Nonlinear Sigma Model Worldsheet Action

• Nonlinear Sigma Model (NLSM) is a framework mapping two-dimensional worldsheets to target spaces, capturing kinetic and topological interactions.
• It integrates worldsheet fields with geometric structures, employing Lagrange multipliers and gauge fields to enforce constraints, including supersymmetric and nonrelativistic extensions.
• The formulation supports rigorous quantum analysis via techniques such as BV quantization, background field expansion, and supersymmetric localization.

The nonlinear sigma model (NLSM) worldsheet action defines a broad framework in two-dimensional quantum field theory for describing strings and constrained fields propagating on target-space manifolds with potentially nontrivial geometry. The action is formulated as the integral over a worldsheet, typically a two-dimensional Riemann surface, and encodes both kinetic and topological terms, with possible couplings to background fields, symmetry constraints, and supersymmetric extensions. The model possesses foundational relevance for string theory, statistical mechanics, and gauge/string duality.

1. General Structure and Geometric Foundations

The classical NLSM action couples a two-dimensional worldsheet $\Sigma$ with local coordinates $(\sigma^1, \sigma^2)$ to a target-space manifold $M$ via embedding fields $X^i(\sigma)$, governed by a metric $G_{ij}(X)$ and potentially a Kalb-Ramond two-form $B_{ij}(X)$. The conventional bosonic form is (Šimunić, 2022): $$S_0[X] = \int_{\Sigma} \frac{1}{2} G_{ij}(X)\, dX^i \wedge * dX^j + \frac{1}{2} B_{ij}(X)\, dX^i \wedge dX^j - \int_{\Gamma_3} X^*H$$ where $H = dB$ is the Wess–Zumino term extending over a three-manifold $\Gamma_3$ with $\partial \Gamma_3 = \Sigma$, and $*$ denotes the Hodge dual.

The target-space geometry may range from Riemannian (ordinary metric) to more elaborate structures, e.g., string Newton–Cartan geometry for nonrelativistic string theory (Yan et al., 2019), generalized Kähler manifolds in presence of torsion (Crichigno et al., 2015), or symmetric spaces in O(N) models (Ferrari, 2011).

2. Worldsheet Field Content and Constraints

The fundamental dynamical fields consist of:

• Embedding coordinates $X^i(\sigma)$ mapping $\Sigma \rightarrow M$.
• Worldsheet scalar or fermion multiplets, allowing for supersymmetric extensions ($\mathcal{N}=(2,2)$ or $\mathcal{N}=(0,2)$) (Alekseev et al., 2022, Ireson et al., 2018).

In O(N) models, additional constraints of the form $\phi^i(\sigma) \phi^i(\sigma) = 1$ are imposed, which may be realized via delta functionals or Lagrange multiplier fields $\lambda(\sigma)$ (Ferrari, 2011). In nonrelativistic string theory, two worldsheet scalars $\lambda(\sigma), \bar\lambda(\sigma)$ enforce holomorphicity constraints, determining the propagation of longitudinal directions (Yan et al., 2019).

Standard supersymmetric variants introduce multiplets containing bosonic, fermionic, and auxiliary components, with necessary covariantization via target-space connections and torsion couplings (Alekseev et al., 2022, Ireson et al., 2018).

3. Action Terms: Kinetic, Topological, and Couplings

The NLSM worldsheet action generically receives contributions from:

• Kinetic terms: $G_{ij}(X) \partial_\alpha X^i \partial_\beta X^j$ governing propagation.
• Kalb–Ramond/Wess–Zumino terms: $B_{ij}(X) \partial_\alpha X^i \partial_\beta X^j$ or its extension $\int_{\Gamma_3} X^*H$ (Šimunić, 2022).
• Lagrange multipliers or constraint terms: $\lambda$-dependent factors specifying geometric constraints, e.g., holomorphicity or unit-length requirements (Yan et al., 2019, Ferrari, 2011).
• Dilaton coupling: $\int d^2\sigma \sqrt{h} R[h] \Phi(X)$ for worldsheet curvature (Yan et al., 2019).
• Supersymmetric (fermionic) couplings: kinetic, Yukawa, and curvature-induced four-fermi terms (Alekseev et al., 2022, Ireson et al., 2018).

Generalized models may include torsion via $H = dB$ which modifies both the bosonic and fermionic parts, and nontrivial gauge structure by coupling vector multiplets, seen in gauged linear sigma model flows and Dirac sigma models (Crichigno et al., 2015, Šimunić, 2022).

4. Geometric and Algebraic Structure of Target Space

NLSMs reflect the underlying geometry via specific field content and constraints:

• Riemannian geometry: ordinary NLSM with target metric $G_{ij}$.
• String Newton–Cartan geometry: split of tangent space into longitudinal ($A=0,1$) and transverse ($A'=2,\ldots,d-1$) directions, with vielbeins $\tau_\mu^A$, $E_\mu^{A'}$ and an additional gauge-field $m_\mu^A$, forming boost-invariant symmetric tensors (Yan et al., 2019).
• Generalized Kähler structure: realized for sigma models with torsion; target space admits nontrivial $g_{ij}, B_{ij}$ and $H_{ijk}$, constructed from beta parameter in GLSMs (Crichigno et al., 2015); explicit formulas relate Kähler quotients to generalized potential $K$ and derive metric and b-field components.
• Dirac structure: in gauged models, the graph $L \subset TM \oplus T^*M$ formed by basis $e_a + \theta_a$, required to be an $H$-twisted Dirac structure to maintain invariance under gauge transformations (Šimunić, 2022).

Supersymmetric models with $\mathcal{N}=(2,2)$ or $\mathcal{N}=(0,2)$ require the target to be Kähler or possess specific holomorphic sectional curvature, with Riemann tensor appearing in Yukawa couplings (Alekseev et al., 2022, Ireson et al., 2018).

5. Supersymmetry, Gauging, and BV Formalism

Supersymmetry imposes additional structure:

• $\mathcal{N}=(2,2)$ sigma models on $S^2$ utilize chiral and anti-chiral multiplets; off-shell supersymmetry is maintained by background multiplets and U(1) equivariance; localization to constant maps or holomorphic disks is achievable via Q-exact/cohomological reformulation (Alekseev et al., 2022).
• $\mathcal{N}=(0,2)$ models, e.g., describing semilocal string moduli, exhibit fermion zero-modes, heterotic deformations, and explicit overlap couplings between translational and size sectors (Ireson et al., 2018).
• Gauging vector fields in the target (Dirac sigma models) requires introduction of worldsheet 1-form gauge fields $A^a$, with minimal coupling via target Lie algebroid structures; BV formalism must be used for systems with open gauge algebra, extending BRST quantization to field–antifield pairs and solving the classical master equation for BV action $S_{\rm BV}$—the only nontrivial antifield term emerges from the curvature of induced connections (Šimunić, 2022).

6. Specific Examples and Refinements

NLSM on 2D Cone with Line Defects

Recent analysis of closed bosonic strings on a 2D cone with metric $ds^2 = dr^2 + \beta^2 r^2 d\theta^2$ yields a worldsheet action (Ahmadain et al., 29 Nov 2025): $$S_{\rm ws}[r,\theta] = \frac{1}{4\pi\alpha'} \int_\Sigma d^2\sigma [(\partial_a r)^2 + \beta^2 r^2 (\partial_a \theta)^2]$$ Lattice regularization enables explicit control of winding sectors and line defects, with partition functions exhibiting intricate IR divergence structure and entropy computations dependent on cone angle $\beta$ and boundary conditions. Semiclassical saddle analysis and replica trick/renormalization provide finite entropy expressions for each winding sector $W$ and the total entropy after summation.

O(N) NLSM and Stochastic Process Analogy

The O(N) sigma model imposes $\phi^i \phi^i = 1$ via delta-functional or Lagrange multiplier; the action is: $$S_0[\phi] = \frac{g}{2} \int_\Sigma d^2x\, \partial_a \phi^i \partial_a \phi^i$$ with auxiliary fields or Fourier representation enabling constrained path integral formulation (Ferrari, 2011).

GLSM Flow to NLSM with Torsion

Gauged linear sigma models with semichiral multiplets and constrained vector multiplets result in IR NLSMs with explicit generalized Kähler structure, torsion from $H = dB$, and explicit Kähler potential $K$ encoding the beta-deformed metric and $B$-field (Crichigno et al., 2015).

7. Quantum Aspects and Localization

Quantum corrections are accessible via background field methods and localization:

• Beta-functions for NLSM couplings can be computed in covariant background field methods for nonrelativistic string theories (Yan et al., 2019).
• Supersymmetric localization on $S^2$ reduces path integrals to finite-dimensional integrals over constant maps, with one-loop determinants expressible in terms of Gamma-classes of the target (Alekseev et al., 2022), yielding exact results for partition functions of Calabi–Yau and generalized Kähler targets.
• Renormalization schemes, e.g., relating UV and IR cutoffs or replica method (as in entropy calculation for strings on conical backgrounds), play a key role in ensuring finiteness and extracting physically meaningful observables (Ahmadain et al., 29 Nov 2025).

Table: Worldsheet Field Content and Role in NLSM Variants

Field/Multi.	Occurrence	Role in Action
$X^i$ (embedding)	All NLSMs	Map $\Sigma \rightarrow M$; kinetic and topological terms
$\lambda$ (scalar)	Nonrelativistic, O(N) models	Constraint enforcement (holomorphicity/unit norm)
$A^a$ (gauge field)	Dirac sigma models, GLSMs	Minimal coupling to target vector fields/Lie algebroids
$\psi$ (worldsheet fermions)	Supersymmetric extensions	Kinetic, Yukawa, curvature, and superpotential couplings
Ghosts, antifields	BV formalism, open gauge algebra	BRST/BV quantization, classical master equation

The precise choice and interaction of fields reflect the geometric, symmetry, and supersymmetric structure of the target and the physical problem addressed.

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In summary, the NLSM worldsheet action is a versatile, geometrically controlled framework that underlies much of modern string theory, integrable systems, and quantum field theory. It admits a rich variety of extensions, constraints, and couplings, all encoded in worldsheet fields mapping into target spaces endowed with diverse geometric and algebraic structures. Advanced methodologies such as gauging, supersymmetry, background field expansion, and BV quantization play a vital role in its rigorous definition and physical applications. All aspects articulated herein follow directly from the cited arXiv sources (Yan et al., 2019, Šimunić, 2022, Ireson et al., 2018, Crichigno et al., 2015, Ferrari, 2011, Alekseev et al., 2022, Ahmadain et al., 29 Nov 2025).