# Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces

Manojna Namuduri, Adel Bilal

► To cite this version:

Manojna Namuduri, Adel Bilal. Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces. Journal of High Energy Physics, 2023, 2023 (11), pp.194. 10.1007/JHEP11(2023)194. hal-04186846

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# Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces

## Manojna Namuduri and Adel Bilal

Laboratoire de Physique de l'École Normale Supérieure, PSL University, CNRS,
Sorbonne Université, Université de Paris,
24 rue Lhomond, F-75231 Paris Cedex 05, France

E-mail: manojna.namuduri@ens.psl.eu, adel.bilal@ens.psl.eu

**ABSTRACT:** We explore the effective gravitational action for two-dimensional massive Euclidean Majorana fermions in a small mass expansion, continuing and completing the study initiated in a previous paper [1]. We perform a detailed analysis of local zeta functions, heat kernels, and Green's functions of the Dirac operator on arbitrary Riemann surfaces. We obtain the full expansion of the effective gravitational action to all orders in $m^2$. For genus one and larger, this requires the understanding of the role of the zero-modes of the (massless) Dirac operator which is worked out.

Besides the Liouville action, at order $m^0$, which only involves the background metric and the conformal factor $\sigma$, the various contributions to the effective gravitational action at higher orders in $m^2$ can be expressed in terms of integrals of the renormalized Green's function at coinciding points of the squared (massless) Dirac operator, as well as of higher Green's functions. In particular, at order $m^2$, these contributions can be re-written as a term $\int e^{2\sigma} \sigma$ characteristic of the Mabuchi action, much as for 2D massive scalars, as well as several other terms that are multi-local in the conformal factor $\sigma$ and involve the Green's functions of the massless Dirac operator and the renormalized Green's function, but for the background metric only, and certain area-like parameters related to the zero-modes.

**KEYWORDS:** 2D Gravity, Effective Field Theories, Models of Quantum Gravity

ARXIV EPRINT: 2308.05802
# Contents

## 1 Introduction 1

1.1 Motivation and outline of the paper 1

1.2 Some (more or less) well-known gravitational actions: Liouville, Mabuchi and Aubin-Yau 7

1.3 Recap: the gravitational action for a massless scalar field 9

## 2 The Dirac operator for Majorana spinors on a curved manifold 12

2.1 Majorana spinors in flat 2D space: $\gamma$-matrices and Dirac operator 13

2.2 Curved space: $\gamma$-matrices, covariant derivatives, Dirac operator and spinorial Laplacian 14

2.2.1 Local orthonormal frames and spin connection (in $n$ dimensions) 15

2.2.2 Covariant derivatives (in $n$ dimensions) 16

2.2.3 Back to two dimensions 20

2.3 The example of the flat torus with arbitrary periods 22

2.4 Conformal transformation of the Dirac operator 24

2.4.1 Dirac operator in $n$ dimension 24

2.4.2 Dirac operator in two dimensions 25

2.5 Zero-modes 26

2.5.1 General remarks about zero-modes 26

2.5.2 Zero-modes on a general torus 28

2.5.3 Zero-modes of $i\nabla$ for arbitrary genus $g \ge 1$ 29

2.5.4 Conformal variation of the zero-modes and of the zero-mode projector 30

## 3 Fermionic matter partition function and gravitational action 32

3.1 Fermionic functional integral for Majorana spinors 32

3.2 Zeta function regularization 34

3.2.1 Zeta-functions of positive operators 34

3.2.2 A short discussion of Riemann's zeta function 35

3.2.3 Expressing the gravitational action in terms of $\zeta_D$ 36

3.3 Variation of the eigenvalues: perturbation theory 37

## 4 The tool box: heat kernels, zeta-function and Green's functions 38

4.1 The basic tools: definitions, relations and properties 39

4.1.1 Local zeta-functions 39

4.1.2 Local heat kernels 42

4.1.3 Relating zeta functions and heat kernels 44

4.1.4 Small-$t$ asymptotics and poles of zeta functions at coinciding points 45

4.1.5 Green's functions 49

4.2 Variation of $\zeta(s)$ 50
<table>
  <tbody>
    <tr>
      <td>4.3</td>
      <td>Small-t asymptotics of the heat kernels and singularity structure of the local zeta-functions for the flat torus</td>
      <td>52</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>General statements about the singularity structure for x ≠ y</td>
      <td>55</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Renormalized Green's functions</td>
      <td>58</td>
    </tr>
    <tr>
      <td>4.5.1</td>
      <td>G<sub>R</sub>, G<sub>ζ</sub>, S<sub>R</sub> and S<sub>ζ</sub></td>
      <td>58</td>
    </tr>
    <tr>
      <td>4.5.2</td>
      <td>Green's function G̃(x, y) and G̃<sub>R</sub>(x) on the flat torus for m = 0</td>
      <td>60</td>
    </tr>
    <tr>
      <td><b>5</b></td>
      <td><b>Determining the gravitational action</b></td>
      <td><b>61</b></td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Expressing the variation of S<sub>grav</sub> in terms of G<sub>R</sub></td>
      <td>61</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Conformal variations of the Green's functions</td>
      <td>62</td>
    </tr>
    <tr>
      <td>5.2.1</td>
      <td>Conformal variations of S(x, y), G(x, y) and G<sub>R</sub>(y)</td>
      <td>63</td>
    </tr>
    <tr>
      <td>5.2.2</td>
      <td>Conformal variation of the higher Green's function G<sub>n</sub>, n ≥ 2</td>
      <td>65</td>
    </tr>
    <tr>
      <td>5.2.3</td>
      <td>Variation of the gravitational action to all orders in m<sup>2</sup> for spherical topology</td>
      <td>67</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Variation of the gravitational action for arbitrary genus</td>
      <td>67</td>
    </tr>
    <tr>
      <td>5.3.1</td>
      <td>Zero-mode and non-zero-mode parts of the Green's functions</td>
      <td>67</td>
    </tr>
    <tr>
      <td>5.3.2</td>
      <td>Expressing the variation of the gravitational action in terms of G̃<sub>R</sub></td>
      <td>69</td>
    </tr>
    <tr>
      <td>5.3.3</td>
      <td>Determining δS̃</td>
      <td>71</td>
    </tr>
    <tr>
      <td>5.3.4</td>
      <td>Determining δG̃<sub>R</sub></td>
      <td>72</td>
    </tr>
    <tr>
      <td>5.3.5</td>
      <td>Conformal variation of the higher G̃<sub>n</sub>(x, y), n ≥ 2</td>
      <td>73</td>
    </tr>
    <tr>
      <td>5.3.6</td>
      <td>Variation of the gravitational action to all orders in m<sup>2</sup> for arbitrary genus</td>
      <td>75</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Integrating the variation of the gravitational action</td>
      <td>78</td>
    </tr>
    <tr>
      <td>5.4.1</td>
      <td>The variations of S̃ under finite conformal rescalings for m = 0</td>
      <td>79</td>
    </tr>
    <tr>
      <td>5.4.2</td>
      <td>The variations of G̃ and G̃<sub>R</sub> under finite conformal rescalings for m = 0</td>
      <td>81</td>
    </tr>
    <tr>
      <td>5.4.3</td>
      <td>The gravitational action at order m<sup>2</sup></td>
      <td>83</td>
    </tr>
    <tr>
      <td><b>6</b></td>
      <td><b>Discussion and outlook</b></td>
      <td><b>84</b></td>
    </tr>
    <tr>
      <td><b>A</b></td>
      <td><b>Small-t expansion for heat kernels of type K<sub>+</sub>(t, x, y)</b></td>
      <td><b>85</b></td>
    </tr>
    <tr>
      <td>A.1</td>
      <td>Riemann normal coordinates</td>
      <td>85</td>
    </tr>
    <tr>
      <td>A.2</td>
      <td>Small-t expansion of the heat kernel</td>
      <td>88</td>
    </tr>
    <tr>
      <td><b>B</b></td>
      <td><b>The small-t asymptotics of K<sub>-</sub>(t, x, y) for the flat torus</b></td>
      <td><b>93</b></td>
    </tr>
  </tbody>
</table>

# 1 Introduction

## 1.1 Motivation and outline of the paper

Quantum field theory on a fixed curved Lorentzian (space-time) or Euclidean (space) manifold has been extensively studied for more than half a century. Standard references include [2–4]. One of the most prominent early manifestations was the discovery of Hawking
radiation near a black hole horizon [5]. In a full theory of quantum gravity one should, of course, also do the functional integral over all geometries which means summing over topologies and, for each topology, over the inequivalent metrics. This is a notoriously difficult problem. The usual way to decompose the problem is to first do the functional integral over the “matter” fields giving the so-called matter partition function, or equivalently the effective gravitational action, and deal with the functional integral over the geometries only in a second step. This is, of course, similar to what one often does for non-abelian gauge theories where one first integrates over the matter fields for a fixed gauge field configuration yielding an effective action $S_{\text{eff}}[A]$ of the gauge fields only, and then deals with the problems of gauge fixing only in a second step. In particular, this allows to discuss important issues like anomalies already at the level of this effective action [6]. Here we will be interested similarly in determining an effective action for two-dimensional gravity and matter systems on spaces of Euclidean signature.

More precisely, the functional integral over the matter field(s) which we denote generically by $\Psi$, computed with a fixed metric $g$ on a given manifold $M$, defines the matter partition function as $Z_{\text{mat}}[g] = \int \mathcal{D}\Psi \exp(-S_{\text{mat}}[g, \Psi])$. In terms of this, the effective gravitational action is defined, as usual, as the ratio of $Z_{\text{mat}}[g]$ and $Z_{\text{mat}}[\hat{g}]$, where we consider $\hat{g}$ as a reference metric:

$$ \exp(-S_{\text{grav}}[g, \hat{g}]) = \frac{Z_{\text{mat}}[g]}{Z_{\text{mat}}[\hat{g}]} \qquad (1.1) $$

In two dimensions, the metric depends on 3 functions, but using the diffeomorphism invariance one can essentially fix two of them, leaving only one geometrically relevant function which can be chosen as an overall conformal factor $e^{2\sigma(x)}$. Hence, for fixed topology and finitely many fixed (“modular”) parameters $\tau_i$ we can always pick a reference metric $\hat{g} \equiv \hat{g}[\tau_i]$ such that any other metric (corresponding to the same $\tau_i$) can be written, up to diffeomorphisms, as<fnref n="1" />

$$ g_{\mu\nu}(x) = e^{2\sigma(x)} \hat{g}_{\mu\nu}(x). \qquad (1.2) $$

We may then equivalently consider that $S_{\text{grav}}$ depends on $\hat{g}$ and $\sigma$: $S_{\text{grav}}[g, \hat{g}] \equiv S_{\text{grav}}[\sigma, \hat{g}]$. The definition (1.1) implies that $S_{\text{grav}}$ must satisfy a cocycle identity

$$ S_{\text{grav}}[g_3, g_2] + S_{\text{grav}}[g_2, g_1] = S_{\text{grav}}[g_3, g_1]. \qquad (1.3) $$

The best-known example is the effective gravitational action for conformal matter coupled to 2D gravity which is the Liouville action [8]

$$ S_{\text{Liouville}}[g, \hat{g}] \equiv S_{\text{Liouville}}[\sigma, \hat{g}] = \int d^2x \sqrt{\hat{g}} (\sigma \hat{\Delta}\sigma + \hat{R}\sigma), \quad g = e^{2\sigma} \hat{g}, \qquad (1.4) $$

where $\hat{R}$ is the scalar curvature for the metric $\hat{g}$, and

$$ S_{\text{grav}}[g, \hat{g}] = -\frac{c}{24\pi} S_{\text{Liouville}}[g, \hat{g}], \qquad (1.5) $$

<footnote n="1">¹This is a global statement. Locally one can go further and choose so-called isothermal coordinates [7] which are such that $g_{\mu\nu}(x) = e^{2\lambda(x)} \delta_{\mu\nu}$, but obviously this cannot be done globally unless $\int \sqrt{\hat{g}}\hat{R} = 8\pi(1-g)$ vanishes.</footnote>
where c is the central charge of the conformal matter system, namely c = 1 for a massless scalar and c = 1/2 for a massless Majorana spinor. An even simpler example satisfying this cocycle identity is the “cosmological constant action”

$$S_c[g, \hat{g}] = \mu_0 \int d^2x (\sqrt{\hat{g}} - \sqrt{g}) = \mu_0(A - A_0). \qquad (1.6)$$

This action typically is also present as a counterterm, in addition to $S_{\text{Liouville}}$, to renormalize certain divergences that might be present.

But these are not the only two-dimensional gravitational actions, satisfying the cocycle condition, that can be constructed and have been studied in the mathematical literature. Other appropriate functionals are the Mabuchi and Aubin-Yau actions [9–12], to be defined in the next subsection. These latter functionals involve not only the conformal factor $\sigma$ but also the Kähler potential $\Phi$, and can also be generalized to higher-dimensional Kähler manifolds, where they appear in relation with the characterization of constant scalar curvature metrics [12]. In the context of a two-dimensional effective action, they appeared first in ref. [13] where the matter partition function of non-conformal matter like a massive scalar field was studied. It was shown there that in this case the gravitational action as defined by (1.1) contains these Mabuchi and Aubin-Yau actions as first-order in $m^2$ corrections to the Liouville action. Higher-order corrections in $m^2$ to the effective gravitational action for such massive scalar fields have been obtained<fnref n="2" /> in [14]. The interesting extension to two-dimensional manifolds with boundaries was studied in [15].

The fixed-area partition function of quantum gravity, with a gravitational action being a sum of the Liouville and Mabuchi actions, has been studied at one loop in [16] and at two and three loops in [17]. In a different approach, a rigorous mathematical construction of the functional integral based on the coupling of the Liouville and Mabuchi actions has been obtained in ref [21] by means of probabilistic tools.

So far, references [13–15] considered the effective gravitational action of a two-dimensional massive scalar field. It is natural to try to extend this to the case of two-dimensional massive fermions. The most “elementary” 2D fermion is a Majorana fermion. As we will discuss below, this corresponds to a real anti-commuting spinor field with a Majorana-type mass term. Since the corresponding Dirac operator can be chosen purely imaginary, classically, at the level of the Dirac equation, one can impose a Majorana (reality) condition, on the spinor. In the quantum theory however, we need to expand the Majorana field on the eigenfunctions of this purely imaginary Dirac operator, and then these eigenfunctions are necessarily complex. One might ask whether it is then appropriate to talk about Majorana fermions in the quantum theory. However, we can obtain the partition function in terms of the eigenvalues of the (real) square of the Dirac operator whose eigenfunctions can all be chosen to be real. Moreover, we will show that we can consistently do the functional integral over a real anti-commuting spinor field. If one is interested instead in a (complex) Dirac spinor, the corresponding gravitational action is easily seen to be simply twice the one for a Majorana spinor.

<footnote n="2">²Note that at present we call $S_{\text{grav}}[g, \hat{g}]$ what was called $S_{\text{grav}}[\hat{g}, g]$ in refs [13–15, 17].</footnote>
In a previous paper [1] involving one of the present authors, the study of this effective gravitational action for massive Majorana fermions was initiated.<fnref n="3" /> While at first this appeared to be a simple generalisation of the case of the scalar field, it actually turned out to be technically quite non-trivial. Much as for the scalar case, the renormalized Green's functions at coinciding points played an important role. In order to do a small mass expansion one must be able to define these Green's functions also for the massless theory in which case the contributions of the zero-modes (of the massless Dirac operator) must be subtracted. For spherical topology there are no zero-modes and then this subtlety does not occur. For this reason, [1] established the form of the effective gravitational action only for spherical topology and only up to order $m^2$. Further technical subtleties implied that even in this case the result was more complicated than the one obtained in [13] or [14] for the massive scalar field. Nevertheless, a term characteristic of the Mabuchi action emerged.

In the present paper we will generalise this result and establish the effective gravitational action for arbitrary genus Riemann surfaces (without boundaries), and in an expansion to all orders in $m^2A$. We will show that this expansion has a finite radius of convergence !

An important ingredient which enabled us to go beyond spherical topology was the understanding of the structure of the projectors on the zero-modes of the massless Dirac operator. Indeed, the massless Dirac operator transforms in a particularly simple way under local conformal transformation (which is neither the case for the massive Dirac operator, nor for the non-zero eigenvalue equation of the Dirac operator, even for zero mass). This implies that the zero-modes of the massless Dirac operator for metric $g$ are related by a very simple conformal rescaling<fnref n="4" /> to the zero-modes of the massless Dirac operator for metric $\hat{g}$, and then a similarly simple relation exists between the zero-mode projectors for metrics $g$ and $\hat{g}$. This allows us to characterise the zero-mode contributions in general, valid on Riemann surfaces of arbitrary genus.

The study of the massive and massless Dirac operator $D$ and of its square $D^2$, of their associated Greens's functions and local zeta-functions and local heat kernels occupied a major part in [1] and will again occupy an important part of the present paper. However, here we will be able to do a much more systematic study of these quantities and prove several important relations, some of which were only conjectured in [1]. As a result, we are able to obtain the effective gravitational action as an all-orders expansion in powers of $m^2$ with each term in the expansion expressed as an integral over the manifold of these Green's functions, appropriately renormalized, as well as higher Green's functions at coinciding points. As already mentioned, a detailed knowledge of the contributions of the zero-modes (for zero mass) is required to perform the small-mass expansion correctly.

We will see that besides the leading Liouville term, we get a new order $m^0$ contribution when zero-modes of the massless Dirac operator are present. It might seem surprising to

<footnote n="3">³After the present paper was posted on arXiv, the authors of [22] kindly brought their reference to our attention, where they also study the effective gravitational action, up to terms quadratic in the curvature, for two-dimensional massive scalar, Dirac and vector matter fields, along the heat kernel methods as worked out in [23, 24]. Their emphasis is mostly on obtaining the beta-functions.</footnote>

<footnote n="4">⁴This result is implicit in [25, 26] and was spelled out more explicitly in an early draft of [27].</footnote>
get a “new” order $m^0$ contribution in the massive theory that seemed not to be there in the massless theory. Actually, we will see that this same contribution does arise in the massless theory. However, in the massless theory one could modify or even remove these new terms by appropriately changing the definition of the matter partition function. On the other hand, in the massive theory there is no reason to do such a redefinition and these zero-mode related contributions are genuinely present.

At order $m^2$ we get a cosmological constant action, and a local $\int \sqrt{\hat{g}} \sigma e^{2\sigma}$ term characteristic of the Mabuchi action, as well as some further “non-local terms” involving integrals of the Green’s functions computed with the metric $g$. We have been able to rewrite the latter as some multi-local functionals in the conformal factor $\sigma$ (i.e. multiple integrals involving the conformal factor at the different integration points) and furthermore only involving the Green’s functions computed in the background metric $\hat{g}$, as well as a finite number of area-like parameters coded in the matrix $P_{0,ij}$ and its inverse $(P_0)^{-1}_{ij}$. It would, of course, be desirable to rewrite all terms as purely local (single) integrals expressed in terms of local quantities like the conformal factor, the Kähler potential or some generalisation thereof, and the metric and curvature. However, as usual with effective actions, there is no reason that they should be local, and we think at present that our multi-local form (or some equivalent formulation) is the best one can achieve.

The gravitational action for a massive Majorana fermion has also been studied long ago by Seiberg in a different approach [18]. While the action we start with for the massive Majorana fermion obviously is non-conformal, the full quantum theory of matter coupled to gravity should eventually be conformally invariant. This is the point of view advocated in the approach by David, and Distler and Kawai [19, 20], and in particular also in [18]. Then the massive Majorana fermion has its mass term gravitationally dressed by the Liouville field (i.e. the conformal factor of the metric) so that this appears as a *conformal perturbation* of the massless (conformal) Majorana fermion. To try to make contact with this result, we should not just integrate out the fermions and compute the matter partition function, but we should really try to compute the quantum effective action for the Majorana fermions coupled to the Liouville field. This should involve introducing fermionic sources $\eta$ and then Legendre transforming to some $\psi_0$. It is the quantum effective action for $\psi_0$ which one could then try to compare with the conformal action for Majorana fermions as given in [18]. Keeping track of the dependence on the conformal factor $\sigma$ throughout this computation (in particular, we know how the inverse Dirac operator, i.e. the fermionic Green's function $S$ depends on $\sigma$) should in principle allow us to obtain this effective action and see how the fermionic mass term is gravitationally dressed. However, if we naively go through this computation we do not reproduce the gravitational dressing as contained in ref [18]. We suspect that there are a few more subtleties to be understood before one can correctly compare both approaches. We hope to get back to this in the near future.

We have tried to write the present paper in a hopefully self-contained and pedagogical way. It is organised as follows. In the second part of this introduction we recall a few facts about the Liouville, Mabuchi and Aubin-Yau actions and their stationary points, and then briefly recall, as a warm-up exercise, how one can show that the effective gravitational action for a massless scalar field is the Liouville action. In the next section we introduce the
relevant differential operators: massive Dirac operator $D$, its square $D^2$, and the scalar and spinoral Laplacians. We discuss their eigenvalue problems in some detail and exhibit the eigenvalues and eigenfunctions for the simple example of the flat torus. We work out the transformations of $D$ and of $D^2$ under conformal changes of the metric (both infinitesimal and finite). We also work out exactly how the zero-modes of the (massless) Dirac operator change under the local conformal rescalings, allowing us to precisely define the zero-mode projectors that are crucial ingredients for the small mass expansion. Indeed, as already mentioned, these projectors have to be subtracted later-on from the Green's function in order to define the Green's functions of the massless theory around which the small-mass expansion will be done. An important ingredient in these zero-mode projectors are certain area-like parameters coded in a matrix $P_{0,ij}$ that involves integrals of $e^\sigma$ times the zero-modes in the background metric.

In section 3 we discuss the fermionic functional integral defining the matter partition function and obtain the gravitational action. We show precisely how it is related to the product of the eigenvalues of the Dirac operator $D$ and how this is related to the corresponding zeta-function. We offer also a few remarks about the convergence of these zeta-functions and their definition by analytic continuation. We show how these zeta-functions change under infinitesimal conformal rescalings of the metric.

Then, section 4 contains most of the important technical results: we embark on a detailed study of the different Green's functions, local zeta-functions $\zeta_\pm(s, x, y)$ and local heat kernels $K_\pm(t, x, y)$. We express precisely how the change of the zeta function (expressing the gravitational action) under infinitesimal conformal rescalings $\delta\sigma$ is related to integrals of the local zeta functions times $\delta\sigma$. We prove a precise correspondence between the local zeta function $\zeta_-(s, x, y)$ associated with $D$ and the local zeta function $\zeta_+(s + 1/2, x, y)$ associated with $D^2$. This relation allows us to make precise statements<fnref n="5" /> about the singularity structures of both local zeta functions $\zeta_\pm$, as well as about the small-$t$ asymptotics of the heat kernel $K_-(t, x, y)$, solely from the knowledge of the corresponding asymptotics of $K_+(t, x, y)$ that is worked out in the appendix.

All this is put together in section 5 to obtain the variation of the effective gravitational action. We will be able to isolate the zero-mode contributions for arbitrary topology and thus obtain results valid on Riemann surfaces of arbitrary genus. The variation of the gravitational action is obtained as a total variation of a converging series expansion in powers of $m^2$. “Integrating” these variations at any order in $m^2$ gives the corresponding gravitational action at any order in terms of integrals over higher and higher Green’s function (computed in the metric $g$) at coinciding points. We then work out how the order $m^2$ contribution can be re-expressed in terms of the Mabuchi type term $\int \sqrt{\hat{g}} \sigma e^{2\sigma}$ and integrals and multiple integrals involving the conformal factor $\sigma$ at the different integration points, the Green’s function of $D$ for the background metric $\hat{g}$ and the renormalized Green’s function of $D^2$ at coinciding points, also in the background metric $\hat{g}$, as well as a certain zero-mode projector

<footnote n="5">⁵In particular, this allows us to correct some inexact conjectures made in [1] about the small-$t$ asymptotics of $K_-(t, x, x)$, as well as about the poles of $\zeta_-(s, x, x)$ at negative half-integers which turn out not to be there.</footnote>
$\bar{P}_0$ which involves the inverse of the matrix $P_{0,ij}$ of area-like parameters. We expect that a similar rewriting can also be performed on the contributions of order $m^4$ and higher.

In appendix A we give the small-$t$ asymptotics of the heat kernel $K_+(t, x, y)$ by solving recursively the associated (heat) differential equation. We do this first quite generally for positive second-order differential operators and then specialise our results to the squared Dirac operator. No such recursive solution is available for the corresponding $K_-$ since it does not have a simple initial condition. In appendix B we discuss some properties of this $K_-(t, x, y)$ for the flat torus in which case we have explicit eigenvalues and eigenfunctions at our disposal, confirming the general properties obtained in section 4.

Results largely overlapping with ours, in particular concerning the zero-mode projectors, have been obtained independently in [27]. Their treatment of the orthonormalisation of the zero-modes allowed us to correct a slight error in the first version of the present paper.

## 1.2 Some (more or less) well-known gravitational actions: Liouville, Mabuchi and Aubin-Yau

Besides the Liouville action $S_{\text{Liouville}}$ which we already defined in (1.4), and the somewhat trivial cosmological constant action $S_c$ given in (1.6) there are two other, more or less well-known gravitational actions, namely the Mabuchi and Aubin-Yau actions. Let us briefly recall the definitions and basic properties of these gravitational actions. While the Liouville action (1.4) can be written in terms of $\hat{g}$ and the conformal factor $\sigma$, the Mabuchi and Aubin-Yau actions are formulated using also the Kähler potential $\Phi$. The conformal factor $\sigma$ and the Kähler potential $\Phi$ are related by (we may take this as the definition of the Kähler potential $\Phi$)

$$g = e^{2\sigma} \hat{g}, \quad e^{2\sigma} = \frac{A}{\hat{A}} \left(1 - \frac{1}{2} \hat{A} \hat{\Delta} \Phi\right), \quad \Delta = \frac{1}{\sqrt{\hat{g}}} \partial_{\mu}(\sqrt{\hat{g}} g^{\mu\nu} \partial_{\nu}) = e^{-2\sigma} \hat{\Delta}, \quad (1.7)$$

where $\hat{\Delta}$ denotes the (scalar) Laplacian for the metric $\hat{g}$ with area $\hat{A} = \int d^2x \sqrt{\hat{g}}$. The second relation in this equation (1.7) shows that the Kähler potential describes different metrics of the same total area $A$, and the latter needs to be introduced as an additional variable. This separation becomes particularly clear if we write out how the variations of $\sigma$ are related to the variations of $\Phi$ and $A$. It follows from (1.7) that

$$\delta\sigma = \frac{\delta A}{2A} - \frac{A}{4}\Delta\delta\Phi \Leftrightarrow \delta\left(\frac{e^{2\sigma}}{A}\right) = -\frac{1}{2}\hat{\Delta}\delta\Phi. \quad (1.8)$$

In terms of the Kähler potential $\Phi$, the area $A$ and the conformal factor $\sigma$, the Mabuchi action on a Riemann surface of genus $g$ can then be written as [13]

$$S_M[\hat{g}, g] = \int d^2x \sqrt{\hat{g}} \left[ 2\pi(g-1)\Phi \hat{\Delta} \Phi + \left( \frac{8\pi(1-g)}{\hat{A}} - \hat{\mathcal{R}} \right) \Phi + \frac{4}{A} \sigma e^{2\sigma} \right], \quad (1.9)$$

and the Aubin-Yau action as

$$S_{\text{AY}}[\hat{g}, g] = - \int d^2x \sqrt{\hat{g}} \left[ \frac{1}{4} \Phi \hat{\Delta} \Phi - \frac{\Phi}{\hat{A}} \right]. \quad (1.10)$$
As already mentioned, they both satisfy a cocycle identity analogous to (1.3) and were shown [13] to appear in the effective gravitational action of a (two-dimensional) massive scalar field in the term of first order in an expansion in $m^2A$. Note that we can rewrite the Mabuchi action as

$$S_M = 8\pi(1-g)S_{AY} + \int d^2x \sqrt{\hat{g}} \left(\frac{4}{A}\sigma e^{2\sigma} - \hat{\mathcal{R}}\Phi\right). \quad (1.11)$$

Let us now give the variations of the Liouville, Mabuchi and Aubin-Yau actions under a conformal variation of the metric (equivalently encoded in $\delta\sigma$ or in $\delta\Phi$ and $\delta A$, at fixed $\hat{g}$ and hence fixed $\hat{A}$). First, the Aubin-Yau action only depends on $\Phi$, not on $A$, and

$$\begin{aligned}
\delta S_{AY}[\hat{g}, g] &= -\int d^2x \sqrt{\hat{g}} \left[\frac{1}{2}\delta\Phi \hat{\Delta}\Phi - \frac{\delta\Phi}{\hat{A}}\right] = \frac{1}{A} \int d^2x \sqrt{\hat{g}} \left[-\frac{A}{2}\hat{\Delta}\Phi + \frac{A}{\hat{A}}\right] \delta\Phi \\
&= \frac{1}{A} \int d^2x \sqrt{\hat{g}} e^{2\sigma} \delta\Phi \\
&= \frac{1}{A} \int d^2x \sqrt{g} \delta\Phi.
\end{aligned} \quad (1.12)$$

Next, we have

$$\begin{aligned}
\delta \int d^2x \sqrt{\hat{g}} \left(\frac{4}{A}\sigma e^{2\sigma} - \hat{\mathcal{R}}\Phi\right) &= \int d^2x \sqrt{\hat{g}} \left(\delta\sigma \frac{4}{A}e^{2\sigma} + 4\sigma \delta\left(\frac{e^{2\sigma}}{A}\right) - \hat{\mathcal{R}}\delta\Phi\right) \\
&= 2\frac{\delta A}{A} - \int d^2x \sqrt{\hat{g}} \left(2\hat{\Delta}\sigma + \hat{\mathcal{R}}\right) \delta\Phi,
\end{aligned} \quad (1.13)$$

where we used the fact that the Laplace operator $\hat{\Delta}$ can be freely integrated by parts since we consider only manifolds without boundaries. Finally, as is well-known, in two dimensions the curvature scalars $\mathcal{R}$ and $\hat{\mathcal{R}}$ are related by

$$\mathcal{R} = e^{-2\sigma}(\hat{\mathcal{R}} + 2\hat{\Delta}\sigma). \quad (1.14)$$

This implies, in particular, that $\sqrt{\hat{g}}\mathcal{R} = \sqrt{\hat{g}}(\hat{\mathcal{R}} + 2\hat{\Delta}\sigma)$. We can then rewrite (1.13) as

$$\delta \int d^2x \sqrt{\hat{g}} \left( \frac{4}{A} \sigma e^{2\sigma} - \hat{\mathcal{R}} \Phi \right) = 2 \frac{\delta A}{A} - \int d^2x \sqrt{\hat{g}} \mathcal{R} \delta\Phi. \quad (1.15)$$

Combining this with (1.11) and (1.12) gives the variation of the Mabuchi action as

$$\delta S_M[\hat{g}, g] = 2\frac{\delta A}{A} - \int d^2x \sqrt{\hat{g}} \left(\mathcal{R} - \frac{8\pi(1-g)}{A}\right) \delta\Phi. \quad (1.16)$$

We see that the stationary points of the Mabuchi action are given by metrics of constant area $A$ and constant curvature $\mathcal{R} = \frac{8\pi(1-g)}{A}$. Indeed, any Riemann surface of genus $g$ admits a constant curvature metric and then, since $\int \sqrt{\hat{g}}\mathcal{R} = 8\pi(1-g)$ one necessarily has $\mathcal{R} = \frac{8\pi(1-g)}{A}$.

One has similarly for the variation of the Liouville action $S_{Liouville}$ given in (1.4).

$$\delta S_{\text{Liouville}}[g, \hat{g}] = \delta \int d^2x \sqrt{\hat{g}} (\sigma \hat{\Delta}\sigma + \hat{\mathcal{R}}\sigma) = \int d^2x \sqrt{\hat{g}} (2\delta\sigma \hat{\Delta}\sigma + \hat{\mathcal{R}}\delta\sigma) = \int d^2x \sqrt{\hat{g}} \mathcal{R} \delta\sigma. \quad (1.17)$$
This is the form of the variation which we will use later-on. But we can also re-express $\delta\sigma$ in terms of $\delta A$ and $\delta\Phi$ so that

$$ \delta S_{\text{Liouville}}[g, \hat{g}] = \int d^2x \sqrt{g} R \left( \frac{\delta A}{2A} - \frac{A}{4} \Delta \delta \Phi \right) = 4\pi(1-\hat{g}) \frac{\delta A}{A} - \frac{A}{4} \int d^2x \sqrt{g} \Delta R \delta \Phi, \quad (1.18) $$

which shows that the stationary points of the Liouville action at constant area $A$ are given by metrics such that $\Delta R = 0$, i.e. by constant curvature metrics. As before, then necessarily $R = \frac{8\pi(1-\hat{g})}{A}$.

We have seen that the Liouville and Mabuchi actions admit metrics of constant scalar curvature as “solutions of their equations of motion” under the constraint of fixed area. Although not obvious from the previous equation, the variation of the Aubin-Yau action when restricted to the space of Bergmann metrics is similarly related to metrics of constant scalar curvature [12].

## 1.3 Recap: the gravitational action for a massless scalar field

Before embarking on the technical complications for the Majorana fermions, it is maybe good to quickly remind the reader of (parts of) the much simpler computation of the gravitational action for a massless (i.e. conformal) scalar field, leading to the Liouville action.

For a massless scalar field $\phi$ on a 2 dimensional (euclidean) manifold without boundary, with metric $g_{\mu\nu}$, the action is

$$ S[g, \phi] = \int d^2x \sqrt{g} g^{\mu\nu} \nabla_{\mu} \phi \nabla_{\nu} \phi = - \int d^2x \sqrt{g} \phi \Delta \phi, \quad (1.19) $$

where, as in (1.7), $\Delta = \frac{1}{\sqrt{g}} \partial_{\nu} (\sqrt{g} g^{\mu\nu} \partial_{\mu}) \equiv g^{\mu\nu} \nabla_{\nu} \nabla_{\mu}$ is the scalar Laplace operator. It is hermitian with respect to the usual scalar product $(f, h) = \int d^2x \sqrt{g} f^*(x) h(x)$. Consider the eigenvalue problem of $-\Delta$:

$$ -\Delta f_n(x) = \lambda_n f_n(x). \quad (1.20) $$

Since $\Delta$ is a real operator these eigenfunctions can be chosen to be real. As always they can also be chosen to be orthonormal:

$$ \int d^2x \sqrt{g} f_n(x) f_m(x) = \delta_{nm}. \quad (1.21) $$

Of course, there are infinitely many eigenfunctions and eigenvalues, so that the index $n$ runs over an infinite set (e.g. over the non-negative integers). One can then see, since the action $S$ is non-negative, that all eigenvalues are $\lambda_n \ge 0$. (This is why we consider $-\Delta$ which is a positive operator, rather than $\Delta$.) Note that there is always a single zero-mode, i.e. an eigenfunction $f_0$ with eigenvalue $\lambda_0 = 0$. Indeed, the constant function $f_0 = \frac{1}{\sqrt{A}}$ is such that $\partial_{\nu} f_0 = 0$ and, hence, $-\Delta f_0 = 0$, and moreover $f_0$ is correctly normalised. Conversely, if $\Delta f_0 = 0$ one has $\int \sqrt{g} g^{\mu\nu} \partial_{\mu} f_0 \partial_{\nu} f_0 = 0$ and then $f_0$ must be constant.

One can then develop any $\phi(x)$ on the basis of these eigenfunctions:

$$ \phi(x) = \sum_n c_n f_n(x). \quad (1.22) $$
Inserting this into the action and using the orthonormality of the $f_n$ one gets

$$S[g, \phi] = \int d^2x \sqrt{g} \phi(-\Delta\phi) = \sum_{n \neq 0} \lambda_n c_n^2. \qquad (1.23)$$

To compute the so-called matter partition function (the field $\phi$ is considered to be the matter) for fixed metric $g_{\mu\nu}$:

$$Z_{\text{matter}}[g] = \int \mathcal{D}\phi e^{-S[g,\phi]}, \qquad (1.24)$$

we also need to define the functional integral measure $\mathcal{D}\phi$ in such a way that one integrates over all possible field configurations $\phi(x)$. This is achieved via the expansion (1.22) by integrating over all the expansion coefficients $c_k$:

$$\mathcal{D}\phi = \sqrt{A} \int \prod_{k \neq 0} dc_k, \qquad (1.25)$$

where we “arbitrarily” excluded the integration over the zero-mode coefficient $c_0$ (since it would lead to an infinite factor in (1.24)), and inserted instead a factor $\sqrt{A}$. Then

$$\int \mathcal{D}\phi e^{-S[g,\phi]} = \sqrt{A} \int \left( \prod_{k \neq 0} dc_k \right) e^{-\sum_{n \neq 0} \lambda_n c_n^2} = \sqrt{A} \prod_{n \neq 0} \sqrt{\frac{\pi}{\lambda_n}} = \mathcal{N} \sqrt{A} (\text{Det}'(-\Delta))^{-1/2}, \qquad (1.26)$$

where $\mathcal{N}$ is some (infinite) normalisation coefficient, and $\text{Det}'$ denotes the determinant computed as the product of all non-zero eigenvalues. One thus has

$$Z_{\text{matter}}[g] = \mathcal{N}\sqrt{A}(\text{Det}'(-\Delta_g))^{-1/2}, \qquad (1.27)$$

where we added a subscript $g$ on $\Delta$ to remind us that this is the Laplacian for metric $g$. The gravitational action is defined as

$$e^{-S_{\text{grav}}[\hat{g},g]} = \frac{Z_{\text{matter}}[g]}{Z_{\text{matter}}[\hat{g}]} = \sqrt{\frac{A}{\hat{A}}} \left( \frac{\text{Det}'(-\Delta_g)}{\text{Det}'(-\Delta_{\hat{g}})} \right)^{-1/2}, \qquad (1.28)$$

where the factors $\mathcal{N}$ now have cancelled.

One can go on, and rewrite this again in terms of the eigenvalues as

$$S_{\text{grav}}[\hat{g}, g] = \frac{1}{2} \sum_{n \neq 0} \log \lambda_n(g) - \frac{1}{2} \sum_{n \neq 0} \log \lambda_n(\hat{g}) - \frac{1}{2} \log \frac{A}{\hat{A}}. \qquad (1.29)$$

These sums over eigenvalues typically are divergent but are efficiently regularized (and “renormalized”!) by the well-known technique of zeta-function regularisation [28, 29]: one introduces the zeta-function<fnref n="6" /> $\zeta_g(s)$ of the operator $-\Delta_g$ as

$$\zeta_g(s) = \sum_{n \neq 0} \frac{1}{(\lambda_n(g))^s} \quad \Rightarrow \quad \zeta'_g(0) = - \sum_{n \neq 0} \log \lambda_n(g), \qquad (1.30)$$

<footnote n="6">⁶Since $n = 0$ is excluded from the sum, this rather corresponds to what we would denote as $\tilde{\zeta}_g$ in the main part of this paper.</footnote>
so that

$$S_{\text{grav}}[\hat{g}, g] = -\frac{1}{2}\zeta'_{g}(0) + \frac{1}{2}\zeta'_{\hat{g}}(0) - \frac{1}{2}\log \frac{A}{\hat{A}}. \qquad (1.31)$$

Although this is a nice expression, it is not very explicit unless we know all the eigenvalues and are able to compute (1.30). Instead, one determines the variation of the zeta function under an infinitesimal conformal variation of the metric and then tries to “integrate” these infinitesimal changes to get the finite difference (1.31). Obviously, from (1.30)

$$\delta\zeta_g(s) = -s \sum_{n \neq 0} \frac{\delta\lambda_n}{\lambda_n^{s+1}}, \qquad (1.32)$$

where $\lambda_n = \lambda_n(g)$ and $\delta\lambda_n$ is its variation when the metric is changed as $g \to e^{2\delta\sigma}g$. Under such a variation of the metric, the Laplace operator $\Delta = e^{-2\sigma}\hat{\Delta}$ changes as $\Delta \to e^{-2\sigma-2\delta\sigma}\hat{\Delta} = e^{-2\delta\sigma}\Delta = \Delta - 2\delta\sigma\Delta$, i.e. $\delta\Delta = -2\delta\sigma\Delta$. Then there is a corresponding change $\delta\lambda_n$ of the eigenvalues and change $\delta f_n$ of the eigenfunctions. Comparing the eigenvalue equations for $-\Delta$ and $-(\Delta + \delta\Delta)$ we have

$$-(\Delta + \delta\Delta)(f_n + \delta f_n) = (\lambda_n + \delta\lambda_n)(f_n + \delta f_n) \quad \Rightarrow \quad -\Delta\delta f_n - \delta\Delta f_n = \lambda_n\delta f_n + \delta\lambda_n f_n. \qquad (1.33)$$

We multiply this last equation with $f_n$ and integrate $\int d^2x \sqrt{g} \dots$. Then the first term gives $-\int d^2x \sqrt{g} f_n \Delta \delta f_n = -\int d^2x \sqrt{g} (\Delta f_n) \delta f_n = \lambda_n \int d^2x \sqrt{g} f_n \delta f_n$ which cancels a corresponding term on the right-hand side. Thus we are left with

$$-\int d^2x \sqrt{g} f_n(x) \delta\Delta f_n = \delta\lambda_n \int d^2x \sqrt{g} (f_n(x))^2 = \delta\lambda_n \qquad (1.34)$$

Now, recall that $-(\delta\Delta)f_n = 2\delta\sigma\Delta f_n = -2\delta\sigma\lambda_n f_n$ so that

$$\delta\lambda_n = -2\lambda_n \int d^2x \sqrt{g} \delta\sigma(x) (f_n(x))^2. \qquad (1.35)$$

This is to be inserted into (1.32):

$$\delta\zeta_g(s) = 2s \sum_{n \neq 0} \frac{\int d^2x \sqrt{g} \delta\sigma(x) (f_n(x))^2}{\lambda_n^s} = 2s \int d^2x \sqrt{g} \delta\sigma(x) \sum_{n \neq 0} \frac{(f_n(x))^2}{\lambda_n^s} \qquad (1.36)$$

If we also define the (bi-) local zeta-function as

$$\zeta_g(s, x, y) = \sum_{n \neq 0} \frac{f_n(x) f_n(y)}{\lambda_n^s}, \qquad (1.37)$$

this can be rewritten as

$$\delta\zeta_g(s) = 2s \int d^2x \sqrt{g} \delta\sigma(x) \zeta_g(s, x, x). \qquad (1.38)$$

We now take the derivative with respect to $s$. This yields two terms, but at $s = 0$ only one of them contributes:

$$\delta\zeta'_{g}(0) \equiv \left. \frac{d}{ds}\zeta_{g}(s) \right|_{s=0} = 2 \int d^{2}x \sqrt{g} \delta\sigma(x) \zeta_{g}(0, x, x). \qquad (1.39)$$
There is a standard relation between the heat kernel $K(t, x, y)$ for an operator and the corresponding local zeta function $\zeta(s, x, y)$:

$$K(t, x, y) = \sum_{n \neq 0} e^{-\lambda_n t} f_n(x) f_n(y), \quad \zeta(s, x, y) = \frac{1}{\Gamma(s)} \int_0^\infty dt \, t^{s-1} K(t, x, y). \quad (1.40)$$

Possible singularities of the integral arise from the integration region of small $t$ and one can show that the integral defines an analytic function for $\Re s > 1$ which can be analytically continued to the whole complex $s$-plane except for poles at $s = 1, 0, -1, -2, \dots$. However, $\Gamma(s)$ also has poles at $s = 0, -1, -2, \dots$ and dividing by $\Gamma(s)$ cancels the poles of the integral, except at $s = 1$. We see that $\zeta(s, x, y)$ has a pole at $s = 1$ and that its finite values at $s = 0, -1, -2, \dots$ are determined by the singularities of the integral due to the small-$t$ behaviour of the heat kernel. But the small-$t$ behaviour of the heat kernel is well-known, in particular

$$K(t, x, x) \sim_{t \to 0} \frac{1}{4\pi t} \left( 1 + \frac{\mathcal{R}(x)}{6} t + \mathcal{O}(t^2) \right) - \frac{1}{A}, \quad (1.41)$$

where the $-1/A$ comes from subtracting the zero-mode $f_0^2 = 1/A$ which is not included in the sum (1.40). From this one gets

$$\zeta(0, x, x) = \frac{1}{24\pi} \mathcal{R}(x) - \frac{1}{A}, \quad (1.42)$$

which when inserted in (1.39) gives

$$\delta S_{\text{grav}}[\hat{g}, g] = -\frac{1}{2} \delta \zeta'_{g}(0) - \frac{\delta A}{2A} = -\frac{1}{24\pi} \int d^2 x \sqrt{g} \delta\sigma(x) \mathcal{R}(x). \quad (1.43)$$

We recognise $\delta S_{\text{Liouville}}$ and conclude

$$S_{\text{grav}}[\hat{g}, g] = -\frac{1}{24\pi} S_{\text{Liouville}}[\hat{g}, g]. \quad (1.44)$$

This is a famous result, first obtained (using a different method) by Polyakov in 1981 [8].

## 2 The Dirac operator for Majorana spinors on a curved manifold

In this section we define the relevant Dirac operator $D$ in 2 Euclidean dimensions, with a Majorana mass term, and relate its square $D^2$ to the spinorial Laplace operator and scalar curvature. We discuss the eigenvalue problem of this Dirac operator and how it changes under (local) conformal transformations. Most importantly, we obtain a sufficient characterisation of the zero-modes of the massless Dirac operator and of the corresponding projectors on these zero-modes, which we will need in section 5. We first look at flat space, not only because it is simpler but also because it provides the formula valid in any given local orthonormal frame of the curved space. The material of the first part of this section is standard or already contained in [1] but we re-derive the relevant formula for a self-contained presentation.
## 2.1 Majorana spinors in flat 2D space: γ-matrices and Dirac operator

In two flat Euclidean dimensions we use labels $a = 1, 2$. Recall that the 3 Pauli matrices satisfy $\sigma_j\sigma_k = \delta_{jk}\mathbf{1} + i\epsilon_{jkl}\sigma_l$. We can then choose two real, hermitian $\gamma$-matrices as

$$ \gamma^1 = \sigma_x, \quad \gamma^2 = \sigma_z \quad \Rightarrow \quad \{\gamma^a, \gamma^b\} = 2\delta^{ab}\mathbf{1}, \quad (\gamma^a)^\dagger = \gamma^a, \quad (\gamma^a)^T = \gamma^a, \quad (\gamma^a)^* = \gamma^a. \qquad (2.1) $$

The chirality matrix (the analogue of $\gamma_5$ in 4 dimensions) then is

$$ \gamma_* = i\gamma^1\gamma^2 = \sigma_y \quad \Rightarrow \quad \gamma_*^\dagger = \gamma_*, \quad \gamma_*^T = \gamma_*^* = -\gamma_*, \qquad (2.2) $$

and, of course, $\gamma_*$ anticommutes with $\gamma^a$. These are all $2 \times 2$ matrices and, hence, a spinor $\psi$ has two components that are a priori complex.

In 2 Euclidean dimensions the “Lorentz” group is just the rotation group $SO(2)$ which has a single generator $J^{12}$. In the spin-$\frac{1}{2}$ representation as carried by the Dirac spinor this (hermitian) generator is $J^{12} = -\frac{i}{4}[\gamma^1, \gamma^2] = -\frac{i}{2}\gamma^1\gamma^2 = -\frac{1}{2}\gamma_*$ and the representation of a finite rotation by an angle $\alpha$ on a spinor $\psi$ is given by the matrix $D(\alpha) = e^{i\alpha J^{12}} = e^{-i\alpha\gamma_*/2}$. More precisely, this means that under such a rotation, acting also on the coordinates$^7$ as $x'^a = \Lambda_b^a x^b$, the spinor transforms as $\psi(x) \rightarrow \psi'(x') = D(\alpha)\psi(x)$. Now, with the choice (2.1) of the $\gamma$-matrices, the matrix $D(\alpha)$ is real and $\psi^*$ transforms exactly as $\psi$. Furthermore since $\gamma_*$ commutes with $D(\alpha)$, $\gamma_*\psi$ also transforms as $\psi$. Finally, using the relation $D(\alpha)^{-1}\gamma^\mu D(\alpha) = \Lambda_\rho^\mu \gamma^\rho$ one also shows that $(\partial\psi)'(x') = D(\alpha)\partial\psi(x)$. Hence, all $\psi^*$, $\gamma_*\psi$ and $\partial\psi$ all transform exactly as $\psi$. In particular, it makes sense to impose the Majorana condition $\psi^* = \psi$, imposing that the two components $\psi_1$ and $\psi_2$ of $\psi$ are both real, or write an equation combining $i\partial\psi$ and $m\gamma_*\psi$.

Note that the spinors are anti-commuting objects obeying $\psi_\alpha\psi_\beta = -\psi_\beta\psi_\alpha$, and one has for a complex spinor $\psi^\dagger\psi = (\psi^*)^\dagger\psi = \psi_1^*\psi_1 + \psi_2^*\psi_2$. The complex conjugation of such an expression is defined as the hermitian conjugate and involves reversing the order of the terms, i.e.

$$ (\chi_a\psi_\beta)^* = \psi_\beta^*\chi_a^*. \qquad (2.3) $$

We then see that $(\psi_1^*\psi_1)^* = \psi_1^*(\psi_1^*)^* = \psi_1^*\psi_1$ and $\psi^\dagger\psi$ is indeed real, and $\int m\psi^\dagger\psi$ is the standard (real) mass term in the action for a Dirac spinor

For anti-commuting Majorana, i.e. real spinors, however, a mass term like $\int m\psi^\dagger\psi = \int m\psi^T\psi = m\int(\psi_1\psi_1 + \psi_2\psi_2)$ vanishes. We can, nevertheless, introduce a non-vanishing, real mass-term as $\int \psi^\dagger m\gamma_*\psi = m\int(\psi_1(-i)\psi_2 + \psi_2i\psi_1) = -2im\int \psi_1\psi_2$. This is indeed real since $(\psi_1\psi_2)^* = \psi_2\psi_1 = -\psi_1\psi_2$. The action for such a Majorana spinor then reads

$$ S = \int \psi^\dagger(i\partial\psi + m\gamma_*)\psi. \qquad (2.4) $$

The corresponding Dirac operator

$$ D = i\partial\psi + m\gamma_* \qquad (2.5) $$

<footnote>$^7$Below, on a curved manifold, we will consider separately the notions of covariance under rotations of the spinor and under transformations of the coordinates.</footnote>
is hermitian with respect to the standard inner product on the space of Dirac spinors,

$$ (\chi, \psi) = \int \chi^{\dagger}(x) \psi(x) = \int (\chi_{1}^{*}\psi_{1} + \chi_{2}^{*}\psi_{2}) = (\psi, \chi)^{*}, \quad (O\chi, \psi) = (\chi, O^{\dagger}\psi). \quad (2.6) $$

Indeed, taking the hermitian conjugate of $D$ involves taking the hermitian conjugate of the $\gamma$ matrices, (which we have chosen to be hermitian), complex conjugation of the $i$ and an integration by parts of $\partial_{\mu}$. Hence

$$ D^{\dagger} = D. \quad (2.7) $$

In particular, we also see that $S = (\psi, D\psi) = (D\psi, \psi) = (\psi, D\psi)^* = S^*$ which confirms that the action is real. Next, one has

$$ D^2 = (i\partial\!\!\!/ + m\gamma_*)(i\partial\!\!\!/ + m\gamma_*) = -\partial\!\!\!/^2 + im(\partial\!\!\!/\gamma_* + \gamma_*\partial\!\!\!/ + m^2) = -\partial\!\!\!/^2 + m^2 = -\partial_\mu\partial^\mu + m^2 $$

which incorporates the correct Euclidean continuation of the mass-shell condition $p_\mu p^\mu + m^2 = 0$.

One might ask why we want to study Majorana spinors with the Dirac operator $D$ and not simply (complex) Dirac spinors with an ordinary mass term and corresponding action $S = \int \psi^{\dagger}(i\partial\!\!\!/ + m)\psi$. However, the square of the corresponding Dirac operator $\tilde{D} = i\partial\!\!\!/ + m$ is $\tilde{D}^2 = (i\partial\!\!\!/ + m)^2 = -\partial_\mu\partial^\mu + m^2 + 2im\partial\!\!\!/\partial\!\!\!/ + m^2$ and this does not correspond to anything simple or physical. (It is $(i\partial\!\!\!/ - m)(i\partial\!\!\!/ + m)$ that instead gives the mass-shell condition.) This is one of the reasons we will focus on the action (2.4) and corresponding Dirac operator (2.5). Note that this Dirac operator $D$ is a purely imaginary. Indeed, the $\gamma^\mu$ are real and $\gamma_*$ is purely imaginary. Thus $iD$ is a real operator and the Dirac equation $D\psi = (i\partial\!\!\!/ + m\gamma_*)\psi = 0$ admits real solutions $\psi$. However, we will be interested in the corresponding eigenvalue problem $D\psi_n = (i\partial\!\!\!/ + m\gamma_*)\psi_n = \lambda_n\psi_n$. Since $D$ is hermitian, the eigenvalues $\lambda_n$ are real and then the $\psi_n$ cannot be real, but must be complex. We will discuss this eigenvalue problem more generally below, on curved space.

## 2.2 Curved space: $\gamma$-matrices, covariant derivatives, Dirac operator and spinorial Laplacian

To define spinors on a curved space (-time) one needs to introduce locally orthonormal frames which is done by introducing the so-called viel-bein $e_\mu^a$. To define the Dirac operator one needs to define the covariant derivative $\nabla_\mu$ which involves the spin connection $\omega_\mu^{ab}$ which is related to the $e_\mu^a$ by the zero-torsion condition. In this subsection we recall how this is done and we also establish a few useful relations. Of course, the equations in this subsection are well-known to supergravity practitioners, but some are still somewhat non-trivial, and we find it useful to re-derive them here for further reference. Since most of the discussion is not specific to two dimensions we first discuss arbitrary $n$-dimensional spaces (with Euclidean signature$^8$) and only set $n = 2$ in the end. Our detailed discussion also allows us to make some statements about possible zero-modes of the Dirac operator that will be useful in the sequel.

<footnote>$^8$The whole discussion can also be repeated for Lorentzian signature, but one would need to pay attention to the upper or lower positions of the flat indices $a, b, \dots$ also.</footnote>
### 2.2.1 Local orthonormal frames and spin connection (in n dimensions)

Starting from the metric tensor written using the coordinate one-forms $dx^\mu$, one introduces the one-forms $e^a$ of the local orthonormal frame as

$$g_{\mu\nu}(x) \, dx^\mu \otimes dx^\nu = \delta_{ab} e^a(x) \otimes e^b(x). \qquad (2.8)$$

Both sets of indices $\mu, \nu, \dots$ and $a, b, \dots$ take values 1, 2, \dots, n. We can always express the $e^a$ in terms of the $dx^\mu$ and vice versa:

$$e^a(x) = e^a_\mu(x) \, dx^\mu, \quad dx^\mu = E^\mu_b(x) e^b(x), \quad e^a_\mu(x) E^\mu_b(x) = \delta^a_b, \quad E^\mu_b(x) e^b_\nu(x) = \delta^\mu_\nu. \qquad (2.9)$$

As indicated here, the viel-bein $e^a_\mu$ and inverse viel-bein $E^\mu_a$ depend on the point $x$ on the manifold. Combining (2.9) and (2.8) results in the relations

$$g_{\mu\nu} E^\mu_a E^\nu_b = \delta_{ab}, \quad \delta_{ab} e^a_\mu e^b_\nu = g_{\mu\nu} \quad \text{and} \quad \delta^{ab} E^\mu_a E^\nu_b = g^{\mu\nu}, \quad g^{\mu\nu} e^a_\mu e^b_\nu = \delta^{ab}. \qquad (2.10)$$

The $\delta_{ab}$ is to be interpreted as the metric in the local orthonormal frames. The $e^a_\mu$ and $E^\mu_a$ implement the change of basis between these orthonormal local frames and the coordinate frames but, of course, in general they cannot be interpreted as Jacobian matrices for a change of coordinates. Since the metric in the orthonormal frame is $\delta_{ab}$ and the inverse metric $\delta^{ab} = \delta_{ab}$, raising and lowering the corresponding indices $a, b, \dots$ has no effect and we write indifferently $T^{ab} \equiv T^a_b$ etc. Of course this does not apply to the “coordinate indices” $\mu, \nu, \dots$.

The relevant connection in the local orthonormal frame is the spin-connection $\omega^a_\mu$. It is defined in terms of the connection coefficients $\Gamma^c_{ab}$ in the local orthonormal frame as $\omega^a_{b\mu} = \Gamma^a_{cb} e^c_\mu$. The $\Gamma^c_{ab}$ can be obtained from the $\Gamma^\rho_{\nu\mu}$ in the coordinate frame through the (in-homogeneous) transformation rules for the connection coefficients as $\Gamma^c_{ab} = E^\mu_a E^\nu_b \Gamma^\rho_{\mu\nu} e^c_\rho + E^\mu_a \partial_\mu E^\rho_b e^c_\rho$. However, since we will only work on Riemannian manifolds, where the torsion vanishes and the metric is covariantly constant, the spin-connection can be more easily obtained through the zero-torsion condition which is most compactly written as an equation for the spin-connection one-form $\omega^{ab} = \omega^{ab}_\mu dx^\mu$ using the exterior derivative $d$:

$$d\omega^a + \omega^{ab} \wedge \omega^b = 0 \quad \Leftrightarrow \quad \partial_\mu e^a_\nu - \partial_\nu e^a_\mu + \omega^{ab} e^b_\nu - \omega^{ab} e^b_\mu = 0. \qquad (2.11)$$

Requiring the metric to be covariantly constant implies that the spin-connection is anti-symmetric in $a$ and $b$:

$$\nabla_\mu g_{\nu\rho} = 0 \quad \Leftrightarrow \quad \nabla_\mu \delta_{ab} = 0 \quad \Leftrightarrow \quad \omega_{ab\mu} = -\omega_{ba\mu}. \qquad (2.12)$$

Then there are $n \times \frac{n(n-1)}{2}$ independent components $\omega^a_\mu$, while (2.11) provides exactly the same number of equation, hence determining the spin-connection completely in terms of the $e^a_\mu$. It is not too difficult to derive the following useful relations

$$\begin{aligned} \nabla_\mu e^a_\nu &\equiv \partial_\mu e^a_\nu + \omega^a_\mu e^b_\nu - \Gamma^\rho_{\mu\nu} e^a_\rho = 0, \\ \nabla_\mu E^\nu_a &\equiv \partial_\mu E^\nu_a + \Gamma^\nu_{\mu\rho} E^\rho_a - E^\nu_b \omega^{ba}_\mu = 0. \end{aligned} \qquad (2.13)$$
Finally we note that we obtain the Riemann curvature tensor and curvature 2-form from the spin connection as

$$R^{ab} = \frac{1}{2} R_{\mu\nu}^{ab} dx^{\mu} \wedge dx^{\nu} = d\omega^{ab} + \omega^{ac} \wedge \omega^{cb}, \quad R_{b\mu\nu}^{a} = e_{\lambda}^{a} E_{b}^{\rho} R_{\rho\mu\nu}^{\lambda}. \quad (2.14)$$

Let us now discuss the Dirac matrices. The Dirac-matrices $\gamma^a$ of the previous sub-section satisfied the Clifford algebra $\{\gamma^a, \gamma^b\} = 2\delta^{ab}1$ as appropriate in flat space. This means that they are also the appropriate Dirac matrices in the local orthonormal frames. To avoid confusion, we temporarily denote these Dirac-matrices as $\tilde{\gamma}^a$ so that

$$\{\tilde{\gamma}^a, \tilde{\gamma}^b\} = 2\delta^{ab}1 \qquad (2.15)$$

One then defines the curved-space Dirac matrices as

$$\gamma^{\mu} = E_{a}^{\mu} \tilde{\gamma}^{a} \quad \Rightarrow \quad \{\gamma^{\mu}, \gamma^{\nu}\} = E_{a}^{\mu} E_{b}^{\nu} \{\tilde{\gamma}^{a}, \tilde{\gamma}^{b}\} = E_{a}^{\mu} E_{b}^{\nu} 2\delta^{ab}1 = 2g^{\mu\nu}1. \qquad (2.16)$$

We assume again that all $n$ matrices $\tilde{\gamma}^a$ are chosen to be hermitian so that also

$$(\gamma^{\mu})^{\dagger} = \gamma^{\mu}. \qquad (2.17)$$

To simplify the notation, we will usually not put the tilde over the flat-space $\gamma$-matrices with the convention that $\gamma^\mu, \gamma^\nu, \gamma^\rho$ etc are always curved-space $\gamma$-matrices and $\gamma^a, \gamma^b, \gamma^c$ etc are always flat-space $\gamma$-matrices. The only potential source of confusion comes when the indices take numerical values, but we will always use the convention that $\gamma^1, \gamma^2, \dots$ are flat-space $\gamma$-matrices.

### 2.2.2 Covariant derivatives (in $n$ dimensions)

Let us now recall how to construct the appropriate covariant derivative acting a the spinor $\psi$. A spinor transforms (locally) in the spin representation of the local Lorentz group $SO(n-1, 1)$ or at present - since we are in Euclidean signature - the local rotation group $SO(n)$ as

$$\psi(x) \rightarrow \psi'(x) = L_{\text{Dirac}}(x)\psi(x) \equiv \exp\left(\frac{i}{2}\theta^{cd}(x)J_{cd}^{\text{Dirac}}\right)\psi(x), \qquad (2.18)$$

where the $\theta^{cd}(x) = -\theta^{dc}(x)$ are the (local) rotation angles in the $cd$ plane, and the $J_{cd}^{\text{Dirac}} = -J_{dc}^{\text{Dirac}}$ are the corresponding hermitian $SO(n)$ generators in the spin / Dirac representation

$$J_{cd}^{\text{Dirac}} = -\frac{i}{2}\gamma_{cd} \equiv -\frac{i}{4}[\gamma_c, \gamma_d], \quad (J_{cd}^{\text{Dirac}})^{\dagger} = J_{cd}^{\text{Dirac}} \qquad (2.19)$$

It follows from the Clifford algebra satisfied by the $\gamma^a$ that the $J_{cd}^{\text{Dirac}}$ indeed satisfy the $SO(n)$ algebra

$$[J_{cd}, J_{ef}] = -i(\delta_{de}J_{cf} - \delta_{df}J_{ce} - \delta_{ce}J_{df} + \delta_{cf}J_{de}). \qquad (2.20)$$

Note that the local $SO(n)$ transformations (2.18) do not act on the coordinates: we have the same $x$ before and after the transformation. The corresponding local $SO(n)$ transformation of the basis forms $e^a$ is

$$e^a(x) \rightarrow e'^a(x) = (L_{\text{vect}}(x))^{ab} e^b(x) = \left[ \exp\left(\frac{i}{2}\theta^{cd}(x)J_{cd}^{\text{vect}}\right) \right]^{ab} e^b(x), \qquad (2.21)$$
where

$$ (J_{cd}^{\text{vect}})^{ab} = -i(\delta_c^a \delta_d^b - \delta_c^b \delta_d^a). \qquad (2.22) $$

are the purely imaginary and anti-symmetric, hence hermitian, generators of the vector representation of SO(n). Of course, they also satisfy the SO(n)-algebra (2.20).

How does the spin-connection transform under such a local SO(n) transformation? The torsion being a tensor, its vanishing cannot be affected by a (local) change of basis. Hence, in the new frame we must still have $de^a + \omega^{ab}e^b = 0$. Using $de^a = d(L^{ab}e^b) = dL^{ab}e^b + L^{ab}de^b = (dL^{ac} - L^{ab}\omega^{bc})e^c$, we get

$$ \omega'^{ab} = L^{ac}\omega^{cd}(L^{-1})^{db} - (dL^{ac})(L^{-1})^{cb} = L^{ac}\omega^{cd}(L^{-1})^{db} + L^{ac}(dL^{-1})^{cb}, \qquad (2.23) $$

showing that the $\omega^{ab}$ transform exactly as a gauge field in the adjoint representation of the local rotation group. Indeed, the matrices $L$ appearing in (2.23) are what we above called $L_{\text{vect}}$ and we have $\omega^{ab} = \frac{i}{2}\omega^{cd}(J_{cd}^{\text{vect}})^{ab}$. But the transformation of the $\omega_{\mu}^{cd}$ cannot not depend on the representation we choose, so that we also have

$$ \frac{i}{2}\omega'^{cd}(x)J_{cd}^{\text{Dirac}} = \frac{i}{2}\omega^{cd}(x)L_{\text{Dirac}}(x)J_{cd}^{\text{Dirac}}L_{\text{Dirac}}^{-1}(x) + L_{\text{Dirac}}(x))dL_{\text{Dirac}}^{-1}(x). \qquad (2.24) $$

The covariant derivative $\nabla_{\mu}$ of a spinor $\psi(x)$ should be defined, as usual, such that under a local SO(n) rotation $\nabla_{\mu}\psi(x)$ transforms exactly as $\psi(x)$. If we define

$$ \nabla_{\mu}\psi = \partial_{\mu}\psi + \frac{i}{2}\omega_{\mu}^{cd}J_{cd}^{\text{Dirac}}\psi \quad \Leftrightarrow \quad \nabla_{\mu}\psi_{\alpha} = \partial_{\mu}\psi_{\alpha} + \frac{i}{2}\omega_{\mu}^{cd}(J_{cd}^{\text{Dirac}})_{\alpha\beta}\psi_{\beta} \qquad (2.25) $$

it is straightforward to check, using (2.18) and (2.24) that we have indeed

$$ \nabla'_{\mu}\psi'(x) \equiv (\partial_{\mu} + \frac{i}{2}\omega'_{\mu}^{cd}(x)J_{cd}^{\text{Dirac}})(L_{\text{Dirac}}(x)\psi(x)) = L_{\text{Dirac}}(x)\nabla_{\mu}\psi(x), \qquad (2.26) $$

where $\nabla'_{\mu}$ involves $\omega'_{\mu}^{cd}$. Note again that $x$ is unchanged so that $\nabla'_{\mu}$ involves $\partial_{\mu}$ and not some $\partial'_{\mu}$. Later-on we will also encounter covariant derivatives acting on other objects transforming in different representations and, accordingly, these covariant derivatives will include other connection terms. If we want to insist that we are dealing exactly with the covariant derivative acting on the spinor as just defined we will write $\nabla_{\mu}^{\text{sp}}$ so that

$$ \nabla_{\mu}^{\text{sp}} \equiv \partial_{\mu} + \frac{i}{2} \omega_{\mu}^{cd} J_{cd}^{\text{Dirac}} = \partial_{\mu} + \frac{1}{4} \omega_{\mu}^{cd} \gamma_{cd}, \qquad \nabla_{\mu} \psi = \nabla_{\mu}^{\text{sp}} \psi. \qquad (2.27) $$

As a general rule, a covariant derivative $\nabla_{\mu}$ acting on some quantity $F_{(i)}$ transforming in a representation $\mathcal{R}_i$ of the local rotation group is always defined in such a way that $\nabla_{\mu}F_{(i)}$ transforms in the same representation $\mathcal{R}_i$. Just as we wrote $\nabla^{\text{sp}}$ to emphasise this fact we could write $\nabla_{\mu}^{(i)}$, but one usually doesn't. It follows from this remark, that the covariant derivative obeys the Leibnitz rule,

$$ \nabla_{\mu}(F_{(i)}F_{(j)}) = (\nabla_{\mu}F_{(i)})F_{(j)} + F_{(i)}\nabla_{\mu}F_{(j)}. \qquad (2.28) $$
It similarly follows that in an integral of a quantity that is a scalar under the local rotation group, the covariant derivatives can be freely integrated by parts (possibly up to a boundary term). One application of these rules is the following identity we will need later-on

$$ - \int \sqrt{g} (\nabla_{\mu} f) \psi^{\dagger} \gamma^{\mu} \chi = \int \sqrt{g} f \left( (\nabla_{\mu}^{\text{sp}} \psi)^{\dagger} \gamma^{\mu} \chi + \psi^{\dagger} \gamma^{\mu} \nabla_{\mu}^{\text{sp}} \chi \right), \quad (2.29) $$

where $f = f(x)$ is a scalar (so that $\nabla_{\mu}f = \partial_{\mu}f$) and $\psi = \psi(x)$ and $\chi = \chi(x)$ are Dirac (or Majorana) spinors. It is relatively straightforward to check this identity explicitly by integrating by parts, using $\frac{\partial_{\mu}\sqrt{g}}{\sqrt{g}} = \Gamma_{\mu\sigma}^{\sigma}$, $\gamma^{\mu} = E_{a}^{\mu}\gamma^{a}$ and the relations (2.13), as well as the hermiticity of the $J_{cd}^{\text{Dirac}}$ and the commutation relation of the latter with the $\gamma^{a}$.

Let us insist that we have discussed local, i.e. $x$-dependent frame transformations by some $L(x)$ in the appropriate representation of $SO(n)$. We have not made any coordinate transformation. In the present formalism, local $SO(n)$ transformations and coordinate transformations are totally decoupled. One may, of course, also do a coordinate transformation $x^{\mu} \rightarrow x'^{\mu} = f^{\mu}(x^{\nu})$. Under such a coordinate transformation the spinor $\psi$, as well as all frame vectors like $v^{a}$ are inert, i.e. behave as scalar quantities. For a vector e.g. one goes from the $v^{a}$ to the $v^{\mu}$ by the appropriate change of basis, i.e. $v^{\mu} = E_{a}^{\mu}v^{a}$ and it is the transformation of the $E_{a}^{\mu}$ under coordinate transformations, namely $E'_{a}{}^{\mu} = \frac{\partial x'^{\mu}}{\partial x^{\nu}} E_{a}^{\nu}$, that implies that the $v^{\mu}$ will indeed transform as the components of a coordinate vector, i.e. $v'^{\mu} = \frac{\partial x'^{\mu}}{\partial x^{\nu}} v^{\nu}$.

Obviously, both $\partial_{\mu}$ and $\omega_{\mu}^{ab}$ transform under coordinate transformations as the covariant components of a coordinate vector and, hence, so does $\nabla_{\mu}\psi$. Let us temporarily denote the quantities in the $x'$ coordinates by a tilde (rather than a prime which we used for the local $SO(n)$ transformations). Since $\psi$ behaves as a scalar quantity under these coordinate transformation we then have $\tilde{\psi}(x') = \psi(x)$ and thus indeed

$$ \tilde{\nabla}_{\mu} \tilde{\psi}(x') = \frac{\partial x^{\nu}}{\partial x'^{\mu}} \nabla_{\nu} \psi(x). \quad (2.30) $$

Next, the Dirac operator should be the obvious curved space generalisation of (2.5), i.e. $D = i\mathcal{V} + m\gamma_{*}$ where we define the operator $\mathcal{V}$ as

$$ \mathcal{V} = \gamma^{\mu} \nabla_{\mu} = \gamma^{a} E_{a}^{\mu} \nabla_{\mu} \Rightarrow \mathcal{V} \psi = \mathcal{V}^{\text{sp}} \psi = \gamma^{\mu} \nabla_{\mu}^{\text{sp}} \psi = \gamma^{a} E_{a}^{\mu} \nabla_{\mu}^{\text{sp}} \psi. \quad (2.31) $$

We have just seen that under coordinate transformations $\nabla_{\mu}^{\text{sp}}\psi$ transforms as the covariant components of a coordinate vector, while $E_{a}^{\mu}$ transforms as the contravariant components. Of course, the $\gamma^{a}$ are just numerical matrices that do not change. Hence, $\mathcal{V}\psi$ is a scalar under coordinate transformations, just as $\psi$. Next, under a local Lorentz transformation $L$, we have seen that $\nabla_{\mu}^{\text{sp}}\psi$ transforms with $L_{\text{Dirac}}$ while $E'_{a}{}^{\mu} = (L_{\text{vect}})_{ab}E_{b}^{\mu}$. Hence<fnref n="9" />

$$ \begin{aligned} \mathcal{V}\psi(x) &= \gamma^a E_a^\mu(x) \nabla_\mu^{\text{sp}} \psi(x) \quad \rightarrow \quad \gamma^a E_a^{\mu}(x) \nabla_\mu^{\prime \text{sp}} \psi'(x) = \gamma^a \left( (L_{\text{vect}})_{ab} E_b^\mu \right) (L_{\text{Dirac}} \nabla_\mu^{\text{sp}} \psi(x)) \\ &= (L_{\text{vect}})_{ab} L_{\text{Dirac}} (L_{\text{Dirac}}^{-1} \gamma^a L_{\text{Dirac}}) E_b^\mu \nabla_\mu^{\text{sp}} \psi(x) = (L_{\text{vect}})_{ab} L_{\text{Dirac}} (L_{\text{vect}})_{ac} \gamma^c E_b^\mu \nabla_\mu^{\text{sp}} \psi(x) \\ &= L_{\text{Dirac}} \gamma^b E_b^\mu \nabla_\mu^{\text{sp}} \psi(x) = L_{\text{Dirac}} \mathcal{V}\psi(x). \end{aligned} \quad (2.32) $$

<footnote n="9">⁹Note that $L_{\text{Dirac}}$, just as $\gamma^a$, are matrices while $(L_{\text{vect}})_{ab}$ are the components of the matrix $L_{\text{vect}}$. Hence, $(L_{\text{vect}})_{ab}$ commutes with $L_{\text{Dirac}}$ and with $\gamma^a$. Also, just as in flat space, we have $L_{\text{Dirac}}^{-1} \gamma^a L_{\text{Dirac}} = (L_{\text{vect}})_{ac} \gamma^c$.</footnote>
This means that under both, coordinate transformations and local Lorentz transformations, $\bar{\nabla}^{\text{sp}}\psi$ transforms exactly as $\psi$. In particular, this implies that the appropriate covariant derivative $\nabla_{\nu}$ acting on $\bar{\nabla}\psi$ is again $\nabla_{\nu}^{\text{sp}}$:

$$ \nabla_{\nu}\bar{\nabla}\psi = \nabla_{\nu}^{\text{sp}}\bar{\nabla}\psi = \nabla_{\nu}^{\text{sp}}\bar{\nabla}^{\text{sp}}\psi \quad \Rightarrow \quad \bar{\nabla}\bar{\nabla}\psi = \bar{\nabla}^{\text{sp}}\bar{\nabla}^{\text{sp}}\psi = \gamma^{\mu}\nabla_{\mu}^{\text{sp}}\gamma^{\nu}\nabla_{\nu}^{\text{sp}}\psi. \quad (2.33) $$

However, $\bar{\nabla}\bar{\nabla}\psi$ is not $\gamma^{\mu}\gamma^{\nu}\nabla_{\mu}^{\text{sp}}\nabla_{\nu}^{\text{sp}}\psi$. Instead one has, of course $\bar{\nabla}\bar{\nabla}\psi = \gamma^{\mu}\nabla_{\mu}(\gamma^{\nu}\nabla_{\nu}^{\text{sp}}\psi)$ and the question is how to "commute" the $\nabla_{\mu}$ with the $\gamma^{\nu}$. On the one hand, from the usual rules about how the covariant derivative should act on coordinate and frame indices we have

$$ \nabla_{\mu}\gamma^{a} = \partial_{\mu}\gamma^{a} + \omega_{\mu}^{ab}\gamma^{b} = \omega_{\mu}^{ab}\gamma^{b} \quad \Leftrightarrow \quad \nabla_{\mu}\gamma^{\nu} = \partial_{\mu}\gamma^{\nu} + \Gamma_{\mu\rho}^{\nu}\gamma^{\rho}, \quad (2.34) $$

where the equivalence of both relations is due to $E_{a}^{\mu}$ being covariantly constant, cf (2.13). On the other hand, $\nabla_{\mu}$ also is a matrix, just like the $\gamma^{\nu}$, and one must also take into account the non-vanishing commutator. This then leads to

$$ \nabla_{\mu}(\gamma^{\nu}\nabla_{\nu}\psi) = \gamma^{\nu}\nabla_{\mu}\nabla_{\nu}\psi, \quad (2.35) $$

where on the right-hand side the covariant derivatives are, of course, $\nabla_{\mu}\nabla_{\nu}\psi = \nabla_{\mu}^{\text{sp}}\nabla_{\nu}^{\text{sp}}\psi - \Gamma_{\mu\nu}^{\rho}\nabla_{\rho}^{\text{sp}}\psi$. Formally, the $\nabla_{\mu}$ just passes through the $\gamma^{\nu}$. One could say that the contribution from the commutator cancels the contribution from $(\nabla_{\mu}\gamma^{\nu})!$ In any case, we have

$$ \bar{\nabla}\bar{\nabla}\psi \equiv (\gamma^{\mu}\nabla_{\mu})(\gamma^{\nu}\nabla_{\nu})\psi = \gamma^{\mu}\gamma^{\nu}\nabla_{\mu}\nabla_{\nu}\psi = \gamma^{\mu}\gamma^{\nu}(\nabla_{\mu}^{\text{sp}}\nabla_{\nu}^{\text{sp}} - \Gamma_{\mu\nu}^{\rho}\nabla_{\rho}^{\text{sp}})\psi. \quad (2.36) $$

Next, using $\gamma^{\mu}\gamma^{\nu} = g^{\mu\nu} + \gamma^{\mu\nu}$ (where, of course, $\gamma^{\mu\nu} = \frac{1}{2}[\gamma^{\mu}, \gamma^{\nu}]$) we have

$$ \bar{\nabla}\bar{\nabla}\psi = \gamma^{\mu}\gamma^{\nu}\nabla_{\mu}\nabla_{\nu}\psi = g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\psi + \gamma^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\psi = g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\psi + \frac{1}{2}\gamma^{\mu\nu}[\nabla_{\mu}, \nabla_{\nu}]\psi. \quad (2.37) $$

The commutator of two covariant derivatives is related to the curvature tensor as

$$ [\nabla_{\mu}, \nabla_{\nu}] = \frac{i}{2} R_{\mu\nu}^{cd} J_{cd}, \quad (2.38) $$

where $J_{cd}$ are the SO(n) generators in the appropriate representation.<fnref n="10" /> When acting on $\psi$ the generators are those in the Dirac representation and

$$ [\nabla_{\mu}, \nabla_{\nu}]\psi = \frac{i}{2} R_{\mu\nu}^{cd} (J_{cd}^{\text{Dirac}})\psi = \frac{1}{4} R_{\mu\nu}^{cd} \gamma_{cd}\psi. \quad (2.39) $$

Inserting this into (2.37) we get

$$ \bar{\nabla}\bar{\nabla}\psi = \Delta_{\text{sp}}\psi + \frac{1}{8}\gamma^{\mu\nu}R_{\mu\nu}^{cd}\gamma_{cd}\psi. \quad (2.40) $$

where we defined the spinoral Laplace operator as

$$ \Delta_{\text{sp}} = g^{\mu\nu}(\nabla_{\mu}^{\text{sp}}\nabla_{\nu}^{\text{sp}} - \Gamma_{\mu\nu}^{\rho}\nabla_{\rho}^{\text{sp}}). \quad (2.41) $$

<footnote n="10">¹⁰For a vector e.g. one has $[\nabla_{\mu}, \nabla_{\nu}]v^a = \frac{i}{2}R_{\mu\nu}^{cd}(J_{cd}^{\text{vect}})^{ab}v^b = \frac{1}{2}R_{\mu\nu}^{cd}(\delta_c^a\delta_d^b - \delta_d^a\delta_c^b)v^b = R_{\mu\nu}^{ab}v^b$ which is indeed the familiar formula.</footnote>
Finally, in any even dimension $n$, there is an appropriate chirality matrix $\gamma_* \sim \frac{1}{n!} \epsilon_{\mu_1 \dots \mu_n} \gamma^{\mu_1} \dots \gamma^{\mu_n} = \frac{1}{n!} \epsilon_{a_1 \dots a_n} \gamma^{a_1} \dots \gamma^{a_n}$. This shows that this chirality matrix is the same as one defines in flat space. We have $\{\gamma_*, \gamma^a\} = \{\gamma_*, \gamma^\mu\} = 0$. Including if necessary<fnref n="11" /> a factor of $i$ we have $\gamma_*^2 = \mathbf{1}$ and $\gamma_*^\dagger = \gamma_*$ (provided one has chosen hermitian $\gamma^a$ as we did). One can then define the hermitian, massive Dirac operator as the curved-space generalisation of (2.5):

$$D = i\slashed{\nabla} + m\gamma_*. \qquad (2.42)$$

Since $J_{cd}^{\text{Dirac}}$ commutes with $\gamma_*$ we see that $\nabla_\mu^{\text{sp}}$ commutes with $\gamma_*$ and the operator $\slashed{\nabla} = \gamma^\mu \nabla_\mu^{\text{sp}}$ anticommutes with $\gamma_*$. We then have

$$D^2 = -\slashed{\nabla}\slashed{\nabla} + m^2 = -\Delta_{\text{sp}} - \frac{1}{8}\gamma^{\mu\nu}R_{\mu\nu}^{\text{cd}}\gamma_{cd} + m^2. \qquad (2.43)$$

### 2.2.3 Back to two dimensions

In two dimensions the general formula simplify drastically. The local rotation group SO(2) is abelian and only has a single generator $J_{12}$. In its Dirac representation there is only $J_{12}^{\text{Dirac}} = -\frac{i}{2}\gamma_{12} = -\frac{i}{2}\gamma_1\gamma_2 = -\frac{1}{2}\gamma_*$ as already discussed in subsection 2.1. Similarly, the spin-connection $\omega^{ab}$ also is just a single one-form $\omega^{12} = -\omega^{21}$. We then have

$$\omega_{\mu}^{cd} \gamma_{cd} = 2\omega_{\mu}^{12} \gamma_{12} = -2i\omega_{\mu}^{12} \gamma_{*}. \qquad (2.44)$$

We find it convenient to define $\omega_\mu = 2\omega_\mu^{12}$, so that

$$\nabla_{\mu}^{\text{sp}} = \partial_{\mu} - \frac{i}{4}\omega_{\mu}\gamma_{*}, \quad \omega_{\mu} = 2\omega_{\mu}^{12}. \qquad (2.45)$$

Note that since $\gamma_* = \sigma_y$ is purely imaginary, $\nabla_\mu^{\text{sp}}$ is a real (anti-hermitian) differential operator. Obviously, it is a $2 \times 2$-matrix, and so is $\slashed{\nabla} = \gamma^\mu \nabla_\mu^{\text{sp}} = \gamma^a E_a^\mu \nabla_\mu^{\text{sp}}$.

Moreover, in two dimensions $\gamma^{\mu\nu} R_{\mu\nu}^{cd} \gamma_{cd}$ simplifies as

$$\gamma^{\mu\nu} R_{\mu\nu}^{cd} \gamma_{cd} = \gamma^{ab} R_{ab}^{cd} \gamma_{cd} = 4\gamma^{12} R_{12}^{12} \gamma_{12} = 4(-i\gamma_*)^2 R_{12}^{12} = -4R_{12}^{12} = -2\mathcal{R}. \qquad (2.46)$$

where $\mathcal{R} = R_{12}^{12} + R_{21}^{21} = 2R_{12}^{12}$ is the scalar Ricci curvature. Thus

$$D^2 = -\slashed{\nabla}\slashed{\nabla} + m^2 = -\Delta_{\text{sp}} + \frac{1}{4}\mathcal{R} + m^2, \qquad (2.47)$$

where the spinorial Laplacian $\Delta_{\text{sp}}$ was defined in (2.41). Just as $\nabla_\mu^{\text{sp}}$, the spinorial Laplacian is a $2 \times 2$-matrix differential operator. More precisely, it has a piece proportional to the identity matrix and involving the scalar Laplacian and a piece proportional to $\gamma_*$:

$$\Delta_{\text{sp}} = \mathbf{1}_{2 \times 2} \left( \Delta_{\text{scalar}} - \frac{1}{16} \omega^\mu \omega_\mu \right) - \frac{i}{4} \gamma_* \left( (\nabla_\mu \omega^\mu) + 2 \omega^\mu \partial_\mu \right), \qquad (2.48)$$

where the scalar Laplacian is defined by (1.7), or equivalently by

$$\Delta_{\text{scalar}} = g^{\mu\nu}(\partial_{\mu}\partial_{\nu} - \Gamma_{\mu\nu}^{\rho}\partial_{\rho}). \qquad (2.49)$$

<footnote n="11">¹¹In Euclidean signature one has $\gamma_* = i\gamma^1 \dots \gamma^n$ for $n = 4k + 2$ and $\gamma_* = \gamma^1 \dots \gamma^n$ for $n = 4k$.</footnote>
The hermitian, massive Dirac operator $D = i\nabla\nabla + m\gamma_*$ was already defined in (2.42). With our 2-dimensional real $\gamma^a$ it is moreover purely imaginary. For later reference, let us note that with our conventions it is very explicitly given by

$$D = i\sigma_x D_1 + i\sigma_z D_2 + m\sigma_y, \quad D_1 = E_1^\mu \partial_\mu - \frac{1}{4} E_2^\mu \omega_\mu, \quad D_2 = E_2^\mu \partial_\mu + \frac{1}{4} E_1^\mu \omega_\mu. \quad (2.50)$$

The eigenvalue problem of $D$ is

$$D\psi_n = (i\nabla\nabla + m\gamma_*)\psi_n = \lambda_n\psi_n. \quad (2.51)$$

Since $D$ is hermitian, its eigenvalues $\lambda_n$ are real. On the other hand, $D$ is purely imaginary and then the eigenfunctions $\psi_n$ cannot be real (unless $\lambda_n = 0$). Taking the complex conjugate of (2.51) we see that $\psi_n^*$ is also an eigenfunction but with eigenvalue $-\lambda_n$:

$$D\psi_n^* = -\lambda_n\psi_n^*. \quad (2.52)$$

For all $\lambda_n \neq 0$, within any couple $(\psi_n, \psi_n^*)$ we decide to call $\psi_n$ the eigenfunction with the positive eigenvalue. With this convention we always have

$$\lambda_n \ge 0. \quad (2.53)$$

We let

$$\psi_n = \frac{1}{\sqrt{2}}(\chi_n + i\phi_n), \quad (2.54)$$

with real $\chi_n$ and $\phi_n$ and then taking the real and imaginary parts of (2.51) gives

$$D\chi_n = i\lambda_n\phi_n, \quad D\phi_n = -i\lambda_n\chi_n. \quad (2.55)$$

Of course, the $\chi_n$, $\phi_n$, $\psi_n$ and $\psi_n^*$ are all eigenfunctions of $D^2$ with eigenvalue $\lambda_n^2$:

$$D^2\chi_n = \lambda_n^2\chi_n, \quad D^2\phi_n = \lambda_n^2\phi_n, \quad D^2\psi_n = \lambda_n^2\psi_n, \quad D^2\psi_n^* = \lambda_n^2\psi_n^*. \quad (2.56)$$

The Dirac operator $D$, as well as $i\nabla\nabla$, are hermitian with respect to the inner product

$$(\Psi_1, \Psi_2) = \int d^2x \sqrt{g} \Psi_1^\dagger \Psi_2. \quad (2.57)$$

As usual, the eigenfunctions $\psi_n$ corresponding to different $\lambda_n$ are orthogonal. Similarly, for the $\psi_n^*$. Also all $\psi_n^*$ are orthogonal to all $\psi_k$ (as long as $\lambda_n \neq 0$). Equivalently, assuming the eigenfunctions to be also normalised, we have for the real and imaginary parts

$$(\chi_n, \chi_k) = \delta_{nk}, \quad (\phi_n, \phi_k) = \delta_{nk}, \quad (\chi_n, \phi_k) = 0 \quad (\lambda_n \neq 0). \quad (2.58)$$

Note that the eigenfunctions of $D$ (the $\psi_n$ and $\psi_n^*$) are automatically eigenfunctions of $D^2$, but the converse is not necessarily true as is exemplified by the $\chi_n$ and $\phi_n$. What is true is that within any eigenspace of $D^2$ with eigenvalue $\Lambda_n = \lambda_n^2$ one can find linear combinations (corresponding precisely to the $\psi_n$ and $\psi_n^*$) that are eigenfunctions of $D$ with eigenvalues $+\lambda_n$ and $-\lambda_n$.
## 2.3 The example of the flat torus with arbitrary periods

It will be helpful to be able to check our general statements in the sequel of this paper against at least one example where we explicitly know the eigenvalues and eigenfunctions. The simplest example is the flat torus. Then the metric is just $g_{\mu\nu} = \delta_{\mu\nu}$ and $\nabla = \partial$. Of course, being flat, we will not be able to appreciate any effects of curvature. The relevant Dirac operator then is

$$D = i\partial\!\!\!/ + m\gamma_* = i\sigma_x\partial_1 + i\sigma_z\partial_2 + \sigma_y m = i \begin{pmatrix} \partial_2 & \partial_1 - m \\ \partial_1 + m & -\partial_2 \end{pmatrix}, \quad D^2 = -(\partial_1^2 + \partial_2^2) + m^2. \qquad (2.59)$$

There are still different tori corresponding to different periods.<fnref n="12" />

**The flat “square” torus:** We will shortly consider the flat torus with arbitrary periods, but we begin by looking at the “square” torus corresponding to a square of length $2\pi$ with opposite sides identified. The eigenvalue problem for $D^2$, i.e. $D^2\psi_n = \lambda_n^2\psi_n$ yields already

$$\lambda_n^2 = \lambda_{\vec{n}}^2 = n_1^2 + n_2^2 + m^2, \quad \psi_n \sim e^{in_1 x^1 + in_2 x^2}. \qquad (2.60)$$

We denote by $\lambda_{\vec{n}}$ the positive square root of $\lambda_{\vec{n}}^2$. Inserting the ansatz $\psi_{\vec{n}}(x^1, x^2) = \begin{pmatrix} a \\ b \end{pmatrix} e^{in_1 x^1 + in_2 x^2}$ into $D\psi_{\vec{n}} = \lambda_{\vec{n}}\psi_{\vec{n}}$ gives the two equivalent equations $-(n_1 + im)b = (\lambda_{\vec{n}} + n_2)a$ and $-(n_1 - im)a = (\lambda_{\vec{n}} - n_2)b$. This is easily solved, up to a common normalization, which we choose such that $\int d^2x \psi_{\vec{n}}^{\dagger}\psi_{\vec{n}} = 1$:

$$\psi_{\vec{n}}(\vec{x}) = \frac{1}{2\pi\sqrt{2\lambda_{\vec{n}}(\lambda_{\vec{n}} + n_2)}} \begin{pmatrix} n_1 + im \\ -\lambda_{\vec{n}} - n_2 \end{pmatrix} e^{in_1 x^1 + in_2 x^2}. \qquad (2.61)$$

Since $\lambda_{\vec{n}}$ only depends on $n_1^2 + n_2^2$, each eigenvalue is (at least) fourfold degenerate.<fnref n="13" /> Also

$$\psi_{\vec{n}}^* = -\psi_{-\vec{n}}|_{\lambda_{\vec{n}} \to -\lambda_{\vec{n}}}, \qquad (2.62)$$

which shows that $\psi_{\vec{n}}^*$ is indeed an eigenfunction of $D$ with eigenvalue $-\lambda_{\vec{n}}$. Thus the complete set of eigenfunctions is the set of all $\psi_{\vec{n}}$ and all $\psi_{\vec{n}}^*$.

**Flat torus with arbitrary periods:** a general flat two-dimensional torus is obtained as the quotient of $\mathbb{R}^2$ by a lattice $\Gamma$ where $\Gamma$ is generated by two linearly independent vectors $\vec{\omega}_1$ and $\vec{\omega}_2$, i.e. we identify any two points of $\mathbb{R}^2$ that differ by $p\vec{\omega}_1 + q\vec{\omega}_2$ with integer $p, q$. With respect to an orthonormal basis $\vec{e}_1, \vec{e}_2$ we can write $\vec{x} = x^1\vec{e}_1 + x^2\vec{e}_2$, $\vec{\omega}_1 = \omega_{11}\vec{e}_1 + \omega_{12}\vec{e}_2$ and $\vec{\omega}_2 = \omega_{21}\vec{e}_1 + \omega_{22}\vec{e}_2$, so that one identifies $x^1 \simeq x^1 + p\omega_{11} + q\omega_{21}$ and $x^2 \simeq x^2 + p\omega_{12} + q\omega_{22}$. The formula simplify, if one chooses the basis $\vec{e}_1, \vec{e}_2$ such that

<footnote n="12">¹²For fermions we could also consider different spin structures, i.e. periodic or anti-periodic boundary conditions around one or the other circle of the torus, leading to four different spin structures. Here we will only investigate the doubly periodic boundary conditions, which are the only ones for which zero-modes are present.</footnote>

<footnote n="13">¹³In the present example where both circles have the same radius, there is a further degeneracy under the exchange of $n_1$ and $n_2$.</footnote>
$\vec{e}_1$ is aligned with $\vec{\omega}_1$ which means $\omega_{12} = 0$. Also, by scaling the coordinates appropriately one may assume that $\omega_{11} = 2\pi$. With this choice one writes

$$ \vec{\omega}_1 = 2\pi \vec{e}_1, \quad \vec{\omega}_2 = 2\pi(\tau_1 \vec{e}_1 + \tau_2 \vec{e}_2), \quad x^1 \simeq x^1 + 2\pi(p + \tau_1 q), \quad x^2 \simeq x^2 + 2\pi\tau_2 q, \quad (2.63) $$

with integer $p$ and $q$. If we use a complex coordinate $z = x^1 + ix^2$ instead, the periodicity is

$$ z \simeq z + 2\pi(p + \tau q), \quad \tau = \tau_1 + i\tau_2, \quad \tau_2 > 0. \quad (2.64) $$

The area of the torus now is given by the area of the fundamental cell which is a parallelogram with sides $\vec{\omega}_1$ and $\vec{\omega}_2$. Hence the area is $A = 2\pi \times 2\pi\tau_2$. Note that the case $\tau_1 = 0$ corresponds to a “rectangular” torus. The “wave-vectors” $\vec{k}$ are such that $\vec{k} \cdot (p\vec{\omega}_1 + q\vec{\omega}_2) \in 2\pi\mathbb{Z}$ for all integer $p, q$. They form the “dual lattice” and are given by

$$ \vec{k} = n_1 \vec{e}_1 + \frac{n_2 - \tau_1 n_1}{\tau_2} \vec{e}_2, \quad n_1, n_2 \in \mathbb{Z}, \quad (2.65) $$

as one can easily check. This means that $(x^1 = \frac{z+\bar{z}}{2}, x^2 = \frac{z-\bar{z}}{2i})$

$$ \exp(i\vec{k}_{\vec{n}} \cdot \vec{x}) = \exp(i n_1 x^1 + i \frac{n_2 - \tau_1 n_1}{\tau_2} x^2) = \exp\left(-\frac{1}{2\tau_2}((n\bar{\tau} - m)z - (n\tau - m)\bar{z})\right). \quad (2.66) $$

has the correct periodicity properties (2.63), resp. (2.64).

The Dirac operator $D$ is still the same as in (2.59) since the present torus is still flat, only the periodic boundary conditions are different. Then also $D^2 = -(\partial_1^2 + \partial_2^2) + m^2$ as before and, hence, $D^2 e^{i\vec{k}_{\vec{n}} \cdot \vec{x}} = \lambda_{\vec{n}}^2 e^{i\vec{k}_{\vec{n}} \cdot \vec{x}}$ with

$$ \lambda_{\vec{n}}^2 = \vec{k}_{\vec{n}}^2 + m^2 = n_1^2 + \left(\frac{n_2 - \tau_1 n_1}{\tau_2}\right)^2 + m^2. \quad (2.67) $$

If we denote $k_1 = n_1$ and $k_2 = \frac{n_2 - \tau_1 n_1}{\tau_2}$ then almost exactly as before<fnref n="14" /> we get for the eigenfunction of $D$

$$ \psi_{\vec{n}}(\vec{x}) = \frac{1}{2\pi\sqrt{2\tau_2\lambda_{\vec{n}}(\lambda_{\vec{n}} + k_2)}} \begin{pmatrix} k_1 + im \\ -\lambda_{\vec{n}} - k_2 \end{pmatrix} e^{ik_1 x^1 + ik_2 x^2}, \quad (2.68) $$

together with the complex conjugate functions which still satisfy

$$ \psi_{\vec{n}}^* = -\psi_{-\vec{n}}|_{\lambda_{\vec{n}} \to -\lambda_{\vec{n}}}. \quad (2.69) $$

**Zero-modes of D:** as discussed in general above, for $m \neq 0$ there are no zero-modes of $D$, while for $m = 0$ the zero-modes correspond to $\vec{k}_{\vec{n}} = 0$, i.e. $n_1 = n_2 = 0$. However, in this case (2.61) and (2.68) are indeterminate. We may instead first take to $n_1 = n_2 = 0$ and then consider the limit $m \to 0$ which gives the two zero-modes of definite chirality:

$$ \psi_{\vec{0}} = \frac{1}{2\pi\sqrt{2\tau_2}} \begin{pmatrix} i \\ -1 \end{pmatrix}, \quad \psi_{\vec{0}}^* = \frac{1}{2\pi\sqrt{2\tau_2}} \begin{pmatrix} -i \\ -1 \end{pmatrix}, \quad \gamma_* \psi_{\vec{0}} = \psi_{\vec{0}}, \quad \gamma_* \psi_{\vec{0}}^* = -\psi_{\vec{0}}. \quad (2.70) $$

<footnote n="14">¹⁴The only difference is the appearance of the additional normalisation factor $\frac{1}{\sqrt{\tau_2}}$ since now the area of the torus is $2\pi \times 2\pi\tau_2$ instead of $(2\pi)^2$.</footnote>
Below, we will define the “projector” on the zero-modes in general as $P_0(x, y) = \sum_i \psi_{0,i}(x) \psi_{0,i}^\dagger(y)$. At present, for the general flat torus this simply gives

$$P_0 = \psi_0 \psi_0^\dagger + \psi_0^* \psi_0^{*\dagger} = \frac{1}{(2\pi)^2} \frac{1}{2\tau_2} \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix} + \text{c.c.} = \frac{1}{(2\pi)^2 \tau_2} \mathbf{1}_{2 \times 2} = \frac{1}{A} \mathbf{1}_{2 \times 2}, \quad (2.71)$$

where $A = (2\pi)^2 \tau_2$ is the area of the torus.

## 2.4 Conformal transformation of the Dirac operator

As discussed in the introduction, we will consider two metrics $g_{\mu\nu}$ and $\hat{g}_{\mu\nu}$ that are conformally equivalent, i.e. that are simply related by a local ($x$-dependent) conformal factor which we write as $e^{2\sigma} \equiv e^{2\sigma(x)}$:

$$g_{\mu\nu} = e^{2\sigma} \hat{g}_{\mu\nu}. \quad (2.72)$$

We will consider $\hat{g}_{\mu\nu}$ as a reference metric (also referred to as the background metric) and $g_{\mu\nu}$ as a “general” metric obtained by varying the conformal factor $e^{2\sigma(x)}$. This view is particularly useful in two dimensions where, by a change of coordinates, any metric can be brought into the form $e^{2\sigma(x)} \hat{g}_{\mu\nu}$ with a fixed reference metric $\hat{g}_{\mu\nu}$. In particular, in two dimensions we have the relation (1.14) between the curvature scalar $\mathcal{R}$ for the metric $g$ and the one $\hat{\mathcal{R}}$ for the metric $\hat{g}$, namely $\mathcal{R} = e^{-2\sigma}(\hat{\mathcal{R}} + 2\hat{\Delta}\sigma)$, as well as

$$\sqrt{\hat{g}} = \sqrt{\hat{g}} e^{2\sigma}. \quad (2.73)$$

Since $\sqrt{\hat{g}} \hat{\Delta}_{\text{scalar}} = \partial_\mu (\hat{g}^{\mu\nu} \sqrt{\hat{g}} \partial_\nu)$ we see that $\sqrt{\hat{g}} \mathcal{R} = \sqrt{\hat{g}} \hat{\mathcal{R}} + \text{total derivative}$, which is the reason why the integral $\int \sqrt{\hat{g}} \mathcal{R}$ is independent of the conformal deformation and it is a topological invariant: its value only depends on the topology of the two-dimensional manifold $\mathcal{M}$. Indeed, it equals $4\pi\chi(\mathcal{M})$ where $\chi(\mathcal{M})$ is the Euler characteristic of $\mathcal{M}$. For a manifold without boundaries the latter is given in terms of the genus $g$ as $\chi(\mathcal{M}) = 2(1-g)$, so that $\int_{\mathcal{M}} \sqrt{\hat{g}} \mathcal{R} = 8\pi(1-g)$. We already noted that the Laplace operators $\hat{\Delta}$ and $\hat{\Delta}$ are simply related by $\hat{\Delta} = e^{-2\sigma} \hat{\Delta}$. We now want to establish how the operators $\hat{\nabla}$ and $\hat{\nabla}$ are related.

### 2.4.1 Dirac operator in $n$ dimension

It is not difficult to work this out in general dimension $n$. Below, we will then specialise to $n=2$. Writing the metric in terms of the local frame forms $e^a$ as in (2.8) we see that there is also a corresponding conformal rescaling of the frame forms as $e^a = e^\sigma \hat{e}^a$ and correspondingly for the vielbeins $e_\mu^a$ and their inverses $E_a^\mu$ as

$$e_\mu^a = e^\sigma \hat{e}_\mu^a, \quad E_a^\mu = e^{-\sigma} \hat{E}_a^\mu, \quad (2.74)$$

How are the spin-connections $\hat{\omega}^{ab}$ and $\omega^{ab}$ related? Writing out the zero-torsion condition for $e^a = (e^\sigma \hat{e}^a)$ and $\omega^{ab}$, and using the one for $\hat{e}^a$ and $\hat{\omega}^{ab}$, gives

$$0 = d\hat{e}^a + \omega^{ab}\hat{e}^b = e^\sigma \left[ d\sigma \hat{e}^a - \hat{\omega}^{ab}\hat{e}^b + \omega^{ab}\hat{e}^b \right], \quad (2.75)$$
which is solved as

$$ \omega_{\mu}^{ab} = \hat{\omega}_{\mu}^{ab} + (\hat{e}_{\mu}^{a} \hat{E}^{b\lambda} - \hat{e}_{\mu}^{b} \hat{E}^{a\lambda}) \partial_{\lambda} \sigma. \qquad (2.76) $$

It follows that

$$ \nabla_{\mu}^{\text{sp}} = \partial_{\mu} + \frac{1}{4} \omega_{\mu}^{cd} \gamma_{cd} = \hat{\nabla}_{\mu}^{\text{sp}} + \frac{1}{2} \hat{e}_{\mu}^{c} \hat{E}^{d\lambda} \partial_{\lambda} \sigma \gamma_{cd} \qquad (2.77) $$

Using $\gamma^{\mu} = E_{a}^{\mu} \gamma^{a} = e^{-\sigma} \hat{E}_{a}^{\mu} \gamma^{a} = e^{-\sigma} \hat{\gamma}^{\mu}$, we arrive at

$$ \nabla = e^{-\sigma} \left( \hat{\nabla} + \frac{1}{2} \gamma^c \hat{E}^{d\lambda} \partial_\lambda \sigma \gamma_{cd} \right) = e^{-\sigma} \left( \hat{\nabla} + \frac{n-1}{2} (\not\partial \sigma) \right), \qquad (2.78) $$

where we used $\gamma^c \gamma_{cd} = (n-1)\gamma_d$. This can also be written as

$$ \nabla = e^{-\frac{n+1}{2}\sigma} \hat{\nabla} e^{\frac{n-1}{2}\sigma}. \qquad (2.79) $$

For an infinitesimal variation $\delta\sigma$ of the conformal factor this yields

$$ \delta \nabla = -\delta \sigma \nabla + \frac{n-1}{2} \gamma^{\lambda} \partial_{\lambda} \delta \sigma \equiv -\delta \sigma \nabla + \frac{n-1}{2} (\partial \delta \sigma). \qquad (2.80) $$

Consider now the Dirac operator $D = i\nabla + m\gamma_*$ as given in (2.42). Obviously, under a conformal rescaling the term $m\gamma_*$ does not change, so that

$$ D = i\nabla + m\gamma_* = e^{-\frac{n+1}{2}\sigma} i\hat{\nabla} e^{\frac{n-1}{2}\sigma} + m\gamma_* \quad \Rightarrow \quad \delta D = i\delta \nabla = -\delta\sigma i\nabla + \frac{n-1}{2} i(\partial\delta\sigma). \qquad (2.81) $$

One can also work out the variation of $D^2$ which requires some care, with the result

$$ \delta D^2 = -2\delta\sigma (D^2 - m^2) + (\partial\delta\sigma)\nabla - (n-1)g^{\mu\nu}(\partial_\mu\delta\sigma)\nabla_\nu^{\text{sp}} - \frac{n-1}{2}(\Delta_{\text{scalar}}\delta\sigma). \qquad (2.82) $$

Actually, we will not need this formula. The relation (2.79) and its direct consequences (2.80) and (2.81) is all we will need.

### 2.4.2 Dirac operator in two dimensions

Let us now specialise to 2 dimensions i.e. $n=2$ in the previous formula. Then

$$ \nabla = e^{-\frac{3}{2}\sigma} \hat{\nabla} e^{\frac{1}{2}\sigma}, \qquad (2.83) $$

as well as

$$ \delta D = -\delta\sigma(D - m\gamma_*) + \frac{i}{2}(\partial\delta\sigma) \qquad (2.84) $$

$$ \delta D^2 = -2\delta\sigma (D^2 - m^2) + \gamma^{\mu\nu}(\partial_\mu\delta\sigma)\nabla_\nu^{\text{sp}} - \frac{1}{2}(\Delta_{\text{scalar}}\delta\sigma), \qquad (2.85) $$

where $\gamma^{\mu\nu} = \epsilon^{\mu\nu} i\gamma_*$.$\text{Let us check that the factors } e^{-\frac{3}{2}\sigma} \text{ and } e^{\frac{1}{2}\sigma} \text{ in (2.83) are exactly what is needed for } i\nabla \text{ to be hermitian: indeed, } \hat{\nabla} \text{ is hermitian for the metric } \hat{g}. \text{ Then}$

$$ \begin{aligned} (\psi_1, i\nabla\psi_2) &\equiv \int d^2z \sqrt{g} \psi_1^\dagger i\nabla\psi_2 = \int d^2z \sqrt{\hat{g}} e^{2\sigma} \psi_1^\dagger e^{-\frac{3}{2}\sigma} i\hat{\nabla}(e^{\frac{1}{2}\sigma}\psi_2) \\ &= \int d^2z \sqrt{\hat{g}} (i\hat{\nabla}(e^{\frac{1}{2}\sigma}\psi_1))^\dagger (e^{\frac{1}{2}\sigma}\psi_2) = \int d^2z \sqrt{g} (i\nabla\psi_1)^\dagger \psi_2 \equiv (i\nabla\psi_1, \psi_2), \end{aligned} \qquad (2.86) $$

so that $i\nabla$ is hermitian for the metric $g$. Actually, the same argument applies in $n$ dimensions with (2.79) and $\sqrt{g} = e^{n\sigma}\sqrt{\hat{g}}$. We see that we could have derived the formula (2.79) and (2.83) solely from the requirement that $i\nabla$ is hermitian for the metric $g$ if $i\hat{\nabla}$ is hermitian for the metric $\hat{g}$.
## 2.5 Zero-modes

We will now collect a few results and remarks one can rather easily derive about the zero-modes of the massless Dirac operator $i\nabla$ on a two-dimensional manifold without boundaries.

We will easily re-derive that there are no zero-modes of $i\nabla$ for spherical topology, and we explicitly obtain the zero-modes for toroidal topology (with doubly periodic spin structure). For genus $g \ge 2$ we do not have explicit formulae for the zero-modes, but we can nevertheless sufficiently characterise their dependences on the conformal factor, as needed later-on. It is actually known [25, 26] that for genus $g \ge 3$, even the number of zero-modes depends on the conformal class of the metric and no simple formula for their number can exist.

### 2.5.1 General remarks about zero-modes

First recall that $i\nabla$ is purely imaginary so that if $i\nabla\psi_0 = 0$ then also $i\nabla\psi_0^* = 0$. Hence, if $\psi_0$ is a complex zero-mode, then $\psi_0^*$ is another zero-mode.

As already noticed, $i\nabla$ anticommutes with $\gamma_*$, Then, for $m \ne 0$, $D = i\nabla + m\gamma_*$ neither commutes nor anti-commutes with $\gamma_*$ and we cannot have eigenfunctions of $D$ of definite chirality (i.e. being also eigenfunctions of $\gamma_*$). However, for $m = 0$, $D = i\nabla$ anti-commutes with $\gamma_*$, so that $\gamma_*\psi_n$ is eigenfunction of $D$ with eigenvalue $-\lambda_n$. Thus for $\lambda_n \ne 0$ (and $m = 0$), $\psi_n$ and $\gamma_*\psi_n$ necessarily are orthogonal. For $\lambda_n = 0$ (and $m = 0$), however, this is not the case. Instead one can always choose a basis of definite chirality eigenfunctions. Indeed, if $\psi_0$ is a zero-mode of $i\nabla$ without being an eigenstate of $\gamma_*$ then $\gamma_*\psi_0$ and $\psi_0$ are two linearly independent zero-modes and one can form the two linear combinations $\psi_{0,\pm} = \frac{1}{2}(1 \pm \gamma_*)\psi_0$. Obviously, the $\psi_{0,\pm}$ now are zero-modes of definite chirality. As is well known, the difference of the number of positive and negative chirality zero-modes of $i\nabla$ is called its index. Since $\gamma_* = \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, the positive chirality eigenfunctions must be $\sim \begin{pmatrix} 1 \\ i \end{pmatrix}$ and the negative chirality eigenfunctions must be $\sim \begin{pmatrix} 1 \\ -i \end{pmatrix}$. We see that they are intrinsically complex, and complex conjugation switches the chirality. Hence there are as many positive chirality as negative chirality zero-modes, i.e. the index of $i\nabla$ is zero. Indeed, this index is related to the Pontryagin invariant (see e.g. [6]) which vanishes in $n = 4k + 2$ dimensions, and in particular in 2 dimensions.

On the other hand, $D^2$ commutes with $\gamma_*$, and one can then take all the eigenfunctions (zero and non-zero modes) of $D^2$ to have definite chirality. From the discussion of the previous paragraph it is then clear that in general these definite chirality eigenfunctions of $D^2$ are not eigenfunctions of $D$.

There are some simple standard arguments about when zero-modes can exist. First of all, assuming the eigenfunctions to be normalised,

$$ \lambda_n^2 - m^2 = (\psi_n, (D^2 - m^2)\psi_n) = (\psi_n, (i\nabla)^2\psi_n) = (i\nabla\psi_n, i\nabla\psi_n) \ge 0. \quad (2.87) $$

We see that

$$ \lambda_n^2 > m^2, \quad \text{unless } i\nabla\psi_0 = 0 \text{ in which case } \lambda_0^2 = m^2. \quad (2.88) $$
In particular, if $m \neq 0$ we always have $\lambda_n^2 \geq m^2 > 0$ and there cannot be any zero-modes of the massive Dirac operator. Furthermore, if $i\nabla$ admits a zero-mode, then the smallest eigenvalue is $\lambda_0 = m$.

Next, we have

$$ (\psi_n, (-\Delta_{\text{sp}})\psi_n) = -\int d^2x \sqrt{g} \psi_n^* g^{\mu\nu} \nabla_\mu \nabla_\nu \psi_n = \int d^2x \sqrt{g} (\nabla_\mu^{\text{sp}} \psi_n)^* g^{\mu\nu} \nabla_\nu^{\text{sp}} \psi_n \equiv \alpha^2 \geq 0. \qquad (2.89) $$

Since the metric $g^{\mu\nu}$ is positive definite, $\alpha = 0$ means that $\nabla_\mu^{\text{sp}}\psi_n = 0$ for both $\mu = 1$ and $\mu = 2$, i.e. $\psi_n$ must be covariantly constant. However, since $[\nabla_1, \nabla_2]\psi \sim \mathcal{R}\gamma_\ast\psi$, we see that a covariantly constant spinor can only exist on a manifold with vanishing curvature, $\mathcal{R} = 0$ everywhere, i.e. only on the flat torus. For all other two-dimensional manifolds without boundary (and metrics) one has $\alpha^2 > 0$. Next, it follows from (2.47) that

$$ \lambda_n^2 - m^2 = \alpha^2 + \frac{1}{4} \int \sqrt{g} \mathcal{R} \psi_n^\dagger \psi_n = \alpha^2 + \frac{1}{4} \langle \mathcal{R} \rangle_n, \qquad (2.90) $$

where $\langle \mathcal{R} \rangle_n$ denotes the “mean curvature in the state $\psi_n$”. If we choose a constant curvature metric, $\mathcal{R} = \frac{8\pi}{A}(1-g)$ where $g$ is the genus of the Riemann surface, this becomes

$$ \lambda_n^2 - m^2 = \alpha^2 + \frac{2\pi}{A}(1-g). \qquad (2.91) $$

We will show soon that there is a simple relation between the zero-modes of the metric $\hat{g}$ and the zero-modes of the metric $g = e^{2\sigma}\hat{g}$. In particular, we may choose $\hat{g}$ to be the constant curvature metric and then use (2.91) to argue about the existence of the corresponding zero-modes or not. In particular, we see that for spherical topology ($g = 0$) the right-hand side is $\alpha^2 + \frac{2\pi}{A} > 0$ so that $\lambda_n^2 > m^2$ and the Dirac operator cannot have any zero-mode even for $m = 0$. For the flat torus we have $\lambda_n^2 = m^2 + \alpha^2$ and for $m = 0$ one can have zero-modes provided $\alpha = 0$, meaning that the zero-modes must be covariantly constant spinors that precisely exist (only) on the flat torus as we have just seen. Then there are also corresponding zero-modes on an arbitrary torus.<fnref n="15" /> Finally, for genus two and larger, the right-hand side of (2.91) can vanish and there are zero-modes of the massless Dirac operator.

**Remarks on comparing eigenvalues for vanishing and non-vanishing mass:** let us denote by $\pm\lambda_n^{(0)}$ the positive and negative eigenvalues of $D$ for $m = 0$, i.e. of $i\nabla$: $i\nabla\psi_{\pm n,i}^{(0)} = \pm\lambda_n^{(0)}\psi_{\pm n,i}^{(0)}$, where the additional index $i$ for the corresponding eigenfunctions takes into account that the eigenvalues $\pm\lambda_n^{(0)}$ can be degenerate. Now, for $\pm\lambda_n^{(0)} \neq 0$ these eigenfunctions are in general not eigenstates of $\gamma_\ast$ and, hence, the $\psi_{\pm n,i}^{(0)}$ are not eigenfunctions of $D = i\nabla+m\gamma_\ast$. Obviously then, there cannot be any linear relation between the $\lambda_n$ and the $\lambda_n^{(0)}$. However, we have $D^2\psi_{\pm n,i}^{(0)} = ((i\nabla)^2+m^2)\psi_{\pm n,i}^{(0)} = ((\pm\lambda_n^{(0)})^2+m^2)\psi_{\pm n,i}^{(0)}$, so that the $\psi_{\pm n,i}^{(0)}$ are eigenfunctions of $D^2$ with eigenvalues

$$ \lambda_n^2 = (\lambda_n^{(0)})^2 + m^2. \qquad (2.92) $$

<footnote n="15">¹⁵Since the latter cannot be covariantly constant and hence must have $\alpha^2 > 0$ they necessarily have a negative mean curvature: $\langle \mathcal{R} \rangle_0 = -4\alpha^2$.</footnote>
This means that the eigenfunctions $\psi_n$ corresponding to $\lambda_n$ can be identified with some linear combination of the $\psi_{\pm n,i}^{(0)}$, and similarly for the $\psi_{-n} = \psi_n^*$ corresponding to $-\lambda_n$.

### 2.5.2 Zero-modes on a general torus

Before discussing the case of arbitrary genus, let us first consider the zero-modes on the torus where we can give very explicit results. The general torus is defined by a) the periodicity conditions of the flat torus and b) a metric given by $g_{\mu\nu} = e^{2\sigma(x)}\delta_{\mu\nu}$ (i.e. $\hat{g}_{\mu\nu} = \delta_{\mu\nu}$), where $\sigma(x) \equiv \sigma(x^1, x^2)$ must also satisfy the appropriate periodicity conditions. Thus from (2.83) we get

$$D = ie^{-3\sigma(x)/2} \partial\bar{\partial} e^{\sigma(x)/2} + m\gamma_* = e^{-3\sigma(x)/2}(i\sigma_x\partial_1 + i\sigma_z\partial_2)e^{\sigma(x)/2} + m\sigma_y. \quad (2.93)$$

The derivative and the mass terms behave differently under the conformal transformations. However, we see that for zero mass, the zero-modes $\psi_{0,i}$ are simply determined by

$$D^{m=0}\psi_{0,i}(x) = i\nabla\bar{\psi}_{0,i}(x) = 0 \quad \Rightarrow \quad i\partial\bar{\partial}(e^{\sigma(x)/2}\psi_{0,i}(x)) = 0, \quad (2.94)$$

and one identifies $e^{\sigma(x)/2}\psi_0(x)$ with the zero-modes $\hat{\psi}_{0,i}$ of $i\partial\bar{\partial}$, i.e. the zero-modes of the flat torus as determined above in (2.70). Actually, this argument generalises to arbitrary genus as we will discuss shortly. What is special for the torus is that we know explicitly the $\hat{\psi}_{0,i}$. Hence the two zero-modes of $i\nabla\bar{\partial}$ are

$$\begin{aligned} \psi_{0,1}(x) &= c e^{-\sigma(x)/2} \hat{\psi}_{0,1}(x) = c \frac{e^{-\sigma(x)/2}}{2\pi\sqrt{2\tau_2}} \begin{pmatrix} i \\ -1 \end{pmatrix}, \\ \psi_{0,2}(x) &= c e^{-\sigma(x)/2} \hat{\psi}_{0,2}(x) = c \frac{e^{-\sigma(x)/2}}{2\pi\sqrt{2\tau_2}} \begin{pmatrix} -i \\ -1 \end{pmatrix}, \end{aligned} \quad (2.95)$$

where $c$ is a normalisation constant to be determined. Indeed, the normalisation now is defined with respect to the metric $g = e^{2\sigma}\hat{g}$ as (no sum over $i$)

$$1 = \int d^2x \sqrt{g(x)} \psi_{0,i}^{\dagger}(x) \psi_{0,i}(x) = \int d^2x \sqrt{\hat{g}} e^{2\sigma(x)} \frac{|c|^2}{(2\pi)^2 \tau_2} e^{-\sigma(x)} = \frac{|c|^2}{(2\pi)^2 \tau_2} \int d^2x \sqrt{\hat{g}} e^{\sigma(x)}. \quad (2.96)$$

Of course, the integral in the last expression is not the area which would be $A = \int d^2x \sqrt{\hat{g}} e^{2\sigma(x)}$. Also note that $(2\pi)^2\tau_2$ is the area of the torus with metric $\hat{g}$, i.e. for $\sigma = 0$. We now define the “area-like” parameters

$$A_{(n)} = \int d^2x \sqrt{\hat{g}} e^{n\sigma(x)} \quad \Rightarrow \quad A_{(2)} = A, \quad A_{(0)} = \hat{A} = (2\pi)^2 \tau_2. \quad (2.97)$$

Hence, up to a choice of phase, $c = \sqrt{\frac{\hat{A}}{A_{(1)}}} = 2\pi\sqrt{\frac{\tau_2}{A_{(1)}}}$, and

$$\psi_{0,1}(x) = \frac{e^{-\sigma(x)/2}}{\sqrt{2A_{(1)}}} \begin{pmatrix} i \\ -1 \end{pmatrix}, \quad \psi_{0,2}(x) = \frac{e^{-\sigma(x)/2}}{\sqrt{2A_{(1)}}} \begin{pmatrix} -i \\ -1 \end{pmatrix}. \quad (2.98)$$
It follows that the projector on the zero-modes is

$$P_0(x, y) = \sum_{i=1}^{2} \psi_{0,i}(x) \psi_{0,i}^{\dagger}(y) = \frac{e^{-(\sigma(x)+\sigma(y))/2}}{A_{(1)}} \mathbf{1}_{2 \times 2} = \frac{e^{-(\sigma(x)+\sigma(y))/2}}{\int d^2 z \sqrt{\hat{g}} e^{\sigma}} \mathbf{1}_{2 \times 2}, \quad (2.99)$$

which now depends on $x$ and $y$, but is still proportional to the unit matrix (as for the flat torus).

The zero-modes $\psi_{0,1}$ and $\psi_{0,2}$ as defined in (2.98) have definite chirality and then also are eigenfunctions of $D$ with non-zero mass. Indeed, we have $\gamma_* \psi_{0,1} = \psi_{0,1}$ and $\gamma_* \psi_{0,2} = -\psi_{0,2}$ and then

$$D\psi_{0,1} = m\psi_{0,1}, \quad D\psi_{0,2} = -m\psi_{0,2}. \qquad (2.100)$$

Thus, the above projector $P_0$ also is the projector on the eigen-modes of $D$, that become zero-modes when the mass is taken to zero.

### 2.5.3 Zero-modes of $i\nabla$ for arbitrary genus $g \ge 1$

We have seen before that for genus zero (spherical topology) there are no zero-modes of $i\nabla$, and we have just discussed the two zero-modes for a general genus-one manifold. Now we want to establish some further general results valid for arbitrary genus. For higher genus surfaces there are more zero-modes, but in any case there are only finitely many of them. As already mentioned, their number depends [25, 26] on the spin structure and, for genus $g \ge 3$ also on the conformal class of the metric, i.e. on the $\hat{g}(\tau_i)$. Of course, within any conformal class this number is fixed, and we have explicitly related the $\psi_{0,i}$ to the $\hat{\psi}_{0,i}$. As usual, we suppose that the zero-modes are all ortho-normalised:

$$i\nabla\psi_{0,i} = 0, \quad \int d^2x \sqrt{g(x)} \psi_{0,i}^{\dagger}(x)\psi_{0,j}(x) = \delta_{ij}. \qquad (2.101)$$

As already discussed, these zero-modes can be chosen to have definite chirality, so that even for non-vanishing mass they continue to be eigenstates of the Dirac operator: $D\psi_{0,i} = m\gamma_*\psi_{0,i} = \pm m\psi_{0,i}$. Recall that the zero-modes of definite chirality necessarily are complex and $\psi_{0,i}$ and $\psi_{0,i}^*$ have opposite chirality and, hence, that there must be an even number $2n_0$ of zero-modes, $n_0$ with positive chirality and $n_0$ with negative chirality, resulting in a vanishing index of $i\nabla$. Let us then choose the labelling of the zero-modes such that $\psi_{0,i}$, $i = 1, \dots, n_0$ have positive chirality and $\psi_{0,i}$, $i = n_0+1, \dots, 2n_0$ have negative chirality, and moreover such that $\psi_{0,i}^* = \psi_{0,i+n_0}$, $i = 1, \dots, n_0$.

The projector on these zero-modes then is

$$P_0(x, y) = \sum_{i=1}^{2n_0} \psi_{0,i}(x) \psi_{0,i}^{\dagger}(y) = \sum_{i=1}^{n_0} (\psi_{0,i}(x) \psi_{0,i}^{\dagger}(y) + \psi_{0,i}^{*}(x) \psi_{0,i}^{*\dagger}(y)). \qquad (2.102)$$

This is indeed a projector in the sense that we have, by (2.101), the relation

$$\int d^2z \sqrt{g(z)} P_0(x, z) P_0(z, y) = P_0(x, y). \qquad (2.103)$$
Since the zero-modes were chosen to have definite chirality $\pm 1$ it follows that

$$ \gamma_* P_0(x, y) \gamma_* = P_0(x, y) \Leftrightarrow [\gamma_*, P_0(x, y)] = 0. \qquad (2.104) $$

Now, the only matrices that commute with $\gamma_*$ are $\mathbf{1}_{2 \times 2}$ and $\gamma_*$ itself. It follows that $P_0$ has the following matrix structure

$$ P_0(x, y) = p_{0,0}(x, y) \mathbf{1}_{2 \times 2} + p_{0,*}(x, y) \gamma_* \qquad (2.105) $$

One can also argue, by choosing a basis of real zero-modes instead, that $P_0(x, x)$ is real and hermitian, excluding the $\gamma_*$-piece for coinciding points. However, this argument does not exclude a $\gamma_*$-piece for $x \neq y$. For an arbitrary torus (with modular parameter $\tau$ and conformal factor $\sigma$), we could explicitly show that the matrix-structure is indeed $P_0(x, y) \sim \mathbf{1}_{2 \times 2}$, and it would be nice to show this for arbitrary genus.

It will also be useful to define similarly

$$ Q_0(x, y) = \sum_{i=1}^{n_0} (\psi_{0,i}(x) \psi_{0,i}^{\dagger}(y) - \psi_{0,i}^{*}(x) \psi_{0,i}^{*\dagger}(y)). \qquad (2.106) $$

Obviously

$$ Q_0(x, y) = \gamma_* P_0(x, y) = P_0(x, y) \gamma_* = p_{0,0}(x, y) \gamma_* + p_{0,*}(x, y) \mathbf{1}_{2 \times 2}, \qquad (2.107) $$

so that

$$ \int d^2 z \sqrt{g(z)} Q_0(x, z) Q_0(z, y) = \int d^2 z \sqrt{g(z)} P_0(x, z) P_0(z, y) = P_0(x, y). \qquad (2.108) $$

### 2.5.4 Conformal variation of the zero-modes and of the zero-mode projector

We will need the variation of the zero-modes under conformal transformations. Fortunately this can be easily obtained exactly, generalising the above discussion for the torus. Recall that under $g = e^{2\sigma}\hat{g}$ we have $\mathcal{W} = e^{-3\sigma/2}\hat{\mathcal{W}}e^{\sigma/2}$. Then $\mathcal{W}\psi_{0,i} = 0$ is equivalent to $\hat{\mathcal{W}}(e^{\sigma/2}\psi_{0,i}) = 0$ so that $e^{\sigma/2}\psi_{0,i}$ is a zero-mode of $\hat{\mathcal{W}}$ and we can identify $e^{\sigma(x)/2}\psi_{0,i}(x) \sim \hat{\psi}_{0,i}(x)$, i.e. the zero-modes of $\mathcal{W}$ are $\psi_{0,i}(x) \sim e^{-\sigma(x)/2}\hat{\psi}_{0,i}(x)$. However, a priori, they are not ortho-normalised. Instead, we can consider the linear combinations of the $n_0$ positive chirality zero-modes

$$ \psi_{0,k}(x) = \sum_{j=1}^{n_0} e^{-\sigma(x)/2} \hat{\psi}_{0,i}(x) c_{ik}, \qquad (2.109) $$

and the corresponding complex conjugate relation for the negative chirality zero-modes. Imposing the ortho-normality of the $\psi_{0,k}$ leads to

$$ \delta_{kl} = \sum_{i,j} c_{ik}^{*} P_{0,ij} c_{jl}, \qquad (2.110) $$

where we have defined the (constant) matrix $P_0$ as

$$ P_{0,ij} = \int d^2 x \sqrt{\hat{g}(x)} e^{\sigma(x)} \hat{\psi}_{0,i}^{\dagger}(x) \hat{\psi}_{0,j}(x). \qquad (2.111) $$
Although the $\mathcal{P}_{0,ij}$ are constants, they are actually functionals of the conformal factor $\sigma$. In matrix notation (2.110) reads $\mathbf{1} = c^\dagger \mathcal{P}_0 c$ so that $\mathcal{P}_0 = (c^{-1})^\dagger c^{-1}$ and $\mathcal{P}_0^{-1} = c c^\dagger$ which shows that $\mathcal{P}_0$ is hermitian.<fnref n="16" /> In components this reads $c_{ki} c_{li}^* = (\mathcal{P}_0)_{kl}^{-1}$, and inserting (2.109) into the above definition of the zero-mode projector $\mathcal{P}_0$ we find

$$P_0(x, y) = \sum_{i=1}^{n_0} \psi_{0,i}(x) \psi_{0,i}^{\dagger}(y) + \text{c.c.} = e^{-(\sigma(x)+\sigma(y))/2} \sum_{i,j=1}^{n_0} ((\mathcal{P}_0^{-1})_{ij} \hat{\psi}_{0,i}(x) \hat{\psi}_{0,j}^{\dagger}(y) + \text{c.c.}). \quad (2.112)$$

Since $Q_0 = \gamma_* P_0 = P_0 \gamma_*$, $Q_0$ obeys a completely analogous relation except that the negative chirality modes contribute with a minus sign to the sum. Hence, it will be enough if we further discuss $P_0$ only. For our later computations it will also be useful to introduce the following notations

$$P_0(x, y) = e^{-\sigma(x)/2} \bar{P}_0(x, y) e^{-\sigma(y)/2}, \quad \bar{P}_0(x, y) = \sum_{i,j=1}^{n_0} ((\mathcal{P}_0^{-1})_{ij} \hat{\psi}_{0,i}(x) \hat{\psi}_{0,j}^{\dagger}(y) + \text{c.c.}), \quad (2.113)$$

$$\hat{P}_0(x, y) = \sum_i (\hat{\psi}_{0,i}(x) \hat{\psi}_{0,i}^{\dagger}(y) + \text{c.c.}). \quad (2.114)$$

One must keep in mind that the quantity $\bar{P}_0(x, y)$ still depends on $\sigma$ due to the appearance of $e^\sigma$ in the ortho-normalisation matrix $\mathcal{P}_0$ of the zero-modes. Moreover, it is the inverse matrix $\mathcal{P}_0^{-1}$ which appears in $\bar{P}_0$. Hence, the overall $\sigma$ dependences of the projectors are rather non-trivial. Under an infinitesimal conformal variation with $\delta\sigma$ we have

$$\delta P_0(x, y) = -\frac{1}{2}(\delta\sigma(x) + \delta\sigma(y))P_0(x, y) - \left[ \sum_{i=1}^{n_0} (c^\dagger \delta \mathcal{P}_0 c)_{ij} \psi_{0,i}(x) \psi_{0,j}^\dagger(y) + \text{c.c.} \right]. \quad (2.115)$$

A somewhat simpler quantity is

$$\int d^2y \sqrt{g} \delta\sigma(y) \operatorname{tr} P_0(y, y) = \sum_{i,j=1}^{n_0} (\mathcal{P}_0^{-1})_{ij} \delta\mathcal{P}_{0,ji} + \text{c.c.} = 2\delta \operatorname{Tr} \log \mathcal{P}_0 = 2\delta \log \det \mathcal{P}_0, \quad (2.116)$$

where here Tr and det denote the trace and determinant of the $n_0 \times n_0$ matrices and the factor 2 occurs since $\mathcal{P}_0$ is hermitian and hence the trace is real.

How shall we interpret the quantities $\mathcal{P}_{0,ij}$? We have seen that for the example of a torus, $n_0 = 1$ and $\hat{\psi}_{0,i}^\dagger(z) \hat{\psi}_{0,i}(z) = \frac{1}{\hat{A}}$, where $\hat{A} = \int d^2z \sqrt{g}$. Hence there is a single $\mathcal{P}_0$ which equals $\mathcal{P}_0 = \frac{A_{(1)}}{\hat{A}}$ with $A_{(1)} = \int d^2z \sqrt{g} e^\sigma$ an area-like constant. Similarly, on a general genus Riemann surface we will interpret the $\mathcal{P}_{0,ij}$ as a finite number of area-like parameters. Of course, the appearance of a term $(\mathcal{P}_0^{-1})_{ij}$ is non-local in $\sigma$ but it is no more non-local than a factor $\frac{1}{\hat{A}}$.

<footnote n="16">¹⁶Of course these relations only determine the matrix $c$ up to right multiplication with any unitary matrix, consistent with the fact that this just leads to another orthonormal basis of the positive chirality zero-modes $\psi_{0,k}$.</footnote>
# 3 Fermionic matter partition function and gravitational action

## 3.1 Fermionic functional integral for Majorana spinors

We define the matter partition function for fermionic matter with action $S = \int d^2x \sqrt{g} \psi^\dagger D\psi$, where $D = i\not\nabla + m\gamma_*$, on a two-dimensional manifold with metric $g$ as the functional integral

$$Z_{\text{mat}}[g] = \int \mathcal{D}\Psi \exp(-S[g, \Psi]), \quad S[g, \Psi] = \int d^2x \sqrt{g} \Psi^\dagger D_g \Psi, \qquad (3.1)$$

where we wrote $D_g$ to insist that this is the Dirac operator $D$ for the metric $g$ (and corresponding vielbein $e$ and spin connection $\omega$). We want to consider here only Majorana spinors $\Psi$, which means that $\Psi(x)$ is an arbitrary anti-commuting real spinor and $\Psi^\dagger \equiv \Psi^T$. The Dirac operator is purely imaginary. Nevertheless, our action is real, as it should. Indeed, recall that for anti-commuting objects $a$ and $b$, complex conjugation is defined as hermitian conjugation and reverses the order, so that $(ab)^* = b^*a^*$. Now $(D_g\Psi)^* = -D_g\Psi$. Then $(S[g, \Psi]^*) = \int (\Psi^T D_g \Psi)^* = \int ((D_g\Psi)^*)^T \Psi = -\int (D_g\Psi)^T \Psi = +\int \Psi^T D_g \Psi = S[g, \Psi]$, where we used the anti-commutativity in the last step.

Before going on, let us show that if we considered instead a Dirac spinor, i.e. an anti-commuting complex spinor $\Psi_{\text{Dirac}} = \Psi + i\tilde{\Psi}$, where $\Psi$ and $\tilde{\Psi}$ are two anti-commuting real (Majorana) spinors, the action for the Dirac spinor (with the same Dirac operator $D$) would simply be the sum of the actions for the real and imaginary parts, $\Psi$ and $\tilde{\Psi}$, separately without any cross-term, namely

$$S[g, \Psi_{\text{Dirac}}] = S[g, \Psi] + S[g, \tilde{\Psi}]. \qquad (3.2)$$

Indeed,

$$ \begin{aligned} S[g, \Psi_{\text{Dirac}}] &= \int \Psi_{\text{Dirac}}^{\dagger} D \Psi_{\text{Dirac}} = \int (\Psi + i\tilde{\Psi})^{\dagger} D (\Psi + i\tilde{\Psi}) \\ &= S[g, \Psi] + S[g, \tilde{\Psi}] - i \int (\tilde{\Psi}^{\dagger} D \Psi - \Psi^{\dagger} D \tilde{\Psi}), \end{aligned} \qquad (3.3) $$

and, using the hermiticity of $D$ and the fact that $\Psi$ and $\tilde{\Psi}$ are real and anti-commuting and that $D$ is purely imaginary, one easily shows that $\int (\tilde{\Psi}^\dagger D\Psi - \Psi^\dagger D\tilde{\Psi}) = 0$. It then follows rather straightforwardly that one has for the corresponding partition functions

$$Z_{\text{mat}}^{\text{Dirac}}[g] = \int \mathcal{D}\Psi_{\text{Dirac}} \exp(-S[g, \Psi_{\text{Dirac}}]) = \int \mathcal{D}\Psi \mathcal{D}\tilde{\Psi} \exp(-S[g, \Psi] - S[g, \tilde{\Psi}]) = (Z_{\text{mat}}[g])^2. \qquad (3.4)$$

Thus the matter partition function for a Dirac spinor is simply the square of the matter partition function for a Majorana spinor, and we may thus consider the latter as the more “fundamental” object to study. Hence, for the remainder of this paper we will consider Majorana spinors only.

We can then expand the anti-commuting real (Majorana) spinor $\Psi$ on a complete set of real eigenfunctions of $D_g$:

$$\Psi(x) = \frac{1}{\sqrt{\mu}} \left( \sum_n (b_n \chi_n(x) + c_n \phi_n(x)) + \sum_i d_i \psi_{0,i}(x) \right). \qquad (3.5)$$
Here and in the following it is understood that the $\phi_n$ and $\chi_n$ are the real eigenfunctions of $D_g^2$, satisfying $D_g\phi_n = -i\lambda_n\chi_n$ and $D_g\chi_n = i\lambda_n\phi_n$ (cf (2.55)) for strictly positive eigenvalues $\lambda_n > 0$. In particular, they are orthonormalised according to (2.58). The orthonormalized zero-modes, if present, are denoted $\psi_{0,i}$. Recall that zero-modes are present only for $m=0$ and genus $g \ge 1$ and then there are only finitely many of them. Of course, they are also orthogonal to the $\phi_n$ and $\chi_n$.

The eigenfunctions $\chi_n$, $\phi_n$ and $\psi_{0,i}$ are real, commuting functions, providing an orthonormalized basis. Since $\psi$ is anticommuting, the expansion coefficients $b_n, c_n, d_i$ must be real and anti-commuting. Then

$$ \Psi^{\dagger}(x) = \frac{1}{\sqrt{\mu}} \left( \sum_{n} (b_n \chi_n^{\dagger}(x) + c_n \phi_n^{\dagger}(x)) + \sum_{i} d_i \psi_{0,i}^{\dagger}(x) \right), \quad (3.6) $$

were actually $\chi_n^\dagger = \chi_n^T, \phi_n^\dagger = \phi_n^T, \psi_{0,i}^\dagger = \psi_{0,i}^T$. We have also introduced an arbitrary mass scale $\mu$ so that these coefficients $b_n, c_n, d_i$ are dimensionless. Indeed, from the normalisation condition of the eigenmodes one sees that the $\chi_n, \phi_n$ and $\psi_{0,i}$ have engineering dimension one, i.e. $\chi_n \sim \phi_n \sim \psi_{0,i} \sim \mu$, and since $\Psi$ must have dimension $\frac{1}{2}$ so that the action $\int d^2x \sqrt{g} \Psi^\dagger D_g \Psi$ is dimensionless, we see from (3.5) that $b_n, c_n, d_i$ are indeed dimensionless thanks to this explicit factor $\frac{1}{\sqrt{\mu}}$.

Inserting the expansions (3.5) and (3.6) into the action $S$ as given in (3.1) we get, thanks to the orthonomality of the eigenfunctions and the anticommutativity of the expansion coefficients $b_n$ and $c_n$

$$ S[g, \Psi] = \int d^2x \sqrt{g} \Psi^\dagger D_g \Psi = 2i \sum_n \frac{\lambda_n}{\mu} c_n b_n, \quad (3.7) $$

where of course only the non-zero modes contribute. Note that, despite the explicit appearance of the factor $i$, this expression is real as we have shown before. Indeed, $(ic_n b_n)^* = -ib_n c_n = +ic_n b_n$.

The functional integral measure $D\Psi$ is defined in terms of grassmann integrals over these coefficients $b_n$ and $c_n$ as $D\Psi = \int \prod_l db_l dc_l$, but not including the $d_i$. For grassmannian variables the integrals are non-vanishing only if each $b_n$ and $c_n$ appears exactly once in the integrand. Expanding the exponential then gives

$$ Z_{\text{mat}}[g] = \int \prod_l db_l dc_l \exp \left( -2i \sum_n \frac{\lambda_n}{\mu} c_n b_n \right) = N \prod_n \frac{\lambda_n}{\mu}, \quad (3.8) $$

where $N$ is some overall (infinite) “normalisation” constant that will drop out in the end. The product is only over all strictly positive eigenvalues $\lambda_n$. In principle one could have also done the integral over the zero-mode coefficients $d_i$ but then one needs to insert an extra factor $\prod_i d_i$ inside the functional integral to get a non-vanishing result. This result would have been the same as in (3.8), where we excluded the zero-mode integration from the beginning. Since there is a certain arbitrariness in dealing with the zero-modes, if present, one can also define $Z_{\text{mat}}$ with some zero-mode related factor included (like e.g. $\sqrt{A}$ in the case of a scalar matter field, see [14]). We will come back to this point later.
The gravitational action was defined in (1.1) as $Z_{\text{mat}}[g] = e^{-S_{\text{grav}}[g, \hat{g}]} Z_{\text{mat}}[\hat{g}]$, or equivalently $S_{\text{grav}}[g, \hat{g}] = -\log \frac{Z_{\text{mat}}[g]}{Z_{\text{mat}}[\hat{g}]}$, so that

$$S_{\text{grav}}[g, \hat{g}] = -\log \prod_n \frac{\lambda_n[g]}{\mu} + \log \prod_n \frac{\lambda_n[\hat{g}]}{\mu}. \qquad (3.9)$$

Since the product over $n$ only involves the non-zero eigenvalues this is sometimes denoted as $\prod'_n$, but in our convention $n$ runs only over the non-zero eigenvalues anyway, so the prime is not necessary. We may also rewrite this in terms of the determinant of $D^2$, which has eigenvalues $\lambda_n^2$ for the $\chi_n$ and $\lambda_n^2$ for the $\phi_n$, and possibly 0 for the $\psi_{0,i}$. The determinant of $D^2$ computed on the space of non-zero-modes is conventionally denoted by $\text{Det}'$ so that here we are interested in $\text{Det}'D^2$:

$$\text{Det}'D^2 = \left(\prod_n \lambda_n^2\right)^2, \qquad (3.10)$$

and

$$S_{\text{grav}}[g, \hat{g}] = -\frac{1}{4} \log \text{Det}'D^2[g] + \frac{1}{4} \log \text{Det}'D^2[\hat{g}]. \qquad (3.11)$$

## 3.2 Zeta function regularization

All these determinants and products of eigenvalues are, of course, ill-defined since the infinite products are divergent, and have to be regularized. We will use the standard tool of regularization via the corresponding zeta-functions [28, 29].

### 3.2.1 Zeta-functions of positive operators

The zeta-function of an operator $\mathcal{O}$ with eigenvalues $\Lambda_n > 0$ is defined as

$$\zeta_{\mathcal{O}}(s) = \sum_{n} \Lambda_{n}^{-s}. \qquad (3.12)$$

Proceeding formally, i.e. disregarding whether or not the series converges, the determinant of $\mathcal{O}$ is related to the derivative of this zeta function at $s = 0$ as follows

$$\zeta'_{\mathcal{O}}(s) \equiv \frac{d}{ds}\zeta_{\mathcal{O}}(s) = -\sum_{n}(\log \Lambda_n)\Lambda_n^{-s} \quad \Rightarrow \quad \zeta'_{\mathcal{O}}(0) = -\sum_{n}(\log \Lambda_n) = -\log \text{Det} \, \mathcal{O}. \qquad (3.13)$$

For the operator $D^2$ we are interested in, the corresponding zeta-function is

$$\zeta_{D^2}(s) = 2 \sum_{n} (\lambda_n^2)^{-s}, \qquad (3.14)$$

since each eigenvalue $\lambda_n^2 > 0$ occurs twice, once with eigenfunction $\chi_n$ and once with $\phi_n$. Again, also, the sum obviously does not include any zero eigenvalue in case there are zero-modes.

The zeta-function regularization is a standard technique to regulate and possibly assign finite values to otherwise diverging sums. The zeta-functions of operators are generalisations of Riemann's zeta function $\zeta_R$, defined as

$$\zeta_R(s) = \sum_{n=1}^{\infty} n^{-s}. \qquad (3.15)$$
### 3.2.2 A short discussion of Riemann's zeta function

It will be useful to briefly discuss some of the well-known properties of Riemann's zeta-function, in view of the generalisation of these properties to the zeta-functions of operators. Riemann's zeta function as defined by the sum (3.15) should be considered as a function of a complex argument $s \in \mathbb{C}$. Then the sum is absolutely convergent for $\Re s > 1$ ($\Re s$ designates the real part of $s$). Indeed, for complex $s$ write $s = s_1 + is_2$ so that $n^{-s} = n^{-s_1}n^{-is_2} = n^{-s_1}e^{-is_2 \log n}$ and $|n^{-s}| = n^{-s_1}$, which shows that the sum indeed converges absolutely for $s_1 > 1$, i.e. $\Re s > 1$. This then defines an analytic function $\zeta_R(s)$ for all $\Re s > 1$. This function can be analytically continued to $\Re s \le 1$. The function that results from analytically continuing $\zeta_R$ will still be still called $\zeta_R$. Riemann has shown that this analytic continuation defines a meromorphic function on the whole complex plane with a single simple pole at $s = 1$ with unit residue. This means that $\zeta_R(s)$ is an analytic function on all of $\mathbb{C}\setminus\{1\}$ and that $\lim_{s\to 1}(s-1)\zeta_R(s) = 1$. In particular, $\zeta_R(0) = -\frac{1}{2}$ is finite.

An important ingredient in this analytic continuation is Euler's Gamma function, defined as

$$ \Gamma(s) = \int_{0}^{\infty} dt \, t^{s-1} e^{-t}, \quad \Re s > 0. \qquad (3.16) $$

Integrating by parts, it is trivial to show that, for $\Re s > 0$, $\Gamma(s)$ satisfies the functional relation $\Gamma(s+1) = s\Gamma(s)$. This relation can then be used to define $\Gamma(s)$ for $\Re s \le 0$, except at the isolated points $s = 0, -1, -2, \dots$. Indeed, the functional relation shows that $\Gamma(s)$ has simple poles at all negative integers and 0, and is otherwise analytic. Similarly, for $\Re s > 1$, the sum (3.15) admits a simple integral representation

$$ \sum_{n=1}^{\infty} n^{-s} = \frac{1}{\Gamma(s)} \int_{0}^{\infty} dt \, t^{s-1} \frac{1}{e^t - 1}, \quad \Re s > 1, \qquad (3.17) $$

as one sees by expanding $(e^t - 1)^{-1} = \sum_{n=1}^{\infty} e^{-nt}$, and interchanging the sum and the integral, which is justified when both sums converge absolutely, i.e. for $\Re s > 1$. For later reference let us denote

$$ K_R(t) = \sum_{n=1}^{\infty} e^{-tn} = \frac{1}{e^t - 1}, \qquad (3.18) $$

and refer to it as the “Riemann heat kernel”. Then the previous equation can be summarised as

$$ \zeta_R(s) = \frac{1}{\Gamma(s)} \int_0^\infty dt \, t^{s-1} K_R(t). \qquad (3.19) $$

We will heavily use a generalisation of this relation later-on.

The basic functional relation used to perform the analytic continuation of $\zeta_R$ is the identity proven by Riemann: the function $\xi(s) = \pi^{-s/2}\Gamma(s/2)\zeta_R(s)$ satisfies $\xi(s) = \xi(1-s)$. The proof of this identity uses various standard techniques including Poisson resummation and Mellin transforms of theta functions (similar to the integral representation (3.17)). We will not go into details here. Instead, we will give a heuristic argument why the integral representation (3.17) or (3.19) of $\zeta_R$ can be analytically continued to yield a meromorphic function on the whole complex plane with a single simple pole at $s = 1$ with residue 1.
Again, this argument will be generalised below to study the zeta-function associated with our operator $D^2$.

First, observe that any singularity of the integral $\int_0^\infty dt \, t^{s-1} \frac{1}{e^{t-1}}$ can only come from the region $t \to 0$ where the integrand behaves as $t^{s-2}$, while for $t \to \infty$ the exponential decrease of the integrand ensures convergence. So let us isolate this small-$t$ region by choosing some small (but finite) positive real $\epsilon > 0$ and split the integral (3.17) into 2 parts:

$$ \frac{1}{\Gamma(s)} \left( \int_0^\epsilon dt \, t^{s-1} \frac{1}{e^{t-1}} + \int_\epsilon^\infty dt \, t^{s-1} \frac{1}{e^{t-1}} \right). \qquad (3.20) $$

In the first integral we can replace $\frac{1}{e^{t-1}} = \frac{1}{t+t^2/2+...}$ by $\frac{1}{t}$ up to terms that contribute at most an order $\epsilon$ correction. Then the first integral is elementary and yields

$$ \int_0^\epsilon dt \, t^{s-2} = \frac{t^{s-1}}{s-1}\Big|_0^\epsilon = \frac{\epsilon^{s-1}-0}{s-1} \quad \text{since } \Re s > 1. \qquad (3.21) $$

This term exhibits the pole at $s=1$. Indeed, letting $s \to 1$ (for fixed $\epsilon$) one has $\epsilon^{s-1} = e^{(s-1)\log \epsilon} = 1 + (s-1)\log \epsilon + ...$ and we get for the first integral $\frac{1}{s-1} + \log \epsilon + \mathcal{O}(s-1)$. On the other hand, the second integral in (3.20) is regular for all $s \in \mathbb{C}$ and defines some analytic function for all $s \in \mathbb{C}$. We see that this allows to define $\zeta_R$ as a meromorphic function on $\mathbb{C}$ with a single simple pole at $s=1$ with residue 1. One might still worry about the order $\epsilon$-corrections to the first integral as they seem to lead to $\int_0^\epsilon dt \, t^{s-1} = \frac{t^s}{s}\Big|_0^\epsilon = \frac{\epsilon^{s-1}}{s}$, i.e. to a pole at $s=0$. However, this pole is cancelled by the zero of $\frac{1}{\Gamma(s)}$ at $s=0$. The same cancellation occurs for all would-be poles at all negative integers. Hence there is only a single pole at $s=1$.

### 3.2.3 Expressing the gravitational action in terms of $\zeta_{D^2}$

Let us now come back to the zeta-function (3.14) of our operator $D^2$. In the next section, we will discuss the convergence properties of the sum, as well as the properties of its analytic continuation, and see that just as $\zeta_R$, our $\zeta_{D^2}(s)$ is analytic, except for a single simple pole at $s=1$. Hence, formally taking the derivative of the sum with respect to $s$ and setting $s=0$ gives as before

$$ \zeta'_{D^2}(0) = -2 \sum_n (\log \lambda_n^2) = -2 \log \prod_n \lambda_n^2. \qquad (3.22) $$

Of course, the expression on the left-hand side is computed from the analytic continuation and thus provides a way to make sense of the otherwise diverging sum on the right-hand side. Similarly, $(\log \mu^2) \zeta_{D^2}(0) = 2 \sum_n \log \mu^2 = -2 \log \prod_n \frac{1}{\mu^2}$. These $\zeta_{D^2}$ depend on the metric $g$ via the dependence of $D^2$ on $g$. Thus, to indicate the dependence on $g$ we should denote them as $\zeta_{D^2}$ but for notational convenience we will simple denote them as $\zeta_g$. Then

$$ -2 \log \prod_n \frac{\lambda_n^2}{\mu^2} = \zeta'_{D^2}(0) + (\log \mu^2) \zeta_{D^2}(0) \equiv \zeta'_g(0) + (\log \mu^2) \zeta_g(0). \qquad (3.23) $$

It follows that the (regularized) gravitational action is

$$ S_{\text{grav}}[g, \hat{g}] = \frac{1}{4} (\zeta'_{g}(0) + (\log \mu^2) \zeta_{g}(0)) - \frac{1}{4} (\zeta'_{\hat{g}}(0) + (\log \mu^2) \zeta'_{\hat{g}}(0)). \qquad (3.24) $$
We want to determine this gravitational action for $g_{\mu\nu}(x) = e^{2\sigma(x)}\hat{g}_{\mu\nu}(x)$. Our strategy will be to first determine $\delta S_{\text{grav}}$ for infinitesimal $\delta\sigma$ (around an arbitrary metric $g$) and then “integrate” this variation to obtain $S_{\text{grav}}[g, \hat{g}]$. Obviously, the variation of the gravitational action is given in terms of the variation of the zeta-function $\zeta_g(s)$ (and of its derivative $\zeta'_g(s)$) around $s = 0$. To obtain this variation, we need to study the variations of the eigenvalues $\lambda_n$ under a corresponding variation of the Dirac operator $D$, which we will do next. To fully exploit these results we will need to study in some detail Green’s functions, local zeta functions and local heat kernels and their variations - which will be the subject of section 4.

### 3.3 Variation of the eigenvalues: perturbation theory

We want to study how the eigenvalues $\lambda_n$ (or $\lambda_n^2$) change under local conformal rescalings of the metric with some $\delta\sigma(x)$. The variation $\delta D = i\delta\nabla\bar{\nabla}$ of the Dirac operator $D$ has been obtained in section 2.4, see eqs (2.80) and (2.84). Recall that $D\chi_n = i\lambda_n\phi_n$ and $D\phi_n = -i\lambda_n\chi_n$. Then under $D \to D+\delta D$ we have $\lambda_n \to \lambda_n+\delta\lambda_n$, as well as $\chi_n \to \chi_n+\delta\chi_n$ and $\phi_n \to \phi_n + \delta\phi_n$:

$$ (D + \delta D)(\chi_n + \delta\chi_n) = i(\lambda_n + \delta\lambda_n)(\phi_n + \delta\phi_n) \quad \Rightarrow \quad \delta D\chi_n + D\delta\chi_n = i\delta\lambda_n\phi_n + i\lambda_n\delta\phi_n. \qquad (3.25) $$

Taking the inner product with $\phi_n$ and using the hermiticity of $D$ one gets

$$ \begin{aligned} i\delta\lambda_n(\phi_n, \phi_n) &= (\phi_n, \delta D\chi_n) + (\phi_n, D\delta\chi_n) - i\lambda_n(\phi_n, \delta\phi_n) \\ &= (\phi_n, \delta D\chi_n) + i\lambda_n(\chi_n, \delta\chi_n) - i\lambda_n(\phi_n, \delta\phi_n) \end{aligned} \qquad (3.26) $$

so that

$$ \delta\lambda_n = -i(\phi_n, \delta D\chi_n) + \lambda_n ((\chi_n, \delta\chi_n) - (\phi_n, \delta\phi_n)). \qquad (3.27) $$

In “ordinary” first order perturbation theory in quantum mechanics one has a relation analogous to $(\chi_n, \delta\chi_n) = (\phi_n, \delta\phi_n) = 0$ that translates that both $\chi_n$ and $\chi_n + \delta\chi_n$ are normalised to one, and similarly for the $\phi_n$. Here, however, one must take into account that the $\chi_n$ and $\phi_n$ are normalised with the metric $g$, while $\chi_n + \delta\chi_n$ and $\phi_n + \delta\phi_n$ are normalised with the metric $e^{2\delta\sigma}g$. Let us write out what this implies (to first order in the perturbation):

$$ 1 = \int d^2x \sqrt{g} e^{2\delta\sigma} (\chi_n + \delta\chi_n)^{\dagger} (\chi_n + \delta\chi_n) = \int d^2x \sqrt{g} (\chi_n^{\dagger}\chi_n + 2\delta\sigma\chi_n^{\dagger}\chi_n + \delta\chi_n^{\dagger}\chi_n + \chi_n^{\dagger}\delta\chi_n) \qquad (3.28) $$

The first term $\int d^2x \sqrt{g} \chi_n^\dagger\chi_n$ is just $(\chi_n, \chi_n) = 1$, while $\delta\chi_n^\dagger\chi_n = \delta\chi_n^T\chi_n = \chi_n^T\delta\chi_n = \chi_n^\dagger\delta\chi_n$ and we get

$$ (\chi_n, \delta\chi_n) = - \int d^2x \sqrt{g} \delta\sigma(x) \chi_n^\dagger(x) \chi_n(x) = - \int \delta\sigma \chi_n^\dagger\chi_n, \qquad (3.29) $$

where $\int \dots$ is short-hand for $\int d^2x \sqrt{g} \dots$. Similarly, $(\phi_n, \delta\phi_n) = - \int \delta\sigma \phi_n^\dagger\phi_n$. Replacing $\delta D = i\delta\nabla\bar{\nabla}$, we can then rewrite (3.27) as

$$ \delta\lambda_n = (\phi_n, \delta\nabla\bar{\nabla}\chi_n) + \lambda_n \int \delta\sigma (\phi_n^\dagger\phi_n - \chi_n^\dagger\chi_n). \qquad (3.30) $$
Next, we insert $\delta\nabla = -\delta\sigma\nabla + \frac{1}{2}\phi\delta\sigma$, cf. (2.84), and integrate by parts (cf (2.29)), so that

$$ 
\begin{aligned}
(\phi_n, \delta\nabla\chi_n) &= -\int \delta\sigma\phi_n^\dagger\nabla\chi_n + \frac{1}{2}\int \partial_\lambda\delta\sigma\phi_n^\dagger\gamma^\lambda\chi_n = -\int \delta\sigma\left(\phi_n^\dagger\nabla\chi_n + \frac{1}{2}(\nabla\phi_n)^\dagger\chi_n + \frac{1}{2}\phi_n\nabla\chi_n\right) \\
&= -\int \delta\sigma\left(-i\phi_n^\dagger(D-m\gamma_*)\chi_n + \frac{i}{2}((D-m\gamma_*)\phi_n)^\dagger\chi_n - \frac{i}{2}\phi_n^\dagger(D-m\gamma_*)\chi_n\right) \\
&= -\int \delta\sigma\left(\frac{3}{2}\lambda_n\phi_n^\dagger\phi_n - \frac{1}{2}\lambda_n\chi_n^\dagger\chi_n + im\phi_n^\dagger\gamma_*\chi_n\right),
\end{aligned} 
\quad (3.31) $$

and then

$$ \delta\lambda_n = -\int \delta\sigma \left( \frac{\lambda_n}{2}(\phi_n^\dagger\phi_n + \chi_n^\dagger\chi_n) + im\phi_n^\dagger\gamma_*\chi_n \right). \quad (3.32) $$

Since $\phi_n$, $\chi_n$ and $i\gamma_*$ are real, we have $\phi_n^\dagger i\gamma_*\chi_n = (\phi_n^\dagger i\gamma_*\chi_n)^\dagger = \chi_n^\dagger(-i)\gamma_*\phi_n$, and we can also rewrite this as

$$ \delta\lambda_n = -\frac{1}{2} \int \delta\sigma \left( \lambda_n (\phi_n^\dagger\phi_n + \chi_n^\dagger\chi_n) + im(\phi_n^\dagger\gamma_*\chi_n - \chi_n^\dagger\gamma_*\phi_n) \right). \quad (3.33) $$

This allows us to express the variation of $\zeta(s)$ as

$$ 
\begin{aligned}
\delta\zeta(s) &= 2\sum_n \delta(\lambda_n^{-2s}) = -4s\sum_n \frac{\delta\lambda_n}{\lambda_n^{2s+1}} \\
&= 2s \int \delta\sigma \sum_n \left( \frac{1}{\lambda_n^{2s}}(\phi_n^\dagger\phi_n + \chi_n^\dagger\chi_n) + im \frac{1}{\lambda_n^{2s+1}}(\phi_n^\dagger\gamma_*\chi_n - \chi_n^\dagger\gamma_*\phi_n) \right).
\end{aligned} 
\quad (3.34) $$

Let us make a remark about zero-modes if they are present. By zero-modes we here mean solutions $\psi_{0,i}$ of $i\nabla\psi_{0,i} = 0$, or in terms of the real and imaginary parts $i\nabla\phi_{0,i} = i\nabla\chi_{0,i} = 0$. Then $i\lambda_0\phi_{0,i} = D\chi_{0,i} = m\gamma_*\chi_{0,i}$ so that necessarily $\lambda_0 = m$ and $\gamma_*\chi_{0,i} = i\phi_{0,i}$. Similarly $\gamma_*\phi_{0,i} = -i\chi_{0,i}$. We have seen that under a conformal transformation the zero-modes change in a rather simple way so as to remain zero-modes of the conformally transformed $i\nabla$. In particular then the new eigenvalue is still $\lambda_0 = m$, so that

$$ \delta\lambda_0 = 0. \quad (3.35) $$

Let us check this from (3.32). One has $\phi_{0,i}^\dagger\phi_{0,i} = -i\phi_{0,i}^\dagger\gamma_*\chi_{0,i}$, and $\chi_{0,i}^\dagger\chi_{0,i} = i\chi_{0,i}^\dagger\gamma_*\phi_{0,i} = (i\chi_{0,i}^\dagger\gamma_*\phi_{0,i})^\dagger = -i\phi_{0,i}^\dagger\gamma_*\chi_{0,i}$. It follows that the bracket on the right-hand side of (3.32) vanishes for $n=0$, confirming $\delta\lambda_0 = 0$. It follows that in (3.34) we can just as well exclude the contributions of the zero-modes and replace $\sum_n \rightarrow \sum_{n \neq 0}$.

## 4 The tool box: heat kernels, zeta-function and Green's functions

In this section we set up and discuss the technical tools we need to obtain the gravitational action. This includes in particular local heat kernels, local zeta functions and Green's functions. As we will see, they are all related through various relations. An important detailed computation about one of the two the heat kernels we introduce, namely $K_+$, is deferred to appendix A. Some partial results about the other heat kernel ($K_-$) are presented in appendix B. General references on the heat kernel are [23, 24, 30, 31].
## 4.1 The basic tools: definitions, relations and properties

Throughout this section we assume the mass $m$ is non-vanishing, so that $D = i\nabla + m\gamma_*$ has no zero-modes. As discussed above, $i\nabla$ has zero-modes for manifolds of genus $g \ge 1$, but the non-vanishing mass implies that $\lambda_n \ge m > 0$. The absence of zero-modes of $D$ is an important assumption. It means in particular that all zeta-functions $\zeta$, Green's functions $G$, determinants $\text{Det}$, etc can be straightforwardly defined in terms of sums or products of all eigenvalues / eigenfunctions. On the other hand, in the zero-mass limit the zero-modes of $i\nabla$ become zero-modes of $D$. One can then still define corresponding quantities $\tilde{\zeta}, \tilde{G}$, $\text{Det}'$ etc by excluding the zero-modes from the sum / product. However, these modified quantities satisfy different relations from those satisfied by $\zeta, G$ and $\text{Det}$. For example, the Green's function $G(x, y)$ of $D^2$ satisfies $D_x^2 G(x, y) = \delta^{(2)}(x-y)\mathbf{1}_{2 \times 2}$ while in the presence of zero-modes, $\tilde{G}(x, y)$ satisfies a similar relation, $D_x^2 \tilde{G}(x, y) = \delta^{(2)}(x-y)\mathbf{1}_{2 \times 2} - P_0(x, y)$ where on the right-hand side one must subtract $P_0(x, y)$, the projector on the zero-modes, which was discussed in subsection 2.5. Since we are interested in the effective gravitational action for massive matter, one might think that this complication just does not occur. However, for massive bosonic (scalar) matter a fruitful method was to do a small mass expansion around zero-mass, and the same will be true here. One must then express everything in terms of quantities that have a smooth limit as $m \to 0$, and these are the $\tilde{G}, \tilde{\zeta}$, etc. On the torus we could compute the zero-mode projector exactly, and on higher genus Riemann surfaces we could still sufficiently characterise its structure. This will allow us in the next section to obtain the variation of the gravitational action in an expansion in $m^2$ to all orders, and on arbitrary genus Riemann surfaces. This is quite some progress with respect to [1]!

Having said this, we emphasize again that throughout this section we assume that $D$ has no zero-modes (i.e. that either $m \ne 0$ or that we deal with spherical topology). Recall that we had defined the complex eigenfunctions $\psi_n$ and $\psi_n^*$ (and hence also the real eigenfunctions $\chi_n$ and $\phi_n$) such that $\lambda_n > 0$ and

$$D\psi_n = \lambda_n\psi_n, \quad D\psi_n^* = -\lambda_n\psi_n^*, \quad D\chi_n = i\lambda_n\phi_n, \quad D\phi_n = -i\lambda_n\chi_n. \quad (4.1)$$

In terms of these eigenfunctions the completeness relation reads

$$\sum_n (\chi_n(x)\chi_n^\dagger(y) + \phi_n(x)\phi_n^\dagger(y)) = \sum_n (\psi_n(x)\psi_n^\dagger(y) + \psi_n^*(x)(\psi_n^*(y)))^\dagger = \frac{\delta^{(2)}(x-y)}{\sqrt{g}} \mathbf{1}_{2 \times 2}. \quad (4.2)$$

### 4.1.1 Local zeta-functions

We define two local zeta-functions $\zeta_+(s, x, y)$ and $\zeta_-(s, x, y)$ as

$$\begin{aligned} \zeta_+(s, x, y) &= \sum_n \lambda_n^{-2s} (\chi_n(x)\chi_n^\dagger(y) + \phi_n(x)\phi_n^\dagger(y)), \\ \zeta_-(s, x, y) &= \sum_n \lambda_n^{-2s} (\chi_n(x)\phi_n^\dagger(y) - \phi_n(x)\chi_n^\dagger(y)), \end{aligned} \quad (4.3)$$

where $x = (x^1, x^2)$ and $y = (y^1, y^2)$ denote points on the manifold. Note that these local zeta-functions are real $2 \times 2$-matrices. They can also be rewritten in terms of the $\psi_n$
and $\psi_n^*$ as

$$ \begin{aligned} \zeta_+(s, x, y) &= \sum_n \lambda_n^{-2s} (\psi_n(x)\psi_n^\dagger(y) + \psi_n^*(x)(\psi_n^*)^\dagger(y)), \\ \zeta_-(s, x, y) &= i \sum_n \lambda_n^{-2s} (\psi_n(x)\psi_n^\dagger(y) - \psi_n^*(x)(\psi_n^*)^\dagger(y)). \end{aligned} \quad (4.4) $$

As we will see below, $\zeta_+(1, x, y)$ is the Green's function of $D^2$, while $-i\zeta_-(\frac{1}{2}, x, y)$ is the Green's function of $D$. It is for this reason that we need to introduce the "strange-looking" $\zeta_-(s, x, y)$.

As mentioned above, we will also encounter the local zeta-functions $\tilde{\zeta}_{\pm}(s, x, y)$ with the zero-modes of $i\mathcal{V}$ (if present) excluded from the sums. The contributions of these zero-modes is precisely given in terms of the zero-mode projectors $P_0$ defined in (2.102) and $Q_0$ defined in (2.106). Hence we obviously have

$$ \tilde{\zeta}_+(s, x, y) = \zeta_+(s, x, y) - \frac{P_0(x, y)}{m^{2s}}, \quad \tilde{\zeta}_-(s, x, y) = \zeta_-(s, x, y) - i \frac{Q_0(x, y)}{m^{2s}}. \quad (4.5) $$

There is an important relation between $\zeta_+$ and $\zeta_-$ which generalises the relation between the corresponding Green's functions. It follows immediately from (4.1) and the definitions (4.3) that

$$ \begin{aligned} D_x \zeta_+(s, x, y) &= -i \zeta_-(s - \frac{1}{2}, x, y), & D_x \zeta_-(s, x, y) &= i \zeta_+(s - \frac{1}{2}, x, y), \\ D_x^2 \zeta_{\pm}(s, x, y) &= \zeta_{\pm}(s - 1, x, y), \end{aligned} \quad (4.6) $$

where the subscript $x$ on $D$ indicates that the derivatives are with respect to $x$. Since $D_x P_0(x, y) = m\gamma_* P_0(x, y) = mQ_0(x, y)$ and $D_x Q_0(x, y) = m\gamma_* Q_0(x, y) = mP_0(x, y)$, these relations (4.6) immediately also carry over to the $\tilde{\zeta}_{\pm}(s, x, y)$.

Denoting the trace over the $2 \times 2$ matrices by tr, and using the orthonormality of the $\phi_n$ and $\chi_n$ we have

$$ \int d^2x \sqrt{g} \operatorname{tr} \zeta_+(s, x, x) = 2 \sum_n \lambda_n^{-2s} \equiv \zeta(s), \quad \int d^2x \sqrt{g} \operatorname{tr} \zeta_-(s, x, x) = 0, \quad (4.7) $$

where $\zeta(s)$ is the zeta function of $D^2$.

The following (anti-) hermiticity relations follow directly from the definitions (4.3):

$$ \begin{aligned} (\zeta_+(s, x, y))^\dagger &= \zeta_+(s, y, x) & \Rightarrow \zeta_+(s, x, x) \text{ is real and hermitian,} \\ (\zeta_-(s, x, y))^\dagger &= -\zeta_-(s, y, x) & \Rightarrow \zeta_-(s, x, x) \text{ is real and anti-hermitian.} \end{aligned} \quad (4.8) $$

There are 3 possible hermitian and real matrices, namely $\sigma_x$, $\sigma_z$ and $\mathbf{1}_{2 \times 2}$, while there is only a single real and anti-hermitian matrix, namely $i\gamma_*$. Hence, we can already assert that

$$ \zeta_-(s, x, x) \sim i\gamma_* \quad (4.9) $$

On the other hand, for $\zeta_+(s, x, x)$ the present argument would allow 3 possible matrix structures, while later-on we will find that $\zeta_+(s, x, x) \sim \mathbf{1}_{2 \times 2}$.
The convergence properties of these local zeta-functions depend essentially on the large-$n$ behaviour of the eigenvalues $\lambda_n^2$. Indeed, equation (4.7) more or less suggests that multiplying the $\lambda_n^{-2s}$ by $\chi_n^\dagger(x)\chi_n(y)$ and $\phi_n^\dagger(x)\phi_n(y)$ does not change the convergence properties.<fnref n="17" /> Of course, discussing the convergence of a sum of functions (or even functions of 2 arguments) needs somewhat more care and requires specifying an appropriate norm in which the convergence is to be appreciated, but without going into these details it is clear that we get a good deal of information about the convergence of the local zeta-functions by studying the behaviour of the large eigenvalues, which amounts to studying the convergence of the (integrated) zeta-function $\zeta(s)$.

The behaviour of the large eigenvalues of $D^2$ is dictated by the leading 2-derivative term in $D^2$ which, by (2.48), is the same as the one of the scalar Laplacian, and which is the same as in flat space (with periodic boundary conditions). This amounts to saying that the large eigenvalue behaviour (“UV-behaviour”) reflects the short-distance properties of the manifold, and at short distances all manifolds are locally flat. Thus the leading large $n$ behaviour of the $\lambda_n$ must be the same as in the case of the flat torus where the eigenvalues depend on two integers $n_1$ and $n_2$. We have seen before that for general periods of the torus the eigenvalues are $\lambda_n^2 = n_1^2 + (\frac{n_2+\tau_1n_1}{\tau_2})^2 + m^2$. Thus, for large eigenvalues we will always have $\lambda_n^2 \equiv \lambda_n^2 \simeq a^2[n_1^2 + (\frac{n_2+\tau_1n_1}{\tau_2})^2]$ for some $a$ which sets the scale. This is essentially the statement of Weyl’s law [32] about the large eigenvalues of the Laplace operator.<fnref n="18" /> Then for large $|n_1|$ and large $|n_2|$ we approximately have

$$ \sum_{(n_1,n_2) \neq (0,0)} \lambda_n^{-2s} \simeq a^{-2s} \sum_{(n_1,n_2) \neq (0,0)} \left[ n_1^2 + \left( \frac{n_2 + \tau_1 n_1}{\tau_2} \right)^2 \right]^{-s} \\ \simeq a^{-2s} \int_{k \notin D} d^2k \left[ k_1^2 + \left( \frac{k_2 + \tau_1 k_1}{\tau_2} \right)^2 \right]^{-s}, \quad (4.10) $$

where $D$ is some domain “of unit radius” around the origin. Assuming convergence, we change variables to $p_1 = k_1$, $p_2 = \frac{k_2+\tau_1k_1}{\tau_2}$ and go to polar coordinates so that

$$ \sum_{n_1,n_2} \lambda_n^{-2s} \simeq 2\pi a^{-2s} \tau_2 \int_1^\infty dp \, p (p^2)^{-s} = \pi a^{-2s} \tau_2 \int_1^\infty d\xi \, \xi^{-s} \\ \simeq \pi a^{-2s} \tau_2 \sum_{n=1}^\infty n^{-s} = \pi a^{-2s} \tau_2 \zeta_R(s). \quad (4.11) $$

We see that the convergence properties of the zeta-functions of Laplace-like operators in two dimensions are exactly the same as for Riemann’s zeta-function! It follows that, just

<footnote n="17">¹⁷This argument is certainly true for the convergence properties of the sum. For large eigenvalues, we expect the corresponding $\psi_n(x)\psi_n^\dagger(y)$ to be rapidly oscillating functions of $x$ and $y$, leading to cancellations between adjacent terms in the sum, and thus to a possible improvement of the convergence of the sum. But this rapidly oscillating behaviour and the corresponding cancellations also mean that one has to examine more carefully the usual argument made for the heat kernel, see below, that for small $t$ the sum is dominated by the large eigenvalues.</footnote>

<footnote n="18">¹⁸We see that the behaviour of the large eigenvalues does not depend much on the details of the geometry. It depends to some extend on the global structure like boundary conditions. And it does, of course, crucially depend on the dimensionality of the manifold.</footnote>
as for the sum defining Riemann's zeta function $\zeta_R(s)$, the zeta function $\zeta(s) \equiv \zeta_{D^2}(s)$ is defined by a convergent sum for $\Re s > 1$. As explained before, we then also expect that the sums defining the local zeta-functions $\zeta_{\pm}(s, x, y)$ are convergent expressions for $\Re s > 1$ and are otherwise defined by analytical continuation. Below, we will indeed confirm this expectation and show that, for $x \neq y$, $\zeta_{\pm}(s, x, y)$ can be defined as analytic functions for all $s \in \mathbb{C}$, and that $\zeta_+(s, x, x)$ has a single pole at $s = 1$ while $\zeta_-(s, x, x)$ has a single pole at $s = \frac{1}{2}$.

### 4.1.2 Local heat kernels

We similarly define local heat kernels that are again $2 \times 2$-matrices:

$$ 
\begin{aligned}
K_+(t, x, y) &= \sum_n e^{-\lambda_n^2 t} (\chi_n(x) \chi_n^\dagger(y) + \phi_n(x) \phi_n^\dagger(y)) \\
&= \sum_n e^{-\lambda_n^2 t} (\psi_n(x) \psi_n^\dagger(y) + \psi_n^*(x) (\psi_n^*)^\dagger(y)), \\
K_-(t, x, y) &= \sum_n e^{-\lambda_n^2 t} (\chi_n(x) \phi_n^\dagger(y) - \phi_n(x) \chi_n^\dagger(y)) \\
&= i \sum_n e^{-\lambda_n^2 t} (\psi_n(x) \psi_n^\dagger(y) - \psi_n^*(x) (\psi_n^*)^\dagger(y)).
\end{aligned}
\qquad (4.12) $$

Again, if necessary, one can also define the corresponding local heat kernels with the zero-modes of $i\mathbb{V}$ excluded as

$$ \widetilde{K}_+(t, x, y) = K_+(t, x, y) - e^{-m^2 t} P_0(x, y), \quad \widetilde{K}_-(t, x, y) = K_-(t, x, y) - i e^{-m^2 t} Q_0(x, y). \qquad (4.13) $$

One could also define

$$ \mathcal{K}(t, x, y) = \frac{1}{2} K_+(t, x, y) - \frac{i}{2} K_-(t, x, y) = \sum_n e^{-\lambda_n^2 t} \psi_n(x) \psi_n^\dagger(y). \qquad (4.14) $$

Then the real functions $K_{\pm}$ constitute (2 times) the real and $(-2$ times) the imaginary parts of $2\mathcal{K}$.

It follows from (4.12) that the large-$t$ behaviour of the heat kernels is controlled by the smallest eigenvalue. Since we assumed that there is no zero-mode, we have $\lambda_n^2 > 0$ and for $t \to \infty$ all our heat kernels vanish (at least) as $e^{-\lambda_{\min}^2 t}$, where $\lambda_{\min}$ denotes the smallest eigenvalue:

$$ K_{\pm}(t, x, y) \sim_{t \to \infty} \mathcal{O}(e^{-\lambda_{\min}^2 t}). \qquad (4.15) $$

From the definitions (4.12) one sees that $K_+$ and $K_-$ (and hence also $\mathcal{K}$) satisfy the “generalised heat equation”

$$ \left( \frac{d}{dt} + D_x^2 \right) K_{\pm}(t, x, y) = 0, \qquad (4.16) $$

where the subscript $x$ on $D^2$ indicates that the derivatives are with respect to $x$. In appendix A we explain how one can obtain the asymptotic small-$t$ expansion of $K_+(t, x, y)$
from this differential equation and an appropriate “initial” condition for $K_+$. Indeed, one
sees from the definition (4.12) and the closure relation (4.2) that for $t \to 0$ we have

$$
K_+(t, x, y) \sim \frac{\delta^{(2)}(x-y)}{\sqrt{g}} \mathbf{1}_{2 \times 2}, \qquad (4.17)
$$

which is the “usual” initial condition for a heat kernel, as also assumed in the appendix.
However, no such simple initial condition holds for $K_-$.
This is why the method of appendix
A yields $K_+$ and not $K_-$.
In particular, in the appendix we work out the relevant first few
terms of the small-$t$ expansion of $K_+$ which we will need below.

Another way to see the differences between $K_+$ and $K_-$ is to note that $K_+$ contains the full sum of all eigenfunctions of $D$, namely the $\psi_n$ and the $\psi_n^*$. Thus if one uses a different basis for these eigenfunctions one still gets the same $K_+$. Moreover, the $\psi_n$ and the $\psi_n^*$ appear symmetrically. We can write $K_+(t, x, y) = \langle x | \sum_{\lambda_n^2} e^{-\lambda_n^2 t} P(\lambda_n^2) | y \rangle$ where $P(\lambda_n^2)$ is the projector on the eigenspace of $D^2$ with eigenvalue $\lambda_n^2$. This makes clear that one could use any basis of eigenfunctions of $D^2$. This also explains why one is able (in principle) to obtain $K_+$ uniquely by solving the heat equation (4.16) for the operator $D^2$ with the appropriately prescribed initial condition. However, this is not true for the imaginary part $K_-$ of $\mathcal{K}$. We see from (4.12) that the definition of $K_-$ is not simply a sum over all eigenfunctions of $D^2$, but that we made a certain distinction between the eigenfunctions of $D$ with eigenvalues $\lambda_n > 0$ and those with eigenvalues $-\lambda_n < 0$. (Recall that we always take $\lambda_n > 0$.) Clearly, the operator $D^2$ does not make this distinction, and hence, one cannot simply get the $K_-$ by solving (4.16).

What can be said about the matrix structure of the heat kernel $K_+(t, x, y)$ ? As just explained, $K_+$ can be obtained by solving the heat equation (4.16) with initial condition (4.17). Now it is clear from (2.43) and (2.48), that the matrix structure of $D^2$ is $\mathbf{1}(\dots) + \gamma_*(\dots)$. It then follows that $K_+(t, x, y)$ must have the same matrix structure:

$$
K_+(t, x, y) = K_+^0(t, x, y) \mathbf{1} + K_+^*(t, x, y) \gamma_*. \quad (4.18)
$$

What can be said about $K_-$? Define an auxiliary quantity $L(t, x, y)$ as

$$
L(t,x,y) = i \sum_n \frac{e^{-\lambda_n^2 t}}{\lambda_n} (\chi_n(x) \chi_n^\dagger(y) + \phi_n(x) \phi_n^\dagger(y)) = i \sum_n \frac{e^{-\lambda_n^2 t}}{\lambda_n} (\psi_n(x) \psi_n^\dagger(y) + \psi_n^*(x) (\psi_n^*)^\dagger(y)), \quad (4.19)
$$

which is now constructed from the eigenfunctions and eigenvalues of $D^2$ only, without any
distinction between positive and negative eigenvalues. We then have

$$
K_{-}(t, x, y) = D_{x}L(t, x, y), \tag{4.20}
$$

Obviously, $L(t, x, y)$ also satisfies the heat equation (4.16). Its initial condition is

$$
L(t, x, y) \sim_{t \to 0} i \sum_n \frac{1}{\lambda_n} (\chi_n(x) \chi_n^\dagger(y) + \phi_n(x) \phi_n^\dagger(y)). \quad (4.21)
$$

However, contrary to the completeness relation, we do not recognise here any known func-
tion (or distribution). This prevents us from obtaining $L$ in analogy to what is done for
$K_+$ in appendix A.
Let us nevertheless see what we can say about $K_-$ and $K_+$ on general grounds. In analogy with the discussion for the local heat kernels, it follows directly from the definitions (4.12) of $K_+$ and $K_-$ in terms of the eigenfunctions $\psi_n$ and $\psi_n^*$ that

$$ (K_+(t, x, y))^{\dagger} = K_+(t, y, x), \quad (K_-(t, x, y))^{\dagger} = -K_-(t, y, x), \qquad (4.22) $$

so that for $x = y$ we simply have

$$ (K_+(t, x, x))^{\dagger} = K_+(t, x, x), \quad (K_-(t, x, x))^{\dagger} = -K_-(t, x, x). \qquad (4.23) $$

This means that $K_+(t, x, x)$ is hermitian and $K_-(t, x, x)$ is anti-hermitian. Moreover, they are both real. The only real and anti-hermitian matrix is $i\gamma_*$ and it follows that:

$$ K_+(t, x, x) \sim \mathbf{1}_{2 \times 2}, \quad K_-(t, x, x) \sim \gamma_* \qquad (4.24) $$

### 4.1.3 Relating zeta functions and heat kernels

There is a well-known relation between zeta-functions and heat kernels which generalises the relation (3.19) for Riemann's zeta function. This relation also generalises to the local zeta-functions and local heat kernels we have defined:

$$ \zeta_{\pm}(s, x, y) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} dt \, t^{s-1} K_{\pm}(t, x, y). \qquad (4.25) $$

Indeed, these relations (4.25) follow trivially (for $\Re s > 1$) from inserting the definitions of $K_\pm$ as a sum over the eigenvalues and eigenfunction, interchanging sum and integration and doing the integrations term per term as $(\Gamma(s))^{-1} \int_0^\infty dt \, t^{s-1} e^{-t\lambda_n^2} = \lambda_n^{-2s}(\Gamma(s))^{-1} \int_0^\infty dt' \, t'^{s-1} e^{-t'} = \lambda_n^{-2s}$. Then, the relation (4.25) can be obtained on all of $\mathbb{C}$ as an identity between meromorphic functions by analytic continuation.

The relation (4.25) is a special case of the Mellin transform. The Mellin transform $\mathcal{M}(f)$ of a (continuous) function $f$ is

$$ \varphi(s) \equiv (\mathcal{M}(f))(s) = \int_{0}^{\infty} dt \, f(t) t^{s-1}. \qquad (4.26) $$

If this integral is absolutely convergent in a strip $a < \Re s < b$, then $\varphi(s)$ is analytic in this strip, and $f$ can be recovered by the inverse Mellin transform defined as

$$ f(t) = (\mathcal{M}^{-1}(\varphi))(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} ds \, \varphi(s) t^{-s}, \quad a < c < b. \qquad (4.27) $$

We can then interpret equation (4.25) by saying that the $\Gamma(s) \zeta_+(s, x, y)$ are the Mellin transforms of the heat kernels $K_\pm(t, x, y)$. The inverse Mellin transform then allows us to get the heat kernels back from the zeta functions as

$$ K_{\pm}(t, x, y) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} ds \, t^{-s} \Gamma(s) \zeta_{\pm}(s, x, y), \qquad (4.28) $$

where the real $c$ must be chosen such that $\Gamma(c + i\sigma) \zeta_+(c + i\sigma, x, y)$ is analytic for all real $\sigma$ and tends uniformly to zero as $\sigma \to \pm\infty$. One can show that any $c > 1$ is a convenient choice. This is discussed in more detail below.
The relation (4.25) translates (4.24) into the corresponding relations for the zeta-functions at coinciding points, confirming the structure of $\zeta_-(s, x, x)$ already obtained earlier, and providing the corresponding result for $\zeta_+(s, x, x)$:

$$ \zeta_+(s, x, x) \sim \mathbf{1}_{2 \times 2}, \quad \zeta_-(s, x, x) \sim \gamma_* \qquad (4.29) $$

As already discussed, one can get quite some information about $K_+$ from the differential equation it satisfies, which is not the case of $K_-$. However, much of the information about $K_+$ translates via (4.25) into corresponding information about $\zeta_+$. We can then use the relations (4.6) to obtain $\zeta_-$ from $\zeta_+$ and obtain the corresponding information about $\zeta_-$. Finally one can use the inverse Mellin transformation to deduce statements about $K_-$. To begin with note that, since $\zeta_+$ can be obtained from $K_+$ by (4.25), its matrix structure must be the same (cf (4.18)):

$$ \zeta_+(s, x, y) = \zeta_+^0(s, x, y) \mathbf{1}_{2 \times 2} + \zeta_+^*(s, x, y) \gamma_* \qquad (4.30) $$

Next, recall from (2.50) that $i\mathcal{W} = i\sigma_x D_1 + i\sigma_z D_2$, where $D_1$ and $D_2$ are some differential operators not involving any matrix. It follows that $i\mathcal{W}\gamma_* = -i\gamma_*\mathcal{W} = -\sigma_z D_1 + \sigma_x D_2$ and

$$ \mathrm{tr} i\mathcal{W} = \mathrm{tr} \gamma_* i\mathcal{W} = \mathrm{tr} i\mathcal{W}\gamma_* = 0. \qquad (4.31) $$

Recall the first relation (4.6) which we rewrite as $\gamma_*\zeta_-(s, x, y) = i\gamma_*(i\mathcal{W}_x + m\gamma_*)\zeta_+(s + \frac{1}{2}, x, y)$. Then taking the Dirac trace of this relation, only the part $\sim m$ survives:

$$ \mathrm{tr} \gamma_* \zeta_-(s, x, y) = im \mathrm{tr} \zeta_+ \left( s + \frac{1}{2}, x, y \right). \qquad (4.32) $$

As we will see in the next subsection, the variation of the gravitational action involves only $\mathrm{tr} \zeta_+(s, x, x)$ and $\mathrm{tr} \gamma_* \zeta_-(s, x, x)$, and the previous relation shows that the knowledge of $\mathrm{tr} \zeta_+(s, x, x)$ should be all we need to know. But we can also easily establish this same relation without the traces: we have seen in (4.29) that $\zeta_-(s, x, x) \sim \gamma_*$ and $\zeta_+(s, x, x) \sim \mathbf{1}_{2 \times 2}$, so that indeed

$$ \zeta_-(s, x, x) = im\gamma_* \zeta_+ \left( s + \frac{1}{2}, x, x \right). \qquad (4.33) $$

We will soon show that $\zeta_+(s, x, x)$ is a meromorphic function with a single pole at $s = 1$. The previous relation should be understood as an identity between meromorphic functions and, in particular, $\zeta_-(s, x, x)$ then has a single pole at $s = \frac{1}{2}$. Let us insist that one should not conclude from this relation that $\zeta_-(s, x, x)$ vanishes for $m = 0$, since our whole discussion assumed that there are no zero-modes of the Dirac operator. It is only for spherical topology where this conclusion would be correct, while for genus larger or equal to one, we expect $\zeta_+(s, x, x)$ to contain a contribution $\sim \frac{1}{m^s}$, so there is a possible zero-mode contribution to $\zeta_-(s, x, x)$ even as $m \to 0$.

### 4.1.4 Small-t asymptotics and poles of zeta functions at coinciding points

Let us now study the possible singularities of the integrals (4.25) in order to establish the analyticity properties of $\zeta_+(s, x, y)$. Since $K_\pm$ vanish exponentially for large $t$, cf (4.15),
the integral always converges at $t \to \infty$. Hence any divergences of the integral must come from the region $t \to 0$ where the $K_\pm$ are singular. Thus any singularities (poles) of the zeta-functions are related to the small-$t$ behaviour of the heat kernel. Furthermore, since $\Gamma(s)$ has poles at $s = 0, -1, -2, \dots$, one sees that $\frac{1}{\Gamma(s)}$ vanishes at $s = 0, -1, -2, \dots$, resulting in finite values of the zeta-functions at $s = 0, -1, -2, \dots$. In particular, these finite values are also determined by the divergences of the integral due to the small-$t$ behaviour of $K$. We will explain this in further detail below. There is quite some abundant literature on the small-$t$ asymptotics of heat kernels associated to various differential operators, in particular at coinciding points $x = y$.

The general expectation is that, due to the presence of $e^{-\lambda_n^2 t}$, the small-$t$ behaviour of the heat kernel is related to the behaviour of the large eigenvalues of order $\lambda_n^2 \sim \frac{1}{t}$, and as discussed above, this is related to the leading-derivative terms of the differential operator $D^2$ and the small-scale structure of the manifold. However, while this expectation turns out to be correct for our heat kernels $K_\pm(t, x, x)$ at coinciding points, as well as for $K_+(t, x, y)$ at $x \neq y$, it will not be true for $K_-(t, x, y)$ at $x \neq y$. Indeed, for $x \neq y$, the local heat kernels not only involve the $e^{-\lambda_n^2 t}$ but also the $\psi_n(x)\psi_n^\dagger(y)$ and the latter typically oscillate rapidly for large $n$ (large $\lambda_n^2$). This can potentially lead to important cancellations and correspondingly the sum no longer is dominated by the eigenvalues of order $\lambda_n^2 \sim \frac{1}{t}$ but by much smaller eigenvalues already. Obviously, this will change the small-$t$ asymptotics of $K_-(t, x, y)$. We will see this very explicitly in appendix B for the $K_-(t, x, y)$ on the flat torus.

As explained above, our heat kernel $K_+$ can, in principle, be entirely determined by solving the heat equation. In particular, its small-$t$ behaviour can be determined in an asymptotic expansion from the differential equation (4.16), with the leading behaviour being the same as in flat space, and the subleading terms being given in terms of the local curvature and derivatives of the curvature. This is studied in some detail in appendix A. Of particular interest will be the expansion at coinciding points $x = y$. It is shown in appendix A eq (A.49) that,<fnref n="19" />

$$K_+(t, x, x) \sim_{t \to 0} \frac{1}{4\pi t} \left( 1 - \left( \frac{\mathcal{R}(x)}{12} + m^2 \right) t + \mathcal{O}(t^2) \right) \mathbf{1}_{2 \times 2}. \quad (4.34)$$

A heuristic way to understand this result is to note that the heat kernel, at coinciding points, for the two-dimensional scalar Laplacian $\Delta_{scalar}$ is well-known (cf (1.41)) to be $K_{\Delta_{scalar}}(t, x, x) = \frac{1}{4\pi t}(1 + \frac{\mathcal{R}}{6}t + \dots)$. Now $K_+$ is the heat kernel for $D^2$, which involves the spinorial Laplacian $\Delta_{sp}$ which, of course, differs from $\Delta_{scalar}$ but has the same leading derivative terms. One might then expect that it also gives the same result to this order.<fnref n="20" />

<footnote n="19">¹⁹In the appendix, we have determined the first few coefficients of the heat kernel $K(t, x, y)$ as follows from the differential equation. As argued above, this actually only determines $K_+(t, x, y)$, so eq (A.49) translates into an equation for $K_+(t, y, y)$. Note that, solely from the differential equation, we could also have expected contributions $\sim \gamma_*$, Indeed, such contributions do appear in the intermediate steps of the computation, in particular for $K_+(t, x, y)$ with $x \neq y$, but they drop out when setting $x = y$ to obtain $K_+(t, x, x)$, in agreement with the general result (4.24).</footnote>

<footnote n="20">²⁰This assumption is not entirely true as one sees from the detailed computations in the appendix. Indeed, the subleading single derivative term gives important contributions to the first subleading term in the small-</footnote>
Moreover, one expects that the additional $\frac{R}{4}+m^2$ in $D^2$ simply gives an extra $e^{-(R/4+m^2)t} \simeq 1-(\frac{R}{4}+m^2)t$, to this order, resulting in $\frac{1}{4\pi t}(1+\frac{R}{6}t+...)(1-\frac{R}{4}t-m^2t+...)=\frac{1}{4\pi t}(1-\frac{R}{12}t-m^2t+...)$, in agreement with (4.34).

Inserting the result (4.34) for $K_+$ into (4.25) allows us to get much information about $\zeta_+(s, x, x)$ as we will explain next. More generally, the small-$t$ expansion is (cf. appendix A)

$$K_+(t, x, x) \sim_{t \to 0} \frac{\mathbf{1}_{2 \times 2}}{4\pi t} \sum_{r=0}^{\infty} F_r(x, x) t^r, \quad F_0(x, x) = 1. \qquad (4.35)$$

This is an asymptotic expansion for small $t$ which means it disregards any exponentially small terms $O(e^{-a/t})$ that can be present but are invisible in such an expansion in powers of $t$. As discussed above, we also know that $K_+(t, x, x)$ vanishes exponentially as $e^{-\lambda_{\min}^2 t}$ as $t \to \infty$. It follows that upon evaluating the integral (4.25) any possible divergences can only come from the small-$t$ region$^{21} \int_0^{\mu^{-2}} \dots$ where the asymptotic expansion (4.35) is valid. Inserting this expansion into this integral, and interchanging the integral and the sum, the $r^{th}$ term in the sum is (this is to be understood as evaluated for $\Re s > 1$ and then analytically continued)

$$\frac{F_r(x, x)}{4\pi} \frac{1}{\Gamma(s)} \int_0^{\mu^{-2}} dt \, t^{s+r-2} = \frac{F_r(x, x)}{4\pi} \frac{1}{\Gamma(s)} \frac{(\mu^{-2})^{s+r-1}}{s+r-1}, \quad r = 0, 1, 2, \dots \qquad (4.36)$$

Let us insist that, since the integral is evaluated for $\Re s > 1$, the contribution from the lower boundary is $0^{s+r-1} = 0$. The expression on the right-hand side of (4.36) can then be analytically continued: it is regular, except possibly as $s \to 1-r$. Indeed, Euler's $\Gamma$-function has no zeros so that $\frac{1}{\Gamma(s)}$ is regular. On the other hand, $\Gamma(s)$ has poles whenever $s \to 0, -1, -2, \dots$ which corresponds to $r=1, 2, \dots$, but not $r=0$. Hence, we must distinguish the cases $r=0$ and $r=1, 2, \dots$. For $r=0$, i.e. $s \to 1$, we have $\Gamma(s) \sim 1-\gamma(s-1)$, where $\gamma \simeq 0.57$ is Euler's constant, and then

$$\frac{F_0(x, x)}{4\pi} \frac{1}{\Gamma(s)} \int_0^{\mu^{-2}} dt \, t^{s-2} \sim_{s \to 1} \frac{F_0(x, x)}{4\pi} \left( \frac{1}{s-1} + \gamma + \log \mu^{-2} + \mathcal{O}(s-1) \right). \qquad (4.37)$$

For $r=1, 2, \dots$, we have $\frac{1}{\Gamma(s)} \sim_{s \to 1-r} (-)^{r-1}(r-1)! (s-1+r)$ so that the “would-be pole” of the integral is cancelled by this zero and one gets

$$\frac{F_r(x, x)}{4\pi} \frac{1}{\Gamma(s)} \int_0^{\mu^{-2}} dt \, t^{s+r-2} \sim_{s \to 1-r} \frac{F_r(x, x)}{4\pi} \left( (-)^{r-1}(r-1)! + \mathcal{O}(s+r-1) \right), \quad r = 1, 2, \dots \qquad (4.38)$$

We conclude that

$$\zeta_+(s, x, x) \sim_{s \to 1} \left( \frac{1}{4\pi(s-1)} + \text{regular} \right) \mathbf{1}_{2 \times 2}, \quad \zeta_+(k, x, x) = \frac{(-)^k}{4\pi} k! F_{1-k}(x, x) \mathbf{1}_{2 \times 2}, \\ k = 0, -1, \dots \qquad (4.39)$$

<footnote>$t$ expansion. Since this first-derivative terms differs between $\Delta_{\text{scalar}}$ and $\Delta_{\text{sp}}$ by $-\frac{i}{2}\gamma_*\omega^\mu\partial_\mu$ one might have expected a different contribution to the first subleading terms in the small-$t$ expansion of $K_+$. However, at this order, these additional contributions cancel.</footnote>

<footnote>$^{21}$We have introduced an arbitrary large mass scale $\mu$ so that $\mu^{-2}$ is small enough for the asymptotic expansion to be applicable. This is similar to the small but finite $\epsilon$ we had introduced when discussing Riemann's zeta function.</footnote>
In particular for $k=0$, we read from (4.34) that $F_1(x, x) = -\left(\frac{\mathcal{R}(x)}{12} + m^2\right) \mathbf{1}_{2\times2}$ which yields

$$ \zeta_+(0, x, x) = -\frac{1}{4\pi} \left( \frac{\mathcal{R}(x)}{12} + m^2 \right) \mathbf{1}_{2\times2}. \qquad (4.40) $$

What about $\zeta_-(s, x, x)$? We know from (4.33) that $\zeta_-(s, x, x) = im\gamma_* \zeta_+(s + \frac{1}{2}, x, x)$. This means that $\zeta_-(s, x, x)$ must be analytic in $s$ except for a simple pole at $s = \frac{1}{2}$. More precisely, (4.39) translates into

$$ \begin{gathered} \zeta_-(s, x, x) \sim_{s \to 1/2} \left( \frac{im}{4\pi(s - \frac{1}{2})} + \text{regular} \right) \gamma_*, \qquad (4.41) \\ \zeta_-(k - \frac{1}{2}, x, x) = (-)^k \frac{im}{4\pi} F_{1-k}(x, x) \gamma_*, \quad k = 0, -1, \dots \end{gathered} $$

We can now use this information, together with the inverse Mellin transform (4.28) to obtain the small-$t$ expansion of $K_-(t, x, x)$ as follows. Choosing e.g. $c > 1$ and assuming that$^{22}$

$$ \lim_{\sigma \to \infty} |\Gamma(c' \pm i\sigma)\zeta_-(c' \pm i\sigma, x, x)| = 0, \forall c' \le c, \qquad (4.44) $$

we can close the integration contour by an infinite rectangle to the left, cf. the figure, so that the integral is given by the sum of the residues of all the poles enclosed.

<figure><img src="image_2.png" /></figure>

The integration contour in the complex $s$-plane There is the pole at $s = \frac{1}{2}$ from $\zeta_-(s, x, x)$

$$ \Gamma(z+1) \sim_{|z| \to \infty} z^z e^{-z} = e^{z(\log z - 1)}, \qquad (4.42) $$

which remains valid even if the argument is complex as long as $\arg z \neq \pm\pi$, since one must avoid the poles on the negative real axis. If we let $z = a \pm i\sigma$ with $\sigma \to \infty$ it is straightforward to see that

$$ \Gamma(a + 1 \pm i\sigma) \sim_{\sigma \to \infty} e^{\pm i \left(\frac{\pi a}{2} + \sigma \log \sigma - \sigma\right)} e^{-\frac{\pi}{2}\sigma + a \log \sigma + \mathcal{O}\left(\frac{1}{\sigma}\right)} \Rightarrow |\Gamma(a + 1 \pm i\sigma)| \sim_{\sigma \to \infty} \sigma^a e^{-\frac{\pi}{2}\sigma}. \qquad (4.43) $$

One can show a similar result if also $a \to -\infty$, so that the $\Gamma$-function vanishes exponentially on the dotted part of the contour in the above figure as this contour is taken to infinity. Note that the left vertical part of the contour must, of course, avoid the poles at the negative integers, but it remains true, even on the negative real axis between any two negative integers $-k-1$ and $-k$, that $|\Gamma|$ vanishes exponentially as $k \to \infty$.

<footnote>$^{22}$This should follow from the fact that $\Gamma(s)$ vanishes exponentially for $s = c \pm i\sigma$ and $\sigma \to \infty$, and, more generally on the dotted part of the integration contour. To estimate the $\Gamma$-function for large arguments we use Stirling's formula</footnote>
and there are the poles from $\Gamma(s)$ at $s = -n, n = 0, 1, 2, \dots$ with residues $\frac{(-)^n}{n!}$. We get

$$ 
\begin{aligned}
K_-(t, x, x) & \sim_{t \to 0} \frac{1}{2\pi i} \int_{\text{infrect.}} \mathrm{d}s \, t^{-s} \Gamma(s) \zeta_-(s, x, x) \\
& = t^{-1/2} \Gamma\left(\frac{1}{2}\right) \frac{im}{4\pi} \gamma_* + \sum_{n=0}^{\infty} t^n \frac{(-)^n}{n!} \zeta_-((-n, x, x)) \\
& = im \gamma_* \left( \frac{1}{4\sqrt{\pi t}} \gamma_* + \sum_{n=0}^{\infty} t^n \frac{(-)^n}{n!} \zeta_+ \left(-n + \frac{1}{2}, x, x\right) \right).
\end{aligned} 
\quad (4.45) $$

Just as for $\zeta_-(s, x, x)$, the explicit factor of $m$ on the right-hand side does not mean that $K_-(t, x, x)$ vanishes for $m = 0$ since, in general, the $\zeta_+(-n + \frac{1}{2}, x, x)$ are expected to contain contributions $\sim m^{2n-1}$. However, we see that for $m = 0$ the only term involving a half-integer power of $t$, namely $\frac{1}{\sqrt{t}}$ is absent (as are probably also the terms with $n \ge 1$).

Below, we will estimate the small-$t$ behaviour of $K_-$ on the flat torus where the eigenvalues and eigenfunctions are explicitly known, and we will find that

$$ K_-(t, x, x) \sim_{t \to 0} \left( \frac{m}{4\sqrt{\pi t}} + \frac{1}{4\pi^2} \right) i\gamma_* + \dots, \quad (4.46) $$

in agreement with the general formula (4.45)!

### 4.1.5 Green's functions

The Green's function $S(x, y)$ of the Dirac operator $D$ is a $2 \times 2$-matrix solution of

$$ D_x S(x, y) = \frac{\delta^{(2)}(x-y)}{\sqrt{g}} \mathbf{1}_{2 \times 2}, \quad (4.47) $$

while we denote $G$ the (also $2 \times 2$-matrix) Green's function of $D^2$:

$$ D_x^2 G(x, y) = \frac{\delta^{(2)}(x-y)}{\sqrt{g}} \mathbf{1}_{2 \times 2}. \quad (4.48) $$

In terms of the eigenfunctions and eigenvalues we have

$$ 
\begin{aligned}
S(x, y) &= -i \sum_n \frac{1}{\lambda_n} (\chi_n(x) \phi_n^\dagger(y) - \phi_n(x) \chi_n^\dagger(y)) = \sum_n \frac{1}{\lambda_n} (\psi_n(x) \psi_n^\dagger(x) - \psi_n^*(x) \psi_n^{*\dagger}(y)), \\
G(x, y) &= \sum_n \frac{1}{\lambda_n^2} (\chi_n(x) \chi_n^\dagger(y) + \phi_n(x) \phi_n^\dagger(y)) = \sum_n \frac{1}{\lambda_n^2} (\psi_n(x) \psi_n^\dagger(x) + \psi_n^*(x) \psi_n^{*\dagger}(y)).
\end{aligned} 
\quad (4.49) $$

They are indeed solutions of (4.47), resp. (4.48) as one sees by applying $D$, resp. $D^2$, and using the completeness relation (4.2). It trivially follows from either (4.47) and (4.48), or from (4.49), that

$$ S(x, y) = D_x G(x, y). \quad (4.50) $$

Comparing with (4.3), one sees that

$$ S(x, y) = -i \zeta_{-} \left( \frac{1}{2}, x, y \right), \quad G(x, y) = \zeta_{+} (1, x, y). \quad (4.51) $$
Note that this is consistent with (4.6) and (4.50). Recall that $\zeta_-(s, x, x)$ has a pole at $s = \frac{1}{2}$ and $\zeta_+(s, x, x)$ at $s = 1$, consistent with the fact that the Green's functions are singular as $x \to y$.

It is also obvious from the definitions that $G(x, y)$ is real and $S(x, y)$ purely imaginary. Furthermore,

$$ (G(x, y))^{\dagger} = G(y, x), \quad (S(x, y))^{\dagger} = S(y, x). \qquad (4.52) $$

It follows from the orthonormality of the $\chi_n$ and $\phi_n$ that

$$ \int d^2 z \sqrt{g(z)} S(x, z) S(z, y) = G(x, y). \qquad (4.53) $$

Note that this is also consistent with the differential equations obeyed by $S$ and $G$. Indeed, we have

$$ \begin{aligned} D_x^2 G(x, y) &= D_x \int d^2 z \sqrt{g(z)} (D_x S(x, z)) S(z, y) = D_x \int d^2 z \sqrt{g(z)} \frac{\delta^{(2)}(x-z)}{\sqrt{g(z)}} S(z, y) \\ &= D_x S(x, y) = \frac{\delta^{(2)}(x-y)}{\sqrt{g}} \mathbf{1}_{2 \times 2}. \end{aligned} \qquad (4.54) $$

What can be said about the matrix structure of these Green's functions? As before, the matrix structure of $D^2$ is $\mathbf{1}(\dots) + \gamma_*(\dots)$ (cf (2.43) and (2.48)), implying that $G$ must have the same structure:

$$ G(x, y) = G_0(x, y) \mathbf{1} + G_*(x, y) \gamma_*, \qquad (4.55) $$

which also follows from (4.51) and (4.30). Combining this with the reality of $G$ and the property (4.52) we see that $G_0(x, y)$ is real and symmetric and $G_*(x, y)$ is imaginary and antisymmetric. The matrix structure of $S$ is somewhat less trivial, but follows from $S(x, y) = (i\nabla_x + m\gamma_*)G(x, y)$ along the same lines as discussed above for the zeta-functions. We find

$$ \mathrm{tr} S(x, y) = m \mathrm{tr} \gamma_* G(x, y), \quad \mathrm{tr} \gamma_* S(x, y) = m \mathrm{tr} G(x, y), \qquad (4.56) $$

which translates the corresponding relation (4.32) between the traces of $\zeta_+$ and $\zeta_-$. For vanishing mass, one must exclude the zero-mode contributions from $S$ and from $G$ (we then call them $\tilde{S}$ and $\tilde{G}$), but one still has the corresponding relation $\tilde{S} = i\nabla\tilde{G}$ and it then follows that, for vanishing mass,

$$ \mathrm{tr} \tilde{S}(x, y) = \mathrm{tr} \gamma_* \tilde{S}(x, y) = 0 \quad \text{for } m = 0. \qquad (4.57) $$

## 4.2 Variation of $\zeta(s)$

Now we have the tools and definitions to express the variation of the zeta-function $\zeta(s)$ that appeared in the variation of the effective gravitational action. In the previous section we did the perturbation theory of the eigenvalues and had obtained the formula (3.34). It can now be nicely rewritten in terms of the local zeta functions $\zeta_{\pm}(s, x, x)$ as (recall that $\int f(x)$ is meant to be $\int d^2x \sqrt{g(x)} f(x)$)

$$ \delta\zeta(s) = 2s \int \delta\sigma \left( \mathrm{tr} \zeta_+(s, x, x) + im \mathrm{tr} \gamma_* \zeta_-(s + \frac{1}{2}, x, x) \right). \qquad (4.58) $$
We may use (4.6) to trade $\zeta_-(s + \frac{1}{2}, x, x)$ for $im\gamma_*\zeta_+(s + 1, x, x)$ so that

$$ \delta\zeta(s) = 2s \int \delta\sigma \left( \text{tr} \zeta_+(s, x, x) - m^2 \text{tr} \zeta_+(s+1, x, x) \right). \qquad (4.59) $$

As we have discussed above, if zero-modes of $i\nabla$ are present, we have $\lambda_0 = m$ and $\delta\lambda_0 = 0$ so that there is no contribution from these zero-modes to the above formula. We can then equivalently rewrite it in terms of the $\tilde{\zeta}_+(s, x, x)$ which are the local zeta functions with the zero-modes of $i\nabla$ excluded from the sum:

$$ \delta\zeta(s) = 2s \int \delta\sigma \left( \text{tr} \tilde{\zeta}_+(s, x, x) - m^2 \text{tr} \tilde{\zeta}_+(s+1, x, x) \right). \qquad (4.60) $$

Note that the zero-modes alone cannot contribute to the singularities and thus $\tilde{\zeta}_+(s, x, x)$ has the same pole and residue at $s = 1$ as $\zeta_+(s, x, x)$, only the regular parts differ by the zero-mode contribution. For the derivative of (4.59) we obviously get

$$ \begin{aligned} \delta\zeta'(s) = 2 \int \delta\sigma \left( \text{tr} \zeta_+(s, x, x) - m^2 \text{tr} \zeta_+(s+1, x, x) \right) \\ +2s \int \delta\sigma \left( \text{tr} \zeta'_+(s, x, x) - m^2 \text{tr} \zeta'_+(s+1, x, x) \right). \end{aligned} \qquad (4.61) $$

We want to evaluate both $\delta\zeta$ and $\delta\zeta'$ at $s = 0$. Now $\zeta_+(0, x, x)$ is regular (cf. e.g. (4.40), and$^{23}$ so is $\zeta'_+(0, x, x)$. It follows that

$$ \begin{aligned} \delta\zeta(0) &= -2m^2 \int \delta\sigma \lim_{s\to 0} s \text{tr} \zeta_+(s+1, x, x), \\ \delta\zeta'(0) &= 2 \int \delta\sigma \text{tr} \zeta_+(0, x, x) - 2m^2 \int \delta\sigma \lim_{s\to 0} \text{tr} \left( \zeta_+(s+1, x, x) + s \zeta'_+(s+1, x, x) \right). \end{aligned} \qquad (4.62) $$

Now $\zeta_+(\tilde{s}, x, x)$ has a pole at $\tilde{s} = 1$: $\zeta_+(\tilde{s}, x, x) \sim (\frac{1}{4\pi}\frac{1}{\tilde{s}-1} + C_+) \mathbf{1}_{2\times2} + \zeta_+^{\text{reg}}(\tilde{s}, x, x)$, where the (finite) value of the constant $C_+$ depends on the exact definition of $\zeta_+^{\text{reg}}$, cf. eqs (4.92) and (4.93) below. It follows that $\zeta_+(s+1, x, x)$ has a pole at $s = 0$ and then

$$ \lim_{s \to 0} s \zeta_+(s+1, x, x) = \lim_{s \to 0} s \tilde{\zeta}_+(s+1, x, x) = \frac{1}{4\pi} \mathbf{1}_{2\times2}. \qquad (4.63) $$

Note that we get the same expression whether we use $\zeta_+$ or $\tilde{\zeta}_+$. In any case

$$ \delta\zeta(0) = -2m^2 \int \delta\sigma \text{tr} \frac{1}{4\pi} \mathbf{1}_{2\times2} = -\frac{m^2}{\pi} \int \delta\sigma = -\frac{m^2}{2\pi} \delta A, \qquad (4.64) $$

where we used that

$$ \int \delta\sigma \equiv \int d^2x \sqrt{g} \delta\sigma = \int d^2x \sqrt{g} e^{2\sigma} \delta\sigma = \frac{1}{2} \delta \int d^2x \sqrt{g} e^{2\sigma} = \frac{1}{2} \delta A. \qquad (4.65) $$

<footnote>$^{23}$If a meromorphic function is regular at a given point, then its derivative necessarily is also regular at this point.</footnote>
Similarly, for the terms appearing in $\delta\zeta'(0)$ we have as $s \to 0$

$$ 
\begin{aligned}
& \lim_{s \to 0} (\zeta_+(s+1, x, x) + s \zeta'_+(s+1, x, x)) \\
&= \lim_{s \to 0} \left[ \left( \frac{1}{4\pi s} + C_+ \right) \mathbf{1}_{2 \times 2} + \zeta_+^{\text{reg}}(s+1, x, x) + s \frac{(-1)}{4\pi s^2} \mathbf{1}_{2 \times 2} + s (\zeta_+^{\text{reg}})'(s+1, x, x) \right] \\
&= \zeta_+^{\text{reg}}(1, x, x) + C_+ \mathbf{1}_{2 \times 2}.
\end{aligned} 
\quad (4.66) $$

(We could have argued similarly about $\zeta_-(s+\frac{1}{2}, x, x) + s \zeta'_-(s+\frac{1}{2}, x, x)$ as $s \to 0$.) Thus

$$ 
\begin{aligned}
\delta\zeta'(0) &= 2 \int \delta\sigma \operatorname{tr} \zeta_+(0, x, x) - 2m^2 \int \delta\sigma \operatorname{tr} \zeta_+^{\text{reg}}(1, x, x) - 2m^2 C_+ \delta A, \\
&= 2 \int \delta\sigma \operatorname{tr} \tilde{\zeta}_+(0, x, x) - 2m^2 \int \delta\sigma \operatorname{tr} \tilde{\zeta}_+^{\text{reg}}(1, x, x) - 2m^2 \tilde{C}_+ \delta A,
\end{aligned} 
\quad (4.67) $$

In the second line we have written $\tilde{C}_+$, but our definitions of $\zeta_+^{\text{reg}}$ and $\tilde{\zeta}_+^{\text{reg}}$ below will be such that $C_+ = \tilde{C}_+ = \frac{\gamma - \log \mu^2}{4\pi}$. Thus we can express $\delta\zeta'(0)$ either in terms of $\zeta_+(0, x, x)$ and $\zeta_+^{\text{reg}}(1, x, x)$ or in terms of $\tilde{\zeta}_+(0, x, x)$ and $\tilde{\zeta}_+^{\text{reg}}(1, x, x)$.

### 4.3 Small-t asymptotics of the heat kernels and singularity structure of the local zeta-functions for the flat torus

We have seen that the singularities (poles) of the local zeta-functions are determined by the small-t asymptotics of the heat kernels. We have also seen that we could relate $\zeta_+$ and $\zeta_-$ and, in particular, at coinciding points we could deduce from this relation the small-t expansion of $K_-(t, x, x)$, see (4.45), from the one of $K_+(t, x, x)$, which is established in appendix A. However, we find it useful (and pedagogical) to try to explicitly compute these quantities for the example of the flat torus and see that we have indeed a perfect agreement with the results from the general statements. In this subsection we will thus establish the results for the flat “square” torus with further detailed computations for $K_-$ presented in appendix B.

From the explicit form of the normalised eigenfunctions $\psi_{\vec{n}}$ of $D$ on the flat square torus (with periods $2\pi$ in both directions) given in (2.61) we find

$$ 
\begin{aligned}
\mathcal{K}(t, x, y) &= \sum_{n_1, n_2} e^{-\lambda_{\vec{n}}^2 t} \psi_{\vec{n}}(x) \psi_{\vec{n}}^{\dagger}(y) \\
&= \frac{1}{8\pi^2} \sum_{n_1, n_2} e^{-(n_1^2+n_2^2+m^2)t} e^{in_1 z^1 + in_2 z^2} \frac{1}{\lambda_{\vec{n}}} \begin{pmatrix} \lambda_{\vec{n}} - n_2 & -n_1 - im \\ -n_1 + im & \lambda_{\vec{n}} + n_2 \end{pmatrix},
\end{aligned} 
\quad (4.68) $$

where here, and throughout this subsection $z^1 = x^1 - y^1$, $z^2 = x^2 - y^2$. Note that the terms odd under $n_1 \to -n_1$ or $n_2 \to -n_2$ drop out of the sum, so that (2 times) the real part is

$$ K_+(t, x, y) = \mathcal{K}(t, x, y) + \mathcal{K}^*(t, x, y) = \frac{1}{4\pi^2} \sum_{n_1, n_2} e^{-(n_1^2+n_2^2+m^2)t} e^{in_1 z^1 + in_2 z^2} \mathbf{1}_{2 \times 2}, \quad (4.69) $$
while (-2 times) the imaginary part of $\mathcal{K}$ is

$$ 
\begin{aligned}
K_-(t, x, y) &= i(\mathcal{K}(t, x, y) - \mathcal{K}^*(t, x, y)) \\
&= -\frac{1}{4\pi^2} \sum_{n_1, n_2} e^{-(n_1^2+n_2^2+m^2)t} \left( \cos(n_1 z^1 + n_2 z^2) \frac{1}{\lambda_{\vec{n}}} \begin{pmatrix} 0 & -m \\ m & 0 \end{pmatrix} \right. \\
&\qquad \left. + \sin(n_1 z^1 + n_2 z^2) \frac{1}{\lambda_{\vec{n}}} \begin{pmatrix} -n_2 & -n_1 \\ -n_1 & n_2 \end{pmatrix} \right) \\
&= \frac{i}{4\pi^2} \sum_{n_1, n_2} e^{-(n_1^2+n_2^2+m^2)t} e^{in_1 z^1 + in_2 z^2} \frac{1}{\lambda_{\vec{n}}} (-n_1\sigma_x - n_2\sigma_z + m\gamma_*) .
\end{aligned} 
\quad (4.70) $$

Note that for $t = 0$ one correctly finds that $K_+(0, x, y) = \delta^{(2)}(x - y)\mathbf{1}_{2 \times 2}$, while nothing like this is true for $K_-$. Also note that the a priori possible $\gamma_*$-terms are absent in (4.69) and, correspondingly, there are no terms $\sim \mathbf{1}_{2 \times 2}$ in (4.70).

Obviously, at coinciding points $x = y$, i.e. $z = 0$, these heat kernels simplify:

$$ 
\begin{aligned}
K_+(t, x, x) &= \frac{1}{4\pi^2} \sum_{n_1, n_2} e^{-(n_1^2+n_2^2+m^2)t} \mathbf{1}_{2 \times 2}, \\
K_-(t, x, x) &= \frac{im}{4\pi^2} \sum_{n_1, n_2} \frac{1}{\lambda_{\vec{n}}} e^{-(n_1^2+n_2^2+m^2)t} \gamma_* .
\end{aligned} 
\quad (4.71) $$

We see that $K_+(t, x, x) \sim \mathbf{1}_{2 \times 2}$ and $K_-(t, x, x) \sim \gamma_*$, in agreement with the general statements above, cf. (4.24). Note that for $K_+(t, x, y)$ the double sum factorises into a sum over $n_1$ and another one over $n_2$, while for $K_-(t, x, y)$ or $K_-(t, x, x)$ there is no such factorisation since $\lambda_{\vec{n}} = \sqrt{n_1^2 + n_2^2 + m^2}$. In fact, $K_+(t, x, y)$ can be expressed in terms of the theta function $\theta_3$, (cf [33]) as

$$ 
\begin{aligned}
K_+(t, x, y) &= \frac{1}{4\pi^2} e^{-m^2 t} \theta_3\left(\frac{z^1}{2\pi} \middle| i \frac{t}{\pi}\right) \theta_3\left(\frac{z^2}{2\pi} \middle| i \frac{t}{\pi}\right) \mathbf{1}_{2 \times 2}, \\
\theta_3(\nu|\tau) &= \sum_{n=-\infty}^{\infty} e^{i\pi\tau n^2 + 2\pi i v n}, \quad \text{Im } \tau > 0.
\end{aligned} 
\quad (4.72) $$

On the other hand, $K_-(t, x, y)$ can be expressed in terms of the auxiliary sum $L$ already defined in (4.19), or equivalently another related sum $\mathcal{L}$, as follows:

$$ K_-(t, x, y) = D_x L(t, x, y) \equiv (i\sigma_x\partial_1 + i\sigma_z\partial_2 + m\gamma_*)L(t, x, y), \quad (4.73) $$

where

$$ L(t, x, y) = \frac{i}{4\pi^2} \mathcal{L}(t, z) \mathbf{1}_{2 \times 2}, \quad \mathcal{L}(t, z) = \sum_{n_1, n_2} \frac{e^{-\lambda_{\vec{n}}^2 t}}{\lambda_{\vec{n}}} e^{i \vec{n} \cdot \vec{z}} \equiv \sum_{n_1, n_2} \frac{e^{-(n_1^2+n_2^2+m^2)t} e^{in_1 z^1 + in_2 z^2}}{\sqrt{n_1^2 + n_2^2 + m^2}}. \quad (4.74) $$

Note that $K_-$ at coinciding points is directly given in terms of this $\mathcal{L}$ as (cf (4.71))

$$ K_-(t, x, x) = \frac{i}{4\pi^2} m\gamma_* \mathcal{L}(t, 0). \quad (4.75) $$
For the present case of the torus, we find even for $x \neq y$ that $L(t, x, y) \sim \mathbf{1}_{2 \times 2}$, a fact that is not necessarily true in general.

The explicit expression of $K_+$ in terms of the theta function $\theta_3$ makes it easy to study its properties. The periodicity property $\theta_3(\nu + 1|\tau) = \theta_3(\nu|\tau)$ reflects the periodicity of the eigenfunctions on the torus. $\theta_3$ has a well-known modular transformation property [33] which can be written as

$$ \theta_3(\frac{z}{2\pi}|i\frac{t}{\pi}) = \sqrt{\frac{\pi}{t}}e^{-z^2/(4t)} \theta_3(-i\frac{z}{2t}, i\frac{\pi}{t}), \quad (4.76) $$

giving immediately the small-$t$ asymptotics as follows:

$$ \theta_3(\frac{z}{2\pi}|i\frac{t}{\pi}) = \sqrt{\frac{\pi}{t}} e^{-z^2/(4t)} \sum_n e^{-\pi^2 n^2/t + \pi z n/t} = \sqrt{\frac{\pi}{t}} \sum_n e^{-\frac{1}{4t}(z-2\pi n)^2}. \quad (4.77) $$

This clearly exhibits again the $2\pi$-periodicity in $z$. Let us then assume that $z \in (-\pi, \pi]$ and consider $t \to 0$. For $z = \pi$, both $n = 0$ and $n = 1$ contribute equally to the sum so that one gets $\sqrt{\frac{\pi}{t}} 2 e^{-\pi^2/(4t)}(1 + \mathcal{O}(e^{-2\pi^2/t}))$. For all other values of $z \in (-\pi, \pi)$ there is a single dominant term that contributes $\sqrt{\frac{\pi}{t}} e^{-z^2/(4t)}$ and all other terms are exponentially smaller by some factor $e^{-a/t}$ for some $a > 0$. Hence the small-$t$ asymptotics is

$$ \theta_3(\frac{z}{2\pi}|i\frac{t}{\pi}) \sim_{t \to 0} \sqrt{\frac{\pi}{t}} e^{-z^2/(4t)} \left(1 + \mathcal{O}(e^{-a/t})\right), \quad a > 0, \quad \text{for } |z| < \pi. \quad (4.78) $$

One immediately deduces the small-$t$ behaviour of $K_+(t, x, y)$ (for $z^i \neq \pm\pi$) as

$$ K_+(t, x, y) = \frac{1}{4\pi t} \exp\left(-\frac{(z^1)^2 + (z^2)^2}{4t}\right) e^{-m^2 t} \mathbf{1}_{2 \times 2} \left(1 + \mathcal{O}(e^{-a/t})\right). \quad (4.79) $$

The leading piece coincides, of course, with the well-known answer on $\mathbf{R}^2$. For $x = y$ one simply has

$$ K_+(t, x, x) = \frac{1}{4\pi t} e^{-m^2 t} \mathbf{1}_{2 \times 2} \left(1 + \mathcal{O}(e^{-a/t})\right). \quad (4.80) $$

On the other hand, the small-$t$ asymptotics of $K_-(t, x, y)$, or equivalently the small-$t$ asymptotics of $L(t, x, y)$ or $\mathcal{L}(t, z)$ is much more difficult to obtain directly. We have devoted some effort to try to estimate their asymptotic behaviour. These results are presented in appendix B. At coinciding points we have been able to estimate rather easily (cf (B.4)) that

$$ \mathcal{L}(t, 0) \approx \sqrt{\frac{\pi^3}{t}} + \frac{1}{m} + \mathcal{O}(t^0, m^0). \quad (4.81) $$

This implies in turn that $K_-(t, x, x)$ is given by (cf (4.75))

$$ K_-(t, x, x) \sim_{t \to 0} \left( \frac{m}{4\sqrt{\pi t}} + \frac{1}{4\pi^2} \right) i\gamma_* + \dots \quad (4.82) $$

where the unwritten terms + ... vanish as $t \to 0$, a result already cited above. To estimate $\mathcal{L}(t, z)$ for $z \neq 0$ turned out to be much more difficult. The reader can find more details in the appendix B. But we will not need them here to proceed further.
As discussed above, the small-$t$ behaviours of $K_±(t, x, x)$ translate into possible poles of the corresponding local zeta-functions $ζ_±(s, x, x)$. Using the above torus results for coinciding points one immediately finds that

$$ \begin{aligned} \zeta_+(s, x, x) &\sim_{s \to 1} \frac{1}{4\pi(s-1)} + \text{finite,} & \zeta_+(0, x, x) &= \text{finite} \times \mathbf{1}, \\ \zeta_-(s, x, x) &\sim_{s \to 1/2} \frac{m i \gamma_*}{4\pi(s-\frac{1}{2})} + \text{finite,} & \zeta_-(0, x, x) &= \text{finite} \times i \gamma_*, \end{aligned} \quad (4.83) $$

again in agreement with our general results.

## 4.4 General statements about the singularity structure for $x \neq y$

In subsection 4.1 we already discussed many general properties of the local heat kernels and local zeta functions, in particular at coinciding point. Now, we will make some further general statements for $x \neq y$. These statements will be based on the small-$t$ behaviour of $K_+(t, x, y)$ as follows from the differential (heat) equation, and worked out in appendix A. For $K_-(t, x, y)$ we can deduce the corresponding statements by using the fundamental relation (4.6) between the corresponding $ζ_+(s, x, y)$ and $ζ_-(s, x, y)$.

As already discussed above, since $D^2$ only contains the matrix structures $\mathbf{1}_{2 \times 2}$ and $γ_*$, the same is true for $G(x, y)$ and $K_+(t, x, y)$. However, the leading small-$t$ singularity of $K_+(t, x, y)$ is only $\sim \mathbf{1}_{2 \times 2}$ and is universal, cf (A.21) of the appendix:<fnref n="24" />

$$ K_+(t, x, y) \sim_{t \to 0} \frac{1}{4\pi t} e^{-\ell^2(x,y)/(4t)} \mathbf{1}_{2 \times 2}, \quad (4.84) $$

where, $ℓ(x, y)$ is the geodesic distance between $x$ and $y$. This generalizes the corresponding formula (4.79) for the torus. We can then deduce the singularity structure of $ζ_+(s, x, y)$ solely from this universal leading behaviour of $K_+$. This universality is due to the (leading) 2-derivative part of $D^2$ being always $-\mathbf{1}_{2 \times 2} g^{\mu\nu} \partial_\mu \partial_\nu$ which is the same as for the scalar Laplacian $-\mathbf{1}_{2 \times 2} \Delta_{\text{scalar}}$. Indeed, as discussed above, we may obtain $K_+(t, x, y)$ solely from the differential equation (4.16).

For $K_-(t, x, y) = D_x L(t, x, y)$ we expect that the leading small-$t$ singularity of $L$ is again generic and hence given by the obvious generalization of the torus result, cf (B.14). However, instead of relying on this “expected” formula, we will instead use the proven relation (4.6) between $ζ_+$ and $ζ_-$, together with (4.84), to deduce the singularity structure of $ζ_-(s, x, y)$.

Let us then establish the singularity structures of both, $ζ_±(s, x, y)$. To begin with, (4.84) defines what we call the singular part of the heat-kernel:

$$ K_+^{\text{sing}}(t, x, y) = \frac{1}{4\pi t} e^{-\ell^2(x,y)/(4t)} \mathbf{1}_{2 \times 2}. \quad (4.85) $$

<footnote n="24">²⁴Actually, in $n=2$ dimensions, equation (A.21) gives $K(t, x, y) = \frac{1}{4\pi t} \exp(-\frac{\ell^2(x,y)}{4t}) (F_0(x, y) + \mathcal{O}(t))$. Now, $F_0(x, y) = 1 + \mathcal{O}(\ell^2)$ and the exponential forces $\ell^2$ to be of order $t$ so that $\mathcal{O}(\ell^2) \simeq \mathcal{O}(t)$ and we indeed have $K(t, x, y) = \frac{1}{4\pi t} \exp(-\frac{\ell^2(x,y)}{4t}) (1 + \mathcal{O}(t))$.</footnote>
From this we define the singular part of the zeta-function as

$$ \zeta_{+}^{\text{sing}}(s, x, y) = \frac{1}{\Gamma(s)} \int_{0}^{1/\mu^2} dt \, t^{s-1} K_{\pm}^{\text{sing}}(t, x, y). \qquad (4.86) $$

Indeed, any singularity of the zeta-function must come from the small-$t$ part of the integral, and as we have seen before, only the $\frac{1}{t}$-part of the heat kernel $K_+$ can lead to a pole of the $\zeta_+(s, \cdot, \cdot)$, while the “would-be” poles due to the sub-leading terms in $t$ are cancelled by the zeros of $\frac{1}{\Gamma(s)}$. For general $x, y$ we explicitly have (setting $\frac{1}{t} = \mu^2 u$)

$$ \zeta_{+}^{\text{sing}}(s, x, y) = \frac{1}{\Gamma(s)} \int_{0}^{1/\mu^2} dt \, t^{s-1} \frac{e^{-\ell^2(x,y)/(4t)}}{4\pi t} \mathbf{1}_{2 \times 2} = \frac{(\mu^2)^{1-s}}{4\pi\Gamma(s)} E_s\left(\frac{\mu^2 \ell^2(x,y)}{4}\right) \mathbf{1}_{2 \times 2}, \qquad (4.87) $$

where $E_s$ is the exponential integral function defined as

$$ E_s(z) = \int_1^\infty du \, u^{-s} e^{-zu}. \qquad (4.88) $$

Its asymptotic expansions are well known and, in particular, if we first set $s = 1$ and then let $x \to y$, i.e. $\ell(x, y) \to 0$, we have

$$ E_1\left(\frac{\mu^2 \ell^2}{4}\right) \sim_{\ell^2 \to 0} -\gamma - \log \frac{\mu^2 \ell^2}{4} + \mathcal{O}(\mu^2 \ell^2), \qquad (4.89) $$

(where $\gamma \simeq 0.57$ is Euler's constant, not to be confused with the matrix $\gamma_*$) so that

$$ \zeta_{+}^{\text{sing}}(1, x, y) \sim_{\ell \to 0} \frac{1}{4\pi} \left( -\log \frac{\mu^2 \ell^2(x, y)}{4} - \gamma + \mathcal{O}(\mu^2 \ell^2) \right) \mathbf{1}_{2 \times 2}, \qquad (4.90) $$

Of course, $\zeta_+(1, x, y)$ is the Green's function $G(x, y)$ and the term $-\frac{1}{4\pi} \log \frac{\mu^2 \ell^2}{4} \mathbf{1}_{2 \times 2}$ is just the (leading) short-distance singularity of $G(x, y)$ which is identical to the well-known short-distance singularity of the Green's function of the scalar Laplace operator in two dimensions. On the other hand, if we first set $x = y$ so that $\ell(x, y) = 0$ we have, assuming $\Re s > 1$, $E_s(0) = \int_0^1 dv \, v^{s-2} = \frac{1}{s-1}$:

$$ E_s(0) = \frac{1}{s-1}. \qquad (4.91) $$

Expanding $\Gamma(s)$ and $(\mu^2)^{1-s}$ we have

$$ \zeta_{+}^{\text{sing}}(s, x, x) = \frac{1}{4\pi} \left( \frac{1}{s-1} - \log \mu^2 + \gamma + \mathcal{O}(s-1) \right) \mathbf{1}_{2 \times 2}. \qquad (4.92) $$

It follows that if we define

$$ \zeta_{+}^{\text{reg}}(s, x, y) = \zeta_{+}(s, x, y) - \zeta_{+}^{\text{sing}}(s, x, y) \qquad (4.93) $$

we have subtracted all the potential singularities of this zeta-function, namely the pole at $s = 1$ that is present for $x = y$ and the short-distance singularity of the Green's function which is present for $s = 1$. In particular, we see that the following limit exists:

$$ \begin{aligned} \lim_{s \to 0} (\zeta_+(s+1, x, x) + s \zeta'_+(s+1, x, x)) &= \zeta_+^{\text{reg}}(1, x, x) + \frac{1}{4\pi}(\gamma - \log \mu^2) \mathbf{1}_{2 \times 2} \\ &\equiv \zeta_+^{\text{reg}}(1, x, x) + C_+ \mathbf{1}_{2 \times 2}. \end{aligned} \qquad (4.94) $$
The existence of this limit was already anticipated above in (4.66), but now we have also determined the value of finite constant $C_+$.

It is straightforward to repeat this reasoning for the $\tilde{\zeta}_{\pm}(s, x, y)$ where the zero-modes are excluded. Since the finitely many zero-modes cannot contribute to the singular parts the latter are unchanged, $\tilde{\zeta}_{\pm}^{\text{sing}}(s, x, y) = \zeta_{\pm}^{\text{sing}}(s, x, y)$. Then whole discussion can be repeated with the result

$$ \lim_{s \to 0} (\tilde{\zeta}_+(s+1, x, x) + s \tilde{\zeta}_+(s+1, x, x)) = \tilde{\zeta}_+^{\text{reg}}(1, x, x) + \frac{1}{4\pi}(\gamma - \log \mu^2) \mathbf{1}_{2 \times 2} \\ \equiv \tilde{\zeta}_+^{\text{reg}}(1, x, x) + C_+ \mathbf{1}_{2 \times 2}, \qquad (4.95) $$

where, in particular, the constant $C_+$ is the same as before.

We can also translate all these statements to $\zeta_-(s, x, y)$ by using (4.6):

$$ \zeta_{-}^{\text{sing}}(s, x, y) = i D_x \zeta_{+}^{\text{sing}} \left( s + \frac{1}{2}, x, y \right) = i \frac{(\mu^2)^{\frac{1}{2}-s}}{4\pi\Gamma(s+\frac{1}{2})} (i \not\nabla_x + m\gamma_*) E_{s+\frac{1}{2}} \left( \frac{\mu^2 \ell^2(x, y)}{4} \right). \qquad (4.96) $$

In particular, for $s = \frac{1}{2}$ we have again $E_1(\frac{\mu^2 \ell^2(x, y)}{4})$ with its small-$\ell$ behaviour given in (4.89) and, for $x = y$, obviously (4.91) translates into $E_{s+\frac{1}{2}}(0) = \frac{1}{s-\frac{1}{2}}$. However, to evaluate $\zeta_{-}^{\text{sing}}(s, x, y)$ at $x = y$ we must first act with $\not\nabla_x$ and then set $x = y$: $\not\nabla_x E_{s+\frac{1}{2}}(\frac{\mu^2 \ell^2(x, y)}{4})|_{x=y} = \frac{\mu^2}{4}(\not\nabla_x \ell^2(x, y))|_{x=y} E'_{s+\frac{1}{2}}(0)$. What does it mean to have a spinorial covariant derivative $\not\nabla = \gamma^\mu \nabla_\mu^{\text{sp}} = \gamma^\mu(\partial_\mu - \frac{i}{4}\omega_\mu\gamma_*)$ acting on $\ell^2(x, y)$? One has $\nabla_\mu^{\text{sp}}\ell^2(x, y) = \partial_\mu\ell^2(x, y) - \frac{i}{4}\omega_\mu\gamma_*\ell^2(x, y)$. This can be evaluated more explicitly, for example, in Riemann normal coordinates around $y$, see the appendix A.1. One then sees that $\partial_\mu\ell^2(x, y) = 2(x-y)^\mu$ and $\omega_\mu = -\frac{\mathcal{R}}{2}\epsilon_{\mu\nu}(x-y)^\nu$ ($\epsilon_{12} = -\epsilon_{21} = 1$). In particular, we see that for $x \to y$ this quantity vanishes. The “invariant” statement we can make is that

$$ \not\nabla_x \ell^2(x, y)|_{x=y} = 0. \qquad (4.97) $$

It follows that

$$ \zeta_{-}^{\text{sing}}(s, x, x) = i m \gamma_{*} \frac{(\mu^2)^{\frac{1}{2}-s}}{4\pi\Gamma(s+\frac{1}{2})} E_{s+\frac{1}{2}}(0). \qquad (4.98) $$

Again, just as for $\zeta_-(s, x, x)$, only the $\gamma_*$-matrix part remains at coinciding points. In particular, for $s \to \frac{1}{2}$ one finds, much as above,

$$ \zeta_{-}^{\text{sing}}(s, x, x) = \frac{i m}{4\pi} \gamma_{*} \left( \frac{1}{s - \frac{1}{2}} - \log \mu^2 + \gamma + \mathcal{O} \left( (s - \frac{1}{2}) \right) \right). \qquad (4.99) $$

On the other hand, if we first set $s = \frac{1}{2}$ and then let $x \to y$, using (4.89), we have

$$ \begin{aligned} \zeta_{-}^{\text{sing}} \left( \frac{1}{2}, x, y \right) &= \frac{i}{4\pi} (i \not\nabla_x + m \gamma_*) E_1 \left( \frac{\mu^2 \ell^2(x, y)}{4} \right) \\ &\sim_{x \to y} \frac{i}{4\pi} (i \not\nabla_x + m \gamma_*) \left( -\gamma - \log \frac{\mu^2 \ell^2}{4} + \mathcal{O}(\mu^2 \ell^2) \right) \\ &\sim_{x \to y} -\frac{i}{4\pi} \left( \frac{i \not\nabla \ell^2}{\ell^2} + m \gamma_* \left( \log \frac{\mu^2 \ell^2}{4} + \gamma \right) \right). \end{aligned} \qquad (4.100) $$