The Topological-Anti-Topological Massive Metric

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Twisted Non-Compact Models

There have been a number of studies on the topological twist of massive N = (2, 2) supersymmetric theories in two dimensions with discrete spectra (e.g. [20, 22–24]). In this chapter we will generalize these analyses to N = (2, 2) supersymmetric two-dimensional theories to theories with a non-compact target space. We will focus on the N = 2 super-Liouville theory which is T-dual to an SL(2, R)/U(1) coset superconfor-mal field theory [25]. This is the conformal field theory side of the holographic pair that will be studied in the next chapter.
We will determine the chiral ring elements in the presence of relevant operator de-formations and then compute the correlation functions to get the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. The topological-anti-topological sec-tor of the deformed N = 2 Liouville theory will also be investigated, where we will study the tt∗ equations for the deformed theory. The following main part of this chapter and appendices A and B will be contents extracted from the paper [26] I coauthored with J. Troost.

The Topological Conformal Field Theory

In this section, we describe topological conformal field theories. Firstly, we recall the topological conformal field theories that arise from twisting the N = 2 minimal model conformal field theories [20, 22–24]. Secondly, we introduce non-compact counterparts (see e.g. [27–32] for necessary background).

The Topological Compact Model

Quantum field theories in two dimensions with N = (2, 2) supersymmetry are sufficiently constrained to frequently allow for the identification of their interacting infrared fixed points. An example of such a succesful identification is the proposal that the N = 2 minimal model conformal field theory is the infrared fixed point of a N = 2 Landau-Ginzburg model with as field content an N = (2, 2) chiral superfield Φ, and interactions dictated by the superpotential W = Φkc /kc. The positive integer power kc is related to the central charge of the N = 2 minimal model by the relation c = 3 − 6/kc. We will refer to the integer kc as the level of the model. 1 Considerable evidence for this identification was amassed over the years [22–24].
Part of the evidence lay in the understanding of the topological subsector of the infrared fixed point, the topological conformal field theory [20]. We briefly review the properties of this topologically twisted compact N = 2 superconformal field theory. Our starting point is the N = 2 minimal model, which can be thought of as a supersym-metric sigma-model on the coset SU(2)/U(1). We concentrate on the A-type diagonal modular invariant. We moreover focus on the (chiral,chiral) ring [16] of the minimal model at central charge c = 3 − 6/kc. It is made up of the operators in the spectrum whose conformal dimension is half their R-charge, both for the left- and the right-movers. In the NSNS sector, they can be identified as arising from the parent SU(2) operators with spin j which have angular momentum m equal to their spin, j = m. The spin j runs over the values j = 0, 1/2, . . . , (kc −2)/2 where kc −2 is the level of the bosonic SU(2) current algebra of the parent theory on the group SU(2). The chiral ring thus has kc −1 elements that have R-charges in the set {0, 1/kc, 2/kc, . . . , 1−2/kc} in the NSNS sector. The ring structure constants agree with those of the polynomial ring C[X] modded out by the ideal generated by Xkc−1, where X has R-charge 1/kc. The ring has a unit. In the Ramond-Ramond sector, after spectral flow by half a unit in the appropriate direction, the chiral primary operators map to ground states as we have seen in chapter 1. These have R-charges shifted down by c/6 = 1/2 − 1/kc and therefore take values in the set {−1/2 + 1/kc, −1/2 + 2/kc, . . . , 1/2 − 1/kc} which runs from −c/6 to +c/6 with increments of 1/kc. The topological metric, defined to be the expectation value of two chiral operators, which we can identify with Xi and Xj, in the topologically twisted theory on the sphere equals (see e.g. [21]) ηij = hXiXji0 = δi+j,kc−2 . (2.1)
We note that the vacuum in this compact theory is normalizable, and that the vacuum state, with conformal dimension and R-charge both equal to zero, survives the topolo-gical twist. The structure constants of the ring are related to the three-point functions via the topological metric, and they equal
cijl = δi+j,l
cijl = hXiXjXli0 = cijmηml = δi+j+l,kc−2 , (2.2)
where i, j, l need to be in the spectrum in order for the structure constants or three-point function to be non-zero. This information is sufficient to describe all correlators of the topological conformal field theory on any Riemann surface, by cutting and sewing. The topological conformal field theory has an alternative description in terms of a twisted N = 2 Landau-Ginzburg model. The Landau-Ginzburg theory has a superpotential W = Xkc (2.3) and the chiral ring is again the polynomial ring C[X] modded out by WX = dWdX = Xkc−1.
1. We denote the level of the compact model by kc, to contrast with the level of the non-compact model which will be denoted k.
Establishing the dictionary for these supersymmetric quantum field theories took several years, and ingenious checks on the identification of the fixed points were perfor-med. We refer to the extensive literature for detailed discussions. Our lightning review mostly serves as a point of reference for the models to come, which are of an entirely different nature, yet show remarkable resemblances.

The Topological Non-Compact Model

More than twenty years ago, it was suggested that the correspondence between Landau-Ginzburg models and conformal field theories could be usefully extended to non-compact models, with continuous spectrum and a central charge c > 3 [33]. See also [34–36]. The N = 2 Liouville theory can indeed be understood as a linear dilaton theory, with an exponential supersymmetric potential which is marginal. Importantly, the linear dilaton profile fixes part of the asymptotics [25] and renders the superpoten-tial term consistent with conformal symmetry. These are two aspects that differ from the compact theories. It was also shown that the N = 2 Liouville theory is T-dual to the N = 2 coset conformal field theory on the cigar SL(2, R)/U(1) [25, 37]. Earlier on, a more rudimentary probe of the N = 2 Liouville theory, namely the Witten index (de-fined as a periodic path integral on the torus), was computed in [38], and demonstrated to depend on the choice of asymptotic radius. Given an N = 2 Liouville theory of cen-tral charge c = 3 + 6/k, where k is a positive integer, the natural choices of asymptotic radii are multiples of pα0/k. The Witten index is equal to the chosen multiple. Since then, our understanding of the spectrum of the SL(2, R)/U(1) conformal field theory has progressed both through an analysis of the bosonic partition function [40] and its supersymmetric counterpart [29, 32], as well as through an analysis of the elliptic genus [31].
We will consider the topologically twisted N = 2 Liouville theory at central charge c = 3 + 6/k and with an asymptotic radius equal to kα0. We choose the level k to be a positive integer. We make this choice of radius since we want the number of ground states to be equal to the level (minus one). The theory is a N = (2, 2) theory with chiral superfield Φ and superpotential W = µ exp(bΦ) (2.4) p where b = k/2, and the theory is supplemented with a linear dilaton at infinity of slope 1/b. Establishing the chiral ring of the theory is subtle. (See also [30] for a discussion.) The Witten index calculation [38] cleverly cancels a volume divergence (as detailed in [31]), and gives a result equal to k (namely, the number of coverings of the minimal circle radius pα0/k). The potential Ramond-Ramond ground states are identified in the NSNS (chiral,chiral) ring of the conformal field theory as follows. The N = 2 Liouville theory under consideration is T-dual to a cigar coset supercon-formal field theory on the coset SL(2, R)k/U(1) [25, 37], modded out by Zk. In the coset conformal field theory we can identify the (chiral,chiral) states of the original Liouville theory as arising from discrete representations Dj+, with lowest weight state with spin component j. The list of possible values for the spin j contains at least the set j = 1, 3/2, . . . , (k − 1)/2, k/2. The supplementary values j = 1/2 and j = (k + 1)/2 correspond to almost, but not quite normalizable states, in the following sense. The discrete lowest weight representation D1+/2 is sometimes referred to as mock discrete. It does not arise in the Plancherel formula for the representation space decomposition of quadratically integrable functions on the group manifold SL(2, R). Intuitively, it is analogous to the ground state of conformal field theories in two dimensions with any non-compact target space, for instance a flat space Rn. The ground state norm is infinite, and proportional to the volume of the target space. Similarly, in Liouville theory, the state at the end of the continuous spectrum (namely in the representation D1+/2) is almost, but not quite, normalizable. Spectral flow then implies that the same type of argument applies to the representation with j = (k + 1)/2 [41]. The bounds on the spin were first suggested in [42] and firmly established in [41] and [36].
This subtlety is also manifest after regularizing the conformal field theory with a linear deformation in Φ, as analyzed in [38] : Wm = µ exp(bΦ) − mΦ . (2.5)
In [38] it is shown that after regularization with the mass term, only k −1 normalizable ground states can be identified. Relatedly, in the calculation of the non-compact elliptic genus [31] a regularization choice decides on whether the j = 1/2 (or j = (k + 1)/2) state contributes to the holomorphic part of the elliptic genus, or not. Thus, the volume divergence contaminates the ground state counting and even the supersymmetric index must be interpreted with care. See [43] for a much more detailed discussion of closely related intricacies.
Despite the subtleties we encounter, we can draw a number of conclusions at this stage. First of all, the unit operator is not normalizable, and is not in the topological cohomology. The chiral ring is naturally without unit. Nevertheless, it is true that all rings can be rendered unital by adding a unit (operator) by hand. If we do, we must remember that there is no (normalizable) state-operator correspondence for the unit operator. Secondly, the ring must contain the elements with R-charge {2/k, 3/k, . . . , 1} which are strictly normalizable. They correspond to the operators exp(2nb Φ) where n = 2, 3, . . . , k. There is another candidate operator at j = 1/2, namely exp(21b Φ) corresponding to (h, q) = (1/2k, 1/k), which permits the interpretation as the state in which to evaluate topological field theory correlators. It is a zero momentum state (since j = 1/2), much as the SL(2, C) invariant vacuum in a conformal field theory with target space Rp. Importantly, it does carry a conformal dimension and R-charge. Finally, it has a counterpart at j = (k + 1)/2 (that arises from spectral flow of a D1−/2 representation in the parent SL(2, R) theory [41]).
To further discuss the operator ring, it is handy to introduce the field theory va-riable Y −1 = e 1 Φ , (2.6) 2b in terms of which the superpotential (2.4) reads W = Y −k , (2.7) k for the choice µ = 1/k. This superpotential has been proposed as a starting point for analysis a while back [33], but it was mostly made sense of directly at the conformal fixed point [25], in the field variable Φ. One of the goals in this chapter is to show that indeed, the negative power superpotential (2.7) leads to a useful and efficient description of the topological model. The variable Y will be handy, despite having a non-canonical kinetic term, and various other perturbing features.
In these variables then, we will formulate and study two rings. The first ring is the (strict) chiral ring R, given by linear combinations of the set {Y −2, Y −3, . . . , Y −k }. We can think of the chiral ring as a subspace of the ring C[Y −1] of polynomials in Y −1. Moreover, the chiral ring is a subspace of the ideal Y −2C[Y −1] which has no unit. In the latter ideal, the chiral ring is the quotient by the ideal hY −k+1i generated by Y −k+1 = −Y 2WY . Indeed, note that we generate only powers Y −k −1 or higher by multiplying Y −k+1 by elements in Y −2C[Y −1]. We can alternatively think of the chiral ring as being obtained by setting Y −k−1 to zero, and we denote these state-ments as R = Y −2C[Y −1]/hY −k+1i = Y − 2C[Y −1]/{Y −k−1 = 0}. The (strict) chiral ring is made up of all linear combinations of the k − 1 monomials in the chiral ring that are strictly normalizable. The Ramond-Ramond sector R-charges of the opera-tors are {−1/2 − 1/k, . . . , 1/2 − 1/k}. 2 The second ring is the extended chiral ring Rext, which consists of operators that are linear combinations of the k + 1 opera-tors {Y −1, Y −2, . . . , Y −k, Y −k−1} , with a standard multiplication rule, and such that Y −k −2 is equivalent to zero. Again we can describe the extended chiral ring abstractly. The ring of polynomials has an ideal Y −1C[Y −1] generated from the monomial Y −1. This ideal is a subring without unity. In the latter ring, we consider the ideal hWY i generated by WY = −Y −k−1. We can then consider the quotient ring of Y −1 C[Y −1] modded out by the ideal hWY i. Again, note that the first element we will put to zero is Y −k−2, since we multiply elements of the ring into the ideal, and the lowest order element in the ring is Y −1. Thus, we can symbolically write the extended chiral ring as Rext = Y −1 C[Y −1]/hWY i = Y −1 C[Y −1]/{Y −k−2 = 0}. The R-charges of the extended chiral ring elements in the NS sector are {1/k, 2/k, . . . , 1, 1 + 1/k}. In the Ramond-Ramond sector the R-charges are {−1/2, −1/2 − 1/k, . . . , 1/2 − 1/k, +1/2}. The set of R-charges is symmetric under charge conjugation. In both rings, there is an operator that allows for supersymmetric marginal deformation of R-charge 1 – it is the original superpotential term. Both the strict and the extended chiral rings will be conceptually useful.
Our next step in defining the non-compact topological conformal field theory is to find a non-zero, non-degenerate symmetric bilinear form η on the ring in the guise of a spherical two-point function for the generating operators Y −i. We wish to define the topological metric ηij again as a two-point function ? , (2.8) ηij = h0|Y −iY −j|0i0 but there are several reasons why this is not entirely trivial to make sense of. The first reason is that the state |0i is the non-normalizable SL(2, C) invariant state, which belongs neither to the original, nor to the twisted theory. Therefore, rather than take the expression (2.8) literally, we interpret the expression as referring to the expectation value ηij = hY −1|Y −i+1Y −j+1|Y −1i0 , (2.9) where the state |Y −1i in which we take the vacuum expectation value is the state cor-responding to the operator Y −1, namely the almost-normalizable state with conformal dimension and R-charge (h, q) = (1/2k, 1/k). Secondly, we analyze R-charge conser-vation in the topologically twisted theory. The charge conservation equation for to-pological correlators evaluated in the SL(2, C) invariant ground state is, as we have discussed in chapter 1,
qi + (qi − 1) = (1 − g) (2.10)    i=1 i=r+1
where g is the genus of the Riemann surface on which we compute the correlators, the charges q1,…,r are the R-charges of the unintegrated vertex operators, and qr+1,…,r+n are the R-charges of the integrated vertex operators. The R-charge at infinity on the right hand side arises from the twisting of the energy momentum tensor by the derivative of the R-current. In our context, the R-charge conservation must moreover be modified appropriately by the R-charge of the states in which we take the expectation value (see equation (2.9)).
In practice, for the spherical two-point function we need qi + qj = c/3 . (2.11)
It is thus natural to propose the result ηij = δi+j,k+2 . (2.12)
Let us briefly discuss this result. One way to view the result is as an expectation value for the operator Y −k−2, which may be surprising since the operator is formally equivalent to zero in the chiral ring. An alternative view on the expectation value is that we evaluate the expectation value of the operators Y −k in the (almost-normalizable) state corresponding to the operator Y −1. Thus, the seemingly surprising feature of giving a vacuum expectation value to the operator Y −k−2, which is set to zero in the ring under the constraint −Y −1WY = Y −k−2 ≡ 0 can either be viewed as due to the strong divergence in the norm of the state |0i or, alternatively, can be read as corresponding to the vacuum expectation value of (non-zero) operator Y −k in the state generated by Y −1. We will provide more detail to the last interpretation shortly.
Meanwhile, we observe that the metric ηij provides a mild cross check on the identification of the extended and strict rings. Indeed, the topological metric is a non-degenerate symmetric bilinear form on the extended ring Rext as well as on the vectorial subspace corresponding to the ring R.
Finally, given the metric η and charge conservation, it is natural to also propose the three-point function cijl = δi+j+l,k+2 . (2.13)
We will further argue these proposals in subsection 2.1.3, and we will derive interesting consequences of accepting them in section 2.2.

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Small Deformations

In this subsection, we study how the topological conformal field theories behave under a small deformation that gives rise to separated massive vacua. The analysis provides extra insight into the features typical of the non-compact model.

Table of contents :

1 Basics 
1.1 Chiral Ring
1.2 Topological Conformal Field Theories
1.3 Summary
2 Twisted Non-Compact Models 
2.1 The Topological Conformal Field Theory
2.1.1 The Topological Compact Model
2.1.2 The Topological Non-Compact Model
2.1.3 Small Deformations
2.2 The Topological Massive Model
2.2.1 The Compact Topological Field Theory
2.2.2 The Non-Compact Topological Field Theory
2.3 The Conformal Metric
2.4 The Topological-Anti-Topological Massive Metric
2.4.1 The tt Equation
2.4.2 A Family of Theories
3 AdS3/Liouville Duality 
3.1 Conformal Field Theory and Metrics
3.1.1 The Energy-momentum Tensor and the Metric
3.1.2 Energy-momentum Tensor Expectation Value
3.1.3 Energy-momentum Tensor on the Cylinder
3.2 Universal Metrics
3.2.1 The BTZ Black Hole Metric
3.2.2 Properties of the Metrics
3.2.3 On Monodromies
3.3 Liouville Quantum Gravity
3.3.1 Properties of Liouville Theory
3.3.2 The Holographic Interpretation
3.3.3 Black Hole Correlators
3.3.4 One Loop Correction
3.4 Supersymmetric AdS3/Liouville
3.4.1 Bulk and Boundary Actions
3.4.2 The Exact Solutions
3.4.3 Supersymmetric Liouville Quantum Gravity
3.5 A Twisted Holography
3.5.1 The Twisted Supergravity Theory
3.5.2 The Topological Conformal Field Theory on the Boundary
3.5.3 The Gravitational Chiral Primaries
3.5.4 The Bulk Chiral Ring
4 Twisted String Theory in Anti-de Sitter Space 
4.1 The Generating Function
4.2 The Three-Dimensional Stringy Geometries
4.3 The twisted background
4.3.1 Degrees of Freedom
4.3.2 Flat Generalizations
4.3.3 The Explicit Boundary Gauge Field Dependence
4.3.4 The Explicit Boundary Metric Dependence
4.3.5 The Asymptotic Twisted Generalization
4.4 The Space-Time Energy-Momentum Operator
4.4.1 The Energy-Momentum
4.4.2 The R-Symmetry Current
4.4.3 The Topological Energy-Momentum Tensor
5 Topological Orbifold Theory 
5.1 The Complex Plane
5.1.1 The Hilbert Scheme of Points on the Complex Plane
5.1.2 The Topological Conformal Field Theory
5.1.3 A Plethora of Results on the Cohomology Ring
5.1.4 The Interaction
5.1.5 The Structure Constants are Hurwitz Numbers
5.1.6 The Broader Context and a Proof of a Conjecture
5.1.7 Summary and Lessons
5.2 Compact Surfaces
5.2.1 The Orbifold Frobenius Algebra
5.2.2 Cup Products of Single Cycle Elements
5.2.3 Cup Products at Low and High Order
5.2.4 Genus One and an Overlap of Three
5.2.5 Further Examples
5.2.6 Remarks


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