite journal |author=R. E. Block |year=1966 |title=On the Mills–Seligman axioms for Lie algebras of classical type |journal=[[Transactions of the American Mathematical Society]] |volume=121 |pages=378–392 |jstor=1994485 |doi=10.1090/S0002-9947-1966-0188356-3 |issue=2|doi-access=free }}</ref> <ref name="gf69">{{cite journal | last1=Gel'fand | first1=I. M. | last2=Fuks | first2=D. B. | title=The cohomologies of the lie algebra of the vector fields in a circle | journal=Functional Analysis and Its Applications | volume=2 | issue=4 | date=1969 | issn=0016-2663 | doi=10.1007/BF01075687 | pages=342–343}}</ref> <ref name="bt71">{{cite journal |author1=R. C. Brower |author2=C. B. Thorn |year=1971 |title=Eliminating spurious states from the dual resonance model |journal=[[Nuclear Physics B]] |volume=31 |issue=1 |pages=163–182 |doi=10.1016/0550-3213(71)90452-4|bibcode = 1971NuPhB..31..163B |url=https://cds.cern.ch/record/352004 }}</ref> <ref name="ken92">{{cite journal |author=A. Kent |year=1991 |title=Singular vectors of the Virasoro algebra |journal=[[Physics Letters B]] |volume=273 |issue=1–2 |pages=56–62 |doi=10.1016/0370-2693(91)90553-3|arxiv = hep-th/9204097 |bibcode = 1991PhLB..273...56K |s2cid=15105921 }}</ref> }} ==Further reading== *{{Citation | last1=Iohara | first1=Kenji | last2=Koga | first2=Yoshiyuki | title=Representation theory of the Virasoro algebra | publisher=Springer-Verlag London Ltd. | location=London | series=Springer Monographs in Mathematics | isbn=978-0-85729-159-2 | doi=10.1007/978-0-85729-160-8 | mr=2744610 | year=2011}} * {{springer|author=Victor Kac|title=Virasoro algebra|id=v/v096710}} *V. G. Kac, A. K. Raina, ''Bombay lectures on highest weight representations'', World Sci. (1987) {{isbn|9971-5-0395-6}}. *{{cite journal | last1 = Dobrev | first1 = V. K. | year = 1986 | title = Multiplet classification of the indecomposable highest weight modules over the Neveu-Schwarz and Ramond superalgebras | journal = Lett. Math. Phys. | volume = 11 | issue = 3| pages = 225–234 | doi=10.1007/bf00400220 | bibcode=1986LMaPh..11..225D| s2cid = 122201087 }} & correction: ibid. '''13''' (1987) 260. *V. K. Dobrev, "Characters of the irreducible highest weight modules over the Virasoro and super-Virasoro algebras", Suppl. ''[[Rendiconti del Circolo Matematico di Palermo]]'', Serie II, Numero 14 (1987) 25-42. *{{cite arXiv|author=Antony Wassermann|title=Lecture notes on Kac-Moody and Virasoro algebras |eprint=1004.1287|class=math.RT|year=2010|author-link=Antony Wassermann }} *{{cite arXiv|author=Antony Wassermann|title=Direct proofs of the Feigin-Fuchs character formula for unitary representations of the Virasoro algebra|eprint=1012.6003|year=2010|class=math.RT|author-link=Antony Wassermann}} {{authority control}} [[Category:Conformal field theory]] [[Category:Lie algebras]] [[Category:Mathematical physics]]
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