Anomalies in Heterotic String and Decoupling Limit

Abstract

We review the recent developments on the decoupling limit zooming in on a BPS string state in heterotic string theory. In this limit, the target-space geometry lacks any ten-dimensional metric description and acquires various non-Lorentzian features. We present the supersymmetric worldsheet sigma model in the decoupling limit. This sigma model describes the heterotic version of non-relativistic string theory, whose second quantization is related to heterotic matrix string theory. We introduce bookkeeping to separate the Yang–Mills gauge and gravitational anomaly analyses. We review the Yang–Mills gauge anomaly analysis in the sigma models before and after the decoupling limit is performed and provide further details on how the quantum calculation commutes with the decoupling limit. We comment on how the analogous calculation can be performed for the gravitational anomaly. This contribution is based on a talk at the University of Science and Technology of China.

1. Introduction

Heterotic string theory [1] is a mechanically consistent quantum framework that unifies gauge and gravitational degrees of freedom. The consistency of this framework relies on the cancellation of anomalies. This cancellation arises from a delicate interplay between the worldsheet sigma model and target-space geometry through the Green–Schwarz mechanism [2]. In connection to matrix theory [3] and holography [4], it is natural to consider how these aspects of quantum consistency behave under decoupling limits that zoom in on Bogomol’nyi–Prasad–Sommerfield (BPS) states, which isolate particular physical sectors of the theory.

In this work, we focus on the particular decoupling limits that zoom in on BPS string states. Such limits lead to string theory in non-Lorentzian target-space geometries that are of the Newton–Cartan type with a stringy generalization, which is therefore referred to as non-relativistic string theory in the literature [5,6,7,8,9,10,11,12] (see [13] for a review). Non-relativistic string theory is unitary and ultraviolet complete, with a (string) Galilei-invariant string spectrum. The second quantization of type IIB non-relativistic string theory [14,15] corresponds to matrix string theory described by two-dimensional $N=8$ super Yang–Mills theory [16,17]. The heterotic analog of non-relativistic string theory was recently developed in [18,19]. As in the relativistic case, understanding anomaly cancellation in the context of the associated limit of heterotic string theory is essential for establishing the quantum consistency of the worldsheet theory.

In this contribution, we discuss the behavior of anomalies in heterotic string theory under such a decoupling limit. We focus on local analyses that exclude, for example, the investigation of modular invariance. A useful way to organize the various theories and their inter-relations is through the diagram shown in Figure 1. This diagram summarizes how classical and quantum (anomaly-free) theories are related—both before and after taking the decoupling limit into consideration.

A central question that we will address in this contribution is whether these two operations of taking the decoupling limit into consideration and performing the anomaly analysis commute. In other words, does one obtain the same anomaly-free non-relativistic theory regardless of the order in which the steps indicated in Figure 1 are performed? Based on [18], we demonstrate that, insofar as the Yang–Mills gauge anomaly is concerned, the answer is yes.

Concerning the gravitational anomalies, we formulate a method that allows for a clean separation between Yang–Mills gauge anomalies and gravitational anomalies in the heterotic sigma model. We will show that the gauge-anomaly cancellation mechanism survives the decoupling limit in a nontrivial way, leading to a consistent heterotic non-relativistic string theory. Furthermore, we will comment on the gravitational anomaly sector, where additional geometric constraints emerge as necessary conditions for consistency. All these results provide further evidence that non-relativistic limits of string theory define well-behaved quantum theories and illuminate the structure of anomalies in non-Lorentzian backgrounds.

This contribution is organized as follows. In Section 2, we present a short review of heterotic gravity and, in particular, point out how the contribution of the Yang–Mills gauge anomalies to the quantum effective action can be separated from a similar contribution of the gravitational anomalies. In Section 3, we discuss the decoupling limit of the heterotic string sigma model introduced in [18]. In particular, we discuss the relation with the $T\overline{T}$ deformation and the supersymmetric generalization. Section 4 is devoted to a detailed calculation of the Yang–Mills gauge anomaly and the emergence of a generalized metric that provides further details in addition to the original discussions presented in [18]. In Section 5, we schematically indicate how a similar calculation of the gravitational anomaly would proceed. Finally, a summary of our results and an outlook on future developments can be found in Section 6.

2. Heterotic Gravity

We define heterotic gravity as the bosonic part of the $10D\mathcal{N}=1$ supergravity theory coupled to Yang–Mills anomalies, including all higher-order ${\alpha }^{\prime }$ corrections following from heterotic string theory. The Lagrangian in the lowest order of derivatives is given by

$${\mathcal{L}}_{\mathrm{heterotic}}=\mathcal{L}\left(R\right)+\beta \mathcal{L}\left(F^2\right),$$

(1)

with

$$\begin{array}{cc}\hfill \mathcal{L}\left(R\right)& ={\displaystyle \frac{1}{2}}\sqrt{-g}{e}^{-2\mathrm{\Phi }}\left[R\left(\omega \left(E\right)\right)+4{\partial }_{\mathrm{M}}\mathrm{\Phi }{\partial }^{\mathrm{M}}\mathrm{\Phi }-\frac{1}{12}{\mathcal{H}}_{\mathrm{MNL}}{\mathcal{H}}^{\mathrm{MNL}}\right],\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill \mathcal{L}\left(F^2\right)& =-{\displaystyle \frac{1}{8}}\sqrt{-g}{e}^{-2\mathrm{\Phi }}\mathrm{Tr}\left(F_{\mathrm{MN}}{F}^{\mathrm{MN}}\right).\hfill \end{array}$$

(2b)

The basic fields are the Zehnbein field $\left({E_{\mathrm{M}}}^a\right)$, the dilaton field ($\mathrm{\Phi }$), the Kalb–Ramond (KR) two-form field ($B_{\mathrm{MN}}$) and the YM gauge field ($V_{\mathrm{M}}^{AB}$) corresponding to a YM group (G). We indicate YM indices with A, curved indices with M and flat Lorentz indices with a. The spin-connection field and the curvatures for the YM gauge field and the KR two-form field are defined as follows (The super-index (n) in ${\mathcal{H}}_{\mathrm{MNL}}^{\left(n\right)}$ indicates that this part of the curvature contains all terms of the n-th order in $\beta$):

$$\begin{array}{cc}\hfill & {{\omega }_{\mathrm{M}}}^{ab}\left(E\right)=2E^{N[a}{\partial }_{[\mathrm{M}}{E_{\mathrm{N}]}}^{b]}-{E_{\mathrm{M}}}^c{E}^{\mathrm{N}a}{E}^{\mathrm{L}b}{\partial }_{[\mathrm{N}}{E}_{\mathrm{L}]c},\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill & F_{\mathrm{MN}}=2{\partial }_{[\mathrm{M}}{V}_{\mathrm{N}]}-i\left[V_{\mathrm{M}},V_{\mathrm{N}}\right],{\mathcal{H}}_{\mathrm{MNP}}={\mathcal{H}}_{\mathrm{MNP}}^{\left(0\right)}+{\mathcal{H}}_{\mathrm{MNP}}^{\left(1\right)},\hfill \end{array}$$

(3b)

with

$$\begin{array}{c}\hfill {\mathcal{H}}_{\mathrm{MNL}}^{\left(0\right)}=3{\partial }_{[\mathrm{M}}{B}_{\mathrm{NL}]},{\mathcal{H}}_{\mathrm{MNL}}^{\left(1\right)}=\beta \mathrm{tr}\left(3V_{[\mathrm{M}}{\partial }_{\mathrm{N}\phantom{]}}{V}_{\mathrm{L}]}-i{V}_{[\mathrm{M}}{V}_{\mathrm{N}\phantom{]}}{V}_{\mathrm{L}]}\right),\end{array}$$

(4)

where ${\mathcal{H}}_{\mathrm{MNL}}^{\left(1\right)}$ represents a YM Chern–Simons term. The $\beta$ parameter can be identified with the YM coupling constant ($g_{\mathrm{YM}}$) or, equivalently, with the ${\alpha }^{\prime }$ parameter of heterotic string theory as follows:

$$\beta =1/g_{\mathrm{YM}}^2={\alpha }^{\prime }.$$

(5)

For $\beta =0$, we obtain the bosonic part of pure $10D\mathcal{N}=1$ supergravity theory. We note that for $\beta \ne 0$, the action can be made supersymmetric to all orders in $\beta$ but that these $\beta$-corrections do not represent all ${\alpha }^{\prime }$ corrections. It turns out that the fully corrected action can only be made supersymmetric order by order in ${\alpha }^{\prime }$.

We now wish to deform ${\mathcal{L}}_{\mathrm{heterotic}}$ by the inclusion of a supersymmetric $\mathcal{L}\left(R^2\right)$ term including a Lorentz Chern–Simons term. From the heterotic-string point of view, this term will be proportional to ${\alpha }^{\prime }$ and, therefore, will be of the same order in $\beta ={\alpha }^{\prime }$ as the YM $F^2$ term. However, from the supersymmetric point of view, the $F^2$ and $R^2$ terms are non-equivalent in the following sense: one can delete the $R^2$ term and remain with an action that is exactly supersymmetric to all orders in $\beta$, corresponding to the Lagrangian expressed as $\mathcal{L}\left(R\right)+\beta \mathcal{L}\left(F^2\right)$. An important feature of this invariant is that under supersymmetry, the KR two-form field ($B_{\mathrm{MN}}$) receives a correction that is linear in $\beta$ and quadratic in the fields of the YM multiplet as follows:

$${\delta }_{\beta }{B}_{\mathrm{MN}}=\beta V_{[\mathrm{M}}^{AB}{\delta }_0{V}_{\mathrm{N}]}^{BA},$$

(6)

where ${\delta }_0{V}_{\mathrm{M}}^{AB}$ denotes the ${\beta }^0$ part of the supersymmetry variation of $V_{\mathrm{M}}^{AB}$. On the other hand, it turns out that one cannot delete the YM $F^2$ term and be left with an action corresponding to the Lagrangian expressed as $\mathcal{L}\left(R\right)+{\alpha }^{\prime }\mathcal{L}\left(R^2\right)$ that is supersymmetric to all orders in ${\alpha }^{\prime }$. In this work, we will only consider a Lagrangian that is supersymmetric to linear order in ${\alpha }^{\prime }$. To deal with this non-equivalence between the YM $F^2$ term and the $R^2$ term, we denote the coefficient in front of the $F^2$ term as $\beta$ and the one in front of $R^2$ with a different coefficient ($\alpha$). We will then consider an action corresponding to the following Lagrangian:

$$\mathcal{L}=\mathcal{L}\left(R\right)+\alpha \mathcal{L}\left(R^2\right)+\beta \mathcal{L}\left(F^2\right),$$

(7)

which is supersymmetric up to linear order in $\alpha$ and $\beta$. Only at the very end, when comparing with the sigma model, we will identify both $\alpha$ and $\beta$ with ${\alpha }^{\prime }$. Using this notation, we can, at every stage, set the deformation parameter to $\alpha =0$ and re-obtain the (supersymmetric version of the) heterotic Lagrangian (1).

To explain the construction of the action corresponding to the Lagrangian (7) that is supersymmetric up to linear order in $\alpha$ and $\beta$, we make use of the following analogy between the supergravity and the YM multiplet [20,21]. It turns out that the lowest-order supersymmetry variation (${\delta }_0{{\mathrm{\Omega }}_{\mathrm{M}}}^{ab}$), independent of $\alpha$ and $\beta$ (From now on, ${\delta }_{{\alpha }^m{\beta }^n}$ refers to a supersymmetry variation of the m-th order in $\alpha$ and the n-th order in $\beta$, and ${\mathcal{H}}_{MNP}^{\left(n\right)}$ refers to all terms in the curvature (${\mathcal{H}}_{MNP}$) that are of the n-th order in ${\alpha }^p{\beta }^q$ with $p+q=n$.) of the torsionful spin-connection, i.e.,

$${{\mathrm{\Omega }}_{\mathrm{M}}}^{ab}\equiv {{\omega }_{\mathrm{M}}}^{ab}+{\displaystyle \frac{1}{2}}({{\mathcal{H}}^{\left(0\right)}{)}_{\mathrm{M}}}^{ab},$$

(8)

is identical to that of an SO(9,1) gauge field ($V_M^{AB}$), where the gaugino ${\chi }^A$ is to be identified with the gravitino curvature (${\mathrm{\Psi }}^{ab}$) via the following supersymmetry rule:

$${\delta }_0{{\mathrm{\Omega }}_{\mathrm{M}}}^{ab}=\frac{1}{2}\overline{ϵ}{\mathrm{\Gamma }}_{\mathrm{M}}{\mathrm{\Psi }}^{ab}.$$

(9)

One can therefore consider the supersymmetric extension of the action corresponding to the Lagrangian (7) as the action of a $G\times$ SO(9,1) YM multiplet coupled to supergravity. This only works to linear order in $\alpha$ and $\beta$ for the following reason. Unlike the YM gauge field, the torsionful spin-connection field is a dependent field whose transformation rule is determined by the supergravity fields it depends on. However, in the construction of the supersymmetric $F^2$ action, one is forced to change the supersymmetry rule of $B_{\mathrm{MN}}$ with a term that is linear in $\beta$ following (6). Accordingly, when constructing a supersymmetric action for a $G\times$ SO(9,1) multiplet, the supersymmetry rule of $B_{\mathrm{MN}}$ receives a linear correction in both $\alpha$ and $\beta$, with ${\delta }_{\beta }{B}_{\mathrm{MN}}$ given in Equation (6) and ${\delta }_{\alpha }{B}_{\mathrm{MN}}$ given by

$${\delta }_{\alpha }{B}_{\mathrm{MN}}=\alpha {{\mathrm{\Omega }}_{[\mathrm{M}}}^{ab}{\delta }_0{\mathrm{\Omega }}_{\mathrm{N}]\mathrm{ab}}.$$

(10)

This leads to a non-zero supersymmetry variation ($\left({\delta }_{\alpha }+{\delta }_{\beta }\right){{\mathrm{\Omega }}_{\mathrm{M}}}^{ab}$) that violates the assumption (9). This explains why the action corresponding to the Lagrangian (7) is only supersymmetric to linear order in $\alpha$ and $\beta$. Note that to this order, the curvature (${\mathcal{H}}_{MNP}$) in $\mathcal{L}\left(R\right)$ contains a Lorentz and YM Chern–Simons term that is linear in $\alpha$ and $\beta$, but the same curvature that occurs in the torsionful spin connection used to define $\alpha \mathcal{L}\left(R{\left(\mathrm{\Omega }\right)}^2\right)$ does not contain these terms in this order. For the purpose of this paper, we do not need to go to higher orders.

As a finishing remark, we note that, based on the above, one would expect that, at the next order, one would find new ${\alpha }^2{R}^3$ and $\alpha \beta R{F}^2$ terms in the action. (We do not consider the higher order in $\beta$ terms that are part of the YM anomaly coupled to a supergravity system.) Surprisingly, it turns out that these new terms do not arise, and only the supersymmetry rules are corrected at this order. As a consequence, at the next quartic orderm one does find new ${\alpha }^3{R}^4$, ${\alpha }^2\beta R^2{F}^2$ and $\alpha {\beta }^2{F}^4$ terms, which we will not consider in this work.

This finishes our discussion of the target-space heterotic gravity action.

3. Decoupling Limit of Heterotic String Sigma Model

We now consider the string worldsheet perspective on heterotic string theory in the Ramond–Neveu–Schwarz (RNS) formalism. Furthermore, we define the complex variable (z) and its complex conjugate ($\overline{z}$) on the Euclidean worldsheet. We also introduce 10 bosonic fields ($X^{\mathrm{M}}(z,\overline{z})$, $\mathrm{M}=0,\cdots ,9$), 10 chiral fermions (${\psi }^{\mathrm{M}}\left(\overline{z}\right)$), and 32 left-moving fermions (${\lambda }^A\left(z\right),A=1,\cdots ,32$). We start with SO(32) or $E_8\times E_8$, which is required by the one-loop modularity invariance [22]. In arbitrary background fields, the heterotic sigma models that are anomaly-free at the lowest loop order are given by

$$\begin{array}{c}\hfill S=\int {\displaystyle \frac{{\mathrm{d}}^{2}z}{4\pi }}\left[\frac{2}{{\alpha }^{\prime }}{\partial }_z{X}^{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{N}}{G}_{\mathrm{MN}}+{\psi }_a{\nabla }_z{\psi }^a+\mathrm{tr}\left(\lambda {\nabla }_{\overline{z}}\lambda \right)+i\sqrt{\frac{\beta }{8}}\mathrm{tr}\left(\lambda {\psi }^{\mathrm{M}}{\psi }^{\mathrm{N}}{F}_{\mathrm{MN}}\lambda \right)\right],\end{array}$$

(11)

where ${\psi }^a={\psi }^{\mathrm{M}}{E_{\mathrm{M}}}^a$ and

$$\begin{array}{cc}\hfill {\nabla }_z{\psi }^a& ={\partial }_z{\psi }^a+\sqrt{\frac{\alpha }{{\alpha }^{\prime }}}{\partial }_z{X}^{\mathrm{M}}{{\mathrm{\Omega }}_{\mathrm{M}}}^{ab}{\psi }^b,{\nabla }_{\overline{z}}\lambda ={\partial }_{\overline{z}}\lambda -i\sqrt{\frac{\beta }{{\alpha }^{\prime }}}{\partial }_{\overline{z}}{X}^{\mathrm{M}}{V}_{\mathrm{M}}\lambda ,\hfill \end{array}$$

(12a)

$$G_{\mathrm{MN}}={E_{\mathrm{M}}}^a{E_{\mathrm{N}}}^b{\eta }_{ab}+B_{\mathrm{MN}}+\frac{1}{4}\alpha {{\mathrm{\Omega }}_{\mathrm{M}}}^{ab}{\mathrm{\Omega }}_{\mathrm{N}ab}+\frac{1}{4}\beta \mathrm{tr}\left(V_{\mathrm{M}}{V}_{\mathrm{N}}\right),$$

(12b)

$$\begin{array}{cc}\hfill F_{\mathrm{MN}}& ={\partial }_{\mathrm{M}}{V}_{\mathrm{N}}-{\partial }_{\mathrm{N}}{V}_{\mathrm{M}}-i\sqrt{\frac{\beta }{2}}\left[V_{\mathrm{M}},V_{\mathrm{N}}\right].\hfill \end{array}$$

(12c)

It is required that $\alpha ={\alpha }^{\prime }$ for the theory to be anomaly-free at the lowest loop order and to be supersymmetric. It was shown in [23] that the path integral associated with this action is free of both YM gauge and gravitational anomaly with respect to the lowest-order quantum corrections. In addition, the cancellation of conformal anomalies, which is not considered in this contribution, leads to the equations of motion governing the effective low-energy target-space dynamics of heterotic gravity [24].

• Limiting prescription.

In order to introduce the decoupling limit, we first focus on the sector without ${\psi }^a$. For convenience, we also set ${\alpha }^{\prime }=2$. We also set $\alpha =0$ so that we do not yet consider the gravitational anomaly. Now, the action (11) reduces to

$$\begin{array}{c}\hfill S={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\left[\frac{2}{{\alpha }^{\prime }}{\partial }_z{X}^{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{N}}{G}_{\mathrm{MN}}+\mathrm{tr}\left(\lambda {\nabla }_{\overline{z}}\lambda \right)\right],\end{array}$$

(13)

with a decoupled ${\psi }_a{\partial }_z{\psi }^a$ term and

$$\begin{array}{c}\hfill G_{\mathrm{MN}}={E_{\mathrm{M}}}^a{E_{\mathrm{N}}}^b{\eta }_{ab}+B_{\mathrm{MN}}+\frac{1}{4}\beta \mathrm{tr}\left(V_{\mathrm{M}}{V}_{\mathrm{N}}\right).\end{array}$$

(14)

This theory is free of YM gauge anomaly with respect to the lowest-order quantum corrections. The decoupling limit of interest is defined by decomposing the ten-dimensional flat index (a) into two lightlike directions and an eight-dimensional spacelike transverse part as $a=\{+,-,i\}$ and using the following reparametrization of the background fields [18,19], i.e.,

$$\begin{array}{cc}\hfill {E_{\mathrm{M}}}^i& ={e_{\mathrm{M}}}^i,\hfill \end{array}$$

(15a)

$${E_{\mathrm{M}}}^{+}=\omega {{\tau }_{\mathrm{M}}}^{+},$$

(15b)

$${E_{\mathrm{M}}}^{-}=\frac{1}{4}{\omega }^3\beta \mathrm{tr}\left(v_{+}^2\right){{\tau }_{\mathrm{M}}}^{+}+\omega \left[{{\tau }_{\mathrm{M}}}^{-}+\frac{1}{2}\beta \mathrm{tr}\left(v_{+}{u}_{\mathrm{M}}\right)\right],$$

(15c)

$$B_{\mathrm{MN}}=-{\omega }^2{{\tau }_{[\mathrm{M}}}^{-}{{\tau }_{\mathrm{N}]}}^{+}+b_{\mathrm{MN}},$$

(15d)

$$V_{\mathrm{M}}={\omega }^2{{\tau }_{\mathrm{M}}}^{+}{v}_{+}+u_{\mathrm{M}},$$

(15e)

followed by the sending of $\omega \to \infty$ in heterotic string theory.

• $\mathit{T}\overline{\mathit{T}}$  deformation.

The BPS nature of the above decoupling limit can be understood more easily by considering the bosonic string in the Nambu–Goto formulation in the absence of the target-space YM gauge field. We show that the $T\overline{T}$ deformation of non-relativistic string theory leads to the conventional relativistic string theory [14,25]. With respect to the Minkowskian worldsheet coordinates (${\sigma }^{\alpha }=(\tau ,\sigma )$), the Nambu–Goto action reads

$$\begin{array}{cc}\hfill S& =-\int {\displaystyle \frac{{\mathrm{d}}^2\sigma }{2\pi {\alpha }^{\prime }}}\left\{\sqrt{-\det \left[{\partial }_{\alpha }{X}^{\mathrm{M}}{\partial }_{\beta }{X}^{\mathrm{N}}\left(-{E_{(\mathrm{M}}}^{+}{E_{\mathrm{N})}}^{-}+{E_{\mathrm{M}}}^i{E_{\mathrm{N}}}^i\right)\right]}+{ϵ}^{\alpha \beta }{\partial }_{\alpha }{X}^{\mathrm{M}}{\partial }_{\beta }{X}^{\mathrm{N}}{B}_{\mathrm{MN}}\right\}\hfill \\ \hfill & =-\int {\displaystyle \frac{{\mathrm{d}}^2\sigma }{2\pi {\alpha }^{\prime }}}\left(\mathcal{L}\left(\mathbf{t}\right)+{ϵ}^{\alpha \beta }{b}_{\alpha \beta }\right).\hfill \end{array}$$

(16)

Here, we define $b_{\alpha \beta }={\partial }_{\alpha }{X}^{\mathrm{M}}{\partial }_{\beta }{X}^{\mathrm{N}}{b}_{\mathrm{MN}}$ and

$$\begin{array}{c}\hfill \mathcal{L}\left(\mathbf{t}\right)={\displaystyle \frac{1}{\mathbf{t}}}\left[\sqrt{-\det \left({\tau }_{\alpha \beta }+\mathbf{t}{\partial }_{\alpha }{X}^{\mathrm{M}}{\partial }_{\beta }{X}^{\mathrm{N}}{e}_{\mathrm{MN}}\right)}-\sqrt{-det\tau }\right],\end{array}$$

(17)

with $\mathbf{t}={\omega }^{-2}$, $e_{\mathrm{MN}}={e_{\mathrm{M}}}^i{e_{\mathrm{N}}}^i$, where ${\tau }_{\alpha \beta }={\partial }_{\alpha }{X}^{\mathrm{M}}{\partial }_{\alpha }{X}^{\mathrm{N}}{\tau }_{\mathrm{MN}}$ is the induced metric on the worldsheet. Defining the stress-energy tensor as

$$\begin{array}{c}\hfill T_{\alpha \beta }=-{\displaystyle \frac{2}{\sqrt{-det\tau }}}{\displaystyle \frac{\partial \mathcal{L}}{\partial {\tau }^{\alpha \beta }}},\end{array}$$

(18)

we realize that $\mathbf{t}$ corresponds to a $T\overline{T}$ deformation with

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}\mathcal{L}\left(\mathbf{t}\right)}{\mathrm{d}\mathbf{t}}}=-{\displaystyle \frac{detT}{2\sqrt{-det\tau }}}.\end{array}$$

(19)

In the $\omega \to \infty$ limit, i.e., $\mathbf{t}\to 0$, we find

$$\mathcal{L}\left(0\right)={\displaystyle \frac{1}{2}}\sqrt{-det\tau }{\tau }^{\alpha \beta }{\partial }_{\alpha }{X}^{\mathrm{M}}{\partial }_{\beta }{X}^{\mathrm{N}}{e}_{\mathrm{MN}}.$$

(20)

In the above calculation, the string charge is fine-tuned to cancel the divergent string tension in the $\omega \to \infty$ limit, which leaves us with a finite Lagrangian term (20) that describes the non-relativistic string, which is invariant under the following string Galilei boost transformation with parameters of $({{\ell }^i}_0,{{\ell }^i}_1)$:

$$\delta {e_{\mathrm{M}}}^i={{\ell }^i}_0{{\tau }_{\mathrm{M}}}^0+{{\ell }^i}_1{{\tau }_{\mathrm{M}}}^1,\delta b_{\mathrm{MN}}=2\left({{\ell }^i}_0{e_{[\mathrm{M}}}^i{{\tau }_{\mathrm{N}]}}^1+{{\ell }^i}_1{e_{[\mathrm{M}}}^i{{\tau }_{\mathrm{N}]}}^0\right).$$

(21)

Background fields ${{\tau }_{\mathrm{M}}}^{0,1}$ and ${e_{\mathrm{M}}}^i$ can be respectively viewed as the longitudinal and transverse vielbein fields in the target space, and they constitute a Newton–Cartan-like formalism of the ten-dimensional non-Lorentzian geometry.

• Heterotic non-relativistic string theory.

We now return to the heterotic string ansatz (15) defining the decoupling limit and plug this ansatz into $G_{\mathrm{MN}}$. This gives

$$G_{\mathrm{MN}}=-{\omega }^2{{\tau }_{\mathrm{M}}}^{-}{{\tau }_{\mathrm{N}}}^{+}+{\mathcal{E}}_{\mathrm{MN}},$$

(22)

with

$${\mathcal{E}}_{\mathrm{MN}}={e_{\mathrm{M}}}^i{e_{\mathrm{N}}}^i+b_{\mathrm{MN}}+\frac{1}{2}\beta \mathrm{tr}\left(u_{\mathrm{M}}{u}_{\mathrm{N}}\right).$$

(23)

Note that the $O\left({\omega }^2\right)$ contributions in the action are canceled out. The resulting action is

$$\begin{array}{cc}S={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z& \{\frac{2}{{\alpha }^{\prime }}{\partial }_z{X}^{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{N}}{\mathcal{E}}_{\mathrm{M}\mathrm{N}}+\mathrm{tr}\left[\lambda \left({\partial }_{\overline{z}}-i\sqrt{\frac{\beta }{{\alpha }^{\prime }}}{u}_{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{M}}\right)\lambda \right]\\ & +{\chi }_{+}{\partial }_{\overline{z}}{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{+}+{\chi }_{-}\left[{\partial }_z{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{-}+i\sqrt{\frac{\beta }{{\alpha }^{\prime }}}\mathrm{tr}\left(\lambda v_{+}\lambda \right)\right]{\omega }^{-2}{\chi }_{+}{\chi }_{-}\},\end{array}$$

(24)

where we introduce the Lagrange multipliers (${\chi }_{\pm }$) to write the original action in an equivalent way. In the $\omega \to \infty$ limit, we are led to the sigma model describing the so-called heterotic non-relativistic string, i.e.,

$$\begin{array}{cc}\hfill S={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\{\frac{2}{{\alpha }^{\prime }}{\partial }_z{X}^{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{N}}{\mathcal{E}}_{\mathrm{M}\mathrm{N}}& +\mathrm{tr}\left[\lambda \left({\partial }_{\overline{z}}-i\sqrt{\frac{\beta }{{\alpha }^{\prime }}}{u}_{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{M}}\right)\lambda \right]\hfill \\ \hfill +{\chi }_{+}{\partial }_{\overline{z}}{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{+}& +{\chi }_{-}\left[{\partial }_z{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{-}+i\sqrt{\frac{\beta }{{\alpha }^{\prime }}}\mathrm{tr}\left(\lambda v_{+}\lambda \right)\right]\}.\hfill \end{array}$$

(25)

This non-relativistic theory is expected to be free of YM gauge anomaly, assuming that the decoupling limit and the quantum computation commute with each other. We will demonstrate this anomaly-free condition for the non-relativistic theory in Section 4. The same action can be alternatively derived by starting with the discrete light-cone quantization of the conventional heterotic string sigma model, where there is a lightlike isometry defined with respect to the generalized metric ($G_{\mathrm{MN}}$) in Equation (14). One then performs a T-duality transformation along this lightlike isometry.

• Supersymmetric generalization.

Next, we return to the same decoupling limit but now applied to the full supersymmetric sigma model (11). It is convenient to switch to the superspace formalism by introducing a supercoordinate ($\overline{\theta }$) such that the worldsheet action (11) can be written equivalently as

$$\begin{array}{c}\hfill S=\int {\displaystyle \frac{{\mathrm{d}}^{2}z\mathrm{d}\overline{\theta }}{4\pi }}\left\{\frac{2}{{\alpha }^{\prime }}{\partial }_z{Y}^{\mathrm{M}}{D}_{\overline{\theta }}{Y}^{\mathrm{N}}{G}_{\mathrm{MN}}\left(Y\right)+\mathrm{tr}\left[\mathrm{\Lambda }{D}_{\overline{\theta }}\mathrm{\Lambda }-i\sqrt{\frac{\beta }{{\alpha }^{\prime }}}\mathrm{\Lambda }{D}_{\overline{\theta }}{Y}^{\mathrm{M}}{V}_{\mathrm{M}}\left(Y\right)\mathrm{\Lambda }\right]\right\},\end{array}$$

(26)

with

$$G_{\mathrm{MN}}\left(Y\right)={E_{\mathrm{M}}}^a\left(Y\right){E_{\mathrm{N}}}^b\left(Y\right){\eta }_{ab}+B_{\mathrm{MN}}\left(Y\right)+\frac{\alpha }{4}{{\mathrm{\Omega }}_{\mathrm{M}}}^{ab}\left(Y\right){\mathrm{\Omega }}_{\mathrm{N}ab}\left(Y\right)+\frac{\beta }{4}\mathrm{tr}\left[V_{\mathrm{M}}\left(Y\right)V_{\mathrm{N}}\left(Y\right)\right].$$

We define the superspace fields as

$$\begin{array}{c}\hfill Y^{\mathrm{M}}=X^{\mathrm{M}}+i\sqrt{\frac{{\alpha }^{\prime }}{2}}\overline{\theta }{\psi }^{\mathrm{M}},{\mathrm{\Lambda }}^A={\lambda }^A+i\overline{\theta }{F}^A,\end{array}$$

(27)

with $F^A$ representing an auxiliary field and the supercovariant derivative expressed as $D_{\overline{\theta }}={\partial }_{\overline{\theta }}+\overline{\theta }{\partial }_{\overline{z}}$. Setting $\alpha =0$ and in terms of the superspace field ($Y^{\mathrm{M}}$), in this case, redefinitions (15) and (22) become

$$G_{\mathrm{MN}}\left(Y\right)=-{\omega }^2{{\tau }_{\mathrm{M}}}^{-}\left(Y\right){{\tau }_{\mathrm{N}}}^{+}\left(Y\right)+{\mathcal{E}}_{\mathrm{MN}}\left(Y\right),$$

(28a)

$$V_{\mathrm{M}}\left(Y\right)={\omega }^2\sqrt{\frac{\beta }{2}}{{\tau }_{\mathrm{M}}}^{+}\left(Y\right)v_{+}\left(Y\right)+u_{\mathrm{M}}\left(Y\right).$$

(28b)

When $\alpha \ne 0$, extra divergences occur when taking $\omega \to \infty$ into consideration due to the presence of the ${\mathrm{\Omega }}^2$ term; these divergences have to be treated with care. We will return to this issue in Section 5. Analogous to Equation (25), we find that the $\omega \to \infty$ limit of the supersymmetric action (26) is given by

$$\begin{array}{cc}\hfill S& ={\displaystyle \frac{1}{4\pi }}\int d^{2}z\mathrm{d}\overline{\theta }\{\frac{2}{{\alpha }^{\prime }}{\partial }_z{Y}^{\mathrm{M}}{D}_{\overline{\theta }}{Y}^{\mathrm{N}}{\mathcal{E}}_{\mathrm{M}\mathrm{N}}\left(Y\right)+\mathrm{tr}\left[\mathrm{\Lambda }{D}_{\overline{\theta }}\mathrm{\Lambda }-i\sqrt{\frac{\beta }{{\alpha }^{\prime }}}\mathrm{\Lambda }{D}_{\overline{\theta }}{Y}^{\mathrm{M}}{u}_{\mathrm{M}}\left(Y\right)\mathrm{\Lambda }\right]\hfill \\ \hfill & +{\mathrm{\Gamma }}_{+}{D}_{\overline{\theta }}{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{+}\left(Y\right)+{\mathrm{\Gamma }}_{-}\left[{\partial }_z{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{-}\left(Y\right)+i\frac{\beta }{2}\mathrm{tr}\left(\mathrm{\Lambda }{v}_{+}\left(Y\right)\mathrm{\Lambda }\right)\right]\},\hfill \end{array}$$

(29)

where ${\mathrm{\Gamma }}_{+}={\chi }_{+}-i\overline{\theta }{\gamma }_{+}$ and ${\mathrm{\Gamma }}_{-}={\gamma }_{-}+i\overline{\theta }{\chi }_{-}$, with ${\gamma }_{\pm }$ representing Grassmannian fields. After integrating out $\overline{\theta }$, we find the following supersymmetric action taken at $\alpha =0$:

$$\begin{array}{cc}\hfill S={\displaystyle \frac{1}{4\pi }}\int d^{2}z[{\partial }_z{X}^{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{N}}{\mathcal{E}}_{\mathrm{M}\mathrm{N}}& +{\psi }^i{\nabla }_{\overline{z}}{\psi }^i+\mathrm{tr}\left(\lambda {\partial }_z\lambda -i\lambda {\partial }_{\overline{z}}{X}^{\mathrm{M}}{u}_{\mathrm{M}}\lambda \right)\hfill \\ \hfill & +{\chi }_{+}{\mathcal{O}}_X^{+}+{\chi }_{-}{\mathcal{O}}_X^{-}+{\gamma }_{+}{\psi }^{+}+{\gamma }_{-}{\mathcal{O}}_{\psi }^{-}],\hfill \end{array}$$

(30)

where

$${\mathcal{O}}_X^{+}={\partial }_{\overline{z}}{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{+}-{\psi }^{\mathrm{M}}{\psi }^{\mathrm{N}}{\partial }_{\mathrm{M}}{{\tau }_{\mathrm{N}}}^{+},$$

(31a)

$${\mathcal{O}}_X^{-}={\partial }_z{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{-}+i\sqrt{\frac{\beta }{2}}\mathrm{tr}\left(\lambda v_{+}\lambda \right),$$

(31b)

$${\mathcal{O}}_{\psi }^{-}={\partial }_z{\psi }^{-}+2{\psi }^{\mathrm{M}}{\partial }_z{X}^{\mathrm{N}}{\partial }_{[\mathrm{M}}{{\tau }_{\mathrm{N}]}}^{-}-i{\psi }^{\mathrm{M}}\mathrm{tr}\left[\lambda \left(\sqrt{\frac{\beta }{2}}{\partial }_{\mathrm{M}}{v}_{+}+i\frac{\beta }{2}[u_{\mathrm{M}},v_{+}]\right)\lambda \right],$$

(31c)

and

$${\nabla }_z{\psi }^i={\partial }_z{\psi }^i+{{\mathrm{\Omega }}_z}^{i+}{\psi }^i{\psi }^{-}+{{\mathrm{\Omega }}_z}^{ij}{\psi }^i{\psi }^j,$$

(32)

with

$${{\mathrm{\Omega }}_{\mathrm{M}}}^{i\pm }={{\omega }_{\mathrm{M}}}^{i\pm }+\frac{1}{2}{e}^{\mathrm{N}i}{\tau }^{\mathrm{L}\pm }{h}_{\mathrm{MNL}},{{\mathrm{\Omega }}_{\mathrm{M}}}^{ij}={{\omega }_{\mathrm{M}}}^{ij}+\frac{1}{2}{e}^{\mathrm{N}i}{e}^{\mathrm{L}j}{h}_{\mathrm{MNL}},$$

(33a)

$${{\omega }_{\mathrm{M}}}^{i\pm }=-2\left({\tau }^{\mathrm{N}\pm }{\partial }_{[\mathrm{M}}{e_{\mathrm{N}]}}^i-{e_{\mathrm{M}}}^j{\tau }^{\mathrm{N}\pm }{e}^{\mathrm{L}i}{\partial }_{[\mathrm{N}}{e_{\mathrm{L}]}}^j\right),$$

(33b)

$${{\omega }_{\mathrm{M}}}^{ij}=2e^{\mathrm{N}[i}{\partial }_{[\mathrm{M}}{e_{\mathrm{N}]}}^{j]}-{e_{\mathrm{M}}}^k{e}^{\mathrm{N}i}{e}^{\mathrm{L}j}{\partial }_{[\mathrm{N}}{e_{\mathrm{L}]}}^k.$$

(33c)

For the purpose explained in Section 2, we introduce the $\alpha$ parameter in Equation (32) such that

$${\nabla }_z{\psi }^i={\partial }_z{\psi }^i+\sqrt{\frac{\alpha }{{\alpha }^{\prime }}}\left({{\mathrm{\Omega }}_z}^{i+}{\psi }^i{\psi }^{-}+{{\mathrm{\Omega }}_z}^{ij}{\psi }^i{\psi }^j\right).$$

(34)

Consistent with the condition of $\alpha =0$, we set ${\nabla }_z{\psi }^i={\partial }_z{\psi }^i$. This choice turns off any gravitational anomaly.

4. Yang–Mills Gauge Anomaly

We now review the anomaly analysis associated with the sector involving the chiral fermions (${\lambda }^A$). We therefore set $\alpha =0$ and focus on the action (13). We will demonstrate that the decoupling limit commutes with the quantum anomaly analysis illustrated in Figure 1. We already discussed the bottom-right corner of Figure 1 in Section 3, so will not repeat the discussion here.

We start with the classical action of

$$S={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\left[\frac{2}{{\alpha }^{\prime }}{\partial }_z{X}^{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{N}}\left({E_{\mathrm{M}}}^a{E_{\mathrm{N}}}^b{\eta }_{ab}+B_{\mathrm{MN}}\right)+\mathrm{tr}\left(\lambda {\partial }_{\overline{z}}\lambda -i\lambda {\partial }_{\overline{z}}{X}^{\mathrm{M}}{V}_{\mathrm{M}}\lambda \right)\right]$$

(35)

as a conventional heterotic sigma model. For simplicity, we also set $\beta ={\alpha }^{\prime }=2$. Classically, the above sigma model is invariant under the following target-space gauge transformations with parameter ${\xi }^{AB}$ in the adjoint representation:

$${\delta }_{\xi }{V}_{\mathrm{M}}^{AB}={\partial }_{\mathrm{M}}{\xi }^{AB}+i[\xi ,V_{\mathrm{M}}{]}^{AB},{\delta }_{\xi }{\lambda }^A=i{\xi }^{AB}{\lambda }^B.$$

(36)

However, since ${\lambda }^A={\lambda }^A\left(z\right)$ are chiral fermions, this theory contains chiral anomalies. We note that the operator product expansion expressed as

$${\lambda }^A\left(z_1\right){\lambda }^B\left(z_2\right)\sim {\displaystyle \frac{{\delta }^{AB}}{z_1-z_2}}$$

(37)

leads to the following anomalous contribution to the YM gauge transformations of the path integral $(\mathcal{Z}$):

$${\delta }_{\xi }\mathcal{Z}={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\mathrm{tr}\left({\partial }_z\xi V_{\overline{z}}\right).$$

(38)

Here, we have drop higher-order quantum contributions. This gauge anomaly is canceled by turning on the extra operator (${\partial }_z{X}^{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{N}}\mathrm{tr}\left(V_{\mathrm{M}}{V}_{\mathrm{N}}\right)$) in the action and simultaneously assigning the following anomalous gauge transformation to the Kalb–Ramond field ($B_{\mathrm{MN}}$):

$${\delta }_{\xi }{B}_{\mathrm{MN}}=-\frac{1}{2}\mathrm{tr}\left(V_{\mathrm{M}}{\partial }_{\mathrm{N}}\xi -V_{\mathrm{N}}{\partial }_{\mathrm{M}}\xi \right).$$

(39)

Thus, we obtain the following heterotic sigma model whose associated path integral is anomaly-free with respect to the lowest-order quantum corrections:

$$S={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\left[{\partial }_z{X}^{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{N}}{G}_{\mathrm{MN}}+\mathrm{tr}\left(\lambda {\partial }_{\overline{z}}\lambda -i\lambda {\partial }_{\overline{z}}{X}^{\mathrm{M}}{V}_{\mathrm{M}}\lambda \right)\right],$$

(40)

with the generalized metric ($G_{\mathrm{MN}}$) given in Equation (14). This analysis completes the upper-right corner of Figure 1.

Next, we consider the bottom-left corner in Figure 1. We reparameterize the background fields (40) as in Equation (15), which describes the decoupling limit. In Section 3, we showed that this limit leads to the action (25) describing the heterotic non-relativistic string. We now consider the anomaly analysis of this theory. An important difference relative to the anomaly calculation in the previous case is that not only does the kinetic term of the heterotic fermion make an anomalous contribution but also the last term in the action (25). Together, they give

$${\delta }_{\xi }\mathcal{Z}\sim -{\displaystyle \frac{1}{8{\pi }^2}}\int {\displaystyle \frac{{\mathrm{d}}^2{z}_1{\mathrm{d}}^2{z}_2}{{(z_1-z_2)}^2}}\mathrm{tr}\left\{\left[u_{{\overline{z}}_1}\left({\overline{z}}_1\right)-{\chi }_{-}\left({\overline{z}}_1\right)v_{+}\left({\overline{z}}_1\right)\right]{\partial }_{{\overline{z}}_2}\xi \left({\overline{z}}_2\right)\right\}.$$

(41)

Integrating twice by parts, like in the relativistic case, we find that the gauge anomaly is given by

$${\delta }_{\xi }\mathcal{Z}\sim {\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\mathrm{Tr}\left[{\partial }_z\xi \left(u_{\overline{z}}-{\chi }_{-}{v}_{+}\right)\right].$$

(42)

See [18] for a detailed derivation of the above expressions. With the extra $\beta \mathrm{Tr}\left(u_{\mathrm{M}}{u}_{\mathrm{N}}\right)$ term in the definition of ${\mathcal{E}}_{\mathrm{MN}}$, like in Equation (23), this presents the anomaly-free heterotic sigma model as given in Equation (25). To cancel the gauge anomaly, the background fields ($b_{\mathrm{MN}}$ and ${{\tau }_{\mathrm{M}}}^{-}$) must also transform non-trivially. These transformation rules can be derived by taking the $\omega \to \infty$ limit of the YM gauge transformations. Plugging the reparameterization (15) into the gauge transformations in Equations (36) and (39) and using ${\delta }_{\xi }{E_{\mathrm{M}}}^{\pm }=0$, we find

$$0=\omega {\delta }_{\xi }{{\tau }_{\mathrm{M}}}^{+},$$

(43a)

$$0={\omega }^3{{\tau }_{\mathrm{M}}}^{+}\mathrm{tr}\left(v_{+}{\delta }_{\xi }{v}_{+}\right)+\omega \left[{\delta }_{\xi }{{\tau }_{\mathrm{M}}}^{-}+\mathrm{tr}\left(u_{\mathrm{M}}{\delta }_{\xi }{v}_{+}+v_{+}{\delta }_{\xi }{u}_{\mathrm{M}}\right)\right],$$

(43b)

$$-{\omega }^2{\delta }_{\xi }{{\tau }_{[\mathrm{M}}}^{-}{{\tau }_{\mathrm{N}]}}^{+}+{\delta }_{\xi }{b}_{\mathrm{MN}}={\omega }^2\mathrm{tr}\left(v_{+}{\partial }_{[\mathrm{M}}\xi \right){{\tau }_{\mathrm{N}]}}^{+}+\mathrm{tr}\left({\partial }_{[\mathrm{M}}\xi u_{\mathrm{N}]}\right),$$

(43c)

$${\omega }^2{{\tau }_{\mathrm{M}}}^{+}{\delta }_{\xi }{v}_{+}+{\delta }_{\xi }{u}_{\mathrm{M}}=i{\omega }^2{{\tau }_{\mathrm{M}}}^{+}\left[\xi ,v_{+}\right]+{\partial }_{\mathrm{M}}\xi +i\left[\xi ,u_{\mathrm{M}}\right],$$

(43d)

At a finite value of $\omega$, these equations are equivalent to

$${\delta }_{\xi }{{\tau }_{\mathrm{M}}}^{+}=0,$$

(44a)

$${\delta }_{\xi }{{\tau }_{\mathrm{M}}}^{-}=-\mathrm{tr}\left(v_{+}{\partial }_{\mathrm{M}}\xi \right)+{\omega }^{-2}{{\sigma }_{\mathrm{M}}}^{-}+O\left({\omega }^{-4}\right),$$

(44b)

$${\delta }_{\xi }{v}_{+}=i[\xi ,v_{+}]+{\omega }^{-2}{\sigma }_{+}+O\left({\omega }^{-4}\right),$$

(44c)

$${\delta }_{\xi }{u}_{\mathrm{M}}={\partial }_{\mathrm{M}}\xi +i\left[\xi ,u_{\mathrm{M}}\right]-{{\tau }_{\mathrm{M}}}^{+}{\sigma }_{+}+O({\omega }^{-2}),$$

(44d)

$${\delta }_{\xi }{b}_{\mathrm{MN}}=\mathrm{tr}\left({\partial }_{[\mathrm{M}}\xi u_{\mathrm{N}]}\right)+{{\sigma }_{[\mathrm{M}}}^{-}{{\tau }_{\mathrm{N}]}}^{+}+O({\omega }^{-2}).$$

(44e)

Here, the extra ${\sigma }_{\mathrm{M}}^{-}$ and ${\sigma }_{+}$ variables are introduced to parameterize the higher-order terms, which are associated with certain Stueckelberg symmetries that we will discuss later below Equation (51). Moreover, using the action (24) at a finite $\omega$ value, the one-form fields (${\chi }_{\pm }$) satisfy the equations of motion, i.e.,

$${\chi }_{+}=-{\omega }^2\left[{\partial }_z{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{-}+i\mathrm{tr}\left(\lambda v_{+}\lambda \right)\right],{\chi }_{-}=-{\omega }^2{\partial }_{\overline{z}}{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{+},$$

(45)

and it follows that ${\chi }_{\pm }$ transforms non-trivially under the YM gauge as

$${\delta }_{\xi }{\chi }_{+}=({\omega }^2\mathrm{tr}\left(v_{+}{\partial }_{\mathrm{M}}\xi \right)-{{\sigma }_{\mathrm{M}}}^{-}){\partial }_z{X}^{\mathrm{M}}-i\mathrm{tr}\left(\lambda {\sigma }_{+}\lambda \right)+O\left({\omega }^{-2}\right),{\delta }_{\xi }{\chi }_{-}=0.$$

(46)

The $O\left({\omega }^2\right)$ term in ${\delta }_{\xi }{\chi }_{+}$ requires a special treatment before the $\omega \to \infty$ limit can be performed. In order to understand these terms, we return to the anomaly in Equation (38), which is now reparameterized as

$$\begin{array}{cc}\hfill {\delta }_{\xi }\mathcal{Z}={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\mathrm{tr}\left({\partial }_z\xi V_{\overline{z}}\right)& ={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\mathrm{tr}\left({\omega }^2{\partial }_z\xi {{\tau }_{\overline{z}}}^{+}{v}_{+}+{\partial }_z\xi u_{\overline{z}}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\mathrm{tr}\left[{\partial }_z\xi \left(u_{\overline{z}}-{\chi }_{-}{v}_{+}\right)\right]\hfill \end{array}$$

(47)

using Equation (45), which is exactly equivalent to Equation (42). The $O\left({\omega }^2\right)$ contribution in ${\delta }_{\xi }{\chi }_{+}$ precisely cancels the corresponding term in ${\delta }_{\xi }\mathcal{Z}$. In the $\omega \to \infty$ limit, we keep ${\chi }_{\pm }$ finite, which implies the following on-shell conditions:

$${\partial }_z{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{-}+i\mathrm{tr}\left(\lambda v_{+}\lambda \right)=O\left({\omega }^{-2}\right)\to 0,{\partial }_{\overline{z}}{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{+}=O\left({\omega }^{-2}\right)\to 0.$$

(48)

According to our assumptions, in the $\omega \to \infty$ limit, the ${\omega }^{-2}{\chi }_{-}{\chi }_{+}$ term in the action (24) drops out. However, the contribution of this term to the gauge transformation is finite, i.e.,

$${\delta }_{\xi }\left({\omega }^{-2}{\chi }_{+}{\chi }_{-}\right)=-\mathrm{tr}\left(v_{+}{\partial }_z\xi v_{+}\right){\chi }_{-}+O\left({\omega }^{-2}\right).$$

(49)

This inconsistency is fixed by removing the $O\left({\omega }^2\right)$ term in ${\delta }_{\xi }{\chi }_{+}$. We present an argument for why this is allowed. First, note that the YM gauge transformations of the Lagrangian terms, i.e.,

$${\chi }_{+}{\partial }_{\overline{z}}{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{+}{\chi }_{-}{\partial }_z{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{-}{\omega }^{-2}{\chi }_{-}{\chi }_{+},$$

(50)

in the action (24) all contribute $-\mathrm{tr}\left({\partial }_z\xi v_{+}\right){\chi }_{-}$, which matches the ${\chi }_{-}$ term in Equation (47), and these contributions cancel in pairs. Note that the on-shell condition for ${\chi }_{-}$ is required for this cancellation at a finite $\omega$ value. Removing the $O\left({\omega }^2\right)$ contribution in ${\delta }_{\xi }{\chi }_{+}$ removes one of these pairs, leaving only the contribution from ${\chi }_{-}{\partial }_z{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{-}$ canceling the anomaly term containing ${\chi }_{-}$ from Equation (47). This subtlety arises due to the introduction of the auxiliary fields (${\chi }_{\pm }$), which change their nature and turn into Lagrange multipliersonly after the $\omega \to \infty$ limit.

Taking the $\omega \to \infty$ limit, we are led to the following non-trivial YM gauge transformations:

$${\delta }_{\xi }{{\tau }_{\mathrm{M}}}^{-}=-\mathrm{tr}\left(v_{+}{\partial }_{\mathrm{M}}\xi \right),{\delta }_{\xi }{v}_{+}=i[\xi ,v_{+}],{\delta }_{\xi }{b}_{\mathrm{MN}}=\mathrm{tr}\left({\partial }_{[\mathrm{M}}\xi u_{\mathrm{N}]}\right)+{{\sigma }_{[\mathrm{M}}}^{-}{{\tau }_{\mathrm{N}]}}^{+},$$

(51a)

$${\delta }_{\xi }{u}_{\mathrm{M}}={\partial }_{\mathrm{M}}\xi +i\left[\xi ,u_{\mathrm{M}}\right]-{{\tau }_{\mathrm{M}}}^{+}{\sigma }_{+},{\delta }_{\xi }{\chi }_{+}=-i\mathrm{tr}\left(\lambda {\sigma }_{+}\lambda \right)-{{\sigma }_{\mathrm{M}}}^{-}{\partial }_z{X}^{\mathrm{M}}.$$

(51b)

The extra transformations parameterized by ${{\sigma }_{\mathrm{M}}}^{-}$ and ${\sigma }_{+}$ are Stueckelberg transformations that arise from the redefinition of the Lagrange multiplier (${\chi }_{+}$). In particular, the Stueckelberg transformation associated with ${\sigma }_{+}$ can be used to set $u_{+}={{\tau }^{\mathrm{M}}}_{+}{u}_{\mathrm{M}}$ to zero, leaving us with the correct number of gauge components, including $v_{+},u_{-}={{\tau }^{\mathrm{M}}}_{-}{u}_{\mathrm{M}}$, and $u_i={e^{\mathrm{M}}}_i{u}_{\mathrm{M}}$. One can check that the path integral for the heterotic non-relativistic string is, indeed, invariant under the transformation (51) when the lowest quantum corrections are concerned.

Finally, we consider the upper-left corner of Figure 1, in which case we start with the classical heterotic string action (35). In this case, the decoupling limit prescription (15) reduces to

$${E_{\mathrm{M}}}^i={e_{\mathrm{M}}}^i,{E_{\mathrm{M}}}^{+}=\omega {{\tau }_{\mathrm{M}}}^{+},{E_{\mathrm{M}}}^{-}=\omega {{\tau }_{\mathrm{M}}}^{-},$$

(52a)

$$B_{\mathrm{MN}}=-{\omega }^2{{\tau }_{[\mathrm{M}}}^{-}{{\tau }_{\mathrm{N}]}}^{+}+b_{\mathrm{MN}},V_{\mathrm{M}}={\omega }^2{{\tau }_{\mathrm{M}}}^{+}{v}_{+}+u_{\mathrm{M}},$$

(52b)

which we obtained by setting $\beta =0$ in Equation (15). In the $\omega \to \infty$ limit, we are led to the classical sigma models describing the heterotic non-relativistic string:

$$\begin{array}{c}\hfill S={\displaystyle \frac{1}{4\pi }}\int {\mathrm{d}}^{2}z\{\frac{2}{{\alpha }^{\prime }}{\partial }_z{X}^{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{N}}\left({e_{\mathrm{M}}}^i{e_{\mathrm{N}}}^i+b_{\mathrm{MN}}\right)+\mathrm{tr}\left[\lambda \left({\partial }_{\overline{z}}-i{u}_{\mathrm{M}}{\partial }_{\overline{z}}{X}^{\mathrm{M}}\right)\lambda \right]\\ \hfill +{\chi }_{+}{\partial }_{\overline{z}}{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{+}+{\chi }_{-}\left[{\partial }_z{X}^{\mathrm{M}}{{\tau }_{\mathrm{M}}}^{-}+i\mathrm{tr}\left(\lambda v_{+}\lambda \right)\right]\}.\end{array}$$

(53)

We have already carried out the anomaly analysis around Equation (42) and showed that requiring the theory to be free of YM gauge anomaly at the lowest order of quantum corrections leads to the action (25) involving ${\mathcal{E}}_{\mathrm{MN}}$, which depends on the gauge potential ($u_{\mathrm{M}}$), as shown in Equation (23).

This completes our arguments for the commutativity between the decoupling limit and YM gauge anomaly analysis.

5. Comments on Gravitational Anomaly

Next, we move on to briefly comment on the gravitational anomalies. A preliminary discussion on the gravitational anomalies of the sigma models describing the heterotic non-relativistic string was presented in [18].

We now set $\beta =0$ and $\alpha =2$ and focus on the heterotic non-relativistic string action (30). The gravitational anomalies can be computed by using the following operator-product expansions:

$${\psi }^i\left(\overline{z}\right){\psi }^j\left(0\right)\sim {\displaystyle \frac{{\delta }^{ij}}{\overline{z}}},{\gamma }_{-}\left(\overline{z}\right){\psi }^{-}\left(0\right)\sim {\displaystyle \frac{1}{\overline{z}}}.$$

(54)

In order to cancel the gravitational anomalies, it is required that we modify ${\mathcal{E}}_{\mathrm{MN}}$ in Equation (23) further to include the $\alpha$ contributions, which alternatively arise from taking the $\omega \to \infty$ limit of the ${\mathrm{\Omega }}^2$ term in the modified metric (12b). However, as we note below Equation (28), the expansion of ${\mathrm{\Omega }}^2$ contains terms that are quadratic in $\omega$, which diverges when $\omega$ is sent to infinity. For these divergences to vanish, the following geometric constraints are required:

$${{\tau }^{\mathrm{M}}}_{-}{e^{\mathrm{N}}}_j{\partial }_{[\mathrm{M}}{{\tau }_{\mathrm{N}]}}^{+}={e^{\mathrm{M}}}_i{e^{\mathrm{N}}}_j{\partial }_{[\mathrm{M}}{{\tau }_{\mathrm{N}]}}^{+}=0.$$

(55)

The same geometric constraints also appear in other analyses of non-relativistic string theory. Firstly, these constraints are required for the sigma models describing the non-relativistic string to be renormalizable [26] in such a way that the current–current deformation term (${\chi }_{+}{\chi }_{-}$) is not quantum mechanically generated at all loop orders [27]. This non-renormalization theorem is underlain by a symmetry algebra that supposedly arises from a contraction of the $\mathcal{N}=1$ super Poincaré algebra in ten dimensions associated with the BPS decoupling limit (see [14] for a particle analog associated with the BFSS matrix theory [3]). Secondly, the same constraints are essential for the $\omega \to \infty$ limit of the $\mathcal{N}=1$ supersymmetry transformation rules to be finite [28]. Our analysis of the gravitational anomalies in the heterotic non-relativistic string sigma models provide yet another argument for why the geometric constraints in Equation (55) are necessary for the self-consistency of non-relativistic string theory.

6. Outlook

In this contribution, we discussed aspects of heterotic string theory with an emphasis on anomalies. We took the string-worldsheet perspective and first considered the decoupling limit of the heterotic string sigma model. We used notations such that, at a given order of ${\alpha }^{\prime }$, we could distinguish between the contributions of the the gauge and gravitational anomalies to the quantum effective action. Concentrating on the Yang–Mills (YM) gauge anomalies, we showed that canceling the YM gauge anomalies first (at the lowest order) and next taking the decoupling limit achieves the same result as first taking the decoupling limit of the classical sigma model into account, then canceling the YM gauge anomalies after taking the limit into consideration (see Figure 1). Furthermore, we reviewed how a marginal current–current deformation closely related to $T\overline{T}$ deforms the sigma models describing the heterotic non-relativistic string theory for the conventional heterotic sigma models. We proposed a supersymmetric generalization of the heterotic sigma model both in superspace as and in components. We used this to consider not only the coupling of the YM gauge fields to the heterotic string but to also included the coupling to the spin-connection fields.

Concentrating on the YM gauge anomaly only, we calculated, in full detail, the lowest-order contribution to the quantum effective action at different stages and explicitly showed that the two operations of ‘taking the decoupling limit’ and ‘canceling the chiral anomaly’ commute (see also Figure 1). Finally, we briefly discussed how these results could be extended by including the gravitational anomaly. There is one caveat here. It is well known that to express an effective action of supersymmetric relativistic heterotic target space up to linear order in ${\alpha }^{\prime }$, it is very useful to make use of a symmetry between the Yang–Mills group and the 10D Lorentz group and to consider the 10D supergravity multiplet as a SO(9,1) gauge multiplet [20,21]. The same symmetry does not survive the decoupling limit. This is evidenced by the fact that the limiting prescription given in Equation (15) does not treat all Lorenz directions in the same way but, instead, makes use of a light-cone decomposition. This leads to more involved calculations in the case of gravitational anomalies that are addressed further in [18].