What's the geometric (or representation independent) definition of central charge of Lie algebra $\mathfrak{g}$?

Question

There is a common way described in Weinberg's Quantum Field Theory Vol.1 (P83) for introducing the concept of the central charge which I find difficult to grasp. This method uses a unitary projective representation $U(g)$ of a Lie group $G$.

$$U(g_1)U(g_2)=e^{i \phi(g_1,g_2)}U(g_1g_2)\tag{1}$$

Local coordinates $\{x^a \}$ near the identity element yield $$g1\cdot g2=g(x_1).g(x_2)=g(x_3(x_1,x_2))$$ with $$x^a_3(x_1,x_2)=x^a_1+x^a_2+\gamma^{abc}{x^b_1}x_2^c+\cdots\tag{2}$$ $$\phi(g_1,g_2)\equiv\phi(x_1,x_2)=\gamma^{bc}x_1^bx_2^c+\cdots\tag{3}$$ $$U(g(x))=1+ix^aT^a+\frac{1}{2}x^a x^b T^{ab}+\cdots\tag{4}$$ where $T^a$ is Hermitian and $T^{ab}=T^{ba}$.

Substituting equations $(2)$, $(3)$, and $(4)$ into equation $(1)$ yields: $$-T^cT^b= i \gamma^{cb}1+i\gamma^{acb}T^a+T^{cb}\tag{5}$$

By defining, $$f^{abc}\equiv \gamma^{acb}-\gamma^{abc} \quad f^{bc}\equiv \gamma^{cb}-\gamma^{bc}$$ $$[T^b,T^c]=i f^{abc}T^a+i f^{bc}1.\tag{6}$$ the term $f^{bc}$ is referred to as the central charge.

My questions are as follows:

1. This derivation heavily relies on coordinates and representations which I find unfamiliar. From what I understand, given a Lie group $G$, $T^a$ should be the tangent vector at the identity element, that is, $T^a \in T_e G = \mathfrak{g}$. The commutator of a Lie algebra should remain within the Lie algebra. Why then does the term $i f^{bc}1$ appear in equation $(6)$, given that $1$ is not an element of $\mathfrak{g}$?

2. The text appears to discuss a specific representation (1). For a given abstract Lie group $G$, the term $e^{i \phi(g_1,g_2)}$ cannot appear in the group product. Only after a projective representation (1) is found and $\phi(g_1,g_2)$ is non-zero, can the central charge be defined in this manner. However, the text also states that for a simply-connected Lie group, a projective representation can exist when the central charge of the Lie algebra is nontrivial. Isn't there a circular argument here?

Or perhaps the central charge is defined specifically for certain representations. If so, what are the necessary and sufficient conditions for a Lie algebra $\mathfrak{g}$ to have a representation with a nontrivial central charge?

3. What would be the geometric definition of a central charge? Surely there must exist a definition of central charge that does not depend on the representation or coordinates.

Answer 1

A brief summary of central extensions.

• Given a group $G$, a central extension of $G$ is another group, $\widetilde G$, such that $Z\subset \widetilde G$ is a central subgroup of $\widetilde G$, and $G\simeq \widetilde G/Z$. Of course, there is always a trivial example, $\widetilde G = G\times Z$. But we are specifically interested in central extensions which do not have the form of a direct product.

• We are interested in central extensions of groups because ordinary (single-valued) irreducible representations of $\widetilde G$ act as projective representations of $G$. A well-known example is this: we have ${\rm SO(3)}\simeq {\rm SU(2)}/\mathbb{Z}_2$. Here $\rm SU(2)$ is the central extension of $\rm SO(3)$ by $\mathbb{Z}_2$, and ordinary (single-valued) representations of $\rm SU(2)$ act as projective (double-valued, spinorial) representations of $\rm SO(3)$.

• We are now specifically interested in situations where $G$ is a Lie group. And here we must distinguish two possibilities:

• We have $G\simeq \widetilde G/Z$, where $G$ and $\widetilde G$ are both Lie groups, but $Z$ is a discrete subgroup of $\widetilde G$. In that case $G$ and $\widetilde G$ will have the same dimension and the same (i.e. isomorphic) Lie algebras, ${\rm Lie}(G)\simeq{\rm Lie}(\widetilde G)$. Thus the central extension will not be seen on the level of the algebras. This is exactly the case in the example mentioned above, ${\rm SO(3)}\simeq {\rm SU(2)}/\mathbb{Z}_2$. And in fact this is a general case: if a Lie group $G$ is connected but not simply-connected, there exists a "universal covering group" $\widetilde G$ such that $G\simeq \widetilde G/Z$ where $Z\simeq \pi_1(G)$, where $\pi_1(G)$ is the fundamental group of $G$. In fact we often write, with a slight but excusable abuse of notation, $G\simeq \widetilde G/\pi_1(G)$.

• The second possibility is that $G\simeq \widetilde G/Z$, but $Z$ is a Lie group as well. Then the central extension will be seen on the level of the algebras: the Lie algebra of $\widetilde G$ will be bigger than that of $G$. In fact we will have $\dim\widetilde G = \dim G + \dim Z$. The extra elements of ${\rm Lie}(\widetilde G)$ are called "central charges". Let's suppose that there is only one such central charge (this is actually the case in most physical examples). Suppose $G$ has structure relations $$ [T_a, T_b] = f^c{}_{ab}T_c $$ where $\{T_a\}$ are the generators of $G$ (a specific basis of ${\rm Lie}(G)$). Let us denote the additional generator of $\widetilde G$ — the central charge — by $C$. That is, ${\rm Lie}(\widetilde G)$ is spanned by $\{T_a, C\}$. Then the structure relations of $\widetilde G$ must take the form $$ [T_a, T_b] = f^c{}_{ab}T_c + h_{ab}C\,,\qquad [T_a, C] = 0 $$ where $h_{ab}$ must satisfy $h_{ab} = -h_{ba}$. In fact sometimes the constants $h$ and the generator $C$ are combined together, $f_{ab} = h_{ab}C$ and called "the central charge".

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Regarding the last case, a way to see that central extensions of Lie algebras produce projective representations of the groups is the following. Let $u$ be a particular irreducible representation of the centrally-extended Lie algebra; that is, $u(T_a)$ and $u(C)$ are operators acting on some vector space, and we have $[u(T_a), u(T_b)] = f^c{}_{ab}u(T_c) + h_{ab}u(C)$, $[u(T_a),u(C)]=0$. Now consider two group elements of the original group $G$, say $g$ and $g'$; suppose they can be written as $g = \exp \lambda^aT_a$ and $g' = \exp\lambda'^aT_a$. Now define operators $U(g) = \exp \lambda^au(T_a)$ and $U(g') = \exp \lambda'^au(T_a)$. It can then be shown that these operators provide a representation of the group $G$, but it is a projective representation, that is, $$ U(g)U(g') = \omega(g,g')U(gg') $$ where $\omega(g,g')$ is a numerical factor. One way to see this is to make use of the Baker–Campbell–Hausdorff (BCH) formula. Since the representation $u$ is irreducible, and $u(C)$ commutes with everything, by Schur's (2nd) Lemma it is a numerical constant, i.e. $u(C)$ acts as a constant on the representation space. It is then not difficult to ascertain that all factors involving $u(C)$ can be pulled out of the BCH expansion and collected into the term $\omega(g,g')$, which is thus indeed a numerical factor.

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Although the study of central extensions of groups and algebras is relevant in physics because of the connection to representation theory, such central extensions can be studied purely on their own, without any mention of representations. Thus, when studying central extensions of a given Lie algebra $\mathfrak{g}$ we are interested in Lie algebras $\tilde{\mathfrak{g}}$ such that $\mathfrak{g} \simeq \tilde{\mathfrak{g}}/\mathfrak{z}$ where $\mathfrak{z}$ belongs to the centre of $\tilde{\mathfrak{g}}$. Of course there are trivial examples where $\tilde{\mathfrak{g}}$ is a direct sum of $\mathfrak{g}$ and $\mathfrak{z}$, but we are interested in examples which do not have the form of a direct sum. (Note that $\mathfrak{z}$ can in theory have any dimension, which means that there will be more than one central charge; but in fact in most physically relevant examples there is only one central charge.) Note that the existence of such central extensions of Lie algebras is not a trivial subject; thus, it is known that if $\mathfrak{g}$ is semisimple, then $\mathfrak{g}$ does not admit non-trivial central extensions — this is sometimes known as Whitehead's (2nd) Lemma.

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Some references:

• Discussed briefly in Fuchs and Schweigert, Symmetries, Lie algebras and Representations, Cambridge University Press 2003 (see chapter 12, p. 199).

• An advanced treatment is de Azcárraga and Izquierdo, Lie groups, Lie algebras, cohomology and some applications in physics, Cambridge University Press 2008

Answer 2

The very well written summary of the topic by ACuriousMind which he linked to in the comments section should normally answer your question(s). You need to just put it all together, that is do some thinking by yourself. So don't jump to any of the texts below.

You asked me for some concrete literature on the subject which is probably also familiar to ACuriousMind - other than the source you quoted, i.e. the 2nd chapter of Weinberg I.

Let me see.

• I would start with the famous article by Valja Bargmann On Unitary Ray Representations of Continuous Groups officially at http://www.jstor.org/stable/1969831

• Then I should be prepared to do some extra reading from a diff. geom. / Lie group text such as Helgason, S. - Differential geometry, Lie groups and symmetric spaces (GSM 34, AMS, 1978) to be able to tackle the short generic monograph Parthasarathy, K. - Multipliers on Locally Compact Groups (LNM 93, Springer, 1969) and the applied monograph to the restricted Poincaré group which is Simms, D.J. - Lie Groups and Quantum Mechanics (LNM 52, Springer, 1968, ---+).

• Moving to newer literature, we have the monograph by Azcarraga, J., Izquierdo, J. - Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics (CUP, 1995) which could be the sum of all literature quoted in the other two paragraphs above and two below and the famous Geometry of Quantum Theory by Varadarajan (Chapter 7, Springer, 1968).

• A less technical approach to the Poincaré group compared to Simms is offered by B. Thaller in his book The Dirac Equation (Springer, 1992), section 2.4 which starts at page 62.

• The lightest version to the whole topic of projective representations and Lie group/Lie algebra extensions is offered by Morton Hamermesh in his book Group Theory and Its Applications to Physical Problems (Dover, 1962, Chapter 12).

• Last, but not least, it all started with the work by Chevalley and Eilenberg freely available at http://www.ams.org/journals/tran/1948-063-01/S0002-9947-1948-0024908-8/S0002-9947-1948-0024908-8.pdf