Preprints 2021

Authors: Agnese Bissi, Parijat Dey and Giulia Fardelli

Preprint number: UUITP-33/21

Abstract: We review the recent developments in the study of conformal field theories in generic space time dimensions using the methods of the conformal bootstrap, in its analytic aspect. These techniques are based solely on symmetries, in particular in the analytic structure and in the associativity of the operator product expansion. We focus on two applications of the analytic conformal bootstrap: the study of the $\epsilon$ expansion of the Wilson Fisher model via the introduction of a dispersion relation and the large $N$ expansion of maximally supersymmetric Super Yang Mills theory in four dimensions. 

Authors: Luca Cassia and Maxim Zabzine

Preprint number: UUITP-32/21

Abstract: We consider the matrix model of U(N) refined Chern-Simons theory on S³ for the unknot. We derive a q-difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as q-Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent q-difference operator for a similar refinement of ABJ theory and show that the corresponding q-Virasoro constraints are equal to those of refined Chern-Simons for a gauge super-group U(N|M). Our equations and solutions are manifestly symmetric under Langlands duality q ↔ 1/t which correctly reproduces 3d Seiberg duality when q is a specific root of unity.

Authors: Alex Edison, Max Guillen, Henrik Johansson, Oliver Schlotterer and Fei Teng

Preprint number: UUITP-31/21

Abstract: In the low-energy effective action of string theories, non-abelian gauge interactions and supergravity are augmented by infinite towers of higher-mass-dimension operators. We propose a new method to construct one-loop matrix elements with insertions of operators D^{2k} F^n and D^{2k} R^n in the tree-level effective action of type-I and type-II superstrings. Inspired by ambitwistor string theories, our method is based on forward limits of moduli-space integrals using string tree-level amplitudes with two extra points, expanded in powers of the inverse string tension alpha'. Similar to one-loop ambitwistor computations, intermediate steps feature non-standard linearized Feynman propagators which eventually recombine to conventional quadratic propagators. With linearized propagators the loop integrand of the matrix elements obey one-loop versions of the monodromy and KLT relations. We express a variety of four- and five-point examples in terms of quadratic propagators and formulate a criterion on the underlying genus-one correlation functions that should make this recombination possible at all orders in alpha'. The ultraviolet divergences of the one-loop matrix elements are crosschecked against the non-separating degeneration of genus-one integrals in string amplitudes. Conversely, our results can be used as a constructive method to determine degenerations of elliptic multiple zeta values and modular graph forms at arbitrary weight.

Authors: Joseph Minahan, Usman Naseer and Charles Thull

Preprint number: UUITP-30/21

We consider mass-deformed theories with ${\cal N}\geq2$ supersymmetry on round and squashed three-spheres. By embedding the supersymmetric backgrounds in extended supergravity we show that at special values of mass deformations the supersymmetry is enhanced on the squashed spheres. When the $3d$ partition function can be obtained by a limit of a $4d$ index we also show that for these special mass deformations only the states annihilated by extra supercharges contribute to the index. By using an equivalence between partition functions on squashed spheres and ellipsoids,  we explain the recently observed squashing independence of the partition function of mass-deformed ABJM theory  on the  ellipsoid. We provide further examples of such simplification for various $3d$ supersymmetric theories.

Authors: N. Cribiori, D. Junghans, V. Van Hemelryck, T. Van Riet, T. Wrase

Preprint number: UUITP-29/21

Abstract: We revisit various aspects of AdS$_4$ flux vacua with scale separation in type II supergravity and M-theory. We show that massless IIA allows both weakly and strongly coupled solutions for which the classical orientifold backreaction can be tuned small. This is explicitly verified by computing the backreaction at leading order in perturbation theory. We give evidence that the strongly coupled solutions can be lifted to scale-separated and sourceless (but classically singular) geometries in 11D supergravity.

Authors: Andreas P. Braun, Paul-Konstantin Oehlmann, and Magdalena Larfors

Preprint number: UUITP-28/21

Abstract: We study six dimensional supergravity theories with superconformal sectors(SCFTs). Instances of such theories can be engineered using type IIB strings, or more generally F-Theory, which translates field theoretic constraints to geometry. Specifically, we study the fate of the discrete 2-form global symmetries of the SCFT sectors. For both $(2,0)$ and $(1,0)$ theories we show that whenever the charge lattice of the SCFT sectors is non-primitively embedded into the charge lattice of the supergravity theory, there is a subgroup of these 2-form symmetries that remains unbroken by BPS strings. By the absence of global symmetries in quantum gravity, this subgroup much be gauged. Using the embedding of the charge lattices also allows us to determine how the gauged 2-form symmetry embeds into the 2-form global symmetries of the SCFT sectors, and we present several concrete examples, as well as some general observations. As an alternative derivation, we recover our results for a large class of models from a dual perspective upon reduction to five dimensions.

Authors: Parijat Dey and Nirmalya Kajuri

Preprint number: UUITP-27/21

Abstract: In the bulk reconstruction program, one constructs boundary representations of bulk fields. However, the boundary representations derived in global and AdS-Rindler coordinates appear to be inequivalent as the AdS-Rindler smearing function is known to diverge in dimensions greater than two. This is an apparent paradox. We investigate the relation between the two representations for AdS$_2$. We obtain the AdS-Rindler smearing function for massive and massless fields and show that the global and AdS-Rindler boundary representations are related by conformal transformations. We also use the boundary representations of creation and annihilation operators to compute the Bogoliubov transformation relating global modes to AdS-Rindler modes for both massive and massless particles.

Authors: Miguel Montero, Cumrun Vafa, Thomas Van Riet and Gerben Venken

Preprint number: UUITP-26/21

Abstract: Demanding that charged Nariai black holes in (quasi-)de Sitter space decay
without becoming super-extremal implies a lower bound on the masses of charged particles, known as the Festina Lente (FL) bound. In this paper we elucidate various aspects of this bound as well as extensions of it to d > 4 and to situations with scalar potentials and dilatonic couplings. We also discuss phenomenological implications of FL including an explanation of why the Higgs potential cannot have a local minimum at the origin, thus explaining why the weak force must be broken. For constructions of meta-stable dS involving anti-brane uplift scenarios, even though the throat region is consistent with FL, the bound implies that we cannot have any light charged matter elds coming from any far away region in the compactied geometry, contrary to the fact that they are typically expected to arise in these scenarios. This strongly suggests that introduction of warped anti-branes in the throat cannot be decoupled from the bulk dynamics as is commonly assumed. Finally, we provide some evidence that in certain situations the FL bound can have implications even with gravity decoupled and illustrate this in the context of non-compact throats.

Authors: Luca Griguolo, Rodolfo Panerai, Jacopo Papalini, Domenico Seminara

Preprint number: UUITP-25/21

Abstract: We investigate the nonperturbative structure of Jackiw-Teitelboim gravity at finite cutoff, as given by its proposed formulation in terms of a TT-deformed Schwarzian quantum mechanics. Our starting point is a careful computation of the disk partition function to all orders in the perturbative expansion in the cutoff parameter. We show that the perturbative series is asymptotic and that it admits a precise completion exploiting the analytical properties of its Borel transform, as prescribed by resurgence theory. The final result is then naturally interpreted in terms of the nonperturbative branch of the TT-deformed spectrum. The finite-cutoff trumpet partition function is computed by applying the same strategy. In the second part of the paper, we propose an extension of this formalism to arbitrary topologies, using the basic gluing rules of the undeformed case. The Weil-Petersson integrations can be safely performed due to the nonperturbative corrections and give results that are compatible with the flow equation associated with the TT deformation. We derive exact expressions for general topologies and show that these are captured by a suitable deformation of the Eynard-Orantin topological recursion. Finally, we study the "slope" and "ramp" regimes of the spectral form factor as functions of the cutoff parameter.

Authors: S. Ekhammar, B. E. W. Nilsson

Preprint number: UUITP-24/21

Abstract: We derive major parts of the eigenvalue spectrum of the operators on the squashed seven-sphere that appear in the compactification of eleven-dimensional supergravity. These spectra determine the mass spectrum of the fields in AdS₄ and are important for the corresponding N =1 supermultiplet structure. This work is a continuation of the work in [1] where the complete spectrum of irreducible isometry representations of the fields in AdS₄ was derived for this compactification. Some comments are also made concerning the G₂ holonomy and its implications on the structure of the operator equations on the squashed seven-sphere.

Authors: U. H. Danielsson, D. Panizo, R. Tielemans, T. Van Riet

Preprint number: UUITP-22/21

Abstract: We argue that the choice of boundary condition for the wavefunction in quantum cosmology depends on the UV completion of general relativity. We provide an explicit example using a braneworld scenario in which a de Sitter cosmology is induced on the surface of a CDL bubble in a 5-dimensional AdS space. The corresponding boundary conditions are unambigously fixed by demanding consistency with the known physics of bubble nucleation and this selects the Vilenkin choice from a 4D viewpoint.

Authors: Gang Chen, Henrik Johansson, Fei Teng and Tianheng Wang

Preprint number: UUITP-21/21

Abstract: Kinematic numerators of Yang-Mills scattering amplitudes possess a rich Lie algebraic structure that suggest the existence of a hidden infinite-dimensional kinematic algebra. Explicitly realizing such a kinematic algebra is a longstanding open problem that only has had partial success for simple helicity sectors. In past work, we introduced a framework using tensor currents and fusion rules to generate BCJ numerators of a special subsector of NMHV amplitudes in Yang-Mills theory. Here we enlarge the scope and explicitly realize a kinematic algebra for all NMHV amplitudes. Master numerators are obtained directly from the algebraic rules and through commutators and kinematic Jacobi identities other numerators can be generated. Inspecting the output of the algebra, we conjecture a closed-form expression for the master BCJ numerator up to any multiplicity. We also introduce a new method, based on group algebra of the permutation group, to solve for the generalized gauge freedom of BCJ numerators. It uses the recently introduced binary BCJ relations to provide a complete set of NMHV kinematic numerators that consist of pure gauge.

Authors: Ulf Lindström and Özgür Sarioglu

Preprint number: UUITP-20/21

Authors: Simon Ekhammar, Dmytro Volin

Preprint number: UUITP-19/21

Abstract: We propose a gl(r)-covariant parameterisation of Bethe algebra appearing in so(2r) integrable models, demonstrate its geometric origin from a fused flag, and use it to compute the spectrum of periodic rational spin chains, for various choices of the rank r and Drinfeld polynomials.

Authors: Paolo di Vecchia, Carlo Heissenberg, Rodolfo Russo, Gabriele Veneziano

Preprint number: UUITP-18/21

Abstract: Using N=8 supergravity as a theoretical laboratory, we extract the 3PM gravitational eikonal for two colliding massive scalars from the classical limit of the corresponding elastic two-loop amplitude. We employ the eikonal phase to obtain the physical deflection angle and to show how its non-relativistic (NR) and ultra-relativistic (UR) regimes are smoothly connected. Such a smooth interpolation rests on keeping contributions to the loop integrals originating from the full soft region, rather than restricting it to its potential sub-region. This task is efficiently carried out by using the method of differential equations with complete near-static boundary conditions. In contrast to the potential-region result, the physical deflection angle includes radiation-reaction contributions that are essential for recovering the finite and universal UR limit implied by general analyticity and crossing arguments. We finally discuss the real emission of massless states, which accounts for the imaginary part of the 3PM eikonal and for the dissipation of energy-momentum. Adopting a direct approach based on unitarity and on the classical limit of the inelastic tree-level amplitude, we are able to treat N=8 and General Relativity on the same footing, and to complete the conservative 3PM eikonal in Einstein's gravity by the addition of the radiation-reaction contribution. We also show how this approach can be used to compute waveforms, as well as the differential and integrated spectra, for the different radiated massless fields.

Authors: Max Guillen, Henrik Johansson, Renann Lipinski Jusinskas, Oliver Schlotterer

Preprint number: UUITP-17/21

Abstract: We present a new method, exact in alpha', to explicitly compute string tree-level amplitudes involving one massive state and any number of massless ones. This construction relies on the so-called twisted heterotic string, which admits only gauge multiplets, a gravitational multiplet, and a single massive supermultiplet in its spectrum. In this simplified model, we determine the moduli-space integrand of all amplitudes with one massive state using Berends-Giele currents of the gauge multiplet. These integrands are then straightforwardly mapped to gravitational amplitudes in the twisted heterotic string and to the corresponding massive amplitudes of the conventional type-I and type-II superstrings.

Authors: Lorenzo Bianchi, Adam Chalabi, Vladimír Procházka, Brandon Robinson, and
Jacopo Sisti

Preprint number: UUITP 16/21

Abstract: We study co-dimension two monodromy defects in theories of conformally
coupled scalars and free Dirac fermions in arbitrary d dimensions. We characterise this
family of conformal defects by computing the one-point functions of the stress-tensor and
conserved current for Abelian flavour symmetries as well as two-point functions of the
displacement operator. In the case of d = 4, the normalisation of these correlation functions
are related to defect Weyl anomaly coefficients, and thus provide crucial information about
the defect conformal field theory. We provide explicit checks on the values of the defect
central charges by calculating the universal part of the defect contribution to entanglement
entropy. Moreover, we leverage the non-supersymmetric free field results to compute a
novel defect Weyl anomaly coefficient in a d = 4 theory of free N = 2 hypermultiplets.
In carefully studying the defect operator product expansion, we identify notable relevant
operators in the defect theories and use them to study the behaviour of the defect under
renormalisation group flow.

Authors: Souvik Banerjee, Ulf Danielsson, Suvendu Giri

Preprint number: UUITP-15/21

Abstract: In this essay we will take a wonderful ride on a dark bubble with strings attached, which carries our universe out of the swampland and makes it realizable in the landscape of string theory. To find the way to the landscape, we make use of apparently hostile corners of the swampland and their instabilities.

Authors:  Linghui Hou, Song He, Jintian Tian and Yong Zhang

Preprint number:  UUITP-14/21

Abstract: In this note we revisit the problem of explicitly computing tree-level scatter-ing amplitudes in various theories in any dimension from worldsheet formulas. The latterare known to produce cubic-tree expansion of tree amplitudes with kinematic numeratorsautomatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is re-duced to logarithmic functions. We review a natural class of worldsheet functions called“Cayley functions”, which are in one-to-one correspondence with labelled trees, and natu-ral expansions of known half-integrands onto them with coefficients that are particularlycompact building blocks of kinematic numerators. We present a general formula expressingthe kinematic numerator of any cubic tree as a linear combination of these coefficients oflabelled trees, including the usual combination in terms of master numerators as a specialcase. Our results provide an efficient algorithm, which is implemented in aMathemat-icapackage, for computing tree amplitudes in non-linear sigma model, special Galileon,Yang-Mills-scalar, Einstein-Yang-Mills, Dirac-Born-Infeld and so on.

Authors: Lara B. Anderson, James Gray, Magdalena Larfors, Matthew Magill, Robin Schneider

Preprint number: UUITP-13/21

Abstract: Heterotic compactifications on Calabi-Yau threefolds frequently exhibit textures of vanishing Yukawa couplings in their low energy description. The vanishing of these couplings is often not enforced by any obvious symmetry and appears to be topological in nature. Recent results in the literature used differential geometric methods to explain the origin of some of this structure. A vanishing theorem was given which showed that the effect could be attributed, in part, to the embedding of the Calabi-Yau manifolds of interest inside higher dimensional ambient spaces, if the gauge bundles involved descended from vector bundles on those larger manifolds. In this paper, we utilize an algebro-geometric approach to provide an alternative derivation of some of these results, and are thus able to generalize them to a much wider arena than has been considered before. For example, we consider cases where the vector bundles of interest do not descend from bundles on the ambient space. In such a manner we are able to highlight the ubiquity with which textures of vanishing Yukawa couplings can be expected to arise in heterotic compactifications, with multiple different constraints arising from a plethora of different geometric features associated to the gauge bundle.

Authors: Michele del Zotto, Nikita Nekrasov, Nicolo Piazzalunga, Maxim Zabzine

Preprint number: UUITP-12/21

Abstract: Motivated by M-theory, we study rank n K-theoretic Donaldson-Thomas theory on a toric threefold X. In the presence of compact four-cycles, we discuss how to include the contribution of D4-branes wrapping them. Combining this with a simple assumption on the (in)dependence on Coulomb moduli in the 7d theory, we show that the partition function factorizes and, when X is Calabi-Yau and it admits an ADE ruling, it reproduces the 5d master formula for the geometrically engineered theory on A(n-1) ALE space, thus extending the usual geometric engineering dictionary to n>1. We finally speculate about implications for instanton counting on Taub-NUT.

Authors: Paul-Konstantin Oehlmann

Preprint number: UUITP-11/21

Abstract: In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases to be flat over compact Riemann surfaces of genus g in the base. These non-flat fibers contribute Kähler moduli to the four-fold but also add to the three-form cohomology for g>0. In F-/M-theory these sectors are to be interpreted as compactifications of six/five dimensional N=(1,0) superconformal matter theories. The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors. We construct explicit examples for E-string theories as well as higher rank cases. For the E-string theories we further investigate conifold transitions that remove those non-flat fibers. First, we show how non-flat fibers can be deformed from curves down to isolated points in the base. This removes the chiral singlet of the three-forms and leads to non-perturbative four-point couplings among matter fields which can be understood as remnants of the former E-string. Alternatively, the non-flat fibers can be avoided by performing birational base changes, analogous to 6D tensor branches. For compact bases these transitions alternate all Hodge numbers but leave the Euler number invariant.

Authors: Laurentiu Rodina, Zhewei Yin

Preprint number: UUITP-10/21

Abstract: We generalize soft theorems of the nonlinear sigma model beyond the O(p^2) amplitudes and the coset of SU(N)×SU(N)/SU(N). We first discuss the flavor ordering of the amplitudes for the Nambu-Goldstone bosons of a general symmetry group representation, so that we can reinterpret the known O(p^2) single soft theorem for SU(N)×SU(N)/SU(N) in the context of a general group representation. We then investigate the special case of the fundamental representation of SO(N), where a special flavor ordering of the "pair basis" is available. We provide novel amplitude relations and a Cachazo-He-Yuan formula for such a basis, and derive the corresponding single soft theorem. Next, we extend the single soft theorem for a general group representation to O(p^4), where for at least two specific choices of the O(p^4) operators, the leading non-vanishing pieces can be interpreted as new extended theory amplitudes involving bi-adjoint scalars, and the corresponding soft factors are the same as at O(p^2). Finally, we compute the general formula for the double soft theorem, valid to all derivative orders, where the leading part in the soft momenta is fixed by the O(p^2) Lagrangian, while any possible corrections to the subleading part are determined by the O(p^4) Lagrangian alone. Higher order terms in the derivative expansion do not contribute any new corrections to the double soft theorem.

Authors: Luca Cassia, Rebecca Lodin and Maxim Zabzine

Preprint number: UUITP-09/21

Abstract: We revisit the Virasoro constraints and explore the relation to the Hirota bilinear equations. We furthermore investigate and provide the solution to non-homogeneous Virasoro constraints, namely those coming from matrix models whose domain of integration has boundaries. In particular, we provide the example of Hermitean matrices with positive eigenvalues in which case one can find a solution by induction on the rank of the matrix model.

Authors: Ruth Britto, Sebastian Mizera, Carlos Rodriguez, Oliver Schlotterer

Preprint number: UUITP-08/21

Abstract: We investigate configuration-space integrals over punctured Riemann spheres from the viewpoint of the motivic Galois coaction and double-copy structures generalizing the Kawai--Lewellen--Tye relations in string theory. For this purpose, explicit bases of twisted cycles and cocycles are worked out whose orthonormality simplifies the coaction. We present methods to efficiently perform and organize the expansions of configuration-space integrals in the inverse string tension alpha' or the dimensional-regularization parameter epsilon. Generating-function techniques open up a new perspective on the coaction of multiple polylogarithms in any number of variables and analytic continuations in the unintegrated punctures. We present a compact recursion for a generalized KLT kernel and discuss its origin from intersection numbers of Stasheff polytopes and its implications for correlation functions of two-dimensional conformal field theories. We find a non-trivial example of correlation functions in (p,2) minimal models, which can be normalized to become uniformly transcendental in the p -> \infty limit.

Authors: Alexander Söderberg

Preprint number: UUITP-07/21

Abstract: We consider two conformal defects close to each other in a free theory, and study what happens as the distance between them goes to zero. This limit is the same as zooming out, and the two defects have fused to another defect. As we zoom in we find a non-conformal effective action for the fused defect. Among other things this means that we cannot in general decompose the two-point correlator of two defects in terms of other conformal defects. We prove the fusion using the path integral formalism by treating the defects as sources for a scalar in the bulk.

Authors: Xenia de la Ossa, Magdalena Larfors, Matthew Magill

Preprint number: UUITP-06/21

Abstract: We review the construction of almost contact metric (three-) structures on manifolds with a G2​ structure. These are of interest for certain supersymmetric configurations in string and M-theory. We compute the torsion of the SU(3) structure associated to an ACMS and apply these computations to heterotic G2​ systems and supersymmetry enhancement. We initiate the study of the space of ACM3Ss, which is an infinite dimensional space with a local product structure and interesting topological features. Tantalising links between ACM3Ss and associative and coassociative submanifolds are observed.

Authors: Souvik Banerjee, Ulf Danielsson, Suvendu Giri

Preprint number: UUITP-05/21

Abstract: In this paper we study shells of matter and black holes on the expanding bubbles realizing de Sitter space, that were proposed in arXiv:1807.01570. The explicit solutions that we find for the black holes, can also be used to construct Randall-Sundrum braneworld black holes in four dimensions.

Authors: Andrea Manenti, Alessandro Vichi

Preprint number: UUITP-04/21

Abstract: We use numerical bootstrap techniques to study correlation functions of scalars transforming in the adjoint representation of SU(N). We obtain upper bounds on operator dimensions for all the relevant representations and several values of $N$. We discover several families of kinks, which do not correspond to any known model and we discuss possible candidates. We then specialize to the case N=3,4, which has been conjectured to describe a phase transition respectively in the non compact complex projective space NCCP^2 and the antiferromagnetic complex projective model ACP^3. Lattice simulations provide strong evidence for the existence of a second order phase transition, while an effective field theory approach does not predict any fixed point. We identify a set of assumptions that constrain operator dimensions to a closed region overlapping with the lattice prediction.

Authors: Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo, Gabriele Veneziano

Preprint number: UUITP-03/21

Abstract: Radiation reaction (RR) terms at the third post-Minkowskian (3PM) order have recently been found to be instrumental in restoring smooth continuity between the non-relativistic, relativistic, and ultra-relativistic (including the massless) regimes. Here we propose a new and intriguing connection between RR and soft (bremsstrahlung) theorems which short-circuits the more involved conventional loop computations. Although first noticed in the context of the maximally supersymmetric theory, unitarity and analyticity arguments support the general validity of this 3PM-order connection that we apply, in particular, to Einstein's gravity and to its Jordan-Brans-Dicke extension. In the former case we find full agreement with a recent result by Damour obtained through a very different reasoning.

Authors: Thomas Hertog, Rob Tielemans, Thomas van Riet

Preprint number: UUITP-02/21

Abstract: In theories with extra dimensions, the cosmological hierarchy problem can be thought of as the unnaturally large radius of the observable universe in Kaluza-Klein units. We sketch a dynamical mechanism that relaxes this. In the early universe scenario we propose, three large spatial dimensions arise through tunneling from a 'cosmic egg', an effectively one-dimensional configuration with all spatial dimensions compact and of comparable, small size. If the string landscape is dominated by low-dimensional compactifications, cosmic eggs would be natural initial conditions for cosmology. A quantum cosmological treatment of a toy model egg predicts that, in a variant of the Hartle-Hawking state, cosmic eggs break to form higher dimensional universes with a small, but positive cosmological constant or quintessence energy. Hence cosmic egg cosmology yields a scenario in which the seemingly unnaturally small observed value of the vacuum energy can arise from natural initial conditions.

Authors: Philip Argyres, Oleg Chalykh, Yongchao Lu

Preprint number: UUITP-01/21

Abstract: In this work we establish that the Inozemtsev system is the Seiberg-Witten integrable system encoding the Coulomb branch physics of 4d \cN=2 USp(2N) gauge theory with four fundamental and (for N≥2) one antisymmetric tensor hypermultiplets. We describe the transformation from the spectral curves and canonical one-form of the Inozemtsev system in the N=1 and N=2 cases to the Seiberg-Witten curves and differentials explicitly, along with the explicit matching of the modulus of the elliptic curve of spectral parameters to the gauge coupling of the field theory, and of the couplings of the Inozemtsev system to the field theory mass parameters. This result is a particular instance of a more general correspondence between crystallographic elliptic Calogero-Moser systems with Seiberg-Witten integrable systems, which will be explored in future work.