Editing Vertex operator algebra (section)

=== Heisenberg vertex operator algebra ===
A basic example of a noncommutative vertex algebra is the rank 1 free boson, also called the Heisenberg vertex operator algebra.  It is "generated" by a single vector ''b'', in the sense that by applying the coefficients of the field ''b''(''z'') := ''Y''(''b'',''z'') to the vector ''1'', we obtain a spanning set.  The underlying vector space is the infinite-variable [[polynomial ring]] <math>\mathbb{C}[b_{-1}, b_{-2}, \cdots]</math>, where for positive <math>n</math>, <math>b_{-n}</math> acts obviously by multiplication, and <math>b_n</math> acts as <math>n\partial_{b_{-n}}</math>.  The action of ''b''<sub>0</sub> is multiplication by zero, producing the "momentum zero" Fock representation ''V''<sub>0</sub> of the Heisenberg Lie algebra (generated by ''b''<sub>n</sub> for integers ''n'', with commutation relations [''b''<sub>n</sub>,''b''<sub>m</sub>]=''n'' δ<sub>n,–m</sub>), induced by the trivial representation of the subalgebra spanned by ''b''<sub>n</sub>, n ≥ 0.

The Fock space ''V''<sub>0</sub> can be made into a vertex algebra by the following definition of the state-operator map on a basis <math>b_{j_1}b_{j_2}...b_{j_k}</math> with each <math>j_i < 0</math>,

:<math>Y( b_{j_1}b_{j_2}...b_{j_k}, z) := \frac{1}{(-j_1 - 1)!(-j_2 - 1)!\cdots (-j_k - 1)!}:\partial^{-j_1 - 1}b(z)\partial^{-j_2 - 1}b(z)...\partial^{-j_k - 1}b(z):</math>

where <math>:\mathcal{O}:</math> denotes normal ordering of an operator <math>\mathcal{O}</math>. The vertex operators may also be written as a functional of a multivariable function f as:

:<math> Y[f,z] \equiv :f\left(\frac{b(z)}{0!},\frac{b'(z)}{1!},\frac{b''(z)}{2!},...\right): </math>

if we understand that each term in the expansion of f is normal ordered.

The rank ''n'' free boson is given by taking an ''n''-fold tensor product of the rank 1 free boson.  For any vector ''b'' in ''n''-dimensional space, one has a field ''b''(''z'') whose coefficients are elements of the rank ''n'' Heisenberg algebra, whose commutation relations have an extra inner product term: [''b''<sub>n</sub>,''c''<sub>m</sub>]=''n'' (b,c) δ<sub>n,–m</sub>.

The Heisenberg vertex operator algebra has a one-parameter family of conformal vectors with parameter <math>\lambda \in \mathbb{C}</math> of conformal vectors <math>\omega_\lambda</math> given by
:<math>\omega_\lambda = \frac{1}{2}b_{-1}^2 + \lambda b_{-2},</math>
with central charge <math>c_\lambda = 1 - 12\lambda^2</math>.<ref>{{cite book |last1=Ben-Zvi |first1=David |last2=Frenkel |first2=Edward |title=Vertex algebras and algebraic curves |date=2004 |location=[Providence, Rhode Island] |isbn=9781470413156 |page=45 |edition=Second}}</ref>

When <math>\lambda = 0</math>, there is the following formula for the Virasoro [[character theory|character]]:

:<math>Tr_V q^{L_0} := \sum_{n \in \mathbf{Z}} \dim V_n q^n = \prod_{n \geq 1} (1-q^n)^{-1}</math>

This is the [[generating function]] for [[partition (number theory)|partitions]], and is also written as ''q''<sup>1/24</sup> times the weight −1/2 modular form 1/η (the reciprocal of the [[Dedekind eta function]]).  The rank ''n'' free boson then has an ''n'' parameter family of Virasoro vectors, and when those parameters are zero, the character is ''q''<sup>''n''/24</sup> times the weight −''n''/2 modular form η<sup>−''n''</sup>.