Renormalization of Supersymmetric Lifshitz Sigma Models Ziqi Yan Nordita KTH Royal Institute of Technology and Stockholm University

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Renormalization of Supersymmetric Lifshitz Sigma Models

Ziqi Yan

Nordita, KTH Royal Institute of Technology and Stockholm University

Hannes Alfv´ens v¨ag 12, SE-106 91 Stockholm, Sweden

E-mail: ziqi.yan@su.se

Abstract: We study the renormalization of an N= 1 supersymmetric Lifshitz sigma model

in three dimensions. The sigma model exhibits worldvolume anisotropy in space and time

around the high-energy z= 2 Lifshitz point, such that the worldvolume is endowed with

a foliation structure along a preferred time direction. In curved backgrounds, the target-

space geometry is equipped with two distinct metrics, and the interacting sigma model is

power-counting renormalizable. At low energies, the theory naturally ﬂows toward the rel-

ativistic sigma model where Lorentz symmetry emerges. In the superspace formalism, we

develop a heat kernel method that is covariantized with respect to the bimetric target-space

geometry, using which we evaluate the one-loop beta-functions of the Lifshitz sigma model.

This study forms an essential step toward a thorough understanding of the quantum critical

supermembrane as a candidate high-energy completion of the relativistic supermembrane.

arXiv:2210.04950v2 [hep-th] 27 Feb 2023

Contents

1. Introduction 2

2. O(N)Nonlinear Sigma Models at a z= 2 Lifshitz point 4

2.1. Relativistic O(N) Nonlinear Sigma Model 5

2.2. Scalar Field Theories at a Lifshitz Point 5

2.3. O(N) Nonlinear Sigma Model at a z= 2 Lifshitz Point 7

2.4. Lifshitz Nonlinear Sigma Model with N= 1 Supersymmetry 9

3. General Sigma Models at a z= 2 Lifshitz Point 12

3.1. The Classical Theory 12

3.1.1. Supersymmetric sigma models and bimetricity 13

3.1.2. Covariant background ﬁeld method 14

3.1.3. The classical propagator 16

3.1.4. Quaternions and bimetric symmetry 17

3.2. One-Loop Eﬀective Action in Superspace 19

3.2.1. The heat kernel equation 20

3.2.2. Fourier transform in superspace 20

3.2.3. Seeley-Gilkey coeﬃcients 21

3.2.4. Solving the recursion relations 23

3.2.5. Changing the reference metric 25

3.3. Renormalization Group Flows 26

4. Quantum Critical Membranes 27

5. Conclusions 29

– 1 –

1. Introduction

The quantization of supermembranes in eleven-dimensional spacetime is key to a better un-

derstanding of M-theory, which provides a uniﬁcation of diﬀerent superstring theories in a

nonperturbative manner. Supermembranes are described by a three-dimensional sigma model

[1], whose quantization is fundamentally distinct from quantizing strings. One of the major

diﬃculties is that three-dimensional relativistic sigma models are not renormalizable, which

invalidates any perturbative treatment. Nevertheless, it is possible to probe some features

of the supermembrane in the light-cone quantization, at least when a matrix regularization

is employed on the spatial manifold of the worldvolume [2,3]. This regularization restricts

the inﬁnite-dimensional basis on the continuum manifold to a ﬁnite-dimensional Lie group,

and the supermembrane is described as a subtle limit of supersymmetric matrix quantum

mechanics [4], closely related to D0-branes in the inﬁnite momentum frame and the Matrix

theory description of M-theory [5]. 1However, a complete understanding of the continuous

spectrum of Matrix theory is still lacking [12,13], which requires a multiparticle interpreta-

tion and thus a second quantization of the membrane [5]. This light-cone treatment is also

limited to special backgrounds, due to the absence of spacetime covariance. Finally, when the

sigma model is coupled to a dynamical worldvolume described by three-dimensional quantum

supergravity, there still exists neither clear quantization technique nor classical expansion for

the membrane [14]. See, e.g., [15,16] for excellent reviews.

According to the Matrix theory description, a supermembrane is unstable and tends to

be dissolved into dynamical bits [12]. 2This suggests that membranes might not be the funda-

mental objects. However, it is still highly valuable to probe M-theory from the worldvolume

perspectives, especially in regard of the lacking of any covariant formalism of supermem-

branes. Since the three-dimensional sigma model is nonrenormalizable, which suggests that

new physics may arise above a particular cutoﬀ. It is therefore natural to ask whether there

exists a renormalizable QFT at our disposal, which can be regarded as an ultra-violet (UV)

completion of the nonrenormalizable membrane theory. If so, there might exist a stable

notion of membranes that only become strongly coupled at low energies, where the Matrix

theory description becomes valid and the physics is captured by emergent multiparticle states.

Unfortunately, no such renormalizable worldvolume theory seems to be available within the

relativistic framework.

It is pioneered in [17] that the relativistic membrane might admit a UV completion

that exhibits nonrelativistic behaviors on the worldvolume: a renormalizable sigma model

describing membranes propagating in spacetime can be constructed if an anisotropy between

the worldvolume space and time is introduced. Furthermore, this sigma model is required

1Another way of obtaining Matrix theory is by considering the discrete light cone quantization (DLCQ)

of M-theory, which is usually deﬁned as a subtle limit of the compactiﬁcation over a spatial circle [6–9]. An

alternative treatment of the DLCQ of string/M-theory as nonrelativistic string/M-theory is recently revived

in e.g. [10,11].

2This instability of the membrane was resolved in the Matrix theory description by realizing that the Hilbert

space of the matrix quantum mechanics naturally contains multi-particle states of the D0-bits [5].

– 2 –

to satisfy a Lifshitz scaling with the dynamical critical exponent z= 2 , which measures the

anisotropy between the worldvolume time tand space x,

t→b t , x→b1/z x.(1.1)

Consequently, the worldvolume degrees of freedom satisfy a quadratic dispersion relation,

ω2∼ |k|4,(1.2)

where ωdenotes the frequency and kthe two-dimensional spatial momentum. The mem-

brane described by such a Lifshitz-type sigma model is referred to as the membrane at

quantum criticality [17]. On a curved worldvolume, this Lifshitz sigma model is coupled to

three-dimensional Hoˇrava gravity, which geometrizes the foliation-preserving diﬀeomorphisms

compatible with the worldvolume anisotropy. Since the worldvolume is foliated by leaves of

constant time, it is possible to consistently restrict the sum over general three-manifolds in

the membrane theory to foliated manifolds, where the spatial leaves are Riemann surfaces.

This may facilitate a perturbative expansion in membranes, akin to the perturbative expan-

sion with respect to the genera of Riemann surfaces in string theory [17]. At low energies, a

relevant deformation is turned on and modiﬁes the dispersion relation (1.2) to be

ω2∼ |k|4+c2|k|2,(1.3)

with a coupling cthat is dimensionful from the UV perspective around the z= 2 Lifshitz

point. Classically, this relevant deformation ostensibly generates a renormalization group

(RG) ﬂow toward the relativistic, nonperturbative membrane theory in the deep infrared

(IR), where low-energy Lorentz symmetry emerges. Whether and how this RG ﬂow toward

a Lorentzian ﬁxed point could be realized quantum mechanically poses a challenging ques-

tion, which requires detailed analysis of the worldvolume dynamics. If achieved, it is then

tempting to conjecture that the ultimately fundamental objects in M-theory might be such

nonrelativistic membranes at quantum criticality.

In this paper, we study a renormalizable three-dimensional Lifshitz-type sigma model that

exhibits a z= 2 Lifshitz scaling symmetry at the critical point. This sigma model constitutes

an essential ingredient for the construction of the membrane at quantum criticality. We will

focus on the case where the worldvolume is ﬂat. The full-ﬂedged membrane theory, however,

requires us to couple the matter contents of the sigma model to dynamical worldvolume

Hoˇrava gravity. On the other hand, in the absence of worldvolume gravitational dynamics,

the quantization of Lifshitz-type sigma models already imposes intriguing challenges and

requires new techniques to tackle with.

In the bosonic case, the most general three-dimensional sigma model at a z= 2 Lifshitz

point is coupled to higher-rank tensorial background ﬁelds, which makes the quantization

rather diﬃcult. In the simpler case where the target space is O(N) symmetric, the z= 2

nonlinear sigma model (NLSM) already exhibits intriguing RG properties [18–20], for which

– 3 –

we will give a quick review as a preparation for the quantization of Lifshitz sigma models

in arbitrary geometric background ﬁelds. The study of Lifshitz O(N) NLSM bears potential

applications to the condensed matter systems of quantum spin liquids [21–23] and quantum

spherical models [24–26].

In connection to the studies of supermembranes, we are ultimately interested in super-

symmetric sigma models, which bring signiﬁcant simpliﬁcations in comparison to the pure

bosonic case. The three-dimension N= 1 supersymmetric sigma model around a z= 2 Lif-

shitz point has been introduced in [27,28] (also see [29] for a review), which is power-counting

renormalizable. The relevant target-space geometry is Lorentzian and has two independent

metrics, but there is no extra higher-form tensorial structure as in the bosonic case. In

the deep IR, a relevant deformation controlled by the coupling constant associated with the

worldvolume speed of light dominates, which drives the theory toward the relativistic ﬁxed

point satisfying a z= 1 Lifshitz scaling. We will generalize the heat kernel method in [29,30]

to supersymmetric ﬁeld theories on a foliated spacetime manifold, and use it to study the

RG ﬂows of the supersymmetric Lifshitz sigma model. This will form an essential step to-

ward determining whether such a worldvolume theory is qualiﬁed to be a quantum membrane

theory that UV completes the relativistic supermembranes, which also requires coupling the

Lifshitz sigma model to dynamical worldvolume supergravity. At the z= 2 Lifshitz point,

the worldvolume supergravity is also power-counting renormalizable in three dimensions and

possesses a preferred time direction, which makes it possible to study its quantization as a

conventional quantum ﬁeld theory.

The paper is organized as follows. In Section 2, as a warm-up, we discuss the renormaliza-

tion of three-dimensional z= 2 Lifshitz O(N) NLSM. In Section 3, we study three-dimensional

N= 1 supersymmetric z= 2 Lifshitz sigma model, which is described by the action (3.4) and

is coupled to a bimetric target-space geometry. We derive the one-loop beta-functionals for

the bimetric ﬁelds in this Lifshitz sigma model in Eq. (3.84), which form the major results of

this paper. In Section 4, as a preliminary step toward the formulation of the supersymmetric

quantum critical membrane, we couple the bosonic part of the sigma model to dynamical

worldvolume Hoˇrava gravity. We conclude the paper in Section 5.

2. O(N)Nonlinear Sigma Models at a z= 2 Lifshitz point

In this section, we illustrate some of the basic ingredients in our later construction of general

three-dimensional Lifshitz sigma models by a simple example, where the target space of the

sigma model is O(N) symmetric. We parametrize the target space by a vector ﬁeld na,

a= 1,··· , N , which satisﬁes the constraint

n·n= 1 .(2.1)

We will ﬁrst review the relativistic O(N) NLSM in three dimensions, which is not renormaliz-

able. Then, we proceed to the construction of the three-dimensional O(N) NLSM at a z= 2

Lifshitz point, which ﬂows toward the relativistic NLSM in the deep IR.

– 4 –

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RenormalizationofSupersymmetricLifshitzSigmaModelsZiqiYanNordita,KTHRoyalInstituteofTechnologyandStockholmUniversityHannesAlfvensvag12,SE-10691Stockholm,SwedenE-mail:ziqi.yan@su.seAbstract:WestudytherenormalizationofanN=1supersymmetricLifshitzsigmamodelinthreedimensions.Thesigmamodelexhibitsworldv...

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Renormalization of Supersymmetric Lifshitz Sigma Models Ziqi Yan Nordita KTH Royal Institute of Technology and Stockholm University

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