Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)

Takiff superalgebras are a family of non semi-simple Lie superalgebras that are believed to give rise to a rich structure of indecomposable representations of associated conformal field theories. We consider the Takiff superalgebra of gl(1|1), especially we perform harmonic analysis for the corresponding supergroup. We find that every simple module appears as submodule of an infinite-dimensional indecomposable but reducible module. We lift our results to two free field realizations for the corresponding conformal field theory and construct some modules.

The presence of logarithmic singularities is tightly connected to the non semi-simple action of the chiral algebra on some of its modules. In some models [26] of non chiral conformal theory related to string theory on AdS 3 × S 3 non semi-simplicity appears on the level of the assumed symmetry, and not only the representation level. In [27], motivated by this fact, different chiral conformal field theories were considered, where non semi-simplicity appears on the level of symmetry algebra itself. These are conformal field theories based on Takiff (super)algebras, e.g. algebras that are non semi-simple extensions of simple Lie (super)algebras by its adjoint representation. Detailed algebraic structure of its representation theory was then investigated.
In the mathematics literature, this type of non-semisimple Lie superalgebras were introduced by Takiff [28], though not in the super setting, but as part of an investigation of invariant polynomial rings. These algebras have since been considered in a slightly generalised form under the names generalised Takiff algebras [29,30] in which a semisimple Lie algebra is tensored with a polynomial ring in a nilpotent formal variable t, and truncated current algebras [30,31] in which one does the same to an affine Kac-Moody algebra. The algebras considered in [27] correspond to taking t 2 = 0, as in Takiff's original paper, and they were named Takiff superalgebras. These algebras were mainly considered algebraically from the representation theory point of view, in their chiral sector. On the other hand, it is well known that potential additional algebraic structures appear for Wess-Zumino-Novikov-Witten (WZNW) theories when representations are subjected to both chiral and anti chiral action of the group. In the present work, we initiate the study of the WZNW theory for the Takiff Lie supergroup GL(1|1).
Wess-Zumino-Novikov-Witten theories of non-compact Lie groups and Lie supergroups provide a rich source of logarithmic conformal field theories. Usually, one considers such theories based on a semi-simple Lie (super)group or closely related supergroups as GL(n|n) 3 . Experience shows that in such theories most modules are actually completely reducible and only a few non-generic modules are indecomposable but reducible. It is an interesting situation if already the Lie (super)algebra/Lie (super)group on which the conformal field theory is based is itself non semi-simple, as e.g. for Takiff (super)groups. In [27], the first author and David Ridout started to study such algebras from a conformal field theory perspective. Here, we'd like to continue their work with the aim of studying an example of a conformal field theory based on a Takiff superalgebra in detail. We believe that the case of gl (1|1) (the Takiff superalgebra of the Lie superalgebra gl (1|1)) is both treatable and instructive, so that we decided to first focus on this case. The chiral algebra of a WZNW model of a Lie group at level k is the simple affine vertex operator algebra of its Lie algebra g at level k. Frenkel and Zhu [32] have proven that there is a one-to-one correspondence of simple representations of this vertex algebra and simple representations of a quotient of the universal envelopping algebra of the Lie algebra g. If the vacuum Verma module of the affine vertex operator algebra is simple then there is actually a one-to-one correspondence to the simple objects of the universal envelopping algebra. As in the case of gl (1|1), the vacuum Verma module of the vertex algebra of gl (1|1) is simple for non-vanishing levels [27]. So, we expect that representations of gl (1|1) can teach us quite a bit about the conformal field theory. Very natural modules of the Lie superalgebra are functions on the Lie supergroup which are studied using harmonic analysis (often called mini superspace analysis in the CFT literature) on the Lie supergroup. This is actually a very general experience in WZNW theories based on Lie supergroups. The key strategy is to understand a problem of the finite-dimensional Lie superalgebra, to lift this understanding to the full conformal field theory, usually to a free field realization, and then to use the new understanding to derive interesting statements concerning the CFT. This strategy has been initiated by Hubert Saleur, Volker Schomerus and Thomas Quella a decade ago. It has been used extensively by various people, including the second author. For example, the key starting ingredient of the bulk theories of GL(1|1) [2], SU(2|1) [33], PSU(1, 1|2) [3] as well as the more general case [4] has been the harmonic analysis, and then lifting the findings to an appropriate free field realization. These free field realizations in turn were very useful in computing correlation functions [2], proving dualities to super Liouville field theories [34,35] and finding relations to CFTs with N = (2, 2) world-sheet superconformal symmetry [36]. Similarly also boundary theories based on GL(1|1) [37,38] and OSP(1|2) [39] have been studied by first performing an appropriate harmonic analysis. The difference to the bulk theory is, that instead of studying the action of both left-and right-invariant vector fields one investigates a twisted adjoint action. Boundary states are then lifts of distributions localized on twisted superconjugacy classes [37,40].
The by far best understood WZNW theory of a Lie supergroup is the one of GL(1|1). It has been first studied by Rozansky and Saleur a while ago [1]. Then as mentioned, Schomerus and Saleur, were able to understand spectrum and correlation functions using harmonic analysis and free field realization [2]. As the next step boundary states were constructed [37]. Studying the twisted adjoint action of invariant vector fields on functions on the supergroup allowed one to find the boundary states explicitely. Boundary and bulkboundary correlation functions could also be computed using a similar free field realization as in the bulk case [38]. The indecomposability structure of modules is not directly visible in the free field realization used in the just mentioned works. It turns out that there is another free field realization using symplectic fermions that makes this structure much clearer [41,42]. Using then the work of Kausch on symplectic fermions [43] most of the previous results could be obtained fairly directly [41]. This conformal field theory can also be studied from a more algebraic perspective, especially many extended algebras like sl(2|1) at levels −1/2 and 1, can be constructed and studied [44][45][46]. In summary, the GL(1|1) WZNW theory has been thoroughly studied and has revealed a rich structure. We believe that a similar (but much more complex) story holds for its Takiff superalgebra. In [27], representations of both finite-dimensional and affine Takiff superalgebra of gl (1|1) have been studied and a Verlinde fusion ring has been computed. We now would like to proceed to the next step, and ask which representations appear in the full bulk CFT, and wether we can realize them using free fields. Our main result is a thorough harmonic analysis of the Takiff supergroup. We find that functions split into three classes, called typical, semitypical and atypical. All of them are part of infinite-dimensional indecomposable but reducible modules. Nonetheless, in the typical case there is still a decomposition into tensor products of modules for the action of left-and right-invariant vector fields somehow similar as one is used from the Peter-Weyl theorem for compact Lie groups. Both semitypical and especially atypical modules have a rather complicated behaviour under the left-right action. We visualize the structure of these modules in various pictures and exact sequences. Further, already the simple submodules of the typical modules posses Jordan cells of size three under the action of the Laplacian (as expected from [27]). The harmonic analysis tells us nicely how to construct a free field realization including screening charges. As in the case of the GL(1|1) WZNW theory, typical modules can be constructed easily using this free field realization. Again, in analogy to the GL(1|1) WZNW theory we find a second free field realization using symplectic fermions which allows us to also construct semitypical and atypical modules. This puts us now in a position to study (in the near future) these modules further, to ask questions about correlation functions, operator product algebra and extended algebras.
This work is organized as follows: Section 2 is the main part of this paper. We first compute basic objects on the Takiff supergroup, namely invariant vector fields, invariant measure and Laplacians. Then we compute the action of invariant vector fields. The typical case can be performed rather directly, while the structure in the semitypical and atypical cases is very rich and first studying the action of various subalgebras turns out to structure the problem nicely. Section 3 then applies our findings to construct a free field realization including screening charges. But we also add another free field realization using symplectic fermions. Finally, we conclude with describing the expected future use of our findings.

Harmonic Analysis
We start from definition of Takiff superalgebra GL(1|1), and briefly describe its highest weight representations. The main part of this section is devoted to harmonic analysis for GL(1|1) supergroup. The procedure for this follows ideas established in various works on GL(1|1) (see mainly [2]) and goes as follows: (1) Consider a supergroup element g according to a triangular decomposition; (2) compute left-and right-invariant vector fields; (3) compute the invariant measure; (4) compute the Laplacians; As one can see the adjoint action of N, E,Ẽ is diagonalised in the chosen basis of generators, but the adjoint action of N acts non semisimply on adjoint module. The obvious triangular decomposition gives rise to highest weight modules: the highest weight is defined as an eigenvector of N, E,Ñ,Ẽ with eigenvalues n, e,ñ,ẽ correspondingly, which is annihilated by the action of raising generators ψ + ,ψ + . The highest weight Verma module is defined as a free action of lowering generators ψ − ,ψ − on the highest weight. Since ψ − ,ψ − are Grassmann odd and anticommute, the Verma module is four dimensional. All the states of the Verma module with highest weight |v are eigenstates ofÑ except for ψ − |v on which it acts non semisimply: As it was shown in [27], there are three possibilities for irreducible quotients of Verma module. If e =ẽ = 0, the irreducible quotient is one dimensional and was called atypical, ifẽ = 0, but e = 0, it is two dimensional and was called semitypical, and ifẽ = 0, the Verma module is irreducible. In a similar way one can define lowest weight Verma modules and their irreducible quotients.
There are two linearly independent quadratic Casimir operators in the universal enveloping algebra of gl (1|1) (modulo polynomials in central elements) which can be chosen as They act on the highest weight Verma module as multiplication by nẽ +ñe,ñẽ respectively.
An important object for affinization of Lie (super)algebras is a non-degenerate symmetric bilinear form κ on the algebra. Among different possibilities, we choose the lifting of the standard bilinear form κ 0 (X, Y ) of gl(1|1) (defined as supertrace str(XY ) in the defining representation of it) as follows:

Invariant Vector Fields.
We choose a Lie super group parameterization according to above triangular decomposition Invariant-vector fields on Lie super groups have to be treated with attention to signs to (see [47]). We define them by for all X in the complement of the ψ + ,ψ + . This definition guarantees that the invariant vector fields obey the relations of gl (1|1) Then a computation reveals that In the same way for the left action we require L − g = −gψ − ,L − g = −gψ − and L X g = Xg for all X in the complement of the ψ − ,ψ − . Then one gets So that for the left action we have 2.3. The Haar measure. The left-invariant Maurer-Cartan form is ω(g) = g −1 dg, and the right-invariant Maurer-Cartan form is ω(g −1 ). Either one can be taken to compute the Haar measure and we will use the right-invariant one This is a Lie super algebra valued one form, that is it can be written as The components are the dual one-forms in our basis. Using (2.1) they can be easily extracted The right-invariant measure is the wedge product of these dual one forms Here, we used graded anti-symmetry of the wedge product as well as integration with respect to an odd variable is the same as taking the derivative, but the double derivative of an odd variable vanishes, in other words we have used dθ ∧ dθ = 0.
2.4. The Laplace Operators. In [27] two Casimir operators of gl (1|1) were given. We change one of them by the central elementẼ. They are then We compute that We call these operators ∆ as they are the Laplace operators on this supergroup. This finishes our preparations, and we can turn to decomposing functions with respect to the left-right action of invariant vector fields.
relates the action of left-and right-invariant vector fields as It will turn out that there are three-types of modules appearing in the decomposition. We will call these modules typical, semi-typical and atypical as in [27].
then the crucial Lemma is: and the second order differential operator Letẽ = 0, then each row of the matrix V e,n,ẽ,ñ = f e,n,ẽ,ñ exp e iỹ e D e,ẽ M carries the irreducible highest-weight representation of the left regular action of highest-weight (e,ẽ, −n + 2, −ñ). Each column transforms in the irreducible highest-weight representation of the right regular action of highest-weight (−e, −ẽ, n,ñ).

Proof:
We define the function f 4 = θ −θ− f e,n,ẽ,ñ , thenf 4 is a highest-weight vector for both left-and right action. The weight under the right action is (−e, −ẽ, n,ñ) and the weight under the left action is (e,ẽ, −n + 2, −ñ). Then under the left-action, the functions carry the four-dimensional irreducible representation of that highest-weight. Since left-and right-action commute, each of these states must be a highest-weight vector of same weight as before for the right action. For the highest weight f 1 the remaining three states of the typical module I areψ In the same way three states completing the module II with the highest weight f 2 arẽ For the module III with the highest weight f 3 we get n−1,ẽ,ñ θ + −ẽf e,n−1,ẽ,ñθ+ + iẽỹf e,n−1,ẽ,ñ θ + +ẽ 2 f e,n,ẽ,ñθ− θ +θ+ , and for the last module IV with highest weight f 4 The action of R N is the same on all four four modules: The action ofR N is diagonal on the states X = f i ,ψ, b:R N X =ñX, but non diagonal on the states ψ:R N ψ =ñψ −ψ In summary, forẽ = 0, each row of the matrix f e,n,ẽ,ñ carries the irreducible highest-weight representation of the left regular action of highestweight (e,ẽ, −n + 2, −ñ). Each column transforms in the irreducible highest-weight representation of the right regular action of highest-weight (−e, −ẽ, n,ñ). Changing basis ψ → ψ − ẽ eψ , this can be written more symmetrically, Remark. The operator e iy D e,ẽ is the semi-classical analouge of a screening charge. it is related to the Casimir operators acting on V e,n,ẽ,ñ via In other words the difference is a semi-simple operator. The proof of the previous Lemma tells us that under this linear combination of Laplacians the typical modules form Jordan blocks up to size three. e,n,ẽ,ñ has been described in Lemma 2.2 and all factors of the series associated to this long-exact sequence are isomorphic to this submodule.

Semitypical
Modules. The caseẽ = 0 and e = 0 is called semi-typical. Our strategy is to first decompose into irreducible gl (1|1) modules. First, fix e = 0,ñ. In order to avoid to many indices we take the short-hand notation f n := f e,n,0,ñ We find the following list for the left-action L ± on weight modules: We then also define for non-negative integer m TheL ± action gives gl (1|1)-module one-to-one maps, that only change the L N -action, TheL N -action on gl (1|1)-modules is described as follows.L N splits into a semi-simple and a nilpotent part. The semi-simple part acts by multiplication by −ñ, while the nilpotent part acts as follows By this we mean that the gl (1|1) action on the image of the nilpotent part is the right-hand side. We also understand A (−1) i,n = 0. An efficient way to picture these computations is the following  This redefinition preserves the sl (1|1) action. However the nilpotent part ofL N acts as We thus can considerably improve the previous proposition. While L N -acts semi-simple on these modules, the nilpotent part ofL N acts as described in equation (2.2). We can split it into two piecesL nil Figure 1. The solid arrows to the right denote the action ofL − , those to the left ofL + . While the dotted arrows to the right indicate the action ofR + and those to the left the one ofR − .
The long-exact sequence describes the action ofÑ a , In order to understand the left-right action, we first have to understand the left-right gl (1|1) action.
Definition. The highest-weight representation of gl (1|1), with highest weight (e, n) and lowest-weight (e, n − 1) will be denoted by ρ e,n− Let nowẽ = e = 0. We will start by studying the left-right gl (1|1) action. We define w (2,2),n = θ −θ− θ +θ+ f n , w (2,−),n =θ − θ +θ+ f n , Further, define the one-dimensional vector spaces is the tensor product of gl (1|1) projective covers, We call the first one U   Here P 1 n (0) denotes the module generated by P + n (0) and P − n (0) whose intersection is the ideal P 0 (1) . These modules are constructed such that the tilded action is easy to read of, theL ± andR ± are sl (1|1) module homomorphism (they only change the action of L N or R N ) and for m > 0 they act as while in the m = 0 case we have The image of these maps is in the kernel and we inspect that it is precisely the kernel, hence bothL + andR + action is described by the long-exact sequences · · · −→ P n

Free Field Realization
In this section we construct free field realization of the holomorphic and anti-holomorphic currents of affine Takiff super Lie algebraˆ gl (1|1), and its modules. It will be done by affinization of constructions worked out in the previous section. The Takiff algebra admits an automorphism ω α for α in C. It is given by and leaving all others invariant. It allows us to restrict to the bilinear formκ. In order to see this, consider the bilinear form κ β :=κ + βκ 0 , then for all X, Y in gl (1|1). In [27], the affinization of gl (1|1) was discussed and levelsk and k associated to the bilinear formsκ and κ were introduced. Above automorphism relates the affine vertex algebra of gl (1|1) at levels (k, k) to the one at levels (k, 0). and all other OPEs are regular. We also need free bosons (generators of the Heisenberg VOA) taking values in the even part of the Takiff superalgebra. Its components have OPE in their holomorphic part 3.3. Free field realization of typical modules. We can now combine holomorphic and anti-holomorphic ghost fields with the vertex operator introduced above to form bulk free field realization for typical modules. It is convenient to represent it in the matrix form φ e,n,ẽ,ñ (z) =:Ỹ m V e,n,ẽ,ñ : (z,z)     1cccc c cc cc ccc cccccccc cc ccc ccc cccc     : Here each row and column of the matrix C is understood as a span of its elements. Using the OPEs one can easily check that columns carry typical representations for the holomorphic currents and the rows -for the anti-holomorphic ones. The elements C 1,i are the highest weights for the columns, and the elements C i,1 -for the rows. For m = 0 they are Virasoro primary fields with conformal dimension (3.4).

3.4.
A free field realization for atypical and semitypical modules. Consider two pairs of symplectic fermions χ ± ,χ ± with OPE and a pair of pairs of free bosons ∂X, ∂Y, ∂X , ∂Ỹ with OPE Recall, that sympplectic fermions are a logarithmic conformal field theory. Especially, the vacuum is part of a larger indecomposable module whose four composition factors are all isomorphic to the vacuum itself, for details see [43]. The generating fields of these modules are Ω ± (z,z),θ ± (z,z), θ ± (z,z) with the following behaviour under the action of the symplectic fermions: .
So in all these cases, we can use our free field realization to construct indecomposable but reducible modules. In the case of m = 0, we have both highest and lowest weight modules. These modules now deserve further investigation. We plan to study the combined left-right action of currents on these modules in the near future. The results then have to be compared to the semi-typicals of our harmonic analysis. Finally, let us note that there is a third free field realization simply by replacingỸ (z) and ∂X(z) by a βγ bosonic ghost algebra. This one will also be investigated later.

Conclusion
This work together with [27] puts one in the situation to study conformal field theories with chiral algebra the affinization of gl (1|1). We have studied the harmonic analysis on the supergroup in detail and especially we have found that all modules are part of larger indecomposable but reducible modules. We then have found two free field realizations for the associated CFT. The next question is to understand the spectrum of the full CFT better. For this, one needs to study the free field modules in detail. Especially the structure of semi-typicals and atypicals from the symplectic fermion free field realization needs to be worked out. In logarithmic CFT, indecomposable structure of modules is related to logarithmic singularities in correlation functions. Screening charges provide integral formula for correlation functions, so that one can use our findings to compute those. Since all our modules in the harmonic analysis are submodules of indecomposable but reducible ones we expect that logarithmic singularities will be a generic feature. This will be in strong contrast to logarithmic CFTs based on non Takiff supergroups.
There are many future directions as boundary states and boundary correlation functions. These can surely be studied along the lines of their non-Takiff analouges [37,38]. Recall, that in that case also harmonic analysis was the important starting point. We are more interested in Takiff superalgebra CFTs based on more complicated algebras. It was shown that gl (1|1) has many interesting simple current extensions [44,45], including sl (2|1) at levels 1 and −1/2. Characters of all these extensions turned out to be mock Jacobi forms for atypical modules [46]. It is probable that the Takiff superalgebra of sl (2|1) appears in extensions of the gl (1|1) CFT, and interesting mock modular-like objects are expected to appear as characters of extended algebra modules.
It is also obvious that more realistic applications of Takiff affine algebras and superalgebras will require detailed understanding of Takiff affine sl (2). It seems natural to start its investigation from harmonic analysis, as a mini superspace toy model. One can expect here more involved structures of modules under the action of left-and right-invariant vector fields on the group manifold, since the representation theory of sl(2) consists of infinite dimensional modules. It is also natural to expect that its free field realization will contain usual Wakimoto sl(2) realization embedded into it.