Mithat Unsal

Supersymmetry on a spatial lattice

We construct a variety of supersymmetric gauge theories on a spatial lattice, including N = 4 supersymmetric Yang-Mills theory in 3+1 dimensions. Exact lattice supersymmetry greatly reduces or eliminates the need for fine tuning to arrive at the desired continuum limit in these examples.

Supersymmetry on a euclidean spacetime lattice 1. A target theory with four supercharges

We formulate a euclidean spacetime lattice whose continuum limit is (2,2) supersymmetric Yang-Mills theory in two dimensions, a theory which possesses four supercharges and an anomalous global chiral symmetry. The lattice action respects one exact supersymmetry, which allows the target theory to emerge in the continuum limit without fine-tuning. Our method exploits an orbifold construction described previously for spatial lattices in Minkowski space, and can be generalized to more complicated theories with additional supersymmetry and more spacetime dimensions.

A euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges

We formulate supersymmetric euclidean spacetime Ad* lattices whose classical continuum limits are U(N) supersymmetric Yang-Mills theories with sixteen supercharges in d = 1,2,3 and 4 dimensions. This family includes the especially interesting = 4 supersymmetry in four dimensions, as well as a euclidean path integral formulation of Matrix Theory on a one dimensional lattice.

Supersymmetry on a Euclidean space-time lattice. 2. Target theories with eight supercharges

We formulate euclidean spacetime lattices whose continuum limits are supersymmetric Yang-Mills theories with eight supercharges in two and three dimensions. The lattice actions are themselves supersymmetric.

Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions

In recent work, we derived the long-distance confining dynamics of certain QCDlike gauge theories formulated on small S1 × R3 based on symmetries, an index theorem, and Abelian duality. Here, we give the microscopic derivation. The solution reveals a new mechanism of confinement in QCD(adj) in the regime where we have control over both perturbative and nonperturbative aspects. In particular, consider SU(2) QCD(adj) theory with 1 ≤ nf ≤ 4 Majorana fermions, a theory which undergoes gauge symmetry breaking at small S1. If the magnetic charge of the BPS monopole is normalized to unity, we show that confinement occurs due to condensation of objects with magnetic charge 2, not 1. Because of index theorems, we know that such an object cannot be a two identical monopole configuration. Its net topological charge must vanish, and hence it must be topologically indistinguishable from the perturbative vacuum. We construct such non-self-dual topological excitations, the magnetically charged, topologically null molecules of a BPS monopole and KK antimonopole, which we refer to as magnetic bions. An immediate puzzle with this proposal is the apparent Coulomb repulsion between the BPS-KK pair. An attraction which overcomes the Coulomb repulsion between the two is induced by 2nf -fermion exchange. Bion condensation is also the mechanism of confinement in N = 1 SYM on the same four-manifold. The SU(N) generalization hints a possible hidden integrability behind nonsupersymmetric QCD of affine Toda type, and allows us to analytically compute the mass gap in the gauge sector. We currently do not know the extension to R4.

The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects

A bstractWe study the dynamics of four dimensional gauge theories with adjoint fermions for all gauge groups, both in perturbation theory and non-perturbatively, by using circle compactification with periodic boundary conditions for the fermions. There are new gauge phenomena. We show that, to all orders in perturbation theory, many gauge groups are Higgsed by the gauge holonomy around the circle to a product of both abelian and nonabelian gauge group factors. Non-perturbatively there are monopole-instantons with fermion zero modes and two types of monopole-anti-monopole molecules, called bions. One type are magnetic bions which carry net magnetic charge and induce a mass gap for gauge fluctuations. Another type are neutral bions which are magnetically neutral, and their understanding requires a generalization of multi-instanton techniques in quantum mechanics — which we refer to as the Bogomolny-Zinn-Justin (BZJ) prescription — to compactified field theory. The BZJ prescription applied to bion-anti-bion topological molecules predicts a singularity on the positive real axis of the Borel plane (i.e., a divergence from summing large orders in peturbation theory) which is of order N times closer to the origin than the leading 4-d BPST instanton-anti-instanton singularity, where N is the rank of the gauge group. The position of the bion-anti-bion singularity is thus qualitatively similar to that of the 4-d IR renormalon singularity, and we conjecture that they are continuously related as the compactification radius is changed. By making use of transseries and Écalle’s resurgence theory we argue that a non-perturbative continuum definition of a class of field theories which admit semi-classical expansions may be possible.

Twisted supersymmetric gauge theories and orbifold lattices

We examine the relation between twisted versions of the extended supersymmetric gauge theories and supersymmetric orbifold lattices. In particular, for the = 4 SYM in d = 4, we show that the continuum limit of orbifold lattice reproduces the twist introduced by Marcus, and the examples at lower dimensions are usually Blau-Thompson type. The orbifold lattice point group symmetry is a subgroup of the twisted Lorentz group, and the exact supersymmetry of the lattice is indeed the nilpotent scalar supersymmetry of the twisted versions. We also introduce twisting in terms of spin groups of finite point subgroups of R-symmetry and spacetime symmetry.We examine the relation between twisted versions of the extended supersymmetric gauge theories and supersymmetric orbifold lattices. In particular, for the N = 4 SYM in d = 4, we show that the continuum limit of orbifold lattice reproduces the twist introduced by Marcus, and the examples at lower dimensions are usually Blau-Thompson type. The orbifold lattice point group symmetry is a subgroup of the twisted Lorentz group, and the exact supersymmetry of the lattice is indeed the nilpotent scalar supersymmetry of the twisted versions. We also introduce twisting in terms of spin groups of finite point subgroups of R-symmetry and spacetime symmetry.

Semiclassical realization of infrared renormalons.

Perturbation series in quantum field theory (QFT) are generally divergent asymptotic series which are also typically not Borel resummable in the sense that the resummed series is ambiguous. The ambiguity is associated with singularities in the Borel plane on the positive real axis. In quantum mechanics there are cases in which the ambiguity that arises in perturbation theory cancels against a similarly ambiguous contribution from instanton-anti-instanton events. In asymptotically free gauge theories, this mechanism does not suffice because perturbation theory develops ambiguities associated with singularities in the Borel plane which are closer to the origin by a factor of about N (the rank of the gauge group) compared to the singularities realized by instanton events. These are called IR renormalon poles, and on R^{4} they do not possess any known semiclassical realization. By using continuity on R(3) × S(1), and by generalizing the works of Bogomolny and Zinn-Justin to QFT, we identify saddle point field configurations, e.g., bion-antibion events, corresponding to singularities in the Borel plane which are of order N times closer to the origin than the four-dimensional instanton-anti-instanton singularities in the Borel plane. We conjecture that these are the leading singularities in the Borel plane and that they are the incarnation of the elusive renormalons in the weak coupling regime.

QCD-like Theories on R_3\times S_1: a Smooth Journey from Small to Large r(S_1)with Double-Trace Deformations

We consider QCD-like theories with one massless fermion in various representations of the gauge group SU(N). The theories are formulated on R{sub 3} x S{sub 1}. In the decompactification limit of large r(S{sub 1}) all these theories are characterized by confinement, mass gap and spontaneous breaking of a (discrete) chiral symmetry ({chi}SB). At small r(S{sub 1}), in order to stabilize the vacua of these theories at a center-symmetric point, we suggest to perform a double trace deformation. With these deformation, the theories at hand are at weak coupling at small r(S{sub 1}) and yet exhibit basic features of the large-r(S{sub 1}) limit: confinement and {chi}SB. We calculate the string tension, mass gap, bifermion condensates and {theta} dependence. The double-trace deformation becomes dynamically irrelevant at large r(S{sub 1}). Despite the fact that at small r(S{sub 1}) confinement is Abelian, while it is expected to be non-Abelian at large r(S{sub 1}), we argue that small and large-r(S{sub 1}) physics are continuously connected. If so, one can use small-r(S{sub 1}) laboratory to extract lessons about QCD and QCD-like theories on R{sub 4}.

Conformality or confinement: (IR)relevance of topological excitations

What distinguishes two asymptotically-free non-abelian gauge theories on R4, one of which is just below the conformal window boundary and confines, while the other is slightly above the boundary and flows to an infrared conformal field theory? In this work, we aim to answer this question for non-supersymmetric Yang-Mills theories with fermions in arbitrary chiral or vectorlike representations. We use the presence or absence of mass gap for gauge fluctuations as an identifier of the infrared behavior. With the present-day understanding of such gauge theories, the mass gap for gauge fluctuations cannot be computed on R4. However, recent progress allows its non-perturbative computation on R3 × S1 by using either the twisted partition function or deformation theory, for a range of sizes of S1 depending on the theory. For small number of fermions, Nf, we show that the mass gap increases with increasing radius, due to the non-dilution of monopoles and bions — the topological excitations relevant for confinement on R3 × S1. For sufficiently large Nf, we show that the mass gap decreases with increasing radius. In a class of theories, we claim that the decompactification limit can be taken while remaining within the region of validity of semiclassical techniques, giving the first examples of semiclassically solvable Yang-Mills theories at any size S1. For general non-supersymmetric vectorlike or chiral theories, we conjecture that the change in the behavior of the mass gap on R3 × S1 as a function of the radius occurs near the lower boundary of the conformal window and give non-perturbative estimates of its value. For vectorlike theories, we compare our estimates of the conformal window with existing lattice results, truncations of the Schwinger-Dyson equations, NSVZ beta function-inspired estimates, and degree of freedom counting criteria. For multi-generation chiral gauge theories, to the best of our knowledge, our estimates of the conformal window are the only known ones.