Trivialization of the gravitational Green-Schwarz transformation in the non-relativistic limit of string theory

1. Problem Statement

In relativistic heterotic string theory, consistency requires the cancellation of gauge and gravitational anomalies. This is achieved via the Green-Schwarz (GS) mechanism, which introduces a non-trivial transformation of the Kalb-Ramond $B$-field under local Lorentz and gauge transformations. This transformation is essential for the invariance of the effective action but imposes rigid constraints:

• The gauge group must be either $SO(32)$ or $E_8 \times E_8$.
• The anomaly polynomial must factorize as $I_{12} \sim (\text{tr}F^2 - \text{tr}R^2)^2$.
• Global topological constraints (matching Chern classes) are required for compactifications.

The paper addresses the question of whether these constraints persist in the Non-Relativistic (NR) limit of heterotic supergravity, specifically focusing on the gravitational sector with four-derivative ($\alpha'$) corrections. While a previous study [7] showed the trivialization of the gauge GS mechanism in the NR limit, the behavior of the gravitational GS mechanism (involving the Lorentz connection and curvature) remained an open problem.

2. Methodology

The author employs a systematic expansion of ten-dimensional heterotic supergravity in the NR limit ($c \to \infty$), incorporating higher-derivative corrections.

• NR Expansion of Fields:

  • The spacetime metric is expanded using the Newton-Cartan geometry formalism: $g_{\mu\nu} = c^2 \tau_\mu^a \tau_\nu^b \eta_{ab} + h_{\mu\nu}$.
  • The Kalb-Ramond field is expanded as $B_{\mu\nu} = -c^2 \tau_\mu^a \tau_\nu^b \epsilon_{ab} + b_{\mu\nu}$.
  • Crucially, to ensure the effective action remains finite in the $c \to \infty$ limit despite divergent curvature-squared terms, the relativistic string tension parameter is rescaled: $\alpha' \to \alpha'_{NR}/c^2$.

• Analysis of Transformations:

  • The paper analyzes the Lorentz transformation of the $B$-field at order $\alpha'$. In the relativistic limit, this is the standard GS transformation: $\delta_\Lambda B_{\mu\nu} = -\frac{\alpha'}{2} \partial_{[\mu} \Lambda_{AB} \omega_{\nu]}^{AB}$.
  • The author decomposes the Lorentz parameters into longitudinal ($SO(1,1)$), transverse ($SO(8)$), and boost components.
  • In the $c \to \infty$ limit, the dominant contribution to the transformation of the finite field $b_{\mu\nu}$ comes from the transverse $SO(8)$ sector.

• Field Redefinition Strategy:

  • The core methodology involves constructing an explicit field redefinition of the $B$-field ($b_{\mu\nu} \to \tilde{b}_{\mu\nu}$) designed to absorb the non-covariant transformation terms.
  • The author identifies that the non-covariant part of the transformation is proportional to a specific tensor structure involving the spin connection and the Newton-Cartan clock forms, which transforms covariantly under $SO(8)$.

3. Key Contributions and Results

A. Trivialization of the Gravitational GS Mechanism

The primary result is the demonstration that the gravitational GS transformation becomes trivial in the NR limit.

• The transformation of the finite field $b_{\mu\nu}$ under an $SO(8)$ rotation is found to be:
  $$\delta_\lambda b_{\mu\nu} = -\frac{\alpha'_{NR}}{4} \partial_{[\mu} \lambda^{a'b'} \tau_{\nu]d} \eta^{de} \tau_{\rho\sigma e} e^\rho_{a'} e^\sigma_{b'}$$
• By defining a redefined field:
  $$\tilde{b}_{\mu\nu} = b_{\mu\nu} - \frac{\alpha'_{NR}}{4} \omega_{[\mu}^{a'b'} \tau_{\nu]d} \eta^{de} \tau_{\rho\sigma e} e^\rho_{a'} e^\sigma_{b'}$$
  the transformation of $\tilde{b}_{\mu\nu}$ vanishes ($\delta_\lambda \tilde{b}_{\mu\nu} = 0$).
• Consequently, the redefined two-form is manifestly invariant under $SO(8)$ transformations.

B. Exactness of the Three-Form Field Strength

The redefinition leads to a modified three-form field strength $\bar{h}_{\mu\nu\rho}$:
$$\bar{h}_{\mu\nu\rho} = 3\partial_{[\mu} \tilde{b}_{\nu\rho]} = h_{\mu\nu\rho} + C_{\mu\nu\rho}$$
where $C_{\mu\nu\rho}$ is a Chern-Simons-like 3-form arising directly from the field redefinition.

• Result: The Bianchi identity for this new field strength becomes trivial:
  $$d\bar{h} = 0$$
  This holds identically without requiring the standard anomaly cancellation condition $dH = \text{tr}(F \wedge F) - \text{tr}(R \wedge R)$.

C. Simultaneous Trivialization with Gauge Sector

The paper compares this result with the previously established trivialization of the gauge GS mechanism.

• The gauge sector redefinition involves the gauge connection $A_\mu$ and a vector $\alpha_-$.
• Since both the gravitational and gauge redefinitions rely on structures that are $SO(8)$ invariant, they can be implemented simultaneously.
• The combined redefined field $\tilde{b}_{\mu\nu}$ is invariant under both local Lorentz and gauge transformations.

4. Significance and Implications

The trivialization of the GS mechanism in the NR limit has profound theoretical implications:

1. Relaxation of Gauge Group Constraints:
   In relativistic theory, the factorization of the anomaly polynomial restricts gauge groups to $SO(32)$ or $E_8 \times E_8$. In the NR limit, since the GS mechanism is trivialized via field redefinition, this factorization is no longer required. This suggests that consistent NR heterotic theories could exist with arbitrary gauge groups (e.g., $SU(N)$), provided they are treated as intrinsically non-relativistic theories rather than limits of relativistic ones.

2. Topological Freedom in Compactifications:
   The standard Bianchi identity enforces a matching of second Chern classes ($c_2(V) = c_2(TX)$) between the gauge bundle and the tangent bundle. With the trivialized Bianchi identity ($d\bar{h}=0$), this topological constraint disappears. This opens the door to compactifications with:

   • Non-matching Chern classes.
   • Non-Kähler internal geometries.
   • Background torsion.

3. Simplification of Black Hole Thermodynamics:
   The calculation of Wald entropy and the first law of thermodynamics for heterotic black holes is complicated by GS couplings. In the NR limit, the redefined fields are covariant, and the Chern-Simons modifications vanish. This leads to a dramatic simplification of the entropy formula and the momentum maps, allowing for a derivation based on manifestly invariant expressions.

4. Anomaly Cancellation as Automatic:
   The results suggest that in the NR regime, anomaly cancellation may become an automatic consequence of field redefinitions rather than an external consistency condition imposed on the theory's topology or gauge group. This implies that non-Lorentzian limits can fundamentally alter the geometric and topological origins of anomalies in string theory.

5. Conclusion

The paper provides a rigorous proof that the gravitational Green-Schwarz transformation in ten-dimensional heterotic supergravity can be completely removed by a field redefinition in the non-relativistic limit. This, combined with the gauge sector trivialization, implies that the strict consistency constraints of relativistic string theory (specific gauge groups, topological matching) are relaxed in the NR regime. This work paves the way for exploring a broader landscape of consistent non-relativistic string theories and offers new insights into the nature of anomalies in non-Lorentzian geometries. Future work will focus on extending these results to include supersymmetry, fermions, and higher-derivative orders.