Sugawara's Construction of the Virasoro Algebra for C=1,H=0

SUGAWARA’S CONSTRUCTION OF VIRASORO ALGEBRA FOR c =1, h =0

LUKE CYBULSKI

Contents 1. Introduction 1 2. Central Extensions of Lie Algebras 2 3. Witt Algebra 3 4. Virasoro Algebra 6 5. Heisenberg Algebra and its Fock Representation 9 6. Sugawara’s Construction of V (1, 0) 10 7. Generalized Sugawara’s Construction of the Virasoro Algebra 14 References 14 Appendix A. Simple Harmonic Oscillator of Countable Degrees of Freedom 14

1. Introduction Group theory provides a useful way of mathematically studying the symmetries of physical systems. Most common are finite dimensional Lie groups such as SU(n),U(n), and SO(n) which contribute to the framework of rotations. Additionally, finite groups have been used extensively in crystallography (Sternberg [1]). These examples maintain a strong focus on the finite dimensional symmetry and have many applications to quantum mechanics. However, to pass into quantum field theory, where it is important to consider systems with an infinite number of degrees of freedom, algebras such as the Virasoro and Kac-Moody algebras are necessary for infinite dimensional symmetries. The Virasoro algebra arises naturally in various applications of physics. In a mathematical formulation, the Virasoro algebra can be viewed as coming from the algebra of the conformal group in one or two dimensions. Two-dimensional conformally invariant structures are present in many areas of physics, including 2-dimensional statistical systems of spins on lattices and in finding a mathematically consistent theory of particle interactions (Goddard and Olive [2]). Furthermore, the Virasoro algebra becomes useful in analyzing mass-less fermionic and bosonic field theories. Due to its presence in numerous areas of physics, it is important to be able to construct representations for the Virasoro algebra. The Virasoro algebra is often associated with observable quantities. In particular, the algebra determines the mass spectrum in string theory, and also energy spectra in 2-dimensional quantum field theories made up of two copies of the algebra (Goddard and Olive [2]). In this sense, it is important for the representation of the Virasoro algebra to be highest weight, so that the energy spectrum is positive (or at least bounded below). In addition to this requirement, it is necessary that the representations be unitary. In quantum mechanics, unitarity is important as it preserves probability. Furthermore, if unitarity is not imposed in quantum field theories, ghost-states, ie. states with negative norm, may appear and lead to non-physical properties, such as space-like momenta or negative probabilities. From representation theory, it is known that unitary representations may be written as direct integrals of irreducible representations. Hence, it is important to analyze irreducible representations as they provide fundamental building blocks to other representations. Furthermore, in a physical sense, a reducible representation corresponds to non-interacting parts. As many systems in physics deal strictly with interaction of particles, it is thus necessary to consider irreducible representations. This paper will begin with discussing how the Virasoro algebra arises as a nontrivial central extension of the Witt algebra. We will then use the Heisenberg algebra to perform a Sugawara construction of the Virasoro algebra in the case of c =1and h =0(representing a free, mass-less bosonic field theory) that 1 2LUKECYBULSKI

is irreducible, unitary, and highest weight. A brief discussion will then be given on how to generalize the Sugawara construction to other c values.

2. Central Extensions of Lie Algebras Let g, h be Lie algebras over a field F of characteristic zero and a an abelian Lie algebra over F. Definition 2.1. Consider the short exact sequence of Lie algebra homomorphisms 0 a h g 0 ! ! ! ! This sequence is a central extension of g by a if a Z(h)= x h [x, h]=0 , where we identify a as ✓ { 2 | } subalgebra of h. The condition that [a, h]= 0 a then implies that we may regard a as an ideal of h. { } ✓ Definition 2.2. AshortexactsequenceofLiealgebrahomomorphisms p 0 a h g 0 ! ! ! ! splits if there exists a Lie algebra homomorphism s : g h such that p s =1 .Wecalls a splitting map ! g and if such a map exists, then h = g a.Moreover,ifthesequenceisacentralextension,thenwesaythatit is a trivial extension. Definition 2.3. Abilinearmap : g g a that satisfies the following conditions 1 and 2 x, y, z g is ⇥ ! 8 2 called a 2-cocycle. (1) (x, y)= (y, x) (2) (x, [y, z]) + (y, [z,x]) + (z,[x, y]) = 0 Lemma 2.1. The 2-cocycle : g g a generates a central extension h = g a of g by a. ⇥ ! Proof. Define a Lie bracket on h by

[(x1,a1), (x2,a2)] = ([x1,x2], (x1,x2)) This is indeed a Lie bracket as may be readily veriﬁed. (1) [(x ,a ), (x ,a )] = ([x ,x ], (x ,x )) = ( [x ,x ], (x ,x )) 1 1 2 2 1 2 1 2 2 1 2 1 = ([x ,x ], (x ,x )) = [(x ,a ), (x ,a )] 2 1 2 1 2 2 1 1

(2)

[(x1,a1), [(x2,a2), (x3,a3)]] + [(x2,a2), [(x3,a3), (x1,a1)]] + [(x3,a3), [(x1,a1), (x2,a2)]]

=[(x1,a1), ([x2,x3], (x2,x3))] + [(x2,a2), ([x3,x1], (x3,x1))] + [(x3,a3), ([x1,x2], (x1,x2))]

=([x1, [x2,x3]], (x1, [x2,x3])) + ([x3, [x1,x2]], (x3, [x1,x2])) + ([x3, [x1,x2]], (x3, [x1,x2]))

=([x1, [x2,x3]] + [x2, [x3,x1]] + [x3, [x1,x2]], (x1, [x2,x3]) + (x2, [x3,x1]) + (x3, [x1,x2])) =(0, 0)

Clearly a Z(h) when we identify a as a subalgebra of h and thus h is a central extension of g by a. 2 ⇤ Lemma 2.2 (see [3,Lemma4.5]). Let : g g a be a 2-cocycle and h = g a. Then the central extension ⇥ ! h of g by a p 0 a h g 0 ! ! ! ! where p is the projection map onto g, is trivial if and only if there exists a C-linear map f : g a such that ! (x, y)=f([x, y]) SUGAWARA’S CONSTRUCTION OF VIRASORO ALGEBRA FOR c =1, h =0 3

Proof. ( )Supposethereexistsalinearmapf : g a such that (x, y)=f([x, y]). Then, define a map : g h ( ! ! by (x)=(x, f(x)) Clearly p =1 . Now, we define the following Lie bracket on h. g [(x1,a1), (x2,a2)] :=([x1,x2], (x1,x2)) Using this Lie bracket on h we find that is a Lie algebra homomorphism. ([x, y]) = ([x, y],f([x, y])) = ([x, y], (x, y)) = [(x, f(x)), (y, f(y))] = [(x), (y)] Hence, the sequence splits and so the central extension is trivial. ( ) Suppose the central extension is trivial. Then there exists a Lie algebra homomorphism : g h such ) ! that p =1.Let(x)=(x0,a0),forsomex0 g and a0 a. Then, p((x)) = p(x0,a0)=x0 but also g 2 2 p((x)) = x and so we find (x)=(x, ⇡(x)) for ⇡ : g a.Furthermore, being a Lie algebra homomorphism ! imposes the condition that ⇡ is a linear map. Then, from lemma 2.1, generates a central extension h = g a of g by a where the Lie bracket on h is defined by [(x1,a1), (x2,a2)] = ([x1,x2], (x1,x2)]) Using this definition along with the fact that is a Lie algebra homomorphism, we find that for x, y g that 2 (1) [(x), (y)] = ([x, y]) = ([x, y], ⇡([x, y])) (2) [(x), (y)] = [(x, ⇡(x)), (y, ⇡(y))] = ([x, y], (x, y)). Hence, it must be that (x, y)=⇡([x, y]). ⇤ Definition 2.4. A 2-coboundary on g is a 2-cocycle : g g a such that (x, y)=f([x, y]) for some ⇥ ! C-linear map f : g a. ! From the definition of a 2-coboundary on g we see that lemma implies a 2-cocycle on g generates a trivial extension if and only if is a 2-coboundary. Hence, we define C2(g; a) as the set of 2-cocycles on g and B2(g; a) as the set of 2-coboundaries on g. It is then of importance to consider the space of nontrivial extensions of g by a.Hence,weconsidertheequivalenceclassesofcentralextensionsofg by a,specifically the second cohomology group of g by values in a. H2(g; a)=C2(g; a)/B2(g; a)

3. Witt Algebra We define the the Witt algebra W as a subalgebra of the algebra of smooth vector fields on the circle S1, Vect(S1).AnarbitraryelementinVect(S1) is of the form f(✓) d , where f : R R is a smooth periodic d✓ ! function on R. Hence, we we may expand it as a complex Fourier series. To avoid problems with convergence, we then further require that the Fourier series is finite, ie. is represented by a finite sum. With these restrictions, W is defined as d 1 W := f(✓) f(✓)= a ein✓,a =0for all but finitely many n . { d✓ | n n } n= X1 Note that as f : R R we require that f ⇤ = f, which gives an⇤ = a n. ! in✓ d Now define Ln W as Ln = ie .FromthedefinitionofW it is clear that Ln n forms a basis for 2 d✓ { } 2Z W .RecallthedefinitionoftheLiebracketonVect(S1). d d dg df d [f ,g ]= f g d✓ d✓ d✓ d✓ d✓ ✓ ◆ We restrict this Lie bracket to the space W and we find that d d d d [f ,f ]=[iein✓ ,ieim✓ ]= (ein✓imeim✓ inein✓eim✓) =(n m)iei(n+m)✓ =(n m)f . n m d✓ d✓ d✓ d✓ n+m Hence, we see W has a basis fn n such that [fn,fm]=(n m)fn+m. { } 2Z In the contexts of physics, the element f W plays the role of energy, which we force to take positive 0 2 values. In addition, the f0 can determine mass spectrum in string theory, and so we wish it to be non-negative. 4LUKECYBULSKI

For these reasons, we are interested in particular representations ⇢ of W such that the spectrum of ⇢(f0) is non-negative (or at least bounded below). To simplify notation, we shall henceforth denote ⇢(fn)=Ln for an arbitrary representation ⇢.

Definition 3.1. A representation for which L0 has a spectrum that is bounded below, ie. has a state of lowest energy, is called a highest weight representation.Alowestenergystate in such a representation | 0i has the following properties (1) L = h 0| 0i | 0i (2) L =0,n>0 n| 0i Any such state satisfying this property is called a vacuum state. | 0i In addition to highest weight representations, we are further interested in unitary representations. We begin by considering antilinear involutions on a Lie algebra g. Definition 3.2. An antilinear involution on a Lie algebra g is a map ! : g g that satisfies the following ! properties for all x, y g and for all a C. 2 2 (1) !(!(x)) = x (2) !(ax)=a⇤!(x) (3) [!(x), !(y)] = !([y, x]) where ⇤ denotes the complex conjugate. Definition 3.3. A Hermitian form ., . on a vector space V is a map V V C such that h i ⇥ ! (1) v, w = w, v ⇤ h i h i (2) v, au + bw = a v, u + b v, w h i h i h i (3) av + bu, w = a⇤ v, w + b⇤ u, w h i h i h i for all u, v, w V and a, b C.Furthermore, ., . is non-degenerate if 2 2 h i u, v =0 u V v =0 h i 8 2 ) Definition 3.4. Let g be a Lie algebra over a field F with an antilinear involution ! : g g.LetV ! be a vector space equipped with a non-degenerate Hermitian inner product ., . and ⇢ : g gl(V ) be a h i ! representation of g. Then ⇢ is an unitary representation of g if ⇢(x)(u),v = u, ⇢(!(x))(v) h i h i for all x g and u, v V .Equivalently,thisimplies⇢ is unitary if 2 2 ⇢(x)† = ⇢(!(x)) for all x g. 2 Consider the map ! : W W defined by ! !(fn)=f n As W is a R-vector space, it is clear conditions (1) and (2) are satisfied in the definition of an antilinear involution. We further find that condition (3) is also satisfied. !([f ,f ]) = !((n m)f )=(n m)!(f ) n m n+m n+m =(n m)f (n+m) =[f m,f n]=[!(fm), !(fn)] Hence, the condition for a unitary representation of W becomes

Ln† = !(fn)=L n Theorem 3.1. The Witt algebra W has no nontrivial, unitary highest weight representation. We ﬁrst prove that the following Lemma. Lemma 3.1. Let V be a vector space equipped with a Hermitian form ., . .Letu, v V . Then, the matrix h i 2 M given by u, u u, v M := h ih i . 0 v, u v, v 1 h ih i is positive semideﬁnite. @ A SUGAWARA’S CONSTRUCTION OF VIRASORO ALGEBRA FOR c =1, h =0 5

Proof. Let x1,x2 C. Then, 2 u, u u, v x x u, u + x u, v h ih i 1 1h i 2h i x1⇤ x2⇤ = x1⇤ x2⇤ 0 v, u v, v 1 0x 1 0x v, u + x v, v 1 ⇣ ⌘ h ih i 2 ⇣ ⌘ 1h i 2h i @ A @ A 2 @ A 2 = x u, u + x⇤x u, v + x x⇤ v, u + x v, v | 1| h i 1 2h i 1 2h i | 2| h i = x u + x v 2 0 k 1 2 k ⇤ Proof of Theorem 3.1. Consider a unitary highest weight representation for W with vacuum state 0 normalized to unity and m | i highest weight h.Deﬁne n,m = L n 0 .LetN Z>0 and deﬁne the matrix ⇤ as | i | i 2

2N,1 2N,1 N,2 2N,1 ⇤ := h | ih | i . 0 1 h 2N,1| N,2ihN,2| N,2i @ A Using the deﬁnition of the Witt algebra and the condition that Ln† = L n for a unitary representation, we may calculate the matrix elements. (1)

We thus find that 4Nh 6N 2h ⇤ := . 06N 2h 4N 3h +8N 2h21 This matrix must be positive semidefinite by@ the preceding lemma.A A consequence of this is that det ⇤ 0. det(⇤)=4N 3h2(8h 5N) 0 Then, for N>>1,wehavethatdet(⇤) < 0. Thus, it must be that h =0.Withh =0,wehavethat L n 0 =2nh =0for all n Z>0. Combining this with the definition of highest weight representation, k | ik 2 we have Ln 0 =0for all n Z. This means that the representation is the trivial representation. Thus, we | i 2 find there does not exist a nontrivial, unitary highest weight representation of W . ⇤

4. Virasoro Algebra Theorem 3.1 shows that W has no nontrivial, unitary, highest weight representation. Thus, we consider nontrivial central extensions of the W that have a nontrivial unitary highest weight representation. For this purpose deﬁne a map : W W C by ⇥ ! 1 2 (fn,fm)= n(n 1)n, m 12 The following two lemmas will then be suﬃcient to determine the central extensions of W .

Lemma 4.1 (see [4, Theorem 5.1]). The map generates a nontrivial central extension V = W C of W by C. Proof. It is clearly sufficient to prove the following two facts. (1) C2(W ; C) 2 (2) / B2(W ; C) 2 We first consider (1). 1 2 1 2 1 2 (fn,fm)= n(n 1)n, m = ( m)(m 1)m, n = m(m 1)m, n = (fm,fn) • 12 12 12 • (fn, [fm,fk]) + (fm, [fk,fn]) + (fk, [fn,fm]) = (f , (m k)f )+(f , (k n)f )+(f , (n m)f ) n m+k m k+n k n+m 1 2 1 2 1 2 = n(m k)(n 1)n, (m+k) + m(k n)(m 1)m, (k+n)) + k(n m)(k 1)k, (n+m) 12 12 12 1 = [n(m k)(n2 1) + m(k n)(m2 1) + k(n m)(k2 1)] 12 n+m+k,0 1 = [n(n +2m)(n2 1) m(2n + m)(m2 1) (n2 m2)((n + m)2 1)] = 0 12 Hence we find that C2(W ; C).Next,supposethat / B2(W ; C). Then, by Lemma 2, (x, y)=f([x, y]) 2 2 for some linear map p : W C. Then, ! (fn,f n)=p([fn,f n)=2np(f0). Thus, we find that for all n Z 0 2 \{ } 1 p(f )= (n2 1). 0 24 However, this cannot hold for all n Z 0 ,leadingtoacontradiction.Thus,wefindthat / B2(W ; C). 2 \{ } 2 ⇤ Lemma 4.2 (see [4, Theorem 5.1]). Let : W W C be a 2-cocycle on W . Then, it must be that, up to ⇥ ! a 2-coboundary, has the form c 2 (fm,fn)= m(m 1)m, n 12 for some c C. 2 SUGAWARA’S CONSTRUCTION OF VIRASORO ALGEBRA FOR c =1, h =0 7

Proof. Let n, m, k Z.As is a 2-cocycle, we have 2 (fn, [fm,fk]) + (fm, [fk,fn]) + (fk, [fn,fm]) =(m k)(f ,f )+(k n)(f ,f )+(n m)(f ,f ) n m+k m k+n k n+m =0. Letting k =0then gives that m(f ,f ) n(f ,f )+(n m)(f ,f )=(n + m)(f ,f )+(n m)(f ,f )=0. n m m n 0 n+m n m 0 n+m This relation then reduces down to m n (f ,f )= (f ,f ) , for n + m =0 n m n + m 0 n+m 6 Next, we define a C-linear map p : W C by ! 1 2 (f1,f 1) n =0 p(fn)= 1 (f ,f ) n =0 ( n 0 n 6 We then consider the 2-cocycle p : W W C defined by p(fn,fm)=p([fn,fm]),andthemap := + p. ⇥ ! We note that B2(W ; C). Thus, it suffices to look at . 2 Now let n + m =0. Then, 6 e e e (fn,fm)=(fn,fm)+p(fn,fm) m n = (f ,f )+p([f ,f ]) n + m 0 en+m n m m n = (f ,f ) (m n)p(f ) n + m 0 n+m n+m m n m n = (f ,f ) (f ,f ) n + m 0 n+m n + m 0 n+m =0.

Hence, we find that (fn,fm) is proportional to n+m,0 and so we may write (fn,fm) as (f ,f )=d(n, m) d(n) n m n+m,0 ⌘ n+m,0 where d : Z C.First,as is a 2-cocycle, it must be antisymmetric. This then implies ! d(n)n+m,0 = (fn,fm) = (f ,f ) m n = d(m) n+m,0 = d( n) . n+m,0 Hence, we find d(n)= d( n). We may thus consider n Z>0. Then, using this property of d(n),wefind 2 (fn, [fm,fk]) + (fm, [fk,fn]) + (fk, [fn,fm]) =(m k)(f ,f )+(k n)(f ,f )+(n m)(f ,f ) n m+k m k+n k n+m =[(m k)d(n)+(k n)d(m)+(n m)d(k)] n+m+k,0 =(2m n)d(n) (2n + m)d(m)+(n m)d( n m) =(2m n)d(n) (2n + m)d(m) (n m)d(n + m) =0 We then plug in m =1into the above relation to obtain a recursive formula for d(n). (n 1)d(n + 1) (n + 2)d(n)+(2n + 1)d(1) = 0 8LUKECYBULSKI

Evaluating (L1,L 1) explicitly gives (f1,f 1)=(f1,f 1)+p(f1,f 1) = (f1,f 1)+p([f1,f 1]) e = (f1,f 1)+2p(f0) 2 = (f1,f 1) (f1,f 1) 2 =0

Thus, (f1,f 1)=d(1) = 0. The recursive formula then reduces down to (n 1)d(n + 1) (n + 2)d(n)=0 Define d(2) = c C.Wethenclaimthatd(n)= c n(n2 1) for all n Z.Evidentlyd(n)= d( n) and so 2 6 2 we may still consider n Z>0. Thus, we proceede by induction on n.Asd(1) = 0,wefindthisformulais 2 6c satisfied for n =1.Furthermore,d(2) = = c.Supposenowthatthisrelationholdsforn Z>1. Then, e 6 2 e n +2 d(n + 1) = d(n) en 1 n +2 c = n(n2 1) n 1 6 c n +2e = n(n + 1)(n 1) 6 n 1 ec = n(n + 1)(n + 2) 6 c = e(n + 1)[(n2 +2n + 1) 1] 6 c = e(n + 1)[(n + 1)2 1] 6 Thus, we confirm that d(n)= c n(n2 1).Asec C was arbitrary, we relabel c = c/12 for c C. This then 6 2 2 gives that e c (f ,f e)= n(n2 1) e n m 12 n+m,0 ⇤ Theorem 4.1. H2(W ; C) C ' Proof. Consider : W W C, where ⇥ ! 1 2 (fm,fn)= m(m 1)m, n. 12 From Lemma 4.1 this is a nontrivial central extension of W . Then, suppose ⇡ C2(W ; C). As shown in 2 Lemma 4.2, ⇡ = t for some t C,sothatH2(W ; C) C. 2 ' ⇤ Thus, the Witt algebra has a nontrivial central extension, and we call it the Virasoro algebra, V .

Definition 4.1. The Virasoro algebra, V ,hasbasis fn n c such that { } 2Z [ { } c 2 [fn,fm]=(n m)fn+m + n(n 1)n, m 12 [c, fn]=0for all n Z 2 As in the Witt algebra, define ⇢(fn)=Ln for an arbitrary representaton ⇢ of V .Wemayfurtherdefinean antilinear involution ! : V V by ! !(fn)=f n !(c)=c

and thus a representation is unitary if Ln† = L n and c† = c,ie.c R. 2 SUGAWARA’S CONSTRUCTION OF VIRASORO ALGEBRA FOR c =1, h =0 9

We can characterize a highest weight representation of the Virasoro algebra by its central charge c and highest weight h. Hence, it is often convenient to write V (c, h) for the Virasoro algebra. Furthermore, as h is an eigenvalue of L0 and represents energy or some observable quantity, it must be that h R.Byimposing 2 that a highest weight representation of the Virasoro algebra be unitary, we may ﬁnd restrictions on c and h. Proposition 4.1. For a unitary highest weight representation of V (c, h),itmustbethatc 0 and h 0. Proof. Let 0 denote the vacuum state and n Z>0. Then, | i 2 2 L n 0 = 0 L† nL n 0 k | ik h | | i = 0 LnL n 0 h | | i = 0 [Ln,L n]+L nLn 0 h | | i c = 2nL + n(n2 1) h 0| 0 12 | 0i c =(2nh + n(n2 1)) 2 12 k| 0ik This then gives the restriction that c 2nh + n(n2 1) 0 12 First suppose that n =1. Then, the above inequality gives that 2h 0,orthath 0.Supposethenthatc<0. 2 c 2 As h>0 and n(n 1) grows faster than n,theremustexistaN Z>0 such that 2Nh+ N(N 1) < 0, 2 12 leading to a contradiction. Hence, we ﬁnd that c 0. ⇤ Proposition 4.2. Any unitary highest weight representation of V is irreducible. Hence, by considering unitary highest weight representations of V we necessarily have irreducibility.

5. Heisenberg Algebra and its Fock Representation The algebra associated with the simple harmonic oscillator can be made more general into the Heisenberg algebra.

Deﬁnition 5.1. The Heisenberg algebra h has basis tn n Z k such that { | 2 } [ { } [tn,tm]=nkn+m,0

[k, tn]=0, for all n

For an arbitrary representation ⇢ of h we shall henceforth denote ⇢(tn)=n.AsinthecasesforW and V , we may deﬁne an antilinear involution ! on h such that

!(tn)=t n !(k)=k.

Hence, a representation h is unitary if n† = n and k R.AsinAppendixA,definetheFockspaceF by 2 F = N N N N =0for all but finitely many N . {| 1 2 3 ···i | k k} It is possible to construct a representation of h on the space F by identifying n with the creation and annihilation operators from the simple harmonic oscillator as discussed in Appendix A. Let n Z>0 and 2 identify n and n as p n = nk an† n = pnk an. Furthermore, define N =0and k = id,forastate N F . It is easy to check that this identification 0| i | i2 yields the appropriate Lie brackets on h.Letn, m Z>0. 2 (1) [n, m]=kpnm [an,am]=0

(2) [n, m]=kpnm [an,am† ]=kpnmn,m = knn,m (3) [n,k]=kpnk [an, 1] = 0

(4) [ n,k]=kpnk [an† , 1] = 0 10 LUKE CYBULSKI

And as 0 annihilates each state, it evidently commutes with all n and k.Hence,wehavethat[n, m]= knn+m,0 and [n,k]=0, which is precisely the Heisenberg Algebra. With this identification we find that for n Z>0 and N1N2 F 2 | ···i2 (1) N N 0 =0 n| 1 2 ··· n ···i (2) N N N = pnkpN N N N 1 n| 1 2 ··· n ···i n | 1 2 ··· n ···i (3) n N1N2 Nn = pnkpNn +1 N1N2 Nn +1 | ··· ···i | ··· ···i Using these actions on F ,wehavearepresentationofh on F ,calledtheFock representation.Moreover,as in the simple harmonic oscillator case, we may construct an arbitrary state in F using an† .Wemaysimilarly do so with n for n Z>0. 2 Nn 1 n N1N2 | ···i p Nn n=1 pNn! nk Y Proposition 5.1. The representation constructed for h is unitary. Proof. We need only show that n† = n for all n Z 0 . 2 \{ } Case 5.1.1. Suppose n Z>0. Then, as k R, 2 2 p p n† =( nk an)† = nk an† = n Case 5.1.2. Suppose n = m for m Z<0. Then 2 p p n† = † m =( mk am† )† = mk am = m = n ⇤ Now recall the definition of a nilpotent Lie algebra. Definition 5.2. ALiealgebrag is nilpotent if for some fixed n Z , 2 n>0 ad ad ad y =0 x1 x2 ··· xn for all x ,x ,...,x ,y g. 1 2 n 2 Proposition 5.2. The Heisenberg algebra h is nilpotent. Specifically, all double brackets vanish. Proof. It clearly suffices to show that [[n, m], s]=0for any n, m, s Z.Butthisiseasyask commutes with all 2 n. [[n, m], s]=[nkn+m,0, s]=nkn+m,0[k, s]=0 ⇤

6. Sugawara’s Construction of V (1, 0) We will use a specific case of Sugawara’s construction to formulate the Fock representation of V (1, 0) from the Heisenberg algebra representation already constructed (see Goddard and Olive [2],Kac[5],Schlichenmaier [6],SchlichenmaierandSheinman[7]). Thus, consider the Heisenberg algebra spanned by n n Z k { | 2 } [ { } as above. We then define elements Ln, known as the Sugawara operators, as 1 Ln = : jj+n : 2k j X2Z where : jj+n : denotes normal ordering,definedby

jj+n j j + n : jj+n :=  . (j+n j j j + n To see the importance of normal ordering, consider L0 without normal ordering acting on the vacuum state denoted by 0 in the space F (ie. the state for which N =0for all j). | i j SUGAWARA’S CONSTRUCTION OF VIRASORO ALGEBRA FOR c =1, h =0 11

1 L0 0 = jj 0 | i 2k | i j X2Z 1 1 1 = jj 0 + jj 0 2k 0 | i | i1 j=0 j= X X1 1 @1 A = j j 0 2k | i j=1 X pk 1 = j 00 1 2k j| ··· j ···i j=1 X p 1 1 = j 0 0 2 | i!1| i j=1 X Thus, we find that without normal ordering we get an eigenvalue of which is not desirable. Conversely, 1 with normal ordering incorporated we find that 1 L0 0 = : jj : 0 | i 2k | i j X2Z 2 1 = jj 0 =0 2k | i j=1 X We will now show that the definition of Ln above gives rise to the Virasoro algebra V (1, 0).

1 2 Proposition 6.1. [Ln,Lm]=(n m)Ln+m + n(n 1)n, m 12 Proof. First, from Proposition 5.2, the Heisenberg algebra is nilpotent and all double brackets vanish. Using this, we ﬁnd for a, b, c, d Z, 2 [ab, cd]=[[a, b]+ba, cd]

=[[a, b], cd]+[ba, cd]

= c[[ab], d]+[[a, b], c]d +[ba, cd]

=[ba, cd]

Aconsequenceofthisresultisthatwemayignorenormalorderingin[Ln,Lm].Wenowcomputeexplicitly [Ln,Lm]. Case 6.1.1. Suppose that m + n =0. 6 1 1 [Ln,Lm]=[ : ii+n : , : jj+m :] 2k 2k i j X2Z X2Z 1 = [: ii+n : , : jj+m :] 4k2 i j X2Z X2Z 1 = [ ii+n, jj+m] 4k2 i j X2Z X2Z 1 = i( j[i+n, j+m]+[i+n, j]j+m)+( j[ i, j+m]+[ i, j]j+m)i+n 4k2 i j X2Z X2Z 1 = i j(i + n)ki+n, j m + ij+m(i + n)ki+n,j ji+nik i, j m j+mi+nik i,j 4k2 i j X2Z X2Z 1 = ( (i + n) ii+n+m i i+mi+n) 2k i i X2Z X2Z 12 LUKE CYBULSKI

Then, replace i by i + m in the second summation to obtain n m [Ln,Lm]= ii+n+m. 2k i X2Z However, as n + m =0we have that [ i, i+n+m]=0and so the summation is equivalent to the normal 6 ordered form. Hence, n m [Ln,Lm]= : ii+n+m :=(n m)Ln+m 2k i X2Z Case 6.1.2. Now suppose that m + n =0.Noticethat[L0,L0]=0and [L n,Ln]= [Ln,L n] so we may assume that n>0 without loss of generality. Then, we may write Ln as 1 Ln = ( jj+n + j+nj). 2k j 0 j>0 X X From a similar calculation as above, we find 2k [Ln,L n]= (j + n) jj j j nj+n + j j+nj n +(n j) jj 4k2 { } j>0 X 1 = (2knL0 + j j+nj n j j mj+m) 2k j>0 j>0 X X Replacing j by j + n in first summation and j and j n in the second summation, we obtain 1 [Ln,L n]= (2knL0 + (j + n) jj + (n j) jj) 2k j> n j>n X X 1 n 1 1 1 1 = (2knL0 + (j + n) jj + (j + n) jj + (n j) jj (n j) jj) 2k j=0 j= n+1 j=1 j=1 X X X X n 1 1 1 1 = (2knL0 + (n j)[j, j]+ (j + n) jj + (j + n)j j) 2k j=1 j=0 j= X X X1 n 1 1 = (2knL0 + k j(n j)+ : jj :) 2k j=1 j X X2Z n 1 1 = (4knL + k j(n j)). 2k 0 j=1 X The finite sum is calculated as n 1 n 1 n 1 j(n j)=n j j2 j=1 j=0 j=0 X X X n2(n 1) n(n 1)(2n 1) = 2 6 n(n 1)(n + 1) = 6 n(n2 1) = . 6 Hence, we obtain 1 2 [Ln,L n]=2nL0 + n(n 1). 12 Combining the two above cases gives that

1 2 [Ln,Lm]=(n m)Ln+m + n(n 1)n, m 12 ⇤ SUGAWARA’S CONSTRUCTION OF VIRASORO ALGEBRA FOR c =1, h =0 13

Then, deﬁning c = id on F and using Ln deﬁned in the above way, we obtain a representation for the Virasoro algebra with c =1on the space (known as the Fock representation).

Proposition 6.2. The representation constructed for V is a unitary highest weight representation with vacuum state 0 and h =0. | i Proof. First, by the above computation, we found that L0 0 =0. Then, let n Z>0 and consider Ln 0 . | i 2 | i Case 6.2.1. Suppose n is even. Then, from the deﬁnition of normal ordering, we ﬁnd 1 Ln 0 = : jj+n : 0 | i 2k | i j X2Z n/2 1 1 = ( j+n j 0 + jj+n 0 ) 2k | i | i j= j= n/2+1 X1 X n/2 1 1 = ( j+n(0) + j(0)) = 0 2k j= j= n/2+1 X1 X where the second sum goes to zero since j n/2+1and so j + n n/2+1> 0. Case 6.2.2. Suppose now that n is odd. Then, 1 Ln 0 = : jj+n : 0 | i 2k | i j X2Z n 1 2 1 1 = ( j+n j 0 + jj+n 0 ) 2k | i n 1 | i j= j= +1 X1 X2 n 1 2 1 1 = ( j+n(0) + j(0)) = 0 2k n 1 j= j= +1 X1 X2 where again the second sum goes to zero since j (n 1)/2+1and so j + n 0. Thus, this is a highest weight representation with h =0. To illustrate unitarity of the representation, recall the following antilinear involution on h.

!(tn)=t n !(k)=k

By replacing k with c and tn with fn we also get the antilinear involution on V from before. Hence, it suﬃces to show that Ln† = L n.Butthisistrivialsince: jj+n : = : j+n j : and k R. 2 1 Ln† =( : jj+n :)† 2k j X2Z 1 1 = : ( jj+n)† : = : j†+n† j : 2k 2k j j X2Z X2Z 1 1 = : j nj : = : j n j : 2k 2k j j X2Z X2Z 1 = : jj n : = L n 2k j X2Z ⇤ 14 LUKE CYBULSKI

7. Generalized Sugawara’s Construction of the Virasoro Algebra It is possible to derive the above construction of V (1, 0) from a more general Sugawara construction (see Goddard and Olive [2],Kac[5],Schlichenmaier[6])ofaffine Lie algebras. This construction will then give rise to the Virasoro algebra for other c values. Let g be a Lie algebra with dim g = d< .Supposeg has a symmetric, non-degenerate bilinear form 1 1 ., . .Definetheaffine Lie algebra gˆ as a central extension of g C[z,z ] by C,ie.asavectorspacegˆ is h i ⌦ 1 gˆ =(g C[z,z ]) Ct. ⌦ Suppose V is a vector space such that v V is annihilated by y zn for all y g for n large enough. A vector 2 ⌦ 2 space satisfying this property is an admissible representation of gˆ.Furthermore,supposethattv = cv for d i d some scalar c and u and u are bases for g and g⇤,respectively(⇤ represents dual space). Recall { i}i=1 { }i=1 that a Casimir element of a Lie algebra is an element in the center and that the Casimir element of g is given by d i Cg = uiu i=1 X Further suppose its action is scalar multiplication on the adjoint representation. In this case we define  (the dual Coxeter number)as 1 d  = ad ad i . 2 ui u i=1 X Suppose that c +  =0and define operators T by 6 n d 1 i(j+n) Tn := : ui( j)u : 2(c + ) j i=1 X2Z X where normal ordering is as before.

Theorem 7.1. Consider the Virasoro algebra with basis fn t . Then, the mapping deﬁned by { }2Z [ { } f T n 7! n t id 7! V is a representation for the Virasoro algebra with central charge c dim g c + 

References

[1] Sternberg, S., Group Theory and Physics, Cambridge University Press, Cambridge, pp. 16-21. [2] Goddard, P., Olive, D., Kac-Moody and Virasoro Algebras in Relation to Quantum Physics, Int. J. Mod. Phys. A 1 (1986), pp. 303-414 [3] Schottenloher, M., Central Extensions of Lie Algebras and Bargmann’s Theorem, Lect. Notes Phys. 759 (2008), pp. 63-73. [4] Schottenloher, M., The Virasoro Algebra, Lect. Notes Phys. 759 (2008), pp. 75-85. [5] Kac, V.G., 1990, Inﬁnite Dimensional Lie Algebras, Cambridge University Press, Cambridge, pp. 228-234. [6] Schlichenmaier, M., Sugawara Construction for Higher Genus Riemann Surfaces, Rept. Math. Phys. 43 (1999), pp. 323-339. [7] Schlichenmaier, M., Sheinman, O.K., Sugawara Construction and Casimir Operators for Krichever-Novikov Algebras, Journal of Mathematical Sciences, 92 (1998), pp. 3807-3834.

Appendix A. Simple Harmonic Oscillator of Countable Degrees of Freedom Consider a simple harmonic oscillator system of countable degrees of freedom. It is then possible to deﬁne annihilation and creation operators ak and a† ,respectively,fork Z>0. k 2 m! i a = (x + p ) k 2 k m! k r } m! i a† = (x p ) k 2 k m! k r } SUGAWARA’S CONSTRUCTION OF VIRASORO ALGEBRA FOR c =1, h =0 15

Using these definitions of the annihilation and creation operators, we find that the Hamiltonian Hk for the kth degree of freedom is thus given by 2 pk 1 2 1 Hk = + m!x = }!(a† ak + ) 2m 2 k k 2 where the total Hamiltonian is H = Hk. However, with this definition of the Hamiltonian, the ground 1 state energy is already infinite (each degree of freedom contributes }!). Hence we simply rescale Hk so that P 2 1 1 H = Hk = }!ak† ak kX=1 kX=1 It is clear that this rescaling is acceptable as only the energy difference between states is of interest. Furthermore, using the canonical commutation relations [xi,pj]=i}i,j and [xi,xj]=[pi,pj]=0,wefind the only non-vanishing commutators are

[ak,al†]=k,l Then, let Nk Z 0 and consider the set of states F given by 2 F = N N N N =0for all but ﬁnitely many N {| 1 2 3 ···i | k k} This space is known as the oscillator Fock space. It is then well known from quantum theory that ak and ak† acting on a given state in F gives (1) a N N N = pN N N N 1 k| 1 2 ··· k ···i k| 1 2 ··· k ···i (2) a† N N N = pN +1N N N +1 k| 1 2 ··· k ···i k | 1 2 ··· k ···i Furthermore, for a state N N 0 , where 0 represents a zero in the kth component, | 1 2 ··· k ···i k a N N 0 =0. k| 1 2 ··· k ···i The state for which all Ni =0is denoted the ground state of the system. It is then possible to build each

state in F from the ground state using the action of ak† on arbitrary states.

1 Nk (ak† ) N1N2 = 00 | ···i pNk! | ···i kY=1 As this state belongs to F ,onlyﬁnitelymanyNk are nonzero.