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A fundamental domain for a Schottky group. Each generator γ i , i = 1,. .. , g, with fixed points a i , r i , maps the circle C −i to the circle C i .
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A systematic analysis of the genus two vacuum amplitudes of chiral self-dual
conformal field theories is performed. It is explained that the existence of a
modular invariant genus two partition function implies infinitely many
relations among the structure constants of the theory. All of these relations
are shown to be a consequence of the associat...
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In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of int...
Citations
... There have been numerous works on the relation between extremal CFTs and pure gravity. See, e.g.,[68][69][70][71][72][73][74][75][76][77][78][79] for the developments. ...
A bstract
We consider chiral fermionic conformal field theories constructed from classical error-correcting codes and provide a systematic way of computing their elliptic genera. We exploit the U(1) current of the $$ \mathcal{N} $$ N = 2 superconformal algebra to obtain the U(1)-graded partition function that is invariant under the modular transformation and the spectral flow. We demonstrate our method by constructing extremal $$ \mathcal{N} $$ N = 2 elliptic genera from classical codes for relatively small central charges. Also, we give near-extremal elliptic genera and decompose them into $$ \mathcal{N} $$ N = 2 superconformal characters.
... As in the chiral case, the partition function will be constructed by a sum over vectors in a lattice defined by the code, and it will be described in terms of higher-genus theta functions with known modular properties. The theory of such functions is less developed compared the chiral case, where indeed the theory of Siegel modular forms has led to strong constraints on meromorphic CFTs at low c [20][21][22][23][24]. Nevertheless, we shall see that the set of partition functions in the form dictated by (1.3) does capture some interesting theories also outside the class of code CFTs, 4 to be discussed more in section 5.3, which may hint at the possibility to develop a theory of non-chiral modular forms. ...
A bstract
Code CFTs are 2d conformal field theories defined by error-correcting codes. Recently, Dymarsky and Shapere generalized the construction of code CFTs to include quantum error-correcting codes. In this paper, we explore this connection at higher genus. We prove that the higher-genus partition functions take the form of polynomials of higher-weight theta functions, and that the higher-genus modular group acts as simple linear transformations on these polynomials. We explain how to solve the modular constraints explicitly, which we do for genus 2. The result is that modular invariance at genus 1 and genus 2 is much more constraining than genus 1 alone. This allows us to drastically reduce the space of possible code CFTs. We also consider a number of examples of “isospectral theories” — CFTs with the same genus 1 partition function — and we find that they have different genus 2 partition functions. Finally, we make connection to some 2d CFTs known from the modular bootstrap. The n = 4 theory conjectured to have the largest possible gap in Virasoro characters, the SO(8) WZW model, is a code CFT, allowing us to give an expression for its genus 2 partition function. We also find some other known CFTs which are not code theories but whose partition functions satisfy the same simple polynomial ansatz as the code theories. This leads us to speculate about the usefulness of the code polynomial form beyond the study of code CFTs.
... There are some obstructions to the existence of such functions, which we would like to briefly review. For more details and precise statements, we refer to the references [72], [73] and [74]. By construction, the numerator W (g) (ϑ i (Ω)) is a Siegel modular form of degree k = c 2 . ...
... Here the outer product is over primitive elements γ in the Schottky uniformization group. For details on this formula and coordinates on Schottky space, see for instance the appendices of [73] Product formulas like (F.10), but without the "anomaly" term exp( S 12π ), were known from the early days of string theory [80,81], see [82]. The exponential factor of (F.9) contains the classical Liouville action S, which is a real-valued function that was determined in [75,76]. ...
... Example We conclude by giving some explicit expressions in the case of genus g = 2, extracted from [73]. They gave an expansion for F 2 in terms of coordinates on Schottky space, and the relation to the usual modular parameters q 1 = e 2πiΩ 11 , q 2 = e 2πiΩ 22 and r = e 2πiΩ 12 : F 2 (q 1 , q 2 , r) = 1 − (q 1 + q 2 ) + −q 2 q 1 r 2 + 1 r 2 + 6q 2 q 1 r + 1 r − q 2 1 − 9q 2 q 1 − q 2 2 + q 1 q 2 −9 (q 1 + q 2 ) r 2 + 1 r 2 + 40 (q 1 + q 2 ) r + 1 r − 61 (q 1 + q 2 ) + q 1 q 2 − q 2 q 1 r 4 + 1 r 4 + 2 3(q 2 1 + q 2 2 ) + 20q 2 q 1 r 3 + 1 r 3 − 61(q 2 1 + q 2 2 ) + 296q 2 q 1 r 2 + 1 r 2 − 270(q 2 1 + q 2 2 ) − 1181q 2 q 1 + 2 95(q 2 1 + q 2 2 ) + 424q 2 q 1 r + 1 r + O(q 5 i ) (F.11) ...
Error-correcting codes are known to define chiral 2d lattice CFTs where all the $U(1)$ symmetries are enhanced to $SU(2)$. In this paper, we extend this construction to a broader class of length-$n$ codes which define full (non-chiral) CFTs with $SU(2)^n$ symmetry, where $n=c+\bar c$. We show that codes give a natural discrete ensemble of 2d theories in which one can compute averaged observables. The partition functions obtained from averaging over all codes weighted equally is found to be given by the sum over modular images of the vacuum character of the full extended symmetry group, and in this case the number of modular images is finite. This averaged partition function has a large gap, scaling linearly with $n$, in primaries of the full $SU(2)^n$ symmetry group. Using the sum over modular images, we conjecture the form of the genus-2 partition function. This exhibits the connected contributions to disconnected boundaries characteristic of wormhole solutions in a bulk dual.
... This leads to the natural question: to what degree does the set of higher genus partition functions characterize a 2d CFT? The view that a 2d CFT may be defined by its vacuum amplitudes for all genera, originally advanced in [20], has been addressed more recently in [21,22]. Other recent interest in deriving universal bounds from higher genus modular invariance include [23][24][25]. ...
... At higher genus, no simple compact expression is known for the denominator. There exist some formal expressions [37] and at genus 2 a useful series expansion [22]. While these constructions give a denominator with the correct weight to cancel the weight of the modular form in the numerator, the resulting partition function picks up phases under the modular transformations that generalize the T transformation in (2.13). ...
A bstract
Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs, which are constructed using classical error-correcting codes. In this setting, the Sp(2 g, ℤ) modular transformations of genus g Riemann surfaces can be recast as a simple set of linear maps acting on 2 g polynomial variables, which comprise an object called the code enumerator polynomial. The CFT partition function is directly related to the enumerator polynomial, meaning that solutions of the linear constraints from modular invariance immediately give a set of seemingly consistent partition functions at a given genus. We then find that higher genus constraints, plus consistency under degeneration limits of the Riemann surface, greatly reduces the number of possible code CFTs. This work provides a step towards a full understanding of the constraints from higher genus modular invariance on 2d CFTs.
... As in the chiral case, the partition function will be constructed by a sum over vectors in a lattice defined by the code, and it will be described in terms of higher-genus theta functions with known modular properties. The theory of such functions is less developed compared the chiral case, where indeed the theory of Siegel modular forms has led to strong constraints on meromorphic CFTs at low c [21][22][23][24][25]. Nevertheless, we shall see that the set of partition functions in the form dictated by (1.3) does capture some interesting theories also outside the class of code CFTs, 4 to be discussed more in section 5.3, which may hint at the possibility to develop a theory of non-chiral modular forms. ...
Code CFTs are 2d conformal field theories defined by error-correcting codes. Recently, Dymarsky and Shapere generalized the construction of code CFTs to include quantum error-correcting codes. In this paper, we explore this connection at higher genus. We prove that the higher-genus partition functions take the form of polynomials of higher-weight theta functions, and that the higher-genus modular group acts as simple linear transformations on these polynomials. We explain how to solve the modular constraints explicitly, which we do for genus 2. The result is that modular invariance at genus 1 and genus 2 is much more constraining than genus 1 alone. This allows us to drastically reduce the space of possible code CFTs. We also consider a number of examples of "isospectral theories" -- CFTs with the same genus 1 partition function -- and we find that they have different genus 2 partition functions. Finally, we make connection to some 2d CFTs known from the modular bootstrap. The $n = 4$ theory conjectured to have the largest possible gap, the $SO(8)$ WZW model, is a code CFT, allowing us to give an expression for its genus 2 partition function. We also find some other known CFTs which are not code theories but whose partition functions satisfy the same simple polynomial ansatz as the code theories. This leads us to speculate about the usefulness of the code polynomial form beyond the study of code CFTs.
... Motivated by these, we focus on the correlation functions in TT −deformed CFTs live on a higher genus Riemann surface. There are main two ways called by sewing prescriptions to construct high genus Riemann surface, the one is Schottky Uniformization [44][45][46] and the other is offered by [47][48][49]. In the current work, we focus on the second sewing prescription. ...
... In [44,46], four disks are removed from the Riemann sphere, and the boundaries of each pair of disks are identified to obtain two handles. Besides, the annuli A 1 and A 2 mentioned above can be introduced on the same torus and are centered at two separate points. ...
We construct the correlation functions of conformal field theories (CFTs) on genus two Riemann surface with the $T\bar{T}$ deformation in terms of perturbative CFT approach. Thanks to sewing construction to form higher genus Riemann surface from lower genus ones and conformal symmetry, we systematically obtain the first order $T\bar{T}$ correction to the higher genus correlation functions in the $T\bar{T}$ deformed CFTs, e.g. partition function and one/higher-point correlation functions. In the weak coupling limit, the higher genus correlation functions can be decomposed by lower genus correlation functions. Our results offer $T\bar{T}$ deformed field theories data to allow us to extract quantum chaos signals and multiple-interval R\'{e}nyi entropies in deformed field theories.
... This leads to the natural question: to what degree does the set of higher genus partition functions characterize a 2d CFT? The view that a 2d CFT may be defined by its vacuum amplitudes for all genera, originally advanced in [20], has been addressed more recently in [21,22]. Other recent interest in deriving universal bounds from higher genus modular invariance include [23][24][25]. ...
... At higher genus, no simple compact expression is known for the denominator. There exist some formal expressions [38] and at genus 2 a useful series expansion [22]. While these constructions give a denominator with the correct weight to cancel the weight of the modular form in the numerator, the resulting partition function picks up phases under the modular transformations that generalize the T transformation in (2.13). ...
Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs, which are constructed using classical error-correcting codes. In this setting, the $\mathrm{Sp}(2g,\mathbb Z)$ modular transformations of genus $g$ Riemann surfaces can be recast as a simple set of linear maps acting on $2^g$ polynomial variables, which comprise an object called the code enumerator polynomial. The CFT partition function is directly related to the enumerator polynomial, meaning that solutions of the linear constraints from modular invariance immediately give a set of seemingly consistent partition functions at a given genus. We then find that higher genus constraints, plus consistency under degeneration limits of the Riemann surface, greatly reduces the number of possible code CFTs. This work provides a step towards a full understanding of the constraints from higher genus modular invariance on 2d CFTs.
... Later Maloney and Witten computed the 3d gravity partition function as a sum over topologies and found that the result could not be interpreted as a trace over some CFT Hilbert space [25]. Moreover, there are arguments (although no proof) that extremal CFTs could not exist for large central charge, i.e., in the semi-classical regime [26][27][28]. Various steps have been taken toward fixing them [29][30][31][32], but it is still unclear whether pure 3d gravity could make sense as a quantum theory. Few months after Witten's proposal, Li, Song and Strominger suggested an alternative for a fully consistent and unitary gravity theory with partition function that of an extremal CFT under the name chiral gravity [33]. ...
The phase space of three-dimensional gravity with Compere-Song-Strominger (CSS) boundary conditions is endowed with asymptotic symmetries consisting in the semidirect product of a Virasoro and a u^(1) Kač-Moody algebra, and contains Bañados-Teitelboim-Zanelli (BTZ) black holes whose entropy can be accounted for by the degeneracy of states of a warped Conformal Field Theory (CFT). By embedding these boundary conditions in topologically massive gravity, we observe the existence of two special points in the space of couplings parametrized by the AdS3 radius ℓ and the Chern-Simons coupling μ. When μ=±1ℓ, the asymptotic symmetries reduce to either a chiral Virasoro algebra or a pure u^(1) Kač-Moody current algebra. At those points, black holes have positive energy while that of linearized excitations are non-negative.
... By using the above mentioned Taylor expansion and lower bounds on the slope of M g , one can try to understand when linear combination of theta series are solution of the Schottky problem; see Theorem 7.1 and the subsequent discussion. This is also the kind of argument used in [GV09] and [GKV10], see Corollary 6.3. To get a progress on these problems using the ideas of this paper, one should have a better understanding of the coefficients Φ (g) g,k (V ) introduced in Section 1.1 and Definition 3.16. ...
... Relation with other works. This project started in 2013, trying to understand the relation between the papers [GV09,GKV10] and the author's Ph.D. thesis [Cod14]. Another important source of inspiration was [HvH]. ...
We associate to any holomorphic vertex algebra a collection of Teichm\"{u}ller modular forms, one in each genus. In genus one we obtain the character of the vertex algebra, and we thus reprove Zhu's modularity result. We propose applications to the Schottky problem, to the study of the slope of the effective cone of the moduli space of curves, and to the classification of holomorphic vertex algebras.
... A natural extension is to analyze the Ramond sector of N = 1 superconformal theory. The careful analysis of higher genus cases and, in particular, of the genus-two case is desirable (see, e.g., [42,[47][48][49]). ...
We develop a recursive approach to computing Neveu-Schwarz conformal blocks associated with n-punctured Riemann surfaces. This work generalizes the results of [1] obtained recently for the Virasoro algebra. The method is based on the analysis of the analytic properties of the superconformal blocks considered as functions of the central charge c. It consists of two main ingredients: the study of the singular behavior of the conformal blocks and the analysis of their asymptotic properties when c tends to infinity. The proposed construction is applicable for computing multi-point blocks in different topologies. We consider some examples for genus zero and one with different numbers of punctures. As a by-product, we propose a new way to solve the recursion relations, which gives more efficient computational procedure and can be applied to SCFT case as well as to pure Virasoro blocks.
